Integrand size = 27, antiderivative size = 397 \[ \int \frac {\sqrt {a-b x^2}}{(e x)^{7/2} (c+d x)} \, dx=-\frac {2 \sqrt {a-b x^2}}{5 c e (e x)^{5/2}}+\frac {2 d \sqrt {a-b x^2}}{3 c^2 e^2 (e x)^{3/2}}+\frac {2 \left (2 b c^2-5 a d^2\right ) \sqrt {a-b x^2}}{5 a c^3 e^3 \sqrt {e x}}+\frac {2 \sqrt [4]{b} \left (2 b c^2-5 a d^2\right ) \sqrt {1-\frac {b x^2}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )\right |-1\right )}{5 \sqrt [4]{a} c^3 e^{7/2} \sqrt {a-b x^2}}-\frac {2 \sqrt [4]{b} \left (6 b c^2+5 \sqrt {a} \sqrt {b} c d-15 a d^2\right ) \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{15 \sqrt [4]{a} c^3 e^{7/2} \sqrt {a-b x^2}}+\frac {2 \sqrt [4]{a} d \left (b c^2-a d^2\right ) \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^4 e^{7/2} \sqrt {a-b x^2}} \] 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 ^(1/4)*(-5*a*d^2+2*b*c^2)*(1-b*x^2/a)^(1/2)*EllipticE(b^(1/4)*(e*x)^(1/2)/ a^(1/4)/e^(1/2),I)/a^(1/4)/c^3/e^(7/2)/(-b*x^2+a)^(1/2)-2/15*b^(1/4)*(6*b* c^2+5*a^(1/2)*b^(1/2)*c*d-15*a*d^2)*(1-b*x^2/a)^(1/2)*EllipticF(b^(1/4)*(e *x)^(1/2)/a^(1/4)/e^(1/2),I)/a^(1/4)/c^3/e^(7/2)/(-b*x^2+a)^(1/2)+2*a^(1/4 )*d*(-a*d^2+b*c^2)*(1-b*x^2/a)^(1/2)*EllipticPi(b^(1/4)*(e*x)^(1/2)/a^(1/4 )/e^(1/2),-a^(1/2)*d/b^(1/2)/c,I)/b^(1/4)/c^4/e^(7/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 24.10 (sec) , antiderivative size = 344, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {a-b x^2}}{(e x)^{7/2} (c+d x)} \, dx=\frac {2 x \left (3 i \sqrt {b} c \left (2 b c^2-5 a d^2\right ) \sqrt {1-\frac {a}{b x^2}} x^{7/2} E\left (\left .i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right )\right |-1\right )-i \left (6 b^{3/2} c^3+10 \sqrt {a} b c^2 d-15 a \sqrt {b} c d^2-15 a^{3/2} d^3\right ) \sqrt {1-\frac {a}{b x^2}} x^{7/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )+\sqrt {a} \left (\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}} c^2 (3 c-5 d x) \left (a-b x^2\right )+15 i d \left (b c^2-a d^2\right ) \sqrt {1-\frac {a}{b x^2}} x^{7/2} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} c}{\sqrt {a} d},i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )\right )\right )}{15 \left (-\frac {\sqrt {a}}{\sqrt {b}}\right )^{3/2} \sqrt {b} c^4 (e x)^{7/2} \sqrt {a-b x^2}} \] Input:
Integrate[Sqrt[a - b*x^2]/((e*x)^(7/2)*(c + d*x)),x]
Output:
(2*x*((3*I)*Sqrt[b]*c*(2*b*c^2 - 5*a*d^2)*Sqrt[1 - a/(b*x^2)]*x^(7/2)*Elli pticE[I*ArcSinh[Sqrt[-(Sqrt[a]/Sqrt[b])]/Sqrt[x]], -1] - I*(6*b^(3/2)*c^3 + 10*Sqrt[a]*b*c^2*d - 15*a*Sqrt[b]*c*d^2 - 15*a^(3/2)*d^3)*Sqrt[1 - a/(b* x^2)]*x^(7/2)*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[a]/Sqrt[b])]/Sqrt[x]], -1] + Sqrt[a]*(Sqrt[-(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[-((Sqrt[b]*c)/(Sq rt[a]*d)), I*ArcSinh[Sqrt[-(Sqrt[a]/Sqrt[b])]/Sqrt[x]], -1])))/(15*(-(Sqrt [a]/Sqrt[b]))^(3/2)*Sqrt[b]*c^4*(e*x)^(7/2)*Sqrt[a - b*x^2])
Time = 2.05 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.04, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.815, Rules used = {616, 27, 1637, 25, 1543, 1542, 2374, 9, 27, 1605, 25, 27, 1605, 25, 27, 1513, 27, 765, 762, 1390, 1389, 327}
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 |
