Integrand size = 26, antiderivative size = 716 \[ \int \frac {\sqrt {a+b x^2}}{(e x)^{7/2} (c+d x)} \, dx=-\frac {2 \sqrt {a+b x^2}}{5 c e (e x)^{5/2}}+\frac {2 d \sqrt {a+b x^2}}{3 c^2 e^2 (e x)^{3/2}}-\frac {2 \left (2 b c^2+5 a d^2\right ) \sqrt {a+b x^2}}{5 a c^3 e^3 \sqrt {e x}}+\frac {2 \sqrt {b} \left (2 b c^2+5 a d^2\right ) \sqrt {e x} \sqrt {a+b x^2}}{5 a c^3 e^4 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {d^{3/2} \sqrt {b c^2+a d^2} \arctan \left (\frac {\sqrt {b c^2+a d^2} \sqrt {e x}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{c^{7/2} e^{7/2}}-\frac {2 \sqrt [4]{b} \left (2 b c^2+5 a d^2\right ) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{5 a^{3/4} c^3 e^{7/2} \sqrt {a+b x^2}}+\frac {2 \sqrt [4]{b} \left (3 b^{3/2} c^3-8 \sqrt {a} b c^2 d+5 a \sqrt {b} c d^2-15 a^{3/2} d^3\right ) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{15 a^{3/4} c^3 \left (\sqrt {b} c-\sqrt {a} d\right ) e^{7/2} \sqrt {a+b x^2}}+\frac {d \left (\sqrt {b} c+\sqrt {a} d\right ) \left (b c^2+a d^2\right ) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {b} c-\sqrt {a} d\right )^2}{4 \sqrt {a} \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} c^4 \left (\sqrt {b} c-\sqrt {a} d\right ) e^{7/2} \sqrt {a+b x^2}} \] Output:
-2/5*(b*x^2+a)^(1/2)/c/e/(e*x)^(5/2)+2/3*d*(b*x^2+a)^(1/2)/c^2/e^2/(e*x)^( 3/2)-2/5*(5*a*d^2+2*b*c^2)*(b*x^2+a)^(1/2)/a/c^3/e^3/(e*x)^(1/2)+2/5*b^(1/ 2)*(5*a*d^2+2*b*c^2)*(e*x)^(1/2)*(b*x^2+a)^(1/2)/a/c^3/e^4/(a^(1/2)+b^(1/2 )*x)-d^(3/2)*(a*d^2+b*c^2)^(1/2)*arctan((a*d^2+b*c^2)^(1/2)*(e*x)^(1/2)/c^ (1/2)/d^(1/2)/e^(1/2)/(b*x^2+a)^(1/2))/c^(7/2)/e^(7/2)-2/5*b^(1/4)*(5*a*d^ 2+2*b*c^2)*(a^(1/2)+b^(1/2)*x)*((b*x^2+a)/(a^(1/2)+b^(1/2)*x)^2)^(1/2)*Ell ipticE(sin(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))),1/2*2^(1/2))/a^( 3/4)/c^3/e^(7/2)/(b*x^2+a)^(1/2)+2/15*b^(1/4)*(3*b^(3/2)*c^3-8*a^(1/2)*b*c ^2*d+5*a*b^(1/2)*c*d^2-15*a^(3/2)*d^3)*(a^(1/2)+b^(1/2)*x)*((b*x^2+a)/(a^( 1/2)+b^(1/2)*x)^2)^(1/2)*InverseJacobiAM(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1 /4)/e^(1/2)),1/2*2^(1/2))/a^(3/4)/c^3/(b^(1/2)*c-a^(1/2)*d)/e^(7/2)/(b*x^2 +a)^(1/2)+1/2*d*(b^(1/2)*c+a^(1/2)*d)*(a*d^2+b*c^2)*(a^(1/2)+b^(1/2)*x)*(( b*x^2+a)/(a^(1/2)+b^(1/2)*x)^2)^(1/2)*EllipticPi(sin(2*arctan(b^(1/4)*(e*x )^(1/2)/a^(1/4)/e^(1/2))),-1/4*(b^(1/2)*c-a^(1/2)*d)^2/a^(1/2)/b^(1/2)/c/d ,1/2*2^(1/2))/a^(1/4)/b^(1/4)/c^4/(b^(1/2)*c-a^(1/2)*d)/e^(7/2)/(b*x^2+a)^ (1/2)
Result contains complex when optimal does not.
