Integrand size = 19, antiderivative size = 831 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{(d+e x)^2} \, dx=-\frac {3 c d^3 \sqrt {a+c x^4}}{e^5}+\frac {c d^2 x \sqrt {a+c x^4}}{e^4}-\frac {3 c d x^2 \sqrt {a+c x^4}}{2 e^3}+\frac {c x^3 \sqrt {a+c x^4}}{5 e^2}+\frac {6 \sqrt {c} \left (5 c d^4+2 a e^4\right ) x \sqrt {a+c x^4}}{5 e^6 \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {\left (a+\frac {c d^4}{e^4}\right ) x \sqrt {a+c x^4}}{d^2-e^2 x^2}-\frac {d \left (a+c x^4\right )^{3/2}}{e \left (d^2-e^2 x^2\right )}-\frac {3 c d^3 \sqrt {c d^4+a e^4} \text {arctanh}\left (\frac {\sqrt {c d^4+a e^4} x}{d e \sqrt {a+c x^4}}\right )}{e^7}-\frac {3 \sqrt {c} d \left (2 c d^4+a e^4\right ) \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{2 e^7}+\frac {3 c d^3 \sqrt {c d^4+a e^4} \text {arctanh}\left (\frac {a e^2+c d^2 x^2}{\sqrt {c d^4+a e^4} \sqrt {a+c x^4}}\right )}{e^7}-\frac {6 \sqrt [4]{a} \sqrt [4]{c} \left (5 c d^4+2 a e^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 e^6 \sqrt {a+c x^4}}+\frac {2 \sqrt [4]{a} \sqrt [4]{c} \left (15 c^{3/2} d^6+5 \sqrt {a} c d^4 e^2+8 a \sqrt {c} d^2 e^4+3 a^{3/2} e^6\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{5 e^6 \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+c x^4}}+\frac {3 c^{3/4} d^2 \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (c d^4+a e^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} e^8 \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+c x^4}} \] Output:
-3*c*d^3*(c*x^4+a)^(1/2)/e^5+c*d^2*x*(c*x^4+a)^(1/2)/e^4-3/2*c*d*x^2*(c*x^ 4+a)^(1/2)/e^3+1/5*c*x^3*(c*x^4+a)^(1/2)/e^2+6/5*c^(1/2)*(2*a*e^4+5*c*d^4) *x*(c*x^4+a)^(1/2)/e^6/(a^(1/2)+c^(1/2)*x^2)+(a+c*d^4/e^4)*x*(c*x^4+a)^(1/ 2)/(-e^2*x^2+d^2)-d*(c*x^4+a)^(3/2)/e/(-e^2*x^2+d^2)-3*c*d^3*(a*e^4+c*d^4) ^(1/2)*arctanh((a*e^4+c*d^4)^(1/2)*x/d/e/(c*x^4+a)^(1/2))/e^7-3/2*c^(1/2)* d*(a*e^4+2*c*d^4)*arctanh(c^(1/2)*x^2/(c*x^4+a)^(1/2))/e^7+3*c*d^3*(a*e^4+ c*d^4)^(1/2)*arctanh((c*d^2*x^2+a*e^2)/(a*e^4+c*d^4)^(1/2)/(c*x^4+a)^(1/2) )/e^7-6/5*a^(1/4)*c^(1/4)*(2*a*e^4+5*c*d^4)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+ a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4) )),1/2*2^(1/2))/e^6/(c*x^4+a)^(1/2)+2/5*a^(1/4)*c^(1/4)*(15*c^(3/2)*d^6+5* a^(1/2)*c*d^4*e^2+8*a*c^(1/2)*d^2*e^4+3*a^(3/2)*e^6)*(a^(1/2)+c^(1/2)*x^2) *((c*x^4+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(c^(1/4 )*x/a^(1/4)),1/2*2^(1/2))/e^6/(c^(1/2)*d^2+a^(1/2)*e^2)/(c*x^4+a)^(1/2)+3/ 2*c^(3/4)*d^2*(c^(1/2)*d^2-a^(1/2)*e^2)*(a*e^4+c*d^4)*(a^(1/2)+c^(1/2)*x^2 )*((c*x^4+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*EllipticPi(sin(2*arctan(c^(1/4 )*x/a^(1/4))),1/4*(c^(1/2)*d^2+a^(1/2)*e^2)^2/a^(1/2)/c^(1/2)/d^2/e^2,1/2* 2^(1/2))/a^(1/4)/e^8/(c^(1/2)*d^2+a^(1/2)*e^2)/(c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 12.93 (sec) , antiderivative size = 549, normalized size of antiderivative = 0.66 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{(d+e x)^2} \, dx=\frac {12 \sqrt {a} \sqrt {c} e^2 \left (5 c d^4+2 a e^4\right ) (d+e x) \sqrt {1+\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )-4 \sqrt {c} \left (15 i c^{3/2} d^6+15 \sqrt {a} c d^4 e^2+10 i a \sqrt {c} d^2 e^4+6 a^{3/2} e^6\right ) (d+e x) \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} \left (-e^3 \left (a+c x^4\right ) \left (10 a e^4+c \left (30 d^4+10 d^3 e x-5 d^2 e^2 x^2+3 d e^3 x^3-2 e^4 x^4\right )\right )+60 c d^3 e \sqrt {-c d^4-a e^4} (d+e x) \sqrt {a+c x^4} \arctan \left (\frac {\sqrt {c} \left (d^2-e^2 x^2\right )+e^2 \sqrt {a+c x^4}}{\sqrt {-c d^4-a e^4}}\right )+60 \sqrt [4]{-1} \sqrt [4]{a} c^{3/4} d^2 \left (c d^4+a e^4\right ) (d+e x) \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (\frac {i \sqrt {a} e^2}{\sqrt {c} d^2},\arcsin \left (\frac {(-1)^{3/4} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )+15 \sqrt {c} d e \left (2 c d^4+a e^4\right ) (d+e x) \sqrt {a+c x^4} \log \left (-\sqrt {c} x^2+\sqrt {a+c x^4}\right )\right )}{10 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} e^8 (d+e x) \sqrt {a+c x^4}} \] Input:
Integrate[(a + c*x^4)^(3/2)/(d + e*x)^2,x]
Output:
(12*Sqrt[a]*Sqrt[c]*e^2*(5*c*d^4 + 2*a*e^4)*(d + e*x)*Sqrt[1 + (c*x^4)/a]* EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] - 4*Sqrt[c]*((15*I)* c^(3/2)*d^6 + 15*Sqrt[a]*c*d^4*e^2 + (10*I)*a*Sqrt[c]*d^2*e^4 + 6*a^(3/2)* e^6)*(d + e*x)*Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sq rt[a]]*x], -1] + Sqrt[(I*Sqrt[c])/Sqrt[a]]*(-(e^3*(a + c*x^4)*(10*a*e^4 + c*(30*d^4 + 10*d^3*e*x - 5*d^2*e^2*x^2 + 3*d*e^3*x^3 - 2*e^4*x^4))) + 60*c *d^3*e*Sqrt[-(c*d^4) - a*e^4]*(d + e*x)*Sqrt[a + c*x^4]*ArcTan[(Sqrt[c]*(d ^2 - e^2*x^2) + e^2*Sqrt[a + c*x^4])/Sqrt[-(c*d^4) - a*e^4]] + 60*(-1)^(1/ 4)*a^(1/4)*c^(3/4)*d^2*(c*d^4 + a*e^4)*(d + e*x)*Sqrt[1 + (c*x^4)/a]*Ellip ticPi[(I*Sqrt[a]*e^2)/(Sqrt[c]*d^2), ArcSin[((-1)^(3/4)*c^(1/4)*x)/a^(1/4) ], -1] + 15*Sqrt[c]*d*e*(2*c*d^4 + a*e^4)*(d + e*x)*Sqrt[a + c*x^4]*Log[-( Sqrt[c]*x^2) + Sqrt[a + c*x^4]]))/(10*Sqrt[(I*Sqrt[c])/Sqrt[a]]*e^8*(d + e *x)*Sqrt[a + c*x^4])
Leaf count is larger than twice the leaf count of optimal. \(1816\) vs. \(2(831)=1662\).
