\(\int \frac {(d+e x^2)^{5/2}}{a+c x^4} \, dx\) [383]

Optimal result

Integrand size = 21, antiderivative size = 474 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{a+c x^4} \, dx=\frac {e^2 x \sqrt {d+e x^2}}{2 c}+\frac {\sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (2 \sqrt {a} c d^2 e+2 a^{3/2} e^3+\left (c d^2-3 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{2 \sqrt {2} a^{3/4} c^{3/2} \sqrt {c d^2+a e^2}}+\frac {5 d e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c}+\frac {\sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (2 \sqrt {a} c d^2 e+2 a^{3/2} e^3-\left (c d^2-3 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e-\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{2 \sqrt {2} a^{3/4} c^{3/2} \sqrt {c d^2+a e^2}} \] Output:

1/2*e^2*x*(e*x^2+d)^(1/2)/c+1/4*(a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*(2*a 
^(1/2)*c*d^2*e+2*a^(3/2)*e^3+(-3*a*e^2+c*d^2)*(a*e^2+c*d^2)^(1/2))*arctan( 
2^(1/2)*a^(1/4)*c^(1/2)*(a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*x*(e*x^2+d)^ 
(1/2)/(a^(1/2)*(a^(1/2)*e+(a*e^2+c*d^2)^(1/2))-c*d*x^2))*2^(1/2)/a^(3/4)/c 
^(3/2)/(a*e^2+c*d^2)^(1/2)+5/2*d*e^(3/2)*arctanh(e^(1/2)*x/(e*x^2+d)^(1/2) 
)/c+1/4*(-a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*(2*a^(1/2)*c*d^2*e+2*a^(3/2 
)*e^3-(-3*a*e^2+c*d^2)*(a*e^2+c*d^2)^(1/2))*arctanh(2^(1/2)*a^(1/4)*c^(1/2 
)*(-a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*x*(e*x^2+d)^(1/2)/(a^(1/2)*(a^(1/ 
2)*e-(a*e^2+c*d^2)^(1/2))-c*d*x^2))*2^(1/2)/a^(3/4)/c^(3/2)/(a*e^2+c*d^2)^ 
(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.23 (sec) , antiderivative size = 391, normalized size of antiderivative = 0.82 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{a+c x^4} \, dx=\frac {e^{3/2} \left (\sqrt {e} x \sqrt {d+e x^2}-5 d \log \left (-\sqrt {e} x+\sqrt {d+e x^2}\right )+\text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2+16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {3 c d^4 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )-a d^2 e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )-2 c d^3 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}-10 a d e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+3 c d^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-a e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}-8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]\right )}{2 c} \] Input:

Integrate[(d + e*x^2)^(5/2)/(a + c*x^4),x]
 

Output:

(e^(3/2)*(Sqrt[e]*x*Sqrt[d + e*x^2] - 5*d*Log[-(Sqrt[e]*x) + Sqrt[d + e*x^ 
2]] + RootSum[c*d^4 - 4*c*d^3*#1 + 6*c*d^2*#1^2 + 16*a*e^2*#1^2 - 4*c*d*#1 
^3 + c*#1^4 & , (3*c*d^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - # 
1] - a*d^2*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 2*c*d 
^3*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 - 10*a*d*e^2*Log 
[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 3*c*d^2*Log[d + 2*e* 
x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2 - a*e^2*Log[d + 2*e*x^2 - 2*S 
qrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2)/(c*d^3 - 3*c*d^2*#1 - 8*a*e^2*#1 + 3* 
c*d*#1^2 - c*#1^3) & ]))/(2*c)
 

Rubi [A] (verified)

Time = 1.38 (sec) , antiderivative size = 551, normalized size of antiderivative = 1.16, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {1489, 27, 318, 25, 403, 25, 398, 224, 219, 291, 218, 221}

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.

\(\displaystyle \int \frac {\left (d+e x^2\right )^{5/2}}{a+c x^4} \, dx\)

\(\Big \downarrow \) 1489

\(\displaystyle -\frac {\sqrt {c} \int \frac {\left (e x^2+d\right )^{5/2}}{\sqrt {c} \left (\sqrt {-a}-\sqrt {c} x^2\right )}dx}{2 \sqrt {-a}}-\frac {\sqrt {c} \int \frac {\left (e x^2+d\right )^{5/2}}{\sqrt {c} \left (\sqrt {c} x^2+\sqrt {-a}\right )}dx}{2 \sqrt {-a}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {\left (e x^2+d\right )^{5/2}}{\sqrt {-a}-\sqrt {c} x^2}dx}{2 \sqrt {-a}}-\frac {\int \frac {\left (e x^2+d\right )^{5/2}}{\sqrt {c} x^2+\sqrt {-a}}dx}{2 \sqrt {-a}}\)