| \(\displaystyle \frac {2 \int \frac {\sqrt {a-b x^2}}{e^2 x^3 (c e+d x e)}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {\sqrt {a-b x^2}}{e^3 x^3 (c e+d x e)}d\sqrt {e x}\) |
| \(\Big \downarrow \) 1637 |
| \(\displaystyle 2 \left (\frac {d \left (b c^2-a d^2\right ) \int \frac {1}{(c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}}{c^3 e^3}-\frac {d \int -\frac {-\frac {\left (b c^2-a d^2\right ) x^2 e^2}{d}-a c x e^2+\frac {a c^2 e^2}{d}}{e^3 x^3 \sqrt {a-b x^2}}d\sqrt {e x}}{c^3 e^3}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {d \left (b c^2-a d^2\right ) \int \frac {1}{(c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}}{c^3 e^3}+\frac {d \int \frac {-\frac {\left (b c^2-a d^2\right ) x^2 e^2}{d}-a c x e^2+\frac {a c^2 e^2}{d}}{e^3 x^3 \sqrt {a-b x^2}}d\sqrt {e x}}{c^3 e^3}\right )\) |
| \(\Big \downarrow \) 1543 |
| \(\displaystyle 2 \left (\frac {d \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \int \frac {1}{(c e+d x e) \sqrt {1-\frac {b x^2}{a}}}d\sqrt {e x}}{c^3 e^3 \sqrt {a-b x^2}}+\frac {d \int \frac {-\frac {\left (b c^2-a d^2\right ) x^2 e^2}{d}-a c x e^2+\frac {a c^2 e^2}{d}}{e^3 x^3 \sqrt {a-b x^2}}d\sqrt {e x}}{c^3 e^3}\right )\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle 2 \left (\frac {d \int \frac {-\frac {\left (b c^2-a d^2\right ) x^2 e^2}{d}-a c x e^2+\frac {a c^2 e^2}{d}}{e^3 x^3 \sqrt {a-b x^2}}d\sqrt {e x}}{c^3 e^3}+\frac {\sqrt [4]{a} d \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^4 e^{7/2} \sqrt {a-b x^2}}\right )\) |
| \(\Big \downarrow \) 2374 |
| \(\displaystyle 2 \left (\frac {d \left (-\frac {\int \frac {2 \left (5 c e \sqrt {e x} a^2+\left (\frac {2 b c^2}{d}-5 a d\right ) (e x)^{3/2} a\right )}{(e x)^{5/2} \sqrt {a-b x^2}}d\sqrt {e x}}{10 a}-\frac {c^2 e^2 \sqrt {a-b x^2}}{5 d (e x)^{5/2}}\right )}{c^3 e^3}+\frac {\sqrt [4]{a} d \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^4 e^{7/2} \sqrt {a-b x^2}}\right )\) |
| \(\Big \downarrow \) 9 |
| \(\displaystyle 2 \left (\frac {d \left (-\frac {\int \frac {2 a \left (5 a c e+\left (\frac {2 b c^2}{d}-5 a d\right ) x e\right )}{e^2 x^2 \sqrt {a-b x^2}}d\sqrt {e x}}{10 a}-\frac {c^2 e^2 \sqrt {a-b x^2}}{5 d (e x)^{5/2}}\right )}{c^3 e^3}+\frac {\sqrt [4]{a} d \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^4 e^{7/2} \sqrt {a-b x^2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {d \left (-\frac {1}{5} \int \frac {5 a c e+\left (\frac {2 b c^2}{d}-5 a d\right ) x e}{e^2 x^2 \sqrt {a-b x^2}}d\sqrt {e x}-\frac {c^2 e^2 \sqrt {a-b x^2}}{5 d (e x)^{5/2}}\right )}{c^3 e^3}+\frac {\sqrt [4]{a} d \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^4 e^{7/2} \sqrt {a-b x^2}}\right )\) |
| \(\Big \downarrow \) 1605 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {1}{5} \left (\frac {\int -\frac {a \left (3 \left (\frac {2 b c^2}{d}-5 a d\right ) e+5 b c x e\right )}{e^2 x \sqrt {a-b x^2}}d\sqrt {e x}}{3 a}+\frac {5 c e \sqrt {a-b x^2}}{3 (e x)^{3/2}}\right )-\frac {c^2 e^2 \sqrt {a-b x^2}}{5 d (e x)^{5/2}}\right )}{c^3 e^3}+\frac {\sqrt [4]{a} d \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^4 e^{7/2} \sqrt {a-b x^2}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {1}{5} \left (\frac {5 c e \sqrt {a-b x^2}}{3 (e x)^{3/2}}-\frac {\int \frac {a \left (3 \left (\frac {2 b c^2}{d}-5 a d\right ) e+5 b c x e\right )}{e^2 x \sqrt {a-b x^2}}d\sqrt {e x}}{3 a}\right )-\frac {c^2 e^2 \sqrt {a-b x^2}}{5 d (e x)^{5/2}}\right )}{c^3 e^3}+\frac {\sqrt [4]{a} d \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^4 e^{7/2} \sqrt {a-b x^2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {1}{5} \left (\frac {5 c e \sqrt {a-b x^2}}{3 (e x)^{3/2}}-\frac {\int \frac {3 \left (\frac {2 b c^2}{d}-5 a d\right ) e+5 b c x e}{e x \sqrt {a-b x^2}}d\sqrt {e x}}{3 e}\right )-\frac {c^2 e^2 \sqrt {a-b x^2}}{5 d (e x)^{5/2}}\right )}{c^3 e^3}+\frac {\sqrt [4]{a} d \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^4 e^{7/2} \sqrt {a-b x^2}}\right )\) |
| \(\Big \downarrow \) 1605 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {1}{5} \left (\frac {5 c e \sqrt {a-b x^2}}{3 (e x)^{3/2}}-\frac {-\frac {\int -\frac {b \left (5 a c e-3 \left (\frac {2 b c^2}{d}-5 a d\right ) e x\right )}{e \sqrt {a-b x^2}}d\sqrt {e x}}{a}-\frac {3 e \sqrt {a-b x^2} \left (\frac {2 b c^2}{d}-5 a d\right )}{a \sqrt {e x}}}{3 e}\right )-\frac {c^2 e^2 \sqrt {a-b x^2}}{5 d (e x)^{5/2}}\right )}{c^3 e^3}+\frac {\sqrt [4]{a} d \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^4 e^{7/2} \sqrt {a-b x^2}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {1}{5} \left (\frac {5 c e \sqrt {a-b x^2}}{3 (e x)^{3/2}}-\frac {\frac {\int \frac {b \left (5 a c e-3 \left (\frac {2 b c^2}{d}-5 a d\right ) e x\right )}{e \sqrt {a-b x^2}}d\sqrt {e x}}{a}-\frac {3 e \sqrt {a-b x^2} \left (\frac {2 b c^2}{d}-5 a d\right )}{a \sqrt {e x}}}{3 e}\right )-\frac {c^2 e^2 \sqrt {a-b x^2}}{5 d (e x)^{5/2}}\right )}{c^3 e^3}+\frac {\sqrt [4]{a} d \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^4 e^{7/2} \sqrt {a-b x^2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {1}{5} \left (\frac {5 c e \sqrt {a-b x^2}}{3 (e x)^{3/2}}-\frac {\frac {b \int \frac {5 a c e-3 \left (\frac {2 b c^2}{d}-5 a d\right ) e x}{\sqrt {a-b x^2}}d\sqrt {e x}}{a e}-\frac {3 e \sqrt {a-b x^2} \left (\frac {2 b c^2}{d}-5 a d\right )}{a \sqrt {e x}}}{3 e}\right )-\frac {c^2 e^2 \sqrt {a-b x^2}}{5 d (e x)^{5/2}}\right )}{c^3 e^3}+\frac {\sqrt [4]{a} d \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^4 e^{7/2} \sqrt {a-b x^2}}\right )\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {1}{5} \left (\frac {5 c e \sqrt {a-b x^2}}{3 (e x)^{3/2}}-\frac {\frac {b \left (\sqrt {a} e \left (\frac {3 \left (\frac {2 b c^2}{d}-5 a d\right )}{\sqrt {b}}+5 \sqrt {a} c\right ) \int \frac {1}{\sqrt {a-b x^2}}d\sqrt {e x}-\frac {3 \sqrt {a} e \left (\frac {2 b c^2}{d}-5 a d\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a} e \sqrt {a-b x^2}}d\sqrt {e x}}{\sqrt {b}}\right )}{a e}-\frac {3 e \sqrt {a-b x^2} \left (\frac {2 b c^2}{d}-5 a d\right )}{a \sqrt {e x}}}{3 e}\right )-\frac {c^2 e^2 \sqrt {a-b x^2}}{5 d (e x)^{5/2}}\right )}{c^3 e^3}+\frac {\sqrt [4]{a} d \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^4 e^{7/2} \sqrt {a-b x^2}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {1}{5} \left (\frac {5 c e \sqrt {a-b x^2}}{3 (e x)^{3/2}}-\frac {\frac {b \left (\sqrt {a} e \left (\frac {3 \left (\frac {2 b c^2}{d}-5 a d\right )}{\sqrt {b}}+5 \sqrt {a} c\right ) \int \frac {1}{\sqrt {a-b x^2}}d\sqrt {e x}-\frac {3 \left (\frac {2 b c^2}{d}-5 a d\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a-b x^2}}d\sqrt {e x}}{\sqrt {b}}\right )}{a e}-\frac {3 e \sqrt {a-b x^2} \left (\frac {2 b c^2}{d}-5 a d\right )}{a \sqrt {e x}}}{3 e}\right )-\frac {c^2 e^2 \sqrt {a-b x^2}}{5 d (e x)^{5/2}}\right )}{c^3 e^3}+\frac {\sqrt [4]{a} d \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^4 e^{7/2} \sqrt {a-b x^2}}\right )\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {1}{5} \left (\frac {5 c e \sqrt {a-b x^2}}{3 (e x)^{3/2}}-\frac {\frac {b \left (\frac {\sqrt {a} e \sqrt {1-\frac {b x^2}{a}} \left (\frac {3 \left (\frac {2 b c^2}{d}-5 a d\right )}{\sqrt {b}}+5 \sqrt {a} c\right ) \int \frac {1}{\sqrt {1-\frac {b x^2}{a}}}d\sqrt {e x}}{\sqrt {a-b x^2}}-\frac {3 \left (\frac {2 b c^2}{d}-5 a d\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a-b x^2}}d\sqrt {e x}}{\sqrt {b}}\right )}{a e}-\frac {3 e \sqrt {a-b x^2} \left (\frac {2 b c^2}{d}-5 a d\right )}{a \sqrt {e x}}}{3 e}\right )-\frac {c^2 e^2 \sqrt {a-b x^2}}{5 d (e x)^{5/2}}\right )}{c^3 e^3}+\frac {\sqrt [4]{a} d \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^4 e^{7/2} \sqrt {a-b x^2}}\right )\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {1}{5} \left (\frac {5 c e \sqrt {a-b x^2}}{3 (e x)^{3/2}}-\frac {\frac {b \left (\frac {a^{3/4} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (\frac {3 \left (\frac {2 b c^2}{d}-5 a d\right )}{\sqrt {b}}+5 \sqrt {a} c\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}-\frac {3 \left (\frac {2 b c^2}{d}-5 a d\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a-b x^2}}d\sqrt {e x}}{\sqrt {b}}\right )}{a e}-\frac {3 e \sqrt {a-b x^2} \left (\frac {2 b c^2}{d}-5 a d\right )}{a \sqrt {e x}}}{3 e}\right )-\frac {c^2 e^2 \sqrt {a-b x^2}}{5 d (e x)^{5/2}}\right )}{c^3 e^3}+\frac {\sqrt [4]{a} d \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^4 e^{7/2} \sqrt {a-b x^2}}\right )\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {1}{5} \left (\frac {5 c e \sqrt {a-b x^2}}{3 (e x)^{3/2}}-\frac {\frac {b \left (\frac {a^{3/4} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (\frac {3 \left (\frac {2 b c^2}{d}-5 a d\right )}{\sqrt {b}}+5 \sqrt {a} c\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}-\frac {3 \sqrt {1-\frac {b x^2}{a}} \left (\frac {2 b c^2}{d}-5 a d\right ) \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {1-\frac {b x^2}{a}}}d\sqrt {e x}}{\sqrt {b} \sqrt {a-b x^2}}\right )}{a e}-\frac {3 e \sqrt {a-b x^2} \left (\frac {2 b c^2}{d}-5 a d\right )}{a \sqrt {e x}}}{3 e}\right )-\frac {c^2 e^2 \sqrt {a-b x^2}}{5 d (e x)^{5/2}}\right )}{c^3 e^3}+\frac {\sqrt [4]{a} d \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^4 e^{7/2} \sqrt {a-b x^2}}\right )\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {1}{5} \left (\frac {5 c e \sqrt {a-b x^2}}{3 (e x)^{3/2}}-\frac {\frac {b \left (\frac {a^{3/4} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (\frac {3 \left (\frac {2 b c^2}{d}-5 