Time = 5.37 (sec) , antiderivative size = 1816, normalized size of antiderivative = 2.19, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {2584, 2255, 27, 1577, 492, 591, 719, 224, 219, 488, 219, 2259, 2009}
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.
| \(\displaystyle \int \frac {\left (a+c x^4\right )^{3/2}}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 2584 |
| \(\displaystyle \int \frac {\left (a+c x^4\right )^{3/2} \left (d^2-2 d e x+e^2 x^2\right )}{\left (d^2-e^2 x^2\right )^2}dx\) |
| \(\Big \downarrow \) 2255 |
| \(\displaystyle \int -\frac {2 d e x \left (c x^4+a\right )^{3/2}}{\left (d^2-e^2 x^2\right )^2}dx+\int \frac {\left (d^2+e^2 x^2\right ) \left (c x^4+a\right )^{3/2}}{\left (d^2-e^2 x^2\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {\left (d^2+e^2 x^2\right ) \left (c x^4+a\right )^{3/2}}{\left (d^2-e^2 x^2\right )^2}dx-2 d e \int \frac {x \left (c x^4+a\right )^{3/2}}{\left (d^2-e^2 x^2\right )^2}dx\) |
| \(\Big \downarrow \) 1577 |
| \(\displaystyle \int \frac {\left (d^2+e^2 x^2\right ) \left (c x^4+a\right )^{3/2}}{\left (d^2-e^2 x^2\right )^2}dx-d e \int \frac {\left (c x^4+a\right )^{3/2}}{\left (d^2-e^2 x^2\right )^2}dx^2\) |
| \(\Big \downarrow \) 492 |
| \(\displaystyle \int \frac {\left (d^2+e^2 x^2\right ) \left (c x^4+a\right )^{3/2}}{\left (d^2-e^2 x^2\right )^2}dx-d e \left (\frac {\left (a+c x^4\right )^{3/2}}{e^2 \left (d^2-e^2 x^2\right )}-\frac {3 c \int \frac {x^2 \sqrt {c x^4+a}}{d^2-e^2 x^2}dx^2}{e^2}\right )\) |
| \(\Big \downarrow \) 591 |
| \(\displaystyle \int \frac {\left (d^2+e^2 x^2\right ) \left (c x^4+a\right )^{3/2}}{\left (d^2-e^2 x^2\right )^2}dx-d e \left (\frac {\left (a+c x^4\right )^{3/2}}{e^2 \left (d^2-e^2 x^2\right )}-\frac {3 c \left (\frac {\int \frac {a d^2 e^2+\left (2 c d^4+a e^4\right ) x^2}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx^2}{2 e^4}-\frac {\sqrt {a+c x^4} \left (2 d^2+e^2 x^2\right )}{2 e^4}\right )}{e^2}\right )\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle \int \frac {\left (d^2+e^2 x^2\right ) \left (c x^4+a\right )^{3/2}}{\left (d^2-e^2 x^2\right )^2}dx-d e \left (\frac {\left (a+c x^4\right )^{3/2}}{e^2 \left (d^2-e^2 x^2\right )}-\frac {3 c \left (\frac {\frac {2 d^2 \left (a e^4+c d^4\right ) \int \frac {1}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx^2}{e^2}-\frac {\left (a e^4+2 c d^4\right ) \int \frac {1}{\sqrt {c x^4+a}}dx^2}{e^2}}{2 e^4}-\frac {\sqrt {a+c x^4} \left (2 d^2+e^2 x^2\right )}{2 e^4}\right )}{e^2}\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \int \frac {\left (d^2+e^2 x^2\right ) \left (c x^4+a\right )^{3/2}}{\left (d^2-e^2 x^2\right )^2}dx-d e \left (\frac {\left (a+c x^4\right )^{3/2}}{e^2 \left (d^2-e^2 x^2\right )}-\frac {3 c \left (\frac {\frac {2 d^2 \left (a e^4+c d^4\right ) \int \frac {1}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx^2}{e^2}-\frac {\left (a e^4+2 c d^4\right ) \int \frac {1}{1-c x^4}d\frac {x^2}{\sqrt {c x^4+a}}}{e^2}}{2 e^4}-\frac {\sqrt {a+c x^4} \left (2 d^2+e^2 x^2\right )}{2 e^4}\right )}{e^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \int \frac {\left (d^2+e^2 x^2\right ) \left (c x^4+a\right )^{3/2}}{\left (d^2-e^2 x^2\right )^2}dx-d e \left (\frac {\left (a+c x^4\right )^{3/2}}{e^2 \left (d^2-e^2 x^2\right )}-\frac {3 c \left (\frac {\frac {2 d^2 \left (a e^4+c d^4\right ) \int \frac {1}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx^2}{e^2}-\frac {\text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right ) \left (a e^4+2 c d^4\right )}{\sqrt {c} e^2}}{2 e^4}-\frac {\sqrt {a+c x^4} \left (2 d^2+e^2 x^2\right )}{2 e^4}\right )}{e^2}\right )\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \int \frac {\left (d^2+e^2 x^2\right ) \left (c x^4+a\right )^{3/2}}{\left (d^2-e^2 x^2\right )^2}dx-d e \left (\frac {\left (a+c x^4\right )^{3/2}}{e^2 \left (d^2-e^2 x^2\right )}-\frac {3 c \left (\frac {-\frac {2 d^2 \left (a e^4+c d^4\right ) \int \frac {1}{c d^4+a e^4-x^4}d\frac {-a e^2-c d^2 x^2}{\sqrt {c x^4+a}}}{e^2}-\frac {\text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right ) \left (a e^4+2 c d^4\right )}{\sqrt {c} e^2}}{2 e^4}-\frac {\sqrt {a+c x^4} \left (2 d^2+e^2 x^2\right )}{2 e^4}\right )}{e^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \int \frac {\left (d^2+e^2 x^2\right ) \left (c x^4+a\right )^{3/2}}{\left (d^2-e^2 x^2\right )^2}dx-d e \left (\frac {\left (a+c x^4\right )^{3/2}}{e^2 \left (d^2-e^2 x^2\right )}-\frac {3 c \left (\frac {-\frac {\text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right ) \left (a e^4+2 c d^4\right )}{\sqrt {c} e^2}-\frac {2 d^2 \sqrt {a e^4+c d^4} \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{e^2}}{2 e^4}-\frac {\sqrt {a+c x^4} \left (2 d^2+e^2 x^2\right )}{2 e^4}\right )}{e^2}\right )\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \int \left (\frac {c^2 x^6}{e^2 \sqrt {c x^4+a}}+\frac {3 c^2 d^2 x^4}{e^4 \sqrt {c x^4+a}}+\frac {c \left (5 c d^4+2 a e^4\right ) x^2}{e^6 \sqrt {c x^4+a}}+\frac {c d^2 \left (7 c d^4+6 a e^4\right )}{e^8 \sqrt {c x^4+a}}+\frac {8 c d^4 \left (c d^4+a e^4\right )}{e^8 \left (e^2 x^2-d^2\right ) \sqrt {c x^4+a}}+\frac {\left (c d^4+a e^4\right )^2}{2 e^8 (e x-d)^2 \sqrt {c x^4+a}}+\frac {\left (c d^4+a e^4\right )^2}{2 e^8 (d+e x)^2 \sqrt {c x^4+a}}\right )dx-d e \left (\frac {\left (a+c x^4\right )^{3/2}}{e^2 \left (d^2-e^2 x^2\right )}-\frac {3 c \left (\frac {-\frac {\text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right ) \left (a e^4+2 c d^4\right )}{\sqrt {c} e^2}-\frac {2 d^2 \sqrt {a e^4+c d^4} \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{e^2}}{2 e^4}-\frac {\sqrt {a+c x^4} \left (2 d^2+e^2 x^2\right )}{2 e^4}\right )}{e^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 c^{5/4} \left (c d^4+a e^4\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) d^4}{\sqrt [4]{a} e^8 \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {c x^4+a}}-\frac {3 c \sqrt {c d^4+a e^4} \text {arctanh}\left (\frac {\sqrt {c d^4+a e^4} x}{d e \sqrt {c x^4+a}}\right ) d^3}{e^7}+\frac {c^{3/4} \left (7 c d^4+6 a e^4\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) d^2}{2 \sqrt [4]{a} e^8 \sqrt {c x^4+a}}-\frac {a^{3/4} c^{3/4} \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) d^2}{2 e^4 \sqrt {c x^4+a}}+\frac {2 c^{3/4} \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (c d^4+a e^4\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )^2}{4 \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) d^2}{\sqrt [4]{a} e^8 \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {c x^4+a}}-\frac {c^{3/4} \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (c d^4+a e^4\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) d^2}{2 \sqrt [4]{a} e^8 \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {c x^4+a}}+\frac {c x \sqrt {c x^4+a} d^2}{e^4}-e \left (\frac {\left (c x^4+a\right )^{3/2}}{e^2 \left (d^2-e^2 x^2\right )}-\frac {3 c \left (\frac {-\frac {2 \sqrt {c d^4+a e^4} \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {c d^4+a e^4} \sqrt {c x^4+a}}\right ) d^2}{e^2}-\frac {\left (2 c d^4+a e^4\right ) \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {c x^4+a}}\right )}{\sqrt {c} e^2}}{2 e^4}-\frac {\left (2 d^2+e^2 x^2\right ) \sqrt {c x^4+a}}{2 e^4}\right )}{e^2}\right ) d-\frac {\sqrt [4]{a} \sqrt [4]{c} \left (c d^4+a e^4\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{e^6 \sqrt {c x^4+a}}-\frac {\sqrt [4]{a} \sqrt [4]{c} \left (5 c d^4+2 a e^4\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{e^6 \sqrt {c x^4+a}}+\frac {3 a^{5/4} \sqrt [4]{c} \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 e^2 \sqrt {c x^4+a}}+\frac {\sqrt [4]{c} \left (c d^4+a e^4\right )^2 \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} e^8 \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {c x^4+a}}+\frac {\sqrt [4]{a} \sqrt [4]{c} \left (5 c d^4+2 a e^4\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 e^6 \sqrt {c x^4+a}}-\frac {3 a^{5/4} \sqrt [4]{c} \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{10 e^2 \sqrt {c x^4+a}}+\frac {c x^3 \sqrt {c x^4+a}}{5 e^2}+\frac {\left (c d^4+a e^4\right ) \sqrt {c x^4+a}}{2 e^5 (d-e x)}-\frac {\left (c d^4+a e^4\right ) \sqrt {c x^4+a}}{2 e^5 (d+e x)}+\frac {\sqrt {c} \left (c d^4+a e^4\right ) x \sqrt {c x^4+a}}{e^6 \left (\sqrt {c} x^2+\sqrt {a}\right )}+\frac {\sqrt {c} \left (5 c d^4+2 a e^4\right ) x \sqrt {c x^4+a}}{e^6 \left (\sqrt {c} x^2+\sqrt {a}\right )}-\frac {3 a \sqrt {c} x \sqrt {c x^4+a}}{5 e^2 \left (\sqrt {c} x^2+\sqrt {a}\right )}\) |
Input:
Int[(a + c*x^4)^(3/2)/(d + e*x)^2,x]
Output:
(c*d^2*x*Sqrt[a + c*x^4])/e^4 + (c*x^3*Sqrt[a + c*x^4])/(5*e^2) + ((c*d^4 + a*e^4)*Sqrt[a + c*x^4])/(2*e^5*(d - e*x)) - ((c*d^4 + a*e^4)*Sqrt[a + c* x^4])/(2*e^5*(d + e*x)) - (3*a*Sqrt[c]*x*Sqrt[a + c*x^4])/(5*e^2*(Sqrt[a] + Sqrt[c]*x^2)) + (Sqrt[c]*(c*d^4 + a*e^4)*x*Sqrt[a + c*x^4])/(e^6*(Sqrt[a ] + Sqrt[c]*x^2)) + (Sqrt[c]*(5*c*d^4 + 2*a*e^4)*x*Sqrt[a + c*x^4])/(e^6*( Sqrt[a] + Sqrt[c]*x^2)) - (3*c*d^3*Sqrt[c*d^4 + a*e^4]*ArcTanh[(Sqrt[c*d^4 + a*e^4]*x)/(d*e*Sqrt[a + c*x^4])])/e^7 - d*e*((a + c*x^4)^(3/2)/(e^2*(d^ 2 - e^2*x^2)) - (3*c*(-1/2*((2*d^2 + e^2*x^2)*Sqrt[a + c*x^4])/e^4 + (-((( 2*c*d^4 + a*e^4)*ArcTanh[(Sqrt[c]*x^2)/Sqrt[a + c*x^4]])/(Sqrt[c]*e^2)) - (2*d^2*Sqrt[c*d^4 + a*e^4]*ArcTanh[(-(a*e^2) - c*d^2*x^2)/(Sqrt[c*d^4 + a* e^4]*Sqrt[a + c*x^4])])/e^2)/(2*e^4)))/e^2) + (3*a^(5/4)*c^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*Arc Tan[(c^(1/4)*x)/a^(1/4)], 1/2])/(5*e^2*Sqrt[a + c*x^4]) - (a^(1/4)*c^(1/4) *(c*d^4 + a*e^4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[ c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(e^6*Sqrt[a + c* x^4]) - (a^(1/4)*c^(1/4)*(5*c*d^4 + 2*a*e^4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[ (a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1 /4)], 1/2])/(e^6*Sqrt[a + c*x^4]) - (a^(3/4)*c^(3/4)*d^2*(Sqrt[a] + Sqrt[c ]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^( 1/4)*x)/a^(1/4)], 1/2])/(2*e^4*Sqrt[a + c*x^4]) - (3*a^(5/4)*c^(1/4)*(S...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (c + d*x)^(n + 1)*((a + b*x^2)^p/(d*(n + 1))), x] - Simp[2*b*(p/(d*(n + 1)) ) Int[x*(c + d*x)^(n + 1)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && GtQ[p, 0] && (IntegerQ[p] || LtQ[n, -1]) && NeQ[n, -1] && !IL tQ[n + 2*p + 1, 0] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
Int[(x_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c + d*x)^(n + 1))*(a + b*x^2)^p*((c*(2*p + 1) - d*(n + 2*p + 1)*x)/ (d^2*(n + 2*p + 1)*(n + 2*p + 2))), x] + Simp[2*(p/(d^2*(n + 2*p + 1)*(n + 2*p + 2))) Int[(c + d*x)^n*(a + b*x^2)^(p - 1)*Simp[a*c*d*n + (b*c^2*(2*p + 1) + a*d^2*(n + 2*p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && GtQ[p, 0] && LeQ[-1, n, 0] && !ILtQ[n + 2*p, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; Free Q[{a, c, d, e, p, q}, x]
Int[(Pr_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{r = Expon[Pr, x], k}, Int[Sum[Coeff[Pr, x, 2*k]*x^(2*k), {k, 0, r/2}]*(d + e*x^2)^q*(a + c*x^4)^p, x] + Int[x*Sum[Coeff[Pr, x, 2*k + 1]*x^ (2*k), {k, 0, (r - 1)/2}]*(d + e*x^2)^q*(a + c*x^4)^p, x]] /; FreeQ[{a, c, d, e, p, q}, x] && PolyQ[Pr, x] && !PolyQ[Pr, x^2]
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a + c*x^4], Px*(d + e*x^2)^q*(a + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x] && IntegerQ[p + 1/ 2] && IntegerQ[q]
Int[((c_) + (d_.)*(x_)^(n_.))^(q_)*((a_) + (b_.)*(x_)^(nn_.))^(p_), x_Symbo l] :> Int[ExpandToSum[(c - d*x^n)^(-q), x]*((a + b*x^nn)^p/(c^2 - d^2*x^(2* n))^(-q)), x] /; FreeQ[{a, b, c, d, n, nn, p}, x] && !IntegerQ[p] && ILtQ[ q, 0] && IGtQ[Log[2, nn/n], 0]
Result contains complex when optimal does not.
