Integrand size = 26, antiderivative size = 712 \[ \int \frac {1}{(e x)^{3/2} (c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\frac {b (c-d x)}{a \left (b c^2+a d^2\right ) e \sqrt {e x} \sqrt {a+b x^2}}-\frac {\left (3 b c^2+2 a d^2\right ) \sqrt {a+b x^2}}{a^2 c \left (b c^2+a d^2\right ) e \sqrt {e x}}+\frac {\sqrt {b} \left (3 b c^2+2 a d^2\right ) \sqrt {e x} \sqrt {a+b x^2}}{a^2 c \left (b c^2+a d^2\right ) e^2 \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {d^{7/2} \arctan \left (\frac {\sqrt {b c^2+a d^2} \sqrt {e x}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{c^{3/2} \left (b c^2+a d^2\right )^{3/2} e^{3/2}}-\frac {\sqrt [4]{b} \left (3 b c^2+2 a d^2\right ) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{a^{7/4} c \left (b c^2+a d^2\right ) e^{3/2} \sqrt {a+b x^2}}+\frac {\sqrt [4]{b} \left (3 \sqrt {b} c-4 \sqrt {a} d\right ) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 a^{7/4} c \left (\sqrt {b} c-\sqrt {a} d\right ) e^{3/2} \sqrt {a+b x^2}}+\frac {d^3 \left (\sqrt {b} c+\sqrt {a} d\right ) \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {b} c-\sqrt {a} d\right )^2}{4 \sqrt {a} \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} c^2 \left (\sqrt {b} c-\sqrt {a} d\right ) \left (b c^2+a d^2\right ) e^{3/2} \sqrt {a+b x^2}} \] Output:
b*(-d*x+c)/a/(a*d^2+b*c^2)/e/(e*x)^(1/2)/(b*x^2+a)^(1/2)-(2*a*d^2+3*b*c^2) *(b*x^2+a)^(1/2)/a^2/c/(a*d^2+b*c^2)/e/(e*x)^(1/2)+b^(1/2)*(2*a*d^2+3*b*c^ 2)*(e*x)^(1/2)*(b*x^2+a)^(1/2)/a^2/c/(a*d^2+b*c^2)/e^2/(a^(1/2)+b^(1/2)*x) -d^(7/2)*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^(3/2)/(a*d^2+b*c^2)^(3/2)/e^(3/2)-b^(1/4)*(2*a*d^2+3*b*c^ 2)*(a^(1/2)+b^(1/2)*x)*((b*x^2+a)/(a^(1/2)+b^(1/2)*x)^2)^(1/2)*EllipticE(s in(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2))),1/2*2^(1/2))/a^(7/4)/c/( a*d^2+b*c^2)/e^(3/2)/(b*x^2+a)^(1/2)+1/2*b^(1/4)*(3*b^(1/2)*c-4*a^(1/2)*d) *(a^(1/2)+b^(1/2)*x)*((b*x^2+a)/(a^(1/2)+b^(1/2)*x)^2)^(1/2)*InverseJacobi AM(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2)),1/2*2^(1/2))/a^(7/4)/c/(b ^(1/2)*c-a^(1/2)*d)/e^(3/2)/(b*x^2+a)^(1/2)+1/2*d^3*(b^(1/2)*c+a^(1/2)*d)* (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^2/(b^(1/2)*c-a^(1/2) *d)/(a*d^2+b*c^2)/e^(3/2)/(b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 23.55 (sec) , antiderivative size = 343, normalized size of antiderivative = 0.48 \[ \int \frac {1}{(e x)^{3/2} (c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\frac {x \left (-\sqrt {b} c \left (3 b c^2+2 a d^2\right ) \sqrt {1+\frac {a}{b x^2}} x^{3/2} E\left (\left .i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right )\right |-1\right )+\left (3 b^{3/2} c^3-i \sqrt {a} b c^2 d+2 a \sqrt {b} c d^2-2 i a^{3/2} d^3\right ) \sqrt {1+\frac {a}{b x^2}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )+\sqrt {a} \left (\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}} b c^2 (c-d x)+2 i a d^3 \sqrt {1+\frac {a}{b x^2}} x^{3/2} \operatorname {EllipticPi}\left (-\frac {i \sqrt {b} c}{\sqrt {a} d},i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )\right )\right )}{a^{3/2} \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}} c^2 \left (b c^2+a d^2\right ) (e x)^{3/2} \sqrt {a+b x^2}} \] Input:
Integrate[1/((e*x)^(3/2)*(c + d*x)*(a + b*x^2)^(3/2)),x]
Output:
(x*(-(Sqrt[b]*c*(3*b*c^2 + 2*a*d^2)*Sqrt[1 + a/(b*x^2)]*x^(3/2)*EllipticE[ I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/Sqrt[x]], -1]) + (3*b^(3/2)*c^3 - I*Sq rt[a]*b*c^2*d + 2*a*Sqrt[b]*c*d^2 - (2*I)*a^(3/2)*d^3)*Sqrt[1 + a/(b*x^2)] *x^(3/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/Sqrt[x]], -1] + Sqr t[a]*(Sqrt[(I*Sqrt[a])/Sqrt[b]]*b*c^2*(c - d*x) + (2*I)*a*d^3*Sqrt[1 + a/( b*x^2)]*x^(3/2)*EllipticPi[((-I)*Sqrt[b]*c)/(Sqrt[a]*d), I*ArcSinh[Sqrt[(I *Sqrt[a])/Sqrt[b]]/Sqrt[x]], -1])))/(a^(3/2)*Sqrt[(I*Sqrt[a])/Sqrt[b]]*c^2 *(b*c^2 + a*d^2)*(e*x)^(3/2)*Sqrt[a + b*x^2])
Time = 2.42 (sec) , antiderivative size = 857, normalized size of antiderivative = 1.20, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.577, Rules used = {616, 27, 1643, 25, 27, 2221, 2368, 25, 2374, 9, 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.
| \(\displaystyle \int \frac {1}{(e x)^{3/2} \left (a+b x^2\right )^{3/2} (c+d x)} \, dx\) |
| \(\Big \downarrow \) 616 |
| \(\displaystyle \frac {2 \int \frac {1}{x (c e+d x e) \left (b x^2+a\right )^{3/2}}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {1}{e x (c e+d x e) \left (b x^2+a\right )^{3/2}}d\sqrt {e x}\) |
| \(\Big \downarrow \) 1643 |
| \(\displaystyle 2 \left (\frac {d^4 \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a} e (c e+d x e) \sqrt {b x^2+a}}d\sqrt {e x}}{c e \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}-\frac {d \int -\frac {-\frac {b^{3/2} d^2 x^3}{\sqrt {a}}+b \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) d x^2-\frac {\sqrt {b} \left (b c^2-\sqrt {a} \sqrt {b} d c+a d^2\right ) x}{\sqrt {a}}+\frac {\left (\sqrt {b} c-\sqrt {a} d\right ) \left (b c^2+a d^2\right )}{\sqrt {a} d}}{e x \left (b x^2+a\right )^{3/2}}d\sqrt {e x}}{c e \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {d \int \frac {-\frac {b^{3/2} d^2 x^3}{\sqrt {a}}+b \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) d x^2-\frac {\sqrt {b} \left (b c^2-\sqrt {a} \sqrt {b} d c+a d^2\right ) x}{\sqrt {a}}+\frac {\left (\sqrt {b} c-\sqrt {a} d\right ) \left (b c^2+a d^2\right )}{\sqrt {a} d}}{e x \left (b x^2+a\right )^{3/2}}d\sqrt {e x}}{c e \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}+\frac {d^4 \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a} e (c e+d x e) \sqrt {b x^2+a}}d\sqrt {e x}}{c e \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {d \int \frac {-\frac {b^{3/2} d^2 x^3}{\sqrt {a}}+b \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) d x^2-\frac {\sqrt {b} \left (b c^2-\sqrt {a} \sqrt {b} d c+a d^2\right ) x}{\sqrt {a}}+\frac {\left (\sqrt {b} c-\sqrt {a} d\right ) \left (b c^2+a d^2\right )}{\sqrt {a} d}}{e x \left (b x^2+a\right )^{3/2}}d\sqrt {e x}}{c e \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}+\frac {d^4 \int \frac {\sqrt {b} x e+\sqrt {a} e}{(c e+d x e) \sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {a} c e^2 \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}\right )\) |
| \(\Big \downarrow \) 2221 |
| \(\displaystyle 2 \left (\frac {d \int \frac {-\frac {b^{3/2} d^2 x^3}{\sqrt {a}}+b \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) d x^2-\frac {\sqrt {b} \left (b c^2-\sqrt {a} \sqrt {b} d c+a d^2\right ) x}{\sqrt {a}}+\frac {\left (\sqrt {b} c-\sqrt {a} d\right ) \left (b c^2+a d^2\right )}{\sqrt {a} d}}{e x \left (b x^2+a\right )^{3/2}}d\sqrt {e x}}{c e \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}+\frac {d^4 \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{\sqrt {a} c e^2 \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}\right )\) |