a d\right )}{\sqrt {b}}+5 \sqrt {a} c\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}-\frac {3 \sqrt {a} e \sqrt {1-\frac {b x^2}{a}} \left (\frac {2 b c^2}{d}-5 a d\right ) \int \frac {\sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}d\sqrt {e x}}{\sqrt {b} \sqrt {a-b x^2}}\right )}{a e}-\frac {3 e \sqrt {a-b x^2} \left (\frac {2 b c^2}{d}-5 a d\right )}{a \sqrt {e x}}}{3 e}\right )-\frac {c^2 e^2 \sqrt {a-b x^2}}{5 d (e x)^{5/2}}\right )}{c^3 e^3}+\frac {\sqrt [4]{a} d \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^4 e^{7/2} \sqrt {a-b x^2}}\right )\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle 2 \left (\frac {d \left (\frac {1}{5} \left (\frac {5 c e \sqrt {a-b x^2}}{3 (e x)^{3/2}}-\frac {\frac {b \left (\frac {a^{3/4} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (\frac {3 \left (\frac {2 b c^2}{d}-5 a d\right )}{\sqrt {b}}+5 \sqrt {a} c\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^2}}-\frac {3 a^{3/4} e^{3/2} \sqrt {1-\frac {b x^2}{a}} \left (\frac {2 b c^2}{d}-5 a d\right ) E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )\right |-1\right )}{b^{3/4} \sqrt {a-b x^2}}\right )}{a e}-\frac {3 e \sqrt {a-b x^2} \left (\frac {2 b c^2}{d}-5 a d\right )}{a \sqrt {e x}}}{3 e}\right )-\frac {c^2 e^2 \sqrt {a-b x^2}}{5 d (e x)^{5/2}}\right )}{c^3 e^3}+\frac {\sqrt [4]{a} d \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c^4 e^{7/2} \sqrt {a-b x^2}}\right )\) |
Input:
Int[Sqrt[a - b*x^2]/((e*x)^(7/2)*(c + d*x)),x]
Output:
2*((d*(-1/5*(c^2*e^2*Sqrt[a - b*x^2])/(d*(e*x)^(5/2)) + ((5*c*e*Sqrt[a - b *x^2])/(3*(e*x)^(3/2)) - ((-3*((2*b*c^2)/d - 5*a*d)*e*Sqrt[a - b*x^2])/(a* Sqrt[e*x]) + (b*((-3*a^(3/4)*((2*b*c^2)/d - 5*a*d)*e^(3/2)*Sqrt[1 - (b*x^2 )/a]*EllipticE[ArcSin[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], -1])/(b^(3/4 )*Sqrt[a - b*x^2]) + (a^(3/4)*(5*Sqrt[a]*c + (3*((2*b*c^2)/d - 5*a*d))/Sqr t[b])*e^(3/2)*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSin[(b^(1/4)*Sqrt[e*x])/(a^ (1/4)*Sqrt[e])], -1])/(b^(1/4)*Sqrt[a - b*x^2])))/(a*e))/(3*e))/5))/(c^3*e ^3) + (a^(1/4)*d*(b*c^2 - a*d^2)*Sqrt[1 - (b*x^2)/a]*EllipticPi[-((Sqrt[a] *d)/(Sqrt[b]*c)), ArcSin[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], -1])/(b^( 1/4)*c^4*e^(7/2)*Sqrt[a - b*x^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[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
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] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt [a + b*x^4] Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ [b/a] && !GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq rt[a] Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt [1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && !GtQ [a, 0] && !(LtQ[a, 0] && GtQ[c, 0])
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-c/a, 2]}, Simp[(d*q - e)/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]] /; FreeQ[{a, c, d, e}, x] && Neg Q[c/a] && NeQ[c*d^2 + a*e^2, 0]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ {q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x ], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[ Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[1/((d + e*x^2)*Sqrt[1 + c*(x^4/a) ]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && !GtQ[a, 0]
