Integrand size = 26, antiderivative size = 597 \[ \int \frac {\sqrt {e x}}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\frac {\sqrt {e x} (a d+b c x)}{a \left (b c^2+a d^2\right ) \sqrt {a+b x^2}}-\frac {\sqrt {b} c \sqrt {e x} \sqrt {a+b x^2}}{a \left (b c^2+a d^2\right ) \left (\sqrt {a}+\sqrt {b} x\right )}-\frac {\sqrt {c} d^{3/2} \sqrt {e} \arctan \left (\frac {\sqrt {b c^2+a d^2} \sqrt {e x}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{\left (b c^2+a d^2\right )^{3/2}}+\frac {\sqrt [4]{b} c \sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{a^{3/4} \left (b c^2+a d^2\right ) \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 a^{3/4} \sqrt [4]{b} \left (\sqrt {b} c-\sqrt {a} d\right ) \sqrt {a+b x^2}}+\frac {d \left (\sqrt {b} c+\sqrt {a} d\right ) \sqrt {e} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {b} c-\sqrt {a} d\right )^2}{4 \sqrt {a} \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (b c^2+a d^2\right ) \sqrt {a+b x^2}} \] Output:
(e*x)^(1/2)*(b*c*x+a*d)/a/(a*d^2+b*c^2)/(b*x^2+a)^(1/2)-b^(1/2)*c*(e*x)^(1 /2)*(b*x^2+a)^(1/2)/a/(a*d^2+b*c^2)/(a^(1/2)+b^(1/2)*x)-c^(1/2)*d^(3/2)*e^ (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))/(a*d^2+b*c^2)^(3/2)+b^(1/4)*c*e^(1/2)*(a^(1/2)+b^(1/2)*x)*((b* x^2+a)/(a^(1/2)+b^(1/2)*x)^2)^(1/2)*EllipticE(sin(2*arctan(b^(1/4)*(e*x)^( 1/2)/a^(1/4)/e^(1/2))),1/2*2^(1/2))/a^(3/4)/(a*d^2+b*c^2)/(b*x^2+a)^(1/2)- 1/2*e^(1/2)*(a^(1/2)+b^(1/2)*x)*((b*x^2+a)/(a^(1/2)+b^(1/2)*x)^2)^(1/2)*In verseJacobiAM(2*arctan(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2)),1/2*2^(1/2))/a ^(3/4)/b^(1/4)/(b^(1/2)*c-a^(1/2)*d)/(b*x^2+a)^(1/2)+1/2*d*(b^(1/2)*c+a^(1 /2)*d)*e^(1/2)*(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)/(b^(1/2 )*c-a^(1/2)*d)/(a*d^2+b*c^2)/(b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 23.30 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.49 \[ \int \frac {\sqrt {e x}}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\frac {e \left (\sqrt {b} c \sqrt {1+\frac {a}{b x^2}} x^{3/2} E\left (\left .i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right )\right |-1\right )-\left (\sqrt {b} c+i \sqrt {a} d\right ) \sqrt {1+\frac {a}{b x^2}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )+\sqrt {a} \left (\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}} (-c+d x)+2 i d \sqrt {1+\frac {a}{b x^2}} x^{3/2} \operatorname {EllipticPi}\left (-\frac {i \sqrt {b} c}{\sqrt {a} d},i \text {arcsinh}\left (\frac {\sqrt {\frac {i \sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )\right )\right )}{\sqrt {a} \sqrt {\frac {i \sqrt {a}}{\sqrt {b}}} \left (b c^2+a d^2\right ) \sqrt {e x} \sqrt {a+b x^2}} \] Input:
Integrate[Sqrt[e*x]/((c + d*x)*(a + b*x^2)^(3/2)),x]
Output:
(e*(Sqrt[b]*c*Sqrt[1 + a/(b*x^2)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt [a])/Sqrt[b]]/Sqrt[x]], -1] - (Sqrt[b]*c + I*Sqrt[a]*d)*Sqrt[1 + a/(b*x^2) ]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/Sqrt[x]], -1] + Sq rt[a]*(Sqrt[(I*Sqrt[a])/Sqrt[b]]*(-c + d*x) + (2*I)*d*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])))/(Sqrt[a]*Sqrt[(I*Sqrt[a])/Sqrt[b]]*(b*c^2 + a* d^2)*Sqrt[e*x]*Sqrt[a + b*x^2])
Time = 1.35 (sec) , antiderivative size = 741, normalized size of antiderivative = 1.24, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {616, 27, 1639, 25, 27, 2221, 2397, 25, 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.