Time = 23.66 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.49 \[ \int \frac {\sqrt {a+b x^2}}{(e x)^{7/2} (c+d x)} \, dx=\frac {x \left (-6 \sqrt {b} c \left (2 b c^2+5 a d^2\right ) \sqrt {1+\frac {a}{b x^2}} x^{7/2} E\left (\left .i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right )\right |-1\right )+2 \left (6 b^{3/2} c^3-10 i \sqrt {a} b c^2 d+15 a \sqrt {b} c d^2-15 i a^{3/2} d^3\right ) \sqrt {1+\frac {a}{b x^2}} x^{7/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )+2 \sqrt {a} \left (\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}} c^2 (-3 c+5 d x) \left (a+b x^2\right )+15 i d \left (b c^2+a d^2\right ) \sqrt {1+\frac {a}{b x^2}} x^{7/2} \operatorname {EllipticPi}\left (-\frac {i \sqrt {b} c}{\sqrt {a} d},i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )\right )\right )}{15 \sqrt {a} \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}} c^4 (e x)^{7/2} \sqrt {a+b x^2}} \] Input:
Integrate[Sqrt[a + b*x^2]/((e*x)^(7/2)*(c + d*x)),x]
Output:
(x*(-6*Sqrt[b]*c*(2*b*c^2 + 5*a*d^2)*Sqrt[1 + a/(b*x^2)]*x^(7/2)*EllipticE [I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/Sqrt[x]], -1] + 2*(6*b^(3/2)*c^3 - (1 0*I)*Sqrt[a]*b*c^2*d + 15*a*Sqrt[b]*c*d^2 - (15*I)*a^(3/2)*d^3)*Sqrt[1 + a /(b*x^2)]*x^(7/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/Sqrt[x]], -1] + 2*Sqrt[a]*(Sqrt[(I*Sqrt[a])/Sqrt[b]]*c^2*(-3*c + 5*d*x)*(a + b*x^2) + (15*I)*d*(b*c^2 + a*d^2)*Sqrt[1 + a/(b*x^2)]*x^(7/2)*EllipticPi[((-I)*Sq rt[b]*c)/(Sqrt[a]*d), I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/Sqrt[x]], -1]))) /(15*Sqrt[a]*Sqrt[(I*Sqrt[a])/Sqrt[b]]*c^4*(e*x)^(7/2)*Sqrt[a + b*x^2])
Time = 4.17 (sec) , antiderivative size = 873, normalized size of antiderivative = 1.22, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {616, 27, 1635, 25, 27, 2221, 2374, 9, 27, 2374, 9, 27, 1605, 27, 1512, 27, 761, 1510}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {a+b x^2}}{(e x)^{7/2} (c+d x)} \, dx\) |
| \(\Big \downarrow \) 616 |
| \(\displaystyle \frac {2 \int \frac {\sqrt {b x^2+a}}{e^2 x^3 (c e+d x e)}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {\sqrt {b x^2+a}}{e^3 x^3 (c e+d x e)}d\sqrt {e x}\) |
| \(\Big \downarrow \) 1635 |
| \(\displaystyle 2 \left (\frac {d^2 \left (a d^2+b c^2\right ) \int \frac {\left (\sqrt {b} c+\sqrt {a} d\right ) \left (\sqrt {b} x e+\sqrt {a} e\right )}{e (c e+d x e) \sqrt {b x^2+a}}d\sqrt {e x}}{c^3 e^3 \left (b c^2-a d^2\right )}-\frac {d^2 \int -\frac {-\frac {\sqrt {b} \left (\sqrt {b} c+\sqrt {a} d\right ) \left (b c^2+a d^2\right ) e^2 x^3}{d}+\frac {\left (b^2 c^4-a^2 d^4\right ) e^2 x^2}{d^2}-\frac {a c \left (b c^2-a d^2\right ) e^2 x}{d}+\frac {a c^2 \left (b c^2-a d^2\right ) e^2}{d^2}}{e^3 x^3 \sqrt {b x^2+a}}d\sqrt {e x}}{c^3 e^3 \left (b c^2-a d^2\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {d^2 \int \frac {-\frac {\sqrt {b} \left (\sqrt {b} c+\sqrt {a} d\right ) \left (b c^2+a d^2\right ) e^2 x^3}{d}+\frac {\left (b^2 c^4-a^2 d^4\right ) e^2 x^2}{d^2}-\frac {a c \left (b c^2-a d^2\right ) e^2 x}{d}+\frac {a c^2 \left (b c^2-a d^2\right ) e^2}{d^2}}{e^3 x^3 \sqrt {b x^2+a}}d\sqrt {e x}}{c^3 e^3 \left (b c^2-a d^2\right )}+\frac {d^2 \left (a d^2+b c^2\right ) \int \frac {\left (\sqrt {b} c+\sqrt {a} d\right ) \left (\sqrt {b} x e+\sqrt {a} e\right )}{e (c e+d x e) \sqrt {b x^2+a}}d\sqrt {e x}}{c^3 e^3 \left (b c^2-a d^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {d^2 \int \frac {-\frac {\sqrt {b} \left (\sqrt {b} c+\sqrt {a} d\right ) \left (b c^2+a d^2\right ) e^2 x^3}{d}+\frac {\left (b^2 c^4-a^2 d^4\right ) e^2 x^2}{d^2}-\frac {a c \left (b c^2-a d^2\right ) e^2 x}{d}+\frac {a c^2 \left (b c^2-a d^2\right ) e^2}{d^2}}{e^3 x^3 \sqrt {b x^2+a}}d\sqrt {e x}}{c^3 e^3 \left (b c^2-a d^2\right )}+\frac {d^2 \left (\sqrt {a} d+\sqrt {b} c\right ) \left (a d^2+b c^2\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{(c e+d x e) \sqrt {b x^2+a}}d\sqrt {e x}}{c^3 e^4 \left (b c^2-a d^2\right )}\right )\) |
| \(\Big \downarrow \) 2221 |
| \(\displaystyle 2 \left (\frac {d^2 \int \frac {-\frac {\sqrt {b} \left (\sqrt {b} c+\sqrt {a} d\right ) \left (b c^2+a d^2\right ) e^2 x^3}{d}+\frac {\left (b^2 c^4-a^2 d^4\right ) e^2 x^2}{d^2}-\frac {a c \left (b c^2-a d^2\right ) e^2 x}{d}+\frac {a c^2 \left (b c^2-a d^2\right ) e^2}{d^2}}{e^3 x^3 \sqrt {b x^2+a}}d\sqrt {e x}}{c^3 e^3 \left (b c^2-a d^2\right )}+\frac {d^2 \left (\sqrt {a} d+\sqrt {b} c\right ) \left (a d^2+b c^2\right ) \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{c^3 e^4 \left (b c^2-a d^2\right )}\right )\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle 2 \left (\frac {d^2 \left (-\frac {\int \frac {2 \left (\frac {5 a \sqrt {b} \left (\sqrt {b} c+\sqrt {a} d\right ) \left (b c^2+a d^2\right ) (e x)^{5/2}}{d e}-a \left (\frac {2 b^2 c^4}{d^2}+3 a b c^2-5 a^2 d^2\right ) (e x)^{3/2}+5 a^2 c \left (\frac {b c^2}{d}-a d\right ) e \sqrt {e x}\right )}{(e x)^{5/2} \sqrt {b x^2+a}}d\sqrt {e x}}{10 a}-\frac {c^2 e^2 \sqrt {a+b x^2} \left (b c^2-a d^2\right )}{5 d^2 (e x)^{5/2}}\right )}{c^3 e^3 \left (b c^2-a d^2\right )}+\frac {d^2 \left (\sqrt {a} d+\sqrt {b} c\right ) \left (a d^2+b c^2\right ) \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{c^3 e^4 \left (b c^2-a d^2\right )}\right )\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle 2 \left (\frac {d^2 \left (-\frac {\int \frac {2 \left (5 c \left (\frac {b c^2}{d}-a d\right ) e a^2+\frac {5 \sqrt {b} \left (\sqrt {b} c+\sqrt {a} d\right ) \left (b c^2+a d^2\right ) e x^2 a}{d}-\left (\frac {2 b^2 c^4}{d^2}+3 a b c^2-5 a^2 d^2\right ) e x a\right )}{e^2 x^2 \sqrt {b x^2+a}}d\sqrt {e x}}{10 a}-\frac {c^2 e^2 \sqrt {a+b x^2} \left (b c^2-a d^2\right )}{5 d^2 (e x)^{5/2}}\right )}{c^3 e^3 \left (b c^2-a d^2\right )}+\frac {d^2 \left (\sqrt {a} d+\sqrt {b} c\right ) \left (a d^2+b c^2\right ) \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{c^3 e^4 \left (b c^2-a d^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {d^2 \left (-\frac {\int \frac {5 c \left (\frac {b c^2}{d}-a d\right ) e a^2+\frac {5 \sqrt {b} \left (\sqrt {b} c+\sqrt {a} d\right ) \left (b c^2+a d^2\right ) e x^2 a}{d}-\left (\frac {2 b^2 c^4}{d^2}+3 a b c^2-5 a^2 d^2\right ) e x a}{e^2 x^2 \sqrt {b x^2+a}}d\sqrt {e x}}{5 a}-\frac {c^2 e^2 \sqrt {a+b x^2} \left (b c^2-a d^2\right )}{5 d^2 (e x)^{5/2}}\right )}{c^3 e^3 \left (b c^2-a d^2\right )}+\frac {d^2 \left (\sqrt {a} d+\sqrt {b} c\right ) \left (a d^2+b c^2\right ) \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{c^3 e^4 \left (b c^2-a d^2\right )}\right )\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle 2 \left (\frac {d^2 \left (-\frac {-\frac {\int \frac {2 \left (3 a^2 \left (\frac {2 b^2 c^4}{d^2}+3 a b c^2-5 a^2 d^2\right ) \sqrt {e x}-\frac {5 a^2 \sqrt {b} \left (\sqrt {b} c+\sqrt {a} d\right ) \left (2 b c^2+\sqrt {a} \sqrt {b} d c+3 a d^2\right ) (e x)^{3/2}}{d e}\right )}{(e x)^{3/2} \sqrt {b x^2+a}}d\sqrt {e x}}{6 a}-\frac {5 a c e \sqrt {a+b x^2} \left (\frac {b c^2}{d}-a d\right )}{3 (e x)^{3/2}}}{5 a}-\frac {c^2 e^2 \sqrt {a+b x^2} \left (b c^2-a d^2\right )}{5 d^2 (e x)^{5/2}}\right )}{c^3 e^3 \left (b c^2-a d^2\right )}+\frac {d^2 \left (\sqrt {a} d+\sqrt {b} c\right ) \left (a d^2+b c^2\right ) \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{c^3 e^4 \left (b c^2-a d^2\right )}\right )\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle 2 \left (\frac {d^2 \left (-\frac {-\frac {\int \frac {2 a^2 \left (3 d \left (\frac {2 b^2 c^4}{d^2}+3 a b c^2-5 a^2 d^2\right ) e-5 \sqrt {b} \left (\sqrt {b} c+\sqrt {a} d\right ) \left (2 b c^2+\sqrt {a} \sqrt {b} d c+3 a d^2\right ) e x\right )}{d e^2 x \sqrt {b x^2+a}}d\sqrt {e x}}{6 a}-\frac {5 a c e \sqrt {a+b x^2} \left (\frac {b c^2}{d}-a d\right )}{3 (e x)^{3/2}}}{5 a}-\frac {c^2 e^2 \sqrt {a+b x^2} \left (b c^2-a d^2\right )}{5 d^2 (e x)^{5/2}}\right )}{c^3 e^3 \left (b c^2-a d^2\right )}+\frac {d^2 \left (\sqrt {a} d+\sqrt {b} c\right ) \left (a d^2+b c^2\right ) \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{c^3 e^4 \left (b c^2-a d^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {d^2 \left (-\frac {-\frac {a \int \frac {3 d \left (\frac {2 b^2 c^4}{d^2}+3 a b c^2-5 a^2 d^2\right ) e-5 \sqrt {b} \left (\sqrt {b} c+\sqrt {a} d\right ) \left (2 b c^2+\sqrt {a} \sqrt {b} d c+3 a d^2\right ) e x}{e x \sqrt {b x^2+a}}d\sqrt {e x}}{3 d e}-\frac {5 a c e \sqrt {a+b x^2} \left (\frac {b c^2}{d}-a d\right )}{3 (e x)^{3/2}}}{5 a}-\frac {c^2 e^2 \sqrt {a+b x^2} \left (b c^2-a d^2\right )}{5 d^2 (e x)^{5/2}}\right )}{c^3 e^3 \left (b c^2-a d^2\right )}+\frac {d^2 \left (\sqrt {a} d+\sqrt {b} c\right ) \left (a d^2+b c^2\right ) \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{c^3 e^4 \left (b c^2-a d^2\right )}\right )\) |
| \(\Big \downarrow \) 1605 |
| \(\displaystyle 2 \left (\frac {d^2 \left (-\frac {-\frac {a \left (-\frac {\int \frac {\sqrt {b} \left (5 a \left (\sqrt {b} c+\sqrt {a} d\right ) \left (2 b c^2+\sqrt {a} \sqrt {b} d c+3 a d^2\right ) e-3 \sqrt {b} d \left (\frac {2 b^2 c^4}{d^2}+3 a b c^2-5 a^2 d^2\right ) e x\right )}{e \sqrt {b x^2+a}}d\sqrt {e x}}{a}-\frac {3 d e \sqrt {a+b x^2} \left (-5 a^2 d^2+3 a b c^2+\frac {2 b^2 c^4}{d^2}\right )}{a \sqrt {e x}}\right )}{3 d e}-\frac {5 a c e \sqrt {a+b x^2} \left (\frac {b c^2}{d}-a d\right )}{3 (e x)^{3/2}}}{5 a}-\frac {c^2 e^2 \sqrt {a+b x^2} \left (b c^2-a d^2\right )}{5 d^2 (e x)^{5/2}}\right )}{c^3 e^3 \left (b c^2-a d^2\right )}+\frac {d^2 \left (\sqrt {a} d+\sqrt {b} c\right ) \left (a d^2+b c^2\right ) \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{c^3 e^4 \left (b c^2-a d^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {d^2 \left (-\frac {-\frac {a \left (-\frac {\sqrt {b} \int \frac {5 a \left (\sqrt {b} c+\sqrt {a} d\right ) \left (2 b c^2+\sqrt {a} \sqrt {b} d c+3 a d^2\right ) e-3 \sqrt {b} d \left (\frac {2 b^2 c^4}{d^2}+3 a b c^2-5 a^2 d^2\right ) e x}{\sqrt {b x^2+a}}d\sqrt {e x}}{a e}-\frac {3 d e \sqrt {a+b x^2} \left (-5 a^2 d^2+3 a b c^2+\frac {2 b^2 c^4}{d^2}\right )}{a \sqrt {e x}}\right )}{3 d e}-\frac {5 a c e \sqrt {a+b x^2} \left (\frac {b c^2}{d}-a d\right )}{3 (e x)^{3/2}}}{5 a}-\frac {c^2 e^2 \sqrt {a+b x^2} \left (b c^2-a d^2\right )}{5 d^2 (e x)^{5/2}}\right )}{c^3 e^3 \left (b c^2-a d^2\right )}+\frac {d^2 \left (\sqrt {a} d+\sqrt {b} c\right ) \left (a d^2+b c^2\right ) \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{c^3 e^4 \left (b c^2-a d^2\right )}\right )\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle 2 \left (\frac {d^2 \left (-\frac {-\frac {a \left (-\frac {\sqrt {b} \left (3 \sqrt {a} d e \left (-5 a^2 d^2+3 a b c^2+\frac {2 b^2 c^4}{d^2}\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{\sqrt {a} e \sqrt {b x^2+a}}d\sqrt {e x}-\frac {2 \sqrt {a} e \left (-10 a^{3/2} \sqrt {b} c d^3-15 a^2 d^4-5 \sqrt {a} b^{3/2} c^3 d-3 a b c^2 d^2+3 b^2 c^4\right ) \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {e x}}{d}\right )}{a e}-\frac {3 d e \sqrt {a+b x^2} \left (-5 a^2 d^2+3 a b c^2+\frac {2 b^2 c^4}{d^2}\right )}{a \sqrt {e x}}\right )}{3 d e}-\frac {5 a c e \sqrt {a+b x^2} \left (\frac {b c^2}{d}-a d\right )}{3 (e x)^{3/2}}}{5 a}-\frac {c^2 e^2 \sqrt {a+b x^2} \left (b c^2-a d^2\right )}{5 d^2 (e x)^{5/2}}\right )}{c^3 e^3 \left (b c^2-a d^2\right )}+\frac {d^2 \left (\sqrt {a} d+\sqrt {b} c\right ) \left (a d^2+b c^2\right ) \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{c^3 e^4 \left (b c^2-a d^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {d^2 \left (-\frac {-\frac {a \left (-\frac {\sqrt {b} \left (3 d \left (-5 a^2 d^2+3 a b c^2+\frac {2 b^2 c^4}{d^2}\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{\sqrt {b x^2+a}}d\sqrt {e x}-\frac {2 \sqrt {a} e \left (-10 a^{3/2} \sqrt {b} c d^3-15 a^2 d^4-5 \sqrt {a} b^{3/2} c^3 d-3 a b c^2 d^2+3 b^2 c^4\right ) \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {e x}}{d}\right )}{a e}-\frac {3 d e \sqrt {a+b x^2} \left (-5 a^2 d^2+3 a b c^2+\frac {2 b^2 c^4}{d^2}\right )}{a \sqrt {e x}}\right )}{3 d e}-\frac {5 a c e \sqrt {a+b x^2} \left (\frac {b c^2}{d}-a d\right )}{3 (e x)^{3/2}}}{5 a}-\frac {c^2 e^2 \sqrt {a+b x^2} \left (b c^2-a d^2\right )}{5 d^2 (e x)^{5/2}}\right )}{c^3 e^3 \left (b c^2-a d^2\right )}+\frac {d^2 \left (\sqrt {a} d+\sqrt {b} c\right ) \left (a d^2+b c^2\right ) \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{c^3 e^4 \left (b c^2-a d^2\right )}\right )\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle 2 \left (\frac {d^2 \left (-\frac {-\frac {a \left (-\frac {\sqrt {b} \left (3 d \left (-5 a^2 d^2+3 a b c^2+\frac {2 b^2 c^4}{d^2}\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{\sqrt {b x^2+a}}d\sqrt {e x}-\frac {\sqrt [4]{a} \sqrt {e} \left (-10 a^{3/2} \sqrt {b} c d^3-15 a^2 d^4-5 \sqrt {a} b^{3/2} c^3 d-3 a b c^2 d^2+3 b^2 c^4\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{\sqrt [4]{b} d \sqrt {a+b x^2}}\right )}{a e}-\frac {3 d e \sqrt {a+b x^2} \left (-5 a^2 d^2+3 a b c^2+\frac {2 b^2 c^4}{d^2}\right )}{a \sqrt {e x}}\right )}{3 d e}-\frac {5 a c e \sqrt {a+b x^2} \left (\frac {b c^2}{d}-a d\right )}{3 (e x)^{3/2}}}{5 a}-\frac {c^2 e^2 \sqrt {a+b x^2} \left (b c^2-a d^2\right )}{5 d^2 (e x)^{5/2}}\right )}{c^3 e^3 \left (b c^2-a d^2\right )}+\frac {d^2 \left (\sqrt {a} d+\sqrt {b} c\right ) \left (a d^2+b c^2\right ) \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{c^3 e^4 \left (b c^2-a d^2\right )}\right )\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle 2 \left (\frac {\left (-\frac {c^2 \left (b c^2-a d^2\right ) \sqrt {b x^2+a} e^2}{5 d^2 (e x)^{5/2}}-\frac {-\frac {5 a c \left (\frac {b c^2}{d}-a d\right ) \sqrt {b