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 + 1)) Int[1/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] + Simp[(-d/e)^(m/2)/e^(2*p + 1) Int[(x^m /Sqrt[a + c*x^4])*ExpandToSum[((e^(2*p + 1)*(a + c*x^4)^(p + 1/2))/(-d/e)^( m/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] & & NegQ[c/a]
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 = 1.17 (sec) , antiderivative size = 574, normalized size of antiderivative = 1.45
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (-b \,x^{2}+a \right ) e x}\, \left (-\frac {2 \sqrt {-b e \,x^{3}+a e x}}{5 e^{4} c \,x^{3}}+\frac {2 d \sqrt {-b e \,x^{3}+a e x}}{3 e^{4} c^{2} x^{2}}-\frac {2 \left (-b e \,x^{2}+a e \right ) \left (5 a \,d^{2}-2 b \,c^{2}\right )}{5 e^{4} a \,c^{3} \sqrt {x \left (-b e \,x^{2}+a e \right )}}-\frac {d \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{3 e^{3} c^{2} \sqrt {-b e \,x^{3}+a e x}}-\frac {\left (5 a \,d^{2}-2 b \,c^{2}\right ) \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \left (-\frac {2 \sqrt {a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{5 a \,e^{3} c^{3} \sqrt {-b e \,x^{3}+a e x}}-\frac {\left (a \,d^{2}-b \,c^{2}\right ) \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{e^{3} c^{3} b \sqrt {-b e \,x^{3}+a e x}\, \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}\right )}{\sqrt {e x}\, \sqrt {-b \,x^{2}+a}}\) | \(574\) |
| default | \(\text {Expression too large to display}\) | \(1144\) |
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+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)*(-2*(x-1/b*(a*b)^(1/2))* b/(a*b)^(1/2))^(1/2)*(-b*x/(a*b)^(1/2))^(1/2)/(-b*e*x^3+a*e*x)^(1/2)*Ellip ticF(((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(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+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)*(- 2*(x-1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)*(-b*x/(a*b)^(1/2))^(1/2)/(-b*e* x^3+a*e*x)^(1/2)*(-2/b*(a*b)^(1/2)*EllipticE(((x+1/b*(a*b)^(1/2))*b/(a*b)^ (1/2))^(1/2),1/2*2^(1/2))+1/b*(a*b)^(1/2)*EllipticF(((x+1/b*(a*b)^(1/2))*b /(a*b)^(1/2))^(1/2),1/2*2^(1/2)))-1/e^3*(a*d^2-b*c^2)/c^3/b*(a*b)^(1/2)*(( x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)*(-2*(x-1/b*(a*b)^(1/2))*b/(a*b)^(1 /2))^(1/2)*(-b*x/(a*b)^(1/2))^(1/2)/(-b*e*x^3+a*e*x)^(1/2)/(c/d-1/b*(a*b)^ (1/2))*EllipticPi(((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2),-1/b*(a*b)^(1/ 2)/(c/d-1/b*(a*b)^(1/2)),1/2*2^(1/2)))
Timed out. \[ \int \frac {\sqrt {a-b x^2}}{(e x)^{7/2} (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((-b*x^2+a)^(1/2)/(e*x)^(7/2)/(d*x+c),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\sqrt {a-b x^2}}{(e x)^{7/2} (c+d x)} \, dx=\int \frac {\sqrt {a - b x^{2}}}{\left (e x\right )^{\frac {7}{2}} \left (c + d x\right )}\, dx \] 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. \[ \int \frac {\sqrt {a-b x^2}}{(e x)^{7/2} (c+d x)} \, dx=\int \frac {\sqrt {a-b\,x^2}}{{\left (e\,x\right )}^{7/2}\,\left (c+d\,x\right )} \,d x \] 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