| \(\Big \downarrow \) 2368 |
| \(\displaystyle 2 \left (\frac {d \left (-\frac {e^4 \int -\frac {-\frac {c^2 \left (1-\frac {\sqrt {b} c}{\sqrt {a} d}\right ) x^2 b^4}{a e^4}-\frac {\left (\frac {b c^2}{\sqrt {a}}-\sqrt {b} d c+2 \sqrt {a} d^2\right ) x b^{5/2}}{e^4}+\frac {2 \left (\sqrt {b} c-\sqrt {a} d\right ) \left (b c^2+a d^2\right ) b^2}{\sqrt {a} d e^4}}{e x \sqrt {b x^2+a}}d\sqrt {e x}}{2 a b^2}-\frac {b c \sqrt {e x} \left (\sqrt {a} e \left (\sqrt {b} c-\sqrt {a} d\right )-b c e x \left (1-\frac {\sqrt {b} c}{\sqrt {a} d}\right )\right )}{2 a^2 e^2 \sqrt {a+b x^2}}\right )}{c e \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}+\frac {d^4 \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{\sqrt {a} c e^2 \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {e^4 \int \frac {-\frac {c^2 \left (1-\frac {\sqrt {b} c}{\sqrt {a} d}\right ) x^2 b^4}{a e^4}-\frac {\left (\frac {b c^2}{\sqrt {a}}-\sqrt {b} d c+2 \sqrt {a} d^2\right ) x b^{5/2}}{e^4}+\frac {2 \left (\sqrt {b} c-\sqrt {a} d\right ) \left (b c^2+a d^2\right ) b^2}{\sqrt {a} d e^4}}{e x \sqrt {b x^2+a}}d\sqrt {e x}}{2 a b^2}-\frac {b c \sqrt {e x} \left (\sqrt {a} e \left (\sqrt {b} c-\sqrt {a} d\right )-b c e x \left (1-\frac {\sqrt {b} c}{\sqrt {a} d}\right )\right )}{2 a^2 e^2 \sqrt {a+b x^2}}\right )}{c e \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}+\frac {d^4 \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{\sqrt {a} c e^2 \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}\right )\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {e^4 \left (-\frac {\int \frac {2 \left (\frac {\sqrt {a} b^{5/2} \left (b c^2-\sqrt {a} \sqrt {b} d c+2 a d^2\right ) \sqrt {e x}}{e^5}-\frac {b^3 \left (\sqrt {b} c-\sqrt {a} d\right ) \left (3 b c^2+2 a d^2\right ) (e x)^{3/2}}{\sqrt {a} d e^6}\right )}{\sqrt {e x} \sqrt {b x^2+a}}d\sqrt {e x}}{2 a}-\frac {2 b^2 \sqrt {a+b x^2} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (a d^2+b c^2\right )}{a^{3/2} d e^4 \sqrt {e x}}\right )}{2 a b^2}-\frac {b c \sqrt {e x} \left (\sqrt {a} e \left (\sqrt {b} c-\sqrt {a} d\right )-b c e x \left (1-\frac {\sqrt {b} c}{\sqrt {a} d}\right )\right )}{2 a^2 e^2 \sqrt {a+b x^2}}\right )}{c e \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}+\frac {d^4 \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{\sqrt {a} c e^2 \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}\right )\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {e^4 \left (-\frac {\int \frac {2 b^{5/2} \left (a d \left (b c^2-\sqrt {a} \sqrt {b} d c+2 a d^2\right ) e-\sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (3 b c^2+2 a d^2\right ) e x\right )}{\sqrt {a} d e^6 \sqrt {b x^2+a}}d\sqrt {e x}}{2 a}-\frac {2 b^2 \sqrt {a+b x^2} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (a d^2+b c^2\right )}{a^{3/2} d e^4 \sqrt {e x}}\right )}{2 a b^2}-\frac {b c \sqrt {e x} \left (\sqrt {a} e \left (\sqrt {b} c-\sqrt {a} d\right )-b c e x \left (1-\frac {\sqrt {b} c}{\sqrt {a} d}\right )\right )}{2 a^2 e^2 \sqrt {a+b