| \(\displaystyle \int \frac {\sqrt {e x}}{\left (a+b x^2\right )^{3/2} (c+d x)} \, dx\) |
| \(\Big \downarrow \) 616 |
| \(\displaystyle \frac {2 \int \frac {e^2 x}{(c e+d x e) \left (b x^2+a\right )^{3/2}}d\sqrt {e x}}{e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {e x}{(c e+d x e) \left (b x^2+a\right )^{3/2}}d\sqrt {e x}\) |
| \(\Big \downarrow \) 1639 |
| \(\displaystyle 2 \left (\frac {\int -\frac {a d^2+\frac {b^{3/2} c x^2 d}{\sqrt {a}}-b c \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) x}{\left (b x^2+a\right )^{3/2}}d\sqrt {e x}}{\left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}+\frac {c d^2 e \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a} e (c e+d x e) \sqrt {b x^2+a}}d\sqrt {e x}}{\left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {c d^2 e \int \frac {\sqrt {b} x e+\sqrt {a} e}{\sqrt {a} e (c e+d x e) \sqrt {b x^2+a}}d\sqrt {e x}}{\left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}-\frac {\int \frac {a d^2+\frac {b^{3/2} c x^2 d}{\sqrt {a}}-b c \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) x}{\left (b x^2+a\right )^{3/2}}d\sqrt {e x}}{\left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {c d^2 \int \frac {\sqrt {b} x e+\sqrt {a} e}{(c e+d x e) \sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}-\frac {\int \frac {a d^2+\frac {b^{3/2} c x^2 d}{\sqrt {a}}-b c \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) x}{\left (b x^2+a\right )^{3/2}}d\sqrt {e x}}{\left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}\right )\) |
| \(\Big \downarrow \) 2221 |
| \(\displaystyle 2 \left (\frac {c d^2 \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}-\frac {\int \frac {a d^2+\frac {b^{3/2} c x^2 d}{\sqrt {a}}-b c \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) x}{\left (b x^2+a\right )^{3/2}}d\sqrt {e x}}{\left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}\right )\) |
| \(\Big \downarrow \) 2397 |
| \(\displaystyle 2 \left (\frac {c d^2 \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}-\frac {-\frac {e^2 \int -\frac {b \left (\sqrt {a} d \left (\sqrt {b} c+\sqrt {a} d\right ) e+b c \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) x e\right )}{e^3 \sqrt {b x^2+a}}d\sqrt {e x}}{2 a b}-\frac {\sqrt {e x} \left (b c e x \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )+\sqrt {a} d e \left (\sqrt {b} c-\sqrt {a} d\right )\right )}{2 a e \sqrt {a+b x^2}}}{\left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {c d^2 \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}-\frac {\frac {e^2 \int \frac {b \left (\sqrt {a} d \left (\sqrt {b} c+\sqrt {a} d\right ) e+b c \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) x e\right )}{e^3 \sqrt {b x^2+a}}d\sqrt {e x}}{2 a b}-\frac {\sqrt {e x} \left (b c e x \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )+\sqrt {a} d e \left (\sqrt {b} c-\sqrt {a} d\right )\right )}{2 a e \sqrt {a+b x^2}}}{\left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {c d^2 \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}-\frac {\frac {\int \frac {\sqrt {a} d \left (\sqrt {b} c+\sqrt {a} d\right ) e+b c \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) x e}{\sqrt {b x^2+a}}d\sqrt {e x}}{2 a e}-\frac {\sqrt {e x} \left (b c e x \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )+\sqrt {a} d e \left (\sqrt {b} c-\sqrt {a} d\right )\right )}{2 a e \sqrt {a+b x^2}}}{\left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}\right )\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle 2 \left (\frac {c d^2 \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}-\frac {\frac {e \left (a d^2+b c^2\right ) \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {e x}-\sqrt {b} c e \left (\sqrt {b} c-\sqrt {a} d\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{\sqrt {a} e \sqrt {b x^2+a}}d\sqrt {e x}}{2 a e}-\frac {\sqrt {e x} \left (b c e x \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )+\sqrt {a} d e \left (\sqrt {b} c-\sqrt {a} d\right )\right )}{2 a e \sqrt {a+b x^2}}}{\left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {c d^2 \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}-\frac {\frac {e \left (a d^2+b c^2\right ) \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {e x}-\frac {\sqrt {b} c \left (\sqrt {b} c-\sqrt {a} d\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{\sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {a}}}{2 a e}-\frac {\sqrt {e x} \left (b c e x \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )+\sqrt {a} d e \left (\sqrt {b} c-\sqrt {a} d\right )\right )}{2 a e \sqrt {a+b x^2}}}{\left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}\right )\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle 2 \left (\frac {c d^2 \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}-\frac {\frac {\frac {\sqrt {e} \left (a d^2+b c^2\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a+b x^2}}-\frac {\sqrt {b} c \left (\sqrt {b} c-\sqrt {a} d\right ) \int \frac {\sqrt {a} e-\sqrt {b} e x}{\sqrt {b x^2+a}}d\sqrt {e x}}{\sqrt {a}}}{2 a e}-\frac {\sqrt {e x} \left (b c e x \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )+\sqrt {a} d e \left (\sqrt {b} c-\sqrt {a} d\right )\right )}{2 a e \sqrt {a+b x^2}}}{\left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}\right )\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle 2 \left (\frac {c d^2 \left (\frac {\left (\sqrt {a} d+\sqrt {b} c\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )^2}{4 \sqrt {b} c d},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} c d \sqrt {e} \sqrt {a+b x^2}}-\frac {\sqrt {e} \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt {e x} \sqrt {a d^2+b c^2}}{\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a+b x^2}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+b c^2}}\right )}{\sqrt {a} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}-\frac {\frac {\frac {\sqrt {e} \left (a d^2+b c^2\right ) \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a+b x^2}}-\frac {\sqrt {b} c \left (\sqrt {b} c-\sqrt {a} d\right ) \left (\frac {\sqrt [4]{a} \sqrt {e} \left (\sqrt {a} e+\sqrt {b} e x\right ) \sqrt {\frac {a e^2+b e^2 x^2}{\left (\sqrt {a} e+\sqrt {b} e x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right )|\frac {1}{2}\right )}{\sqrt [4]{b} \sqrt {a+b x^2}}-\frac {e^2 \sqrt {e x} \sqrt {a+b x^2}}{\sqrt {a} e+\sqrt {b} e x}\right )}{\sqrt {a}}}{2 a e}-\frac {\sqrt {e x} \left (b c e x \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )+\sqrt {a} d e \left (\sqrt {b} c-\sqrt {a} d\right )\right )}{2 a e \sqrt {a+b x^2}}}{\left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \left (a d^2+b c^2\right )}\right )\) |