\(\Big \downarrow \) 318

\(\displaystyle -\frac {\frac {\int \frac {\sqrt {e x^2+d} \left (e \left (7 \sqrt {c} d-4 \sqrt {-a} e\right ) x^2+d \left (4 \sqrt {c} d-\sqrt {-a} e\right )\right )}{\sqrt {c} x^2+\sqrt {-a}}dx}{4 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {-a}}-\frac {-\frac {\int -\frac {\sqrt {e x^2+d} \left (e \left (7 \sqrt {c} d+4 \sqrt {-a} e\right ) x^2+d \left (4 \sqrt {c} d+\sqrt {-a} e\right )\right )}{\sqrt {-a}-\sqrt {c} x^2}dx}{4 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {-a}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\int \frac {\sqrt {e x^2+d} \left (e \left (7 \sqrt {c} d-4 \sqrt {-a} e\right ) x^2+d \left (4 \sqrt {c} d-\sqrt {-a} e\right )\right )}{\sqrt {c} x^2+\sqrt {-a}}dx}{4 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {-a}}-\frac {\frac {\int \frac {\sqrt {e x^2+d} \left (e \left (7 \sqrt {c} d+4 \sqrt {-a} e\right ) x^2+d \left (4 \sqrt {c} d+\sqrt {-a} e\right )\right )}{\sqrt {-a}-\sqrt {c} x^2}dx}{4 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {-a}}\)

\(\Big \downarrow \) 403

\(\displaystyle -\frac {\frac {\frac {\int \frac {e \left (15 c d^2-20 \sqrt {-a} \sqrt {c} e d-8 a e^2\right ) x^2+d \left (8 c d^2-9 \sqrt {-a} \sqrt {c} e d-4 a e^2\right )}{\left (\sqrt {c} x^2+\sqrt {-a}\right ) \sqrt {e x^2+d}}dx}{2 \sqrt {c}}+\frac {1}{2} e x \sqrt {d+e x^2} \left (7 d-\frac {4 \sqrt {-a} e}{\sqrt {c}}\right )}{4 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {-a}}-\frac {\frac {-\frac {\int -\frac {e \left (15 c d^2+20 \sqrt {-a} \sqrt {c} e d-8 a e^2\right ) x^2+d \left (8 c d^2+9 \sqrt {-a} \sqrt {c} e d-4 a e^2\right )}{\left (\sqrt {-a}-\sqrt {c} x^2\right ) \sqrt {e x^2+d}}dx}{2 \sqrt {c}}-\frac {1}{2} e x \sqrt {d+e x^2} \left (\frac {4 \sqrt {-a} e}{\sqrt {c}}+7 d\right )}{4 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {-a}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\frac {\int \frac {e \left (15 c d^2-20 \sqrt {-a} \sqrt {c} e d-8 a e^2\right ) x^2+d \left (8 c d^2-9 \sqrt {-a} \sqrt {c} e d-4 a e^2\right )}{\left (\sqrt {c} x^2+\sqrt {-a}\right ) \sqrt {e x^2+d}}dx}{2 \sqrt {c}}+\frac {1}{2} e x \sqrt {d+e x^2} \left (7 d-\frac {4 \sqrt {-a} e}{\sqrt {c}}\right )}{4 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {-a}}-\frac {\frac {\frac {\int \frac {e \left (15 c d^2+20 \sqrt {-a} \sqrt {c} e d-8 a e^2\right ) x^2+d \left (8 c d^2+9 \sqrt {-a} \sqrt {c} e d-4 a e^2\right )}{\left (\sqrt {-a}-\sqrt {c} x^2\right ) \sqrt {e x^2+d}}dx}{2 \sqrt {c}}-\frac {1}{2} e x \sqrt {d+e x^2} \left (\frac {4 \sqrt {-a} e}{\sqrt {c}}+7 d\right )}{4 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {-a}}\)