Time = 3.48 (sec) , antiderivative size = 604, normalized size of antiderivative = 0.73
| method | result | size |
| default | \(-\frac {\left (e^{4} a +c \,d^{4}\right ) \sqrt {c \,x^{4}+a}}{e^{5} \left (e x +d \right )}+\frac {c \,x^{3} \sqrt {c \,x^{4}+a}}{5 e^{2}}-\frac {c d \,x^{2} \sqrt {c \,x^{4}+a}}{2 e^{3}}+\frac {c \,d^{2} x \sqrt {c \,x^{4}+a}}{e^{4}}-\frac {2 c \,d^{3} \sqrt {c \,x^{4}+a}}{e^{5}}+\frac {\left (\frac {c \,d^{2} \left (6 e^{4} a +7 c \,d^{4}\right )}{e^{8}}-\frac {d^{2} c \left (e^{4} a +c \,d^{4}\right )}{e^{8}}-\frac {c \,d^{2} a}{e^{4}}\right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {\left (-\frac {2 c d \left (2 e^{4} a +3 c \,d^{4}\right )}{e^{7}}+\frac {c d a}{e^{3}}\right ) \ln \left (2 \sqrt {c}\, x^{2}+2 \sqrt {c \,x^{4}+a}\right )}{2 \sqrt {c}}+\frac {i \left (\frac {c \left (2 e^{4} a +5 c \,d^{4}\right )}{e^{6}}+\frac {c \left (e^{4} a +c \,d^{4}\right )}{e^{6}}-\frac {3 c a}{5 e^{2}}\right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}-\frac {6 c \,d^{3} \left (e^{4} a +c \,d^{4}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+2 a}{2 \sqrt {a +\frac {c \,d^{4}}{e^{4}}}\, \sqrt {c \,x^{4}+a}}\right )}{2 \sqrt {a +\frac {c \,d^{4}}{e^{4}}}}+\frac {e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, e^{2}}{\sqrt {c}\, d^{2}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}+a}}\right )}{e^{9}}\) | \(604\) |
| elliptic | \(-\frac {\left (e^{4} a +c \,d^{4}\right ) \sqrt {c \,x^{4}+a}}{e^{5} \left (e x +d \right )}+\frac {c \,x^{3} \sqrt {c \,x^{4}+a}}{5 e^{2}}-\frac {c d \,x^{2} \sqrt {c \,x^{4}+a}}{2 e^{3}}+\frac {c \,d^{2} x \sqrt {c \,x^{4}+a}}{e^{4}}-\frac {2 c \,d^{3} \sqrt {c \,x^{4}+a}}{e^{5}}+\frac {\left (\frac {c \,d^{2} \left (6 e^{4} a +7 c \,d^{4}\right )}{e^{8}}-\frac {d^{2} c \left (e^{4} a +c \,d^{4}\right )}{e^{8}}-\frac {c \,d^{2} a}{e^{4}}\right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {\left (-\frac {2 c d \left (2 e^{4} a +3 c \,d^{4}\right )}{e^{7}}+\frac {c d a}{e^{3}}\right ) \ln \left (2 \sqrt {c}\, x^{2}+2 \sqrt {c \,x^{4}+a}\right )}{2 \sqrt {c}}+\frac {i \left (\frac {c \left (2 e^{4} a +5 c \,d^{4}\right )}{e^{6}}+\frac {c \left (e^{4} a +c \,d^{4}\right )}{e^{6}}-\frac {3 c a}{5 e^{2}}\right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}-\frac {6 c \,d^{3} \left (e^{4} a +c \,d^{4}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+2 a}{2 \sqrt {a +\frac {c \,d^{4}}{e^{4}}}\, \sqrt {c \,x^{4}+a}}\right )}{2 \sqrt {a +\frac {c \,d^{4}}{e^{4}}}}+\frac {e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, e^{2}}{\sqrt {c}\, d^{2}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}+a}}\right )}{e^{9}}\) | \(604\) |