Input:
Int[Sqrt[e*x]/((c + d*x)*(a + b*x^2)^(3/2)),x]
Output:
2*(-((-1/2*(Sqrt[e*x]*(Sqrt[a]*d*(Sqrt[b]*c - Sqrt[a]*d)*e + b*c*((Sqrt[b] *c)/Sqrt[a] - d)*e*x))/(a*e*Sqrt[a + b*x^2]) + (-((Sqrt[b]*c*(Sqrt[b]*c - Sqrt[a]*d)*(-((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)*Sq rt[e])], 1/2])/(b^(1/4)*Sqrt[a + b*x^2])))/Sqrt[a]) + ((b*c^2 + a*d^2)*Sqr t[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] )/(2*a^(1/4)*b^(1/4)*Sqrt[a + b*x^2]))/(2*a*e))/(((Sqrt[b]*c)/Sqrt[a] - d) *(b*c^2 + a*d^2))) + (c*d^2*(-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]*E llipticPi[-1/4*(Sqrt[a]*((Sqrt[b]*c)/Sqrt[a] - d)^2)/(Sqrt[b]*c*d), 2*ArcT an[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(4*a^(1/4)*b^(1/4)*c*d*Sq rt[e]*Sqrt[a + b*x^2])))/(Sqrt[a]*((Sqrt[b]*c)/Sqrt[a] - d)*(b*c^2 + a*d^2 )))
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[(c*d^2 + a*e^2)^(p + 1/2)/(Rt[c/a, 2]*d - e) Int[(a + c*x^4)^p*Expand ToSum[((Rt[c/a, 2]*d - e)*(c*d^2 + a*e^2)^(-p - 1/2)*x^m + ((-d/e)^(m/2)*(1 + Rt[c/a, 2]*x^2)*(a + c*x^4)^(-p - 1/2))/e^(2*p))/(d + e*x^2), x], x], x] /; FreeQ[{a, c, d, e}, x] && ILtQ[p + 1/2, 0] && IGtQ[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_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x]}, S imp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) Int[(a + b*x^n)^(p + 1)* ExpandToSum[a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x], x]] /; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]
Time = 1.34 (sec) , antiderivative size = 557, normalized size of antiderivative = 0.93
| method | result | size |
| default | \(\frac {\left (\sqrt {2}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a^{2} d^{2} \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}+\sqrt {2}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}-2 \sqrt {2}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a b \,c^{2} \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}+2 \sqrt {2}\, \operatorname {EllipticE}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) a c d \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \sqrt {-a b}-2 \sqrt {2}\, \operatorname {EllipticPi}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {-a b}\, d}{\sqrt {-a b}\, d -b c}, \frac {\sqrt {2}}{2}\right ) a c d \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \sqrt {-a b}+2 b^{2} c^{2} x^{2}-2 b c d \,x^{2} \sqrt {-a b}+2 a b c d x -2 a \,d^{2} x \sqrt {-a b}\right ) \sqrt {e x}}{2 \sqrt {b \,x^{2}+a}\, a \left (b c -\sqrt {-a b}\, d \right ) \left (a \,d^{2}+b \,c^{2}\right ) x}\) | \(557\) |
| elliptic | \(\frac {\sqrt {e x}\, \sqrt {\left (b \,x^{2}+a \right ) e x}\, \left (-\frac {2 b e x \left (-\frac {c x}{2 a \left (a \,d^{2}+b \,c^{2}\right )}-\frac {d}{2 \left (a \,d^{2}+b \,c^{2}\right ) b}\right )}{\sqrt {\left (x^{2}+\frac {a}{b}\right ) b e x}}+\frac {d e \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{2 \left (a \,d^{2}+b \,c^{2}\right ) b \sqrt {b e \,x^{3}+a e x}}-\frac {e c \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{2 a \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {b e \,x^{3}+a e x}}-\frac {c e \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}}{b \left (-\frac {\sqrt {-a b}}{b}+\frac {c}{d}\right )}, \frac {\sqrt {2}}{2}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) b \sqrt {b e \,x^{3}+a e x}\, \left (-\frac {\sqrt {-a b}}{b}+\frac {c}{d}\right )}\right )}{e x \sqrt {b \,x^{2}+a}}\) | \(569\) |