\(\Big \downarrow \) 398

\(\displaystyle -\frac {\frac {\frac {\frac {8 \left (3 \sqrt {-a} c d^2 e-3 a \sqrt {c} d e^2+(-a)^{3/2} e^3+c^{3/2} d^3\right ) \int \frac {1}{\left (\sqrt {-a}-\sqrt {c} x^2\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}-\frac {e \left (20 \sqrt {-a} \sqrt {c} d e-8 a e^2+15 c d^2\right ) \int \frac {1}{\sqrt {e x^2+d}}dx}{\sqrt {c}}}{2 \sqrt {c}}-\frac {1}{2} e x \sqrt {d+e x^2} \left (\frac {4 \sqrt {-a} e}{\sqrt {c}}+7 d\right )}{4 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {-a}}-\frac {\frac {\frac {\frac {8 \left (-3 \sqrt {-a} c d^2 e-3 a \sqrt {c} d e^2+\sqrt {-a} a e^3+c^{3/2} d^3\right ) \int \frac {1}{\left (\sqrt {c} x^2+\sqrt {-a}\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}+\frac {e \left (-20 \sqrt {-a} \sqrt {c} d e-8 a e^2+15 c d^2\right ) \int \frac {1}{\sqrt {e x^2+d}}dx}{\sqrt {c}}}{2 \sqrt {c}}+\frac {1}{2} e x \sqrt {d+e x^2} \left (7 d-\frac {4 \sqrt {-a} e}{\sqrt {c}}\right )}{4 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {-a}}\)

\(\Big \downarrow \) 224

\(\displaystyle -\frac {\frac {\frac {\frac {8 \left (-3 \sqrt {-a} c d^2 e-3 a \sqrt {c} d e^2+\sqrt {-a} a e^3+c^{3/2} d^3\right ) \int \frac {1}{\left (\sqrt {c} x^2+\sqrt {-a}\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}+\frac {e \left (-20 \sqrt {-a} \sqrt {c} d e-8 a e^2+15 c d^2\right ) \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{\sqrt {c}}}{2 \sqrt {c}}+\frac {1}{2} e x \sqrt {d+e x^2} \left (7 d-\frac {4 \sqrt {-a} e}{\sqrt {c}}\right )}{4 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {-a}}-\frac {\frac {\frac {\frac {8 \left (3 \sqrt {-a} c d^2 e-3 a \sqrt {c} d e^2+(-a)^{3/2} e^3+c^{3/2} d^3\right ) \int \frac {1}{\left (\sqrt {-a}-\sqrt {c} x^2\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}-\frac {e \left (20 \sqrt {-a} \sqrt {c} d e-8 a e^2+15 c d^2\right ) \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{\sqrt {c}}}{2 \sqrt {c}}-\frac {1}{2} e x \sqrt {d+e x^2} \left (\frac {4 \sqrt {-a} e}{\sqrt {c}}+7 d\right )}{4 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {-a}}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\frac {\frac {\frac {8 \left (3 \sqrt {-a} c d^2 e-3 a \sqrt {c} d e^2+(-a)^{3/2} e^3+c^{3/2} d^3\right ) \int \frac {1}{\left (\sqrt {-a}-\sqrt {c} x^2\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}-\frac {\sqrt {e} \left (20 \sqrt {-a} \sqrt {c} d e-8 a e^2+15 c d^2\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}-\frac {1}{2} e x \sqrt {d+e x^2} \left (\frac {4 \sqrt {-a} e}{\sqrt {c}}+7 d\right )}{4 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {-a}}-\frac {\frac {\frac {\frac {8 \left (-3 \sqrt {-a} c d^2 e-3 a \sqrt {c} d e^2+\sqrt {-a} a e^3+c^{3/2} d^3\right ) \int \frac {1}{\left (\sqrt {c} x^2+\sqrt {-a}\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}+\frac {\sqrt {e} \left (-20 \sqrt {-a} \sqrt {c} d e-8 a e^2+15 c d^2\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}+\frac {1}{2} e x \sqrt {d+e x^2} \left (7 d-\frac {4 \sqrt {-a} e}{\sqrt {c}}\right )}{4 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {-a}}\)

\(\Big \downarrow \) 291

\(\displaystyle -\frac {\frac {\frac {\frac {8 \left (-3 \sqrt {-a} c d^2 e-3 a \sqrt {c} d e^2+\sqrt {-a} a e^3+c^{3/2} d^3\right ) \int \frac {1}{\sqrt {-a}-\frac {\left (\sqrt {-a} e-\sqrt {c} d\right ) x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{\sqrt {c}}+\frac {\sqrt {e} \left (-20 \sqrt {-a} \sqrt {c} d e-8 a e^2+15 c d^2\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}+\frac {1}{2} e x \sqrt {d+e x^2} \left (7 d-\frac {4 \sqrt {-a} e}{\sqrt {c}}\right )}{4 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {-a}}-\frac {\frac {\frac {\frac {8 \left (3 \sqrt {-a} c d^2 e-3 a \sqrt {c} d e^2+(-a)^{3/2} e^3+c^{3/2} d^3\right ) \int \frac {1}{\sqrt {-a}-\frac {\left (\sqrt {c} d+\sqrt {-a} e\right ) x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{\sqrt {c}}-\frac {\sqrt {e} \left (20 \sqrt {-a} \sqrt {c} d e-8 a e^2+15 c d^2\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}-\frac {1}{2} e x \sqrt {d+e x^2} \left (\frac {4 \sqrt {-a} e}{\sqrt {c}}+7 d\right )}{4 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {-a}}\)