x^2}}\right )}{c e \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}+\frac {d^4 \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{\sqrt {a} c e^2 \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {e^4 \left (-\frac {b^{5/2} \int \frac {a d \left (b c^2-\sqrt {a} \sqrt {b} d c+2 a d^2\right ) e-\sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (3 b c^2+2 a d^2\right ) e x}{\sqrt {b x^2+a}}d\sqrt {e x}}{a^{3/2} d e^6}-\frac {2 b^2 \sqrt {a+b x^2} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (a d^2+b c^2\right )}{a^{3/2} d e^4 \sqrt {e x}}\right )}{2 a b^2}-\frac {b c \sqrt {e x} \left (\sqrt {a} e \left (\sqrt {b} c-\sqrt {a} d\right )-b c e x \left (1-\frac {\sqrt {b} c}{\sqrt {a} d}\right )\right )}{2 a^2 e^2 \sqrt {a+b x^2}}\right )}{c e \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}+\frac {d^4 \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{\sqrt {a} c e^2 \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}\right )\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {e^4 \left (-\frac {b^{5/2} \left (\sqrt {a} e \left (\sqrt {b} c-\sqrt {a} d\right ) \left (2 a d^2+3 b c^2\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{\sqrt {a} e \sqrt {b x^2+a}}d\sqrt {e x}-\sqrt {a} e \left (3 \sqrt {b} c-4 \sqrt {a} d\right ) \left (a d^2+b c^2\right ) \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {e x}\right )}{a^{3/2} d e^6}-\frac {2 b^2 \sqrt {a+b x^2} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (a d^2+b c^2\right )}{a^{3/2} d e^4 \sqrt {e x}}\right )}{2 a b^2}-\frac {b c \sqrt {e x} \left (\sqrt {a} e \left (\sqrt {b} c-\sqrt {a} d\right )-b c e x \left (1-\frac {\sqrt {b} c}{\sqrt {a} d}\right )\right )}{2 a^2 e^2 \sqrt {a+b x^2}}\right )}{c e \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}+\frac {d^4 \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{\sqrt {a} c e^2 \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {e^4 \left (-\frac {b^{5/2} \left (\left (\sqrt {b} c-\sqrt {a} d\right ) \left (2 a d^2+3 b c^2\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{\sqrt {b x^2+a}}d\sqrt {e x}-\sqrt {a} e \left (3 \sqrt {b} c-4 \sqrt {a} d\right ) \left (a d^2+b c^2\right ) \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {e x}\right )}{a^{3/2} d e^6}-\frac {2 b^2 \sqrt {a+b x^2} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (a d^2+b c^2\right )}{a^{3/2} d e^4 \sqrt {e x}}\right )}{2 a b^2}-\frac {b c \sqrt {e x} \left (\sqrt {a} e \left (\sqrt {b} c-\sqrt {a} d\right )-b c e x \left (1-\frac {\sqrt {b} c}{\sqrt {a} d}\right )\right )}{2 a^2 e^2 \sqrt {a+b x^2}}\right )}{c e \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}+\frac {d^4 \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{\sqrt {a} c e^2 \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}\right )\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {e^4 \left (-\frac {b^{5/2} \left (\left (\sqrt {b} c-\sqrt {a} d\right ) \left (2 a d^2+3 b c^2\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{\sqrt {b x^2+a}}d\sqrt {e x}-\frac {\sqrt [4]{a} \sqrt {e} \left (3 \sqrt {b} c-4 \sqrt {a} d\right ) \left (a d^2+b c^2\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{b} \sqrt {a+b x^2}}\right )}{a^{3/2} d e^6}-\frac {2 b^2 \sqrt {a+b x^2} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (a d^2+b c^2\right )}{a^{3/2} d e^4 \sqrt {e x}}\right )}{2 a b^2}-\frac {b c \sqrt {e x} \left (\sqrt {a} e \left (\sqrt {b} c-\sqrt {a} d\right )-b c e x \left (1-\frac {\sqrt {b} c}{\sqrt {a} d}\right )\right )}{2 a^2 e^2 \sqrt {a+b x^2}}\right )}{c e \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}+\frac {d^4 \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{\sqrt {a} c e^2 \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}\right )\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle 2 \left (\frac {\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^4}{\sqrt {a} c \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (b c^2+a d^2\right ) e^2}+\frac {\left (\frac {e^4 \left (-\frac {\left (\left (\sqrt {b} c-\sqrt {a} d\right ) \left (3 b c^2+2 a 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 \sqrt {b} c-4 \sqrt {a} d\right ) \left (b c^2+a d^2\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 )}{2 \sqrt [4]{b} \sqrt {b x^2+a}}\right ) b^{5/2}}{a^{3/2} d e^6}-\frac {2 \left (\sqrt {b} c-\sqrt {a} d\right ) \left (b c^2+a d^2\right ) \sqrt {b x^2+a} b^2}{a^{3/2} d e^4 \sqrt {e x}}\right )}{2 a b^2}-\frac {b c \sqrt {e x} \left (\sqrt {a} \left (\sqrt {b} c-\sqrt {a} d\right ) e-b c \left (1-\frac {\sqrt {b} c}{\sqrt {a} d}\right ) e x\right )}{2 a^2 e^2 \sqrt {b x^2+a}}\right ) d}{c \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (b c^2+a d^2\right ) e}\right )\) |
Input:
Int[1/((e*x)^(3/2)*(c + d*x)*(a + b*x^2)^(3/2)),x]
Output:
2*((d*(-1/2*(b*c*Sqrt[e*x]*(Sqrt[a]*(Sqrt[b]*c - Sqrt[a]*d)*e - b*c*(1 - ( Sqrt[b]*c)/(Sqrt[a]*d))*e*x))/(a^2*e^2*Sqrt[a + b*x^2]) + (e^4*((-2*b^2*(S qrt[b]*c - Sqrt[a]*d)*(b*c^2 + a*d^2)*Sqrt[a + b*x^2])/(a^(3/2)*d*e^4*Sqrt [e*x]) - (b^(5/2)*((Sqrt[b]*c - Sqrt[a]*d)*(3*b*c^2 + 2*a*d^2)*(-((e^2*Sqr t[e*x]*Sqrt[a + b*x^2])/(Sqrt[a]*e + Sqrt[b]*e*x)) + (a^(1/4)*Sqrt[e]*(Sqr t[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*Sqrt[b]*c - 4*Sqrt[a]*d)*(b*c^2 + a*d^2) *Sqrt[e]*(Sqrt[a]*e + Sqrt[b]*e*x)*Sqrt[(a*e^2 + b*e^2*x^2)/(Sqrt[a]*e + S qrt[b]*e*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(2*b^(1/4)*Sqrt[a + b*x^2])))/(a^(3/2)*d*e^6)))/(2*a*b^2)))/(c*((Sqr t[b]*c)/Sqrt[a] - d)*(b*c^2 + a*d^2)*e) + (d^4*(-1/2*((Sqrt[b]*c - Sqrt[a] *d)*Sqrt[e]*ArcTan[(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)/(S qrt[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])))/(Sqrt[a]*c*((Sqrt[b]*c)/Sqrt[a ] - d)*(b*c^2 + a*d^2)*e^2))
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[((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)*(Rt[c/a, 2]*d - e))) Int[(1 + Rt[c/a, 2]*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] + S imp[(-d/e)^(m/2)*((c*d^2 + a*e^2)^(p + 1/2)/(Rt[c/a, 2]*d - e)) Int[x^m*( a + c*x^4)^p*ExpandToSum[(((Rt[c/a, 2]*d - e)*(c*d^2 + a*e^2)^(-p - 1/2))/( -d/e)^(m/2) + ((1 + Rt[c/a, 2]*x^2)*(a + c*x^4)^(-p - 1/2))/(e^(2*p)*x^m))/ (d + e*x^2), x], x], x] /; FreeQ[{a, c, d, e}, x] && ILtQ[p + 1/2, 0] && IL tQ[m/2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[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_)*(x_)^(m_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[a*b^(Floor[(q - 1)/n] + 1)*x^ m*Pq, a + b*x^n, x], R = PolynomialRemainder[a*b^(Floor[(q - 1)/n] + 1)*x^m *Pq, a + b*x^n, x], i}, Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a^2*n*(p + 1)*b^( Floor[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) Int[x^m*(a + b*x^n)^(p + 1)*ExpandToSum[(n*(p + 1)*Q)/x^m + Sum[((n*(p + 1) + i + 1)/a)*Coeff[R, x, i]*x^(i - m), {i, 0, n - 1}], x], x], x]]] /; F reeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]