Input:
int((e*x)^(1/2)/(d*x+c)/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/2*(2^(1/2)*EllipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2) )*a^2*d^2*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*((-b*x+(-a*b)^(1/2))/(-a *b)^(1/2))^(1/2)*(-1/(-a*b)^(1/2)*b*x)^(1/2)+2^(1/2)*EllipticF(((b*x+(-a*b )^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*a*b*c^2*((b*x+(-a*b)^(1/2))/(-a* b)^(1/2))^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-1/(-a*b)^(1/2)* b*x)^(1/2)-2*2^(1/2)*EllipticE(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2 *2^(1/2))*a*b*c^2*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*((-b*x+(-a*b)^(1 /2))/(-a*b)^(1/2))^(1/2)*(-1/(-a*b)^(1/2)*b*x)^(1/2)+2*2^(1/2)*EllipticE(( (b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*a*c*d*((b*x+(-a*b)^(1/ 2))/(-a*b)^(1/2))^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-1/(-a*b )^(1/2)*b*x)^(1/2)*(-a*b)^(1/2)-2*2^(1/2)*EllipticPi(((b*x+(-a*b)^(1/2))/( -a*b)^(1/2))^(1/2),(-a*b)^(1/2)*d/((-a*b)^(1/2)*d-b*c),1/2*2^(1/2))*a*c*d* ((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2)) ^(1/2)*(-1/(-a*b)^(1/2)*b*x)^(1/2)*(-a*b)^(1/2)+2*b^2*c^2*x^2-2*b*c*d*x^2* (-a*b)^(1/2)+2*a*b*c*d*x-2*a*d^2*x*(-a*b)^(1/2))*(e*x)^(1/2)/(b*x^2+a)^(1/ 2)/a/(b*c-(-a*b)^(1/2)*d)/(a*d^2+b*c^2)/x
Timed out. \[ \int \frac {\sqrt {e x}}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((e*x)^(1/2)/(d*x+c)/(b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\sqrt {e x}}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {e x}}{\left (a + b x^{2}\right )^{\frac {3}{2}} \left (c + d x\right )}\, dx \] Input:
integrate((e*x)**(1/2)/(d*x+c)/(b*x**2+a)**(3/2),x)
Output:
Integral(sqrt(e*x)/((a + b*x**2)**(3/2)*(c + d*x)), x)
\[ \int \frac {\sqrt {e x}}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {e x}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}} \,d x } \] Input:
integrate((e*x)^(1/2)/(d*x+c)/(b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate(sqrt(e*x)/((b*x^2 + a)^(3/2)*(d*x + c)), x)
\[ \int \frac {\sqrt {e x}}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {e x}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}} \,d x } \] Input:
integrate((e*x)^(1/2)/(d*x+c)/(b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate(sqrt(e*x)/((b*x^2 + a)^(3/2)*(d*x + c)), x)
Timed out. \[ \int \frac {\sqrt {e x}}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {e\,x}}{{\left (b\,x^2+a\right )}^{3/2}\,\left (c+d\,x\right )} \,d x \] Input:
int((e*x)^(1/2)/((a + b*x^2)^(3/2)*(c + d*x)),x)
Output:
int((e*x)^(1/2)/((a + b*x^2)^(3/2)*(c + d*x)), x)
\[ \int \frac {\sqrt {e x}}{(c+d x) \left (a+b x^2\right )^{3/2}} \, dx=\sqrt {e}\, \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b^{2} d \,x^{5}+b^{2} c \,x^{4}+2 a b d \,x^{3}+2 a b c \,x^{2}+a^{2} d x +a^{2} c}d x \right ) \] Input:
int((e*x)^(1/2)/(d*x+c)/(b*x^2+a)^(3/2),x)
Output:
sqrt(e)*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)