x^2+a} e}{3 (e x)^{3/2}}-\frac {a \left (-\frac {3 d \left (\frac {2 b^2 c^4}{d^2}+3 a b c^2-5 a^2 d^2\right ) \sqrt {b x^2+a} e}{a \sqrt {e x}}-\frac {\sqrt {b} \left (3 d \left (\frac {2 b^2 c^4}{d^2}+3 a b c^2-5 a^2 d^2\right ) \left (\frac {\sqrt [4]{a} \sqrt {e} \left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {\frac {b x^2 e^2+a e^2}{\left (\sqrt {b} x e+\sqrt {a} e\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {b x^2+a}}-\frac {e^2 \sqrt {e x} \sqrt {b x^2+a}}{\sqrt {b} x e+\sqrt {a} e}\right )-\frac {\sqrt [4]{a} \left (3 b^2 c^4-5 \sqrt {a} b^{3/2} d c^3-3 a b d^2 c^2-10 a^{3/2} \sqrt {b} d^3 c-15 a^2 d^4\right ) \sqrt {e} \left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {\frac {b x^2 e^2+a e^2}{\left (\sqrt {b} x e+\sqrt {a} e\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{\sqrt [4]{b} d \sqrt {b x^2+a}}\right )}{a e}\right )}{3 d e}}{5 a}\right ) d^2}{c^3 \left (b c^2-a d^2\right ) e^3}+\frac {\left (\sqrt {b} c+\sqrt {a} d\right ) \left (b c^2+a d^2\right ) \left (\frac {\left (\sqrt {b} c+\sqrt {a} d\right ) \left (\sqrt {b} x e+\sqrt {a} e\right ) \sqrt {\frac {b x^2 e^2+a e^2}{\left (\sqrt {b} x e+\sqrt {a} e\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {b x^2+a}}-\frac {\left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {e} \arctan \left (\frac {\sqrt {b c^2+a d^2} \sqrt {e x}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {b x^2+a}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {b c^2+a d^2}}\right ) d^2}{c^3 \left (b c^2-a d^2\right ) e^4}\right )\) |
Input:
Int[Sqrt[a + b*x^2]/((e*x)^(7/2)*(c + d*x)),x]
Output:
2*((d^2*(-1/5*(c^2*(b*c^2 - a*d^2)*e^2*Sqrt[a + b*x^2])/(d^2*(e*x)^(5/2)) - ((-5*a*c*((b*c^2)/d - a*d)*e*Sqrt[a + b*x^2])/(3*(e*x)^(3/2)) - (a*((-3* d*(3*a*b*c^2 + (2*b^2*c^4)/d^2 - 5*a^2*d^2)*e*Sqrt[a + b*x^2])/(a*Sqrt[e*x ]) - (Sqrt[b]*(3*d*(3*a*b*c^2 + (2*b^2*c^4)/d^2 - 5*a^2*d^2)*(-((e^2*Sqrt[ e*x]*Sqrt[a + b*x^2])/(Sqrt[a]*e + Sqrt[b]*e*x)) + (a^(1/4)*Sqrt[e]*(Sqrt[ a]*e + Sqrt[b]*e*x)*Sqrt[(a*e^2 + b*e^2*x^2)/(Sqrt[a]*e + Sqrt[b]*e*x)^2]* EllipticE[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(b^(1/4)* Sqrt[a + b*x^2])) - (a^(1/4)*(3*b^2*c^4 - 5*Sqrt[a]*b^(3/2)*c^3*d - 3*a*b* c^2*d^2 - 10*a^(3/2)*Sqrt[b]*c*d^3 - 15*a^2*d^4)*Sqrt[e]*(Sqrt[a]*e + Sqrt [b]*e*x)*Sqrt[(a*e^2 + b*e^2*x^2)/(Sqrt[a]*e + Sqrt[b]*e*x)^2]*EllipticF[2 *ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(b^(1/4)*d*Sqrt[a + b*x^2])))/(a*e)))/(3*d*e))/(5*a)))/(c^3*(b*c^2 - a*d^2)*e^3) + (d^2*(Sqrt[ b]*c + Sqrt[a]*d)*(b*c^2 + a*d^2)*(-1/2*((Sqrt[b]*c - Sqrt[a]*d)*Sqrt[e]*A rcTan[(Sqrt[b*c^2 + a*d^2]*Sqrt[e*x])/(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a + b* x^2])])/(Sqrt[c]*Sqrt[d]*Sqrt[b*c^2 + a*d^2]) + ((Sqrt[b]*c + Sqrt[a]*d)*( Sqrt[a]*e + Sqrt[b]*e*x)*Sqrt[(a*e^2 + b*e^2*x^2)/(Sqrt[a]*e + Sqrt[b]*e*x )^2]*EllipticPi[-1/4*(Sqrt[a]*((Sqrt[b]*c)/Sqrt[a] - d)^2)/(Sqrt[b]*c*d), 2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(4*a^(1/4)*b^(1/4)* c*d*Sqrt[e]*Sqrt[a + b*x^2])))/(c^3*(b*c^2 - a*d^2)*e^4))
Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && !MonomialQ[Px, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e Subst[Int[x^(k*(m + 1) - 1)*(c + d*(x^k/e))^n*(a + b*(x^(2*k)/e^2))^p, x], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p}, x] && ILtQ[n, 0] && FractionQ[m]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + c*x^4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, c , d, e}, x] && PosQ[c/a]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_ Symbol] :> Simp[d*(f*x)^(m + 1)*((a + c*x^4)^(p + 1)/(a*f*(m + 1))), x] + S imp[1/(a*f^2*(m + 1)) Int[(f*x)^(m + 2)*(a + c*x^4)^p*(a*e*(m + 1) - c*d* (m + 4*p + 5)*x^2), x], x] /; FreeQ[{a, c, d, e, f, p}, x] && LtQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
Int[((x_)^(m_)*((a_) + (c_.)*(x_)^4)^(p_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-(-d/e)^(m/2))*((c*d^2 + a*e^2)^(p + 1/2)/(e^(2*p)*(c*d^2 - a*e^2 ))) Int[(a*d*Rt[c/a, 2] + a*e + (c*d + a*e*Rt[c/a, 2])*x^2)/((d + e*x^2)* Sqrt[a + c*x^4]), x], x] + Simp[(-d/e)^(m/2)/(e^(2*p)*(c*d^2 - a*e^2)) In t[(x^m/Sqrt[a + c*x^4])*ExpandToSum[((e^(2*p)*(c*d^2 - a*e^2)*(a + c*x^4)^( p + 1/2))/(-d/e)^(m/2) + ((a*d*Rt[c/a, 2] + a*e + (c*d + a*e*Rt[c/a, 2])*x^ 2)*(c*d^2 + a*e^2)^(p + 1/2))/x^m)/(d + e*x^2), x], x], x] /; FreeQ[{a, c, d, e}, x] && IGtQ[p + 1/2, 0] && ILtQ[m/2, 0] && NeQ[c*d^2 - a*e^2, 0] && P osQ[c/a]
Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]) , x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(ArcTan[Rt[c*(d/e ) + a*(e/d), 2]*(x/Sqrt[a + c*x^4])]/(2*d*e*Rt[c*(d/e) + a*(e/d), 2])), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4* d*e*q*Sqrt[a + c*x^4]))*EllipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x ], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && Po sQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && PosQ[c*(d/e) + a*(e/d)]
Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Wit h[{Pq0 = Coeff[Pq, x, 0]}, Simp[Pq0*(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c *(m + 1))), x] + Simp[1/(2*a*c*(m + 1)) Int[(c*x)^(m + 1)*ExpandToSum[2*a *(m + 1)*((Pq - Pq0)/x) - 2*b*Pq0*(m + n*(p + 1) + 1)*x^(n - 1), x]*(a + b* x^n)^p, x], x] /; NeQ[Pq0, 0]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[m, -1] && LeQ[n - 1, Expon[Pq, x]]
Time = 1.81 (sec) , antiderivative size = 595, normalized size of antiderivative = 0.83
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (-\frac {2 \sqrt {b e \,x^{3}+a e x}}{5 e^{4} c \,x^{3}}+\frac {2 d \sqrt {b e \,x^{3}+a e x}}{3 e^{4} c^{2} x^{2}}-\frac {2 \left (b e \,x^{2}+a e \right ) \left (5 a \,d^{2}+2 b \,c^{2}\right )}{5 e^{4} a \,c^{3} \sqrt {x \left (b e \,x^{2}+a e \right )}}+\frac {d \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{3 e^{3} c^{2} \sqrt {b e \,x^{3}+a e x}}+\frac {\left (5 a \,d^{2}+2 b \,c^{2}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{5 a \,e^{3} c^{3} \sqrt {b e \,x^{3}+a e x}}-\frac {\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{b \left (-\frac {\sqrt {-a b}}{b}+\frac {c}{d}\right )}, \frac {\sqrt {2}}{2}\right )}{e^{3} c^{3} b \sqrt {b e \,x^{3}+a e x}\, \left (-\frac {\sqrt {-a b}}{b}+\frac {c}{d}\right )}\right )}{\sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(595\) |
| risch | \(-\frac {2 \sqrt {b \,x^{2}+a}\, \left (15 a \,d^{2} x^{2}+6 b \,c^{2} x^{2}-5 a d x c +3 a \,c^{2}\right )}{15 a \,c^{3} x^{2} e^{3} \sqrt {e x}}+\frac {\left (\frac {6 b \,c^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{\sqrt {b e \,x^{3}+a e x}}+\frac {15 a \,d^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{\sqrt {b e \,x^{3}+a e x}}+\frac {5 a c d \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{\sqrt {b e \,x^{3}+a e x}}-\frac {15 a \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{b \left (-\frac {\sqrt {-a b}}{b}+\frac {c}{d}\right )}, \frac {\sqrt {2}}{2}\right )}{b \sqrt {b e \,x^{3}+a e x}\, \left (-\frac {\sqrt {-a b}}{b}+\frac {c}{d}\right )}\right ) \sqrt {\left (b \,x^{2}+a \right ) e x}}{15 c^{3} a \,e^{3} \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(702\) |