| risch | \(\text {Expression too large to display}\) | \(993\) |
Input:
int((c*x^4+a)^(3/2)/(e*x+d)^2,x,method=_RETURNVERBOSE)
Output:
-(a*e^4+c*d^4)/e^5*(c*x^4+a)^(1/2)/(e*x+d)+1/5*c*x^3*(c*x^4+a)^(1/2)/e^2-1 /2*c*d*x^2*(c*x^4+a)^(1/2)/e^3+c*d^2*x*(c*x^4+a)^(1/2)/e^4-2*c*d^3*(c*x^4+ a)^(1/2)/e^5+(c*d^2*(6*a*e^4+7*c*d^4)/e^8-d^2*c*(a*e^4+c*d^4)/e^8-c*d^2/e^ 4*a)/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*c^(1/2 )*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I*c^(1/2)/a^(1/2))^(1/2) ,I)+1/2*(-2*c*d/e^7*(2*a*e^4+3*c*d^4)+c*d/e^3*a)*ln(2*c^(1/2)*x^2+2*(c*x^4 +a)^(1/2))/c^(1/2)+I*(c/e^6*(2*a*e^4+5*c*d^4)+c*(a*e^4+c*d^4)/e^6-3/5*c/e^ 2*a)*a^(1/2)/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c^(1/2)*x^2/a^(1/2))^(1/2)*(1+ I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2)/c^(1/2)*(EllipticF(x*(I*c^(1/ 2)/a^(1/2))^(1/2),I)-EllipticE(x*(I*c^(1/2)/a^(1/2))^(1/2),I))-6*c*d^3/e^9 *(a*e^4+c*d^4)*(-1/2/(a+c*d^4/e^4)^(1/2)*arctanh(1/2*(2*c*x^2*d^2/e^2+2*a) /(a+c*d^4/e^4)^(1/2)/(c*x^4+a)^(1/2))+1/(I*c^(1/2)/a^(1/2))^(1/2)/d*e*(1-I *c^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2 )*EllipticPi(x*(I*c^(1/2)/a^(1/2))^(1/2),-I/c^(1/2)*a^(1/2)/d^2*e^2,(-I/a^ (1/2)*c^(1/2))^(1/2)/(I*c^(1/2)/a^(1/2))^(1/2)))
Timed out. \[ \int \frac {\left (a+c x^4\right )^{3/2}}{(d+e x)^2} \, dx=\text {Timed out} \] Input:
integrate((c*x^4+a)^(3/2)/(e*x+d)^2,x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\left (a+c x^4\right )^{3/2}}{(d+e x)^2} \, dx=\int \frac {\left (a + c x^{4}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{2}}\, dx \] Input:
integrate((c*x**4+a)**(3/2)/(e*x+d)**2,x)
Output:
Integral((a + c*x**4)**(3/2)/(d + e*x)**2, x)
\[ \int \frac {\left (a+c x^4\right )^{3/2}}{(d+e x)^2} \, dx=\int { \frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate((c*x^4+a)^(3/2)/(e*x+d)^2,x, algorithm="maxima")
Output:
integrate((c*x^4 + a)^(3/2)/(e*x + d)^2, x)
\[ \int \frac {\left (a+c x^4\right )^{3/2}}{(d+e x)^2} \, dx=\int { \frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate((c*x^4+a)^(3/2)/(e*x+d)^2,x, algorithm="giac")
Output:
integrate((c*x^4 + a)^(3/2)/(e*x + d)^2, x)
Timed out. \[ \int \frac {\left (a+c x^4\right )^{3/2}}{(d+e x)^2} \, dx=\int \frac {{\left (c\,x^4+a\right )}^{3/2}}{{\left (d+e\,x\right )}^2} \,d x \] Input:
int((a + c*x^4)^(3/2)/(d + e*x)^2,x)
Output:
int((a + c*x^4)^(3/2)/(d + e*x)^2, x)
\[ \int \frac {\left (a+c x^4\right )^{3/2}}{(d+e x)^2} \, dx=\int \frac {\left (c \,x^{4}+a \right )^{\frac {3}{2}}}{\left (e x +d \right )^{2}}d x \] Input:
int((c*x^4+a)^(3/2)/(e*x+d)^2,x)
Output:
int((c*x^4+a)^(3/2)/(e*x+d)^2,x)