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 = 2.82 (sec) , antiderivative size = 635, normalized size of antiderivative = 0.89
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (-\frac {2 \left (b e \,x^{2}+a e \right )}{e^{2} a^{2} c \sqrt {x \left (b e \,x^{2}+a e \right )}}-\frac {2 b e x \left (\frac {b c x}{2 e^{2} a^{2} \left (a \,d^{2}+b \,c^{2}\right )}+\frac {d}{2 a \,e^{2} \left (a \,d^{2}+b \,c^{2}\right )}\right )}{\sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}-\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 )}{2 a e \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {b e \,x^{3}+a e x}}+\frac {\left (\frac {b}{a^{2} e c}+\frac {b^{2} c}{2 a^{2} e \left (a \,d^{2}+b \,c^{2}\right )}\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 )}{b \sqrt {b e \,x^{3}+a e x}}-\frac {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}}}\, \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 )}{\left (a \,d^{2}+b \,c^{2}\right ) e c 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}}\) | \(635\) |
| risch | \(-\frac {2 \sqrt {b \,x^{2}+a}}{a^{2} c e \sqrt {e x}}+\frac {\left (\frac {\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 {a^{2} 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}}}\, \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 )}{\left (a \,d^{2}+b \,c^{2}\right ) b \sqrt {b e \,x^{3}+a e x}\, \left (-\frac {\sqrt {-a b}}{b}+\frac {c}{d}\right )}-\frac {a b c \left (-\frac {2 x b e \left (-\frac {c x}{2 a e}-\frac {d}{2 e b}\right )}{\sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}+\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 )}{2 b \sqrt {b e \,x^{3}+a e x}}-\frac {c \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 )}{2 a \sqrt {b e \,x^{3}+a e x}}\right )}{a \,d^{2}+b \,c^{2}}\right ) \sqrt {\left (b \,x^{2}+a \right ) e x}}{a^{2} c e \sqrt {e x}\, \sqrt {b \,x^{2}+a}}\) | \(729\) |
| default | \(\text {Expression too large to display}\) | \(1028\) |
Input:
int(1/(e*x)^(3/2)/(d*x+c)/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
((b*x^2+a)*e*x)^(1/2)/(e*x)^(1/2)/(b*x^2+a)^(1/2)*(-2*(b*e*x^2+a*e)/e^2/a^ 2/c/(x*(b*e*x^2+a*e))^(1/2)-2*b*e*x*(1/2/e^2/a^2*b*c/(a*d^2+b*c^2)*x+1/2*d /a/e^2/(a*d^2+b*c^2))/((x^2+a/b)*b*e*x)^(1/2)-1/2/a/e/(a*d^2+b*c^2)*d*(-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)*El lipticF(((x+(-a*b)^(1/2)/b)/(-a*b)^(1/2)*b)^(1/2),1/2*2^(1/2))+(b/a^2/e/c+ 1/2/a^2*b^2*c/e/(a*d^2+b*c^2))*(-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)*(-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)))-d^2/(a*d^2+b*c^2)/e/c* (-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)))
Timed out. \[ \int \frac {1}{(e x)^{3/2} (c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x)^(3/2)/(d*x+c)/(b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{(e x)^{3/2} (c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (e x\right )^{\frac {3}{2}} \left (a + b x^{2}\right )^{\frac {3}{2}} \left (c + d x\right )}\, dx \] Input:
integrate(1/(e*x)**(3/2)/(d*x+c)/(b*x**2+a)**(3/2),x)
Output:
Integral(1/((e*x)**(3/2)*(a + b*x**2)**(3/2)*(c + d*x)), x)
\[ \int \frac {1}{(e x)^{3/2} (c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(e*x)^(3/2)/(d*x+c)/(b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((b*x^2 + a)^(3/2)*(d*x + c)*(e*x)^(3/2)), x)
\[ \int \frac {1}{(e x)^{3/2} (c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )} \left (e x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(e*x)^(3/2)/(d*x+c)/(b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate(1/((b*x^2 + a)^(3/2)*(d*x + c)*(e*x)^(3/2)), x)
Timed out. \[ \int \frac {1}{(e x)^{3/2} (c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (e\,x\right )}^{3/2}\,{\left (b\,x^2+a\right )}^{3/2}\,\left (c+d\,x\right )} \,d x \] Input:
int(1/((e*x)^(3/2)*(a + b*x^2)^(3/2)*(c + d*x)),x)
Output:
int(1/((e*x)^(3/2)*(a + b*x^2)^(3/2)*(c + d*x)), x)
\[ \int \frac {1}{(e x)^{3/2} (c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\frac {\sqrt {e}\, \left (-2 \sqrt {b \,x^{2}+a}-\sqrt {x}\, \left (\int \frac {\sqrt {b \,x^{2}+a}}{\sqrt {x}\, a^{2} c +\sqrt {x}\, a^{2} d x +2 \sqrt {x}\, a b c \,x^{2}+2 \sqrt {x}\, a b d \,x^{3}+\sqrt {x}\, b^{2} c \,x^{4}+\sqrt {x}\, b^{2} d \,x^{5}}d x \right ) a^{2} d -\sqrt {x}\, \left (\int \frac {\sqrt {b \,x^{2}+a}}{\sqrt {x}\, a^{2} c +\sqrt {x}\, a^{2} d x +2 \sqrt {x}\, a b c \,x^{2}+2 \sqrt {x}\, a b d \,x^{3}+\sqrt {x}\, b^{2} c \,x^{4}+\sqrt {x}\, b^{2} d \,x^{5}}d x \right ) a b d \,x^{2}-3 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}\, x}{b^{2} d \,x^{5}+b^{2} c \,x^{4}+2 a b d \,x^{3}+2 a b c \,x^{2}+a^{2} d x +a^{2} c}d x \right ) a b d -3 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}\, x}{b^{2} d \,x^{5}+b^{2} c \,x^{4}+2 a b d \,x^{3}+2 a b c \,x^{2}+a^{2} d x +a^{2} c}d x \right ) b^{2} d \,x^{2}-3 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b^{2} d \,x^{5}+b^{2} c \,x^{4}+2 a b d \,x^{3}+2 a b c \,x^{2}+a^{2} d x +a^{2} c}d x \right ) a b c -3 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b^{2} d \,x^{5}+b^{2} c \,x^{4}+2 a b d \,x^{3}+2 a b c \,x^{2}+a^{2} d x +a^{2} c}d x \right ) b^{2} c \,x^{2}\right )}{\sqrt {x}\, a c \,e^{2} \left (b \,x^{2}+a \right )} \] Input:
int(1/(e*x)^(3/2)/(d*x+c)/(b*x^2+a)^(3/2),x)
Output:
(sqrt(e)*( - 2*sqrt(a + b*x**2) - sqrt(x)*int(sqrt(a + b*x**2)/(sqrt(x)*a* *2*c + sqrt(x)*a**2*d*x + 2*sqrt(x)*a*b*c*x**2 + 2*sqrt(x)*a*b*d*x**3 + sq rt(x)*b**2*c*x**4 + sqrt(x)*b**2*d*x**5),x)*a**2*d - sqrt(x)*int(sqrt(a + b*x**2)/(sqrt(x)*a**2*c + sqrt(x)*a**2*d*x + 2*sqrt(x)*a*b*c*x**2 + 2*sqrt (x)*a*b*d*x**3 + sqrt(x)*b**2*c*x**4 + sqrt(x)*b**2*d*x**5),x)*a*b*d*x**2 - 3*sqrt(x)*int((sqrt(x)*sqrt(a + b*x**2)*x)/(a**2*c + a**2*d*x + 2*a*b*c* x**2 + 2*a*b*d*x**3 + b**2*c*x**4 + b**2*d*x**5),x)*a*b*d - 3*sqrt(x)*int( (sqrt(x)*sqrt(a + b*x**2)*x)/(a**2*c + a**2*d*x + 2*a*b*c*x**2 + 2*a*b*d*x **3 + b**2*c*x**4 + b**2*d*x**5),x)*b**2*d*x**2 - 3*sqrt(x)*int((sqrt(x)*s qrt(a + b*x**2))/(a**2*c + a**2*d*x + 2*a*b*c*x**2 + 2*a*b*d*x**3 + b**2*c *x**4 + b**2*d*x**5),x)*a*b*c - 3*sqrt(x)*int((sqrt(x)*sqrt(a + b*x**2))/( a**2*c + a**2*d*x + 2*a*b*c*x**2 + 2*a*b*d*x**3 + b**2*c*x**4 + b**2*d*x** 5),x)*b**2*c*x**2))/(sqrt(x)*a*c*e**2*(a + b*x**2))