| default | \(\text {Expression too large to display}\) | \(1231\) |
Input:
int((b*x^2+a)^(1/2)/(e*x)^(7/2)/(d*x+c),x,method=_RETURNVERBOSE)
Output:
((b*x^2+a)*e*x)^(1/2)/(e*x)^(1/2)/(b*x^2+a)^(1/2)*(-2/5/e^4/c*(b*e*x^3+a*e *x)^(1/2)/x^3+2/3/e^4*d/c^2*(b*e*x^3+a*e*x)^(1/2)/x^2-2/5*(b*e*x^2+a*e)/e^ 4/a*(5*a*d^2+2*b*c^2)/c^3/(x*(b*e*x^2+a*e))^(1/2)+1/3*d/e^3/c^2*(-a*b)^(1/ 2)*((x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2)*(-2*(x-(-a*b)^(1/2)/b)/(-a*b) ^(1/2)*b)^(1/2)*(-1/(-a*b)^(1/2)*b*x)^(1/2)/(b*e*x^3+a*e*x)^(1/2)*Elliptic F(((x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2),1/2*2^(1/2))+1/5/a*(5*a*d^2+2* b*c^2)/e^3/c^3*(-a*b)^(1/2)*((x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2)*(-2* (x-(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2)*(-1/(-a*b)^(1/2)*b*x)^(1/2)/(b*e* x^3+a*e*x)^(1/2)*(-2*(-a*b)^(1/2)/b*EllipticE(((x+(-a*b)^(1/2)/b)/(-a*b)^( 1/2)*b)^(1/2),1/2*2^(1/2))+(-a*b)^(1/2)/b*EllipticF(((x+(-a*b)^(1/2)/b)/(- a*b)^(1/2)*b)^(1/2),1/2*2^(1/2)))-1/e^3*(a*d^2+b*c^2)/c^3*(-a*b)^(1/2)/b*( (x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2)*(-2*(x-(-a*b)^(1/2)/b)/(-a*b)^(1/ 2)*b)^(1/2)*(-1/(-a*b)^(1/2)*b*x)^(1/2)/(b*e*x^3+a*e*x)^(1/2)/(-(-a*b)^(1/ 2)/b+c/d)*EllipticPi(((x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2),-(-a*b)^(1/ 2)/b/(-(-a*b)^(1/2)/b+c/d),1/2*2^(1/2)))
\[ \int \frac {\sqrt {a+b x^2}}{(e x)^{7/2} (c+d x)} \, dx=\int { \frac {\sqrt {b x^{2} + a}}{{\left (d x + c\right )} \left (e x\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)/(e*x)^(7/2)/(d*x+c),x, algorithm="fricas")
Output:
integral(sqrt(b*x^2 + a)*sqrt(e*x)/(d*e^4*x^5 + c*e^4*x^4), x)
\[ \int \frac {\sqrt {a+b x^2}}{(e x)^{7/2} (c+d x)} \, dx=\int \frac {\sqrt {a + b x^{2}}}{\left (e x\right )^{\frac {7}{2}} \left (c + d x\right )}\, dx \] Input:
integrate((b*x**2+a)**(1/2)/(e*x)**(7/2)/(d*x+c),x)
Output:
Integral(sqrt(a + b*x**2)/((e*x)**(7/2)*(c + d*x)), x)
\[ \int \frac {\sqrt {a+b x^2}}{(e x)^{7/2} (c+d x)} \, dx=\int { \frac {\sqrt {b x^{2} + a}}{{\left (d x + c\right )} \left (e x\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)/(e*x)^(7/2)/(d*x+c),x, algorithm="maxima")
Output:
integrate(sqrt(b*x^2 + a)/((d*x + c)*(e*x)^(7/2)), x)
\[ \int \frac {\sqrt {a+b x^2}}{(e x)^{7/2} (c+d x)} \, dx=\int { \frac {\sqrt {b x^{2} + a}}{{\left (d x + c\right )} \left (e x\right )^{\frac {7}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)/(e*x)^(7/2)/(d*x+c),x, algorithm="giac")
Output:
integrate(sqrt(b*x^2 + a)/((d*x + c)*(e*x)^(7/2)), x)
Timed out. \[ \int \frac {\sqrt {a+b x^2}}{(e x)^{7/2} (c+d x)} \, dx=\int \frac {\sqrt {b\,x^2+a}}{{\left (e\,x\right )}^{7/2}\,\left (c+d\,x\right )} \,d x \] Input:
int((a + b*x^2)^(1/2)/((e*x)^(7/2)*(c + d*x)),x)
Output:
int((a + b*x^2)^(1/2)/((e*x)^(7/2)*(c + d*x)), x)
\[ \int \frac {\sqrt {a+b x^2}}{(e x)^{7/2} (c+d x)} \, dx=\frac {\sqrt {e}\, \left (\int \frac {\sqrt {b \,x^{2}+a}}{\sqrt {x}\, c \,x^{3}+\sqrt {x}\, d \,x^{4}}d x \right )}{e^{4}} \] Input:
int((b*x^2+a)^(1/2)/(e*x)^(7/2)/(d*x+c),x)
Output:
(sqrt(e)*int(sqrt(a + b*x**2)/(sqrt(x)*c*x**3 + sqrt(x)*d*x**4),x))/e**4