\(\Big \downarrow \) 218

\(\displaystyle -\frac {\frac {\frac {\frac {8 \left (3 \sqrt {-a} c d^2 e-3 a \sqrt {c} d e^2+(-a)^{3/2} e^3+c^{3/2} d^3\right ) \int \frac {1}{\sqrt {-a}-\frac {\left (\sqrt {c} d+\sqrt {-a} e\right ) x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{\sqrt {c}}-\frac {\sqrt {e} \left (20 \sqrt {-a} \sqrt {c} d e-8 a e^2+15 c d^2\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}-\frac {1}{2} e x \sqrt {d+e x^2} \left (\frac {4 \sqrt {-a} e}{\sqrt {c}}+7 d\right )}{4 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {-a}}-\frac {\frac {\frac {\frac {8 \left (-3 \sqrt {-a} c d^2 e-3 a \sqrt {c} d e^2+\sqrt {-a} a e^3+c^{3/2} d^3\right ) \arctan \left (\frac {x \sqrt {\sqrt {c} d-\sqrt {-a} e}}{\sqrt [4]{-a} \sqrt {d+e x^2}}\right )}{\sqrt [4]{-a} \sqrt {c} \sqrt {\sqrt {c} d-\sqrt {-a} e}}+\frac {\sqrt {e} \left (-20 \sqrt {-a} \sqrt {c} d e-8 a e^2+15 c d^2\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}+\frac {1}{2} e x \sqrt {d+e x^2} \left (7 d-\frac {4 \sqrt {-a} e}{\sqrt {c}}\right )}{4 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {-a}}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\frac {\frac {\frac {8 \left (-3 \sqrt {-a} c d^2 e-3 a \sqrt {c} d e^2+\sqrt {-a} a e^3+c^{3/2} d^3\right ) \arctan \left (\frac {x \sqrt {\sqrt {c} d-\sqrt {-a} e}}{\sqrt [4]{-a} \sqrt {d+e x^2}}\right )}{\sqrt [4]{-a} \sqrt {c} \sqrt {\sqrt {c} d-\sqrt {-a} e}}+\frac {\sqrt {e} \left (-20 \sqrt {-a} \sqrt {c} d e-8 a e^2+15 c d^2\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}+\frac {1}{2} e x \sqrt {d+e x^2} \left (7 d-\frac {4 \sqrt {-a} e}{\sqrt {c}}\right )}{4 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {-a}}-\frac {\frac {\frac {\frac {8 \left (3 \sqrt {-a} c d^2 e-3 a \sqrt {c} d e^2+(-a)^{3/2} e^3+c^{3/2} d^3\right ) \text {arctanh}\left (\frac {x \sqrt {\sqrt {-a} e+\sqrt {c} d}}{\sqrt [4]{-a} \sqrt {d+e x^2}}\right )}{\sqrt [4]{-a} \sqrt {c} \sqrt {\sqrt {-a} e+\sqrt {c} d}}-\frac {\sqrt {e} \left (20 \sqrt {-a} \sqrt {c} d e-8 a e^2+15 c d^2\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}-\frac {1}{2} e x \sqrt {d+e x^2} \left (\frac {4 \sqrt {-a} e}{\sqrt {c}}+7 d\right )}{4 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{3/2}}{4 \sqrt {c}}}{2 \sqrt {-a}}\)

Input:

Int[(d + e*x^2)^(5/2)/(a + c*x^4),x]
 

Output:

-1/2*((e*x*(d + e*x^2)^(3/2))/(4*Sqrt[c]) + ((e*(7*d - (4*Sqrt[-a]*e)/Sqrt 
[c])*x*Sqrt[d + e*x^2])/2 + ((8*(c^(3/2)*d^3 - 3*Sqrt[-a]*c*d^2*e - 3*a*Sq 
rt[c]*d*e^2 + Sqrt[-a]*a*e^3)*ArcTan[(Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*x)/((-a 
)^(1/4)*Sqrt[d + e*x^2])])/((-a)^(1/4)*Sqrt[c]*Sqrt[Sqrt[c]*d - Sqrt[-a]*e 
]) + (Sqrt[e]*(15*c*d^2 - 20*Sqrt[-a]*Sqrt[c]*d*e - 8*a*e^2)*ArcTanh[(Sqrt 
[e]*x)/Sqrt[d + e*x^2]])/Sqrt[c])/(2*Sqrt[c]))/(4*Sqrt[c]))/Sqrt[-a] - (-1 
/4*(e*x*(d + e*x^2)^(3/2))/Sqrt[c] + (-1/2*(e*(7*d + (4*Sqrt[-a]*e)/Sqrt[c 
])*x*Sqrt[d + e*x^2]) + (-((Sqrt[e]*(15*c*d^2 + 20*Sqrt[-a]*Sqrt[c]*d*e - 
8*a*e^2)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/Sqrt[c]) + (8*(c^(3/2)*d^3 
+ 3*Sqrt[-a]*c*d^2*e - 3*a*Sqrt[c]*d*e^2 + (-a)^(3/2)*e^3)*ArcTanh[(Sqrt[S 
qrt[c]*d + Sqrt[-a]*e]*x)/((-a)^(1/4)*Sqrt[d + e*x^2])])/((-a)^(1/4)*Sqrt[ 
c]*Sqrt[Sqrt[c]*d + Sqrt[-a]*e]))/(2*Sqrt[c]))/(4*Sqrt[c]))/(2*Sqrt[-a])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 219

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])
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 224

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]
 

rule 291

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst 
[Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, 
d}, x] && NeQ[b*c - a*d, 0]
 

rule 318

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S 
imp[1/(b*(2*(p + q) + 1))   Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b 
*c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 
 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G 
tQ[q, 1] && NeQ[2*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntBinomialQ[a, b, c, 
d, 2, p, q, x]
 

rule 398

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) 
, x_Symbol] :> Simp[f/b   Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ 
b   Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} 
, x]
 

rule 403

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + 
 q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1))   Int[(a + b*x^2)^p*(c 
+ d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + 
f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, 
 d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
 

rule 1489

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{r 
= Rt[(-a)*c, 2]}, Simp[-c/(2*r)   Int[(d + e*x^2)^q/(r - c*x^2), x], x] - S 
imp[c/(2*r)   Int[(d + e*x^2)^q/(r + c*x^2), x], x]] /; FreeQ[{a, c, d, e, 
q}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[q]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(825\) vs. \(2(378)=756\).

Time = 1.05 (sec) , antiderivative size = 826, normalized size of antiderivative = 1.74

method	result	size
pseudoelliptic	\(-\frac {3 \left (\frac {\sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, \left (\left (\frac {\sqrt {e}\, c \,d^{2}}{3}-\frac {2 \sqrt {a \,e^{2}+c \,d^{2}}\, e^{\frac {3}{2}} \sqrt {a}}{3}-e^{\frac {5}{2}} a \right ) \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-\frac {e^{\frac {3}{2}} a c \,d^{2}}{3}+a^{2} e^{\frac {7}{2}}+\frac {2 \sqrt {a \,e^{2}+c \,d^{2}}\, e^{\frac {5}{2}} a^{\frac {3}{2}}}{3}\right ) \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}\, \ln \left (\frac {\sqrt {a}\, \left (e \,x^{2}+d \right )-\sqrt {e \,x^{2}+d}\, \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x +\sqrt {a \,e^{2}+c \,d^{2}}\, x^{2}}{x^{2}}\right )}{4}-\frac {\sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, \left (\left (\frac {\sqrt {e}\, c \,d^{2}}{3}-\frac {2 \sqrt {a \,e^{2}+c \,d^{2}}\, e^{\frac {3}{2}} \sqrt {a}}{3}-e^{\frac {5}{2}} a \right ) \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-\frac {e^{\frac {3}{2}} a c \,d^{2}}{3}+a^{2} e^{\frac {7}{2}}+\frac {2 \sqrt {a \,e^{2}+c \,d^{2}}\, e^{\frac {5}{2}} a^{\frac {3}{2}}}{3}\right ) \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}\, \ln \left (\frac {\sqrt {a}\, \left (e \,x^{2}+d \right )+\sqrt {e \,x^{2}+d}\, \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x +\sqrt {a \,e^{2}+c \,d^{2}}\, x^{2}}{x^{2}}\right )}{4}+\left (-\frac {a^{\frac {3}{2}} \left (5 \,\operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right ) d \,e^{2}+\sqrt {e \,x^{2}+d}\, e^{\frac {5}{2}} x \right ) \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}}{3}+d \left (\arctan \left (\frac {\sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x -2 \sqrt {a}\, \sqrt {e \,x^{2}+d}}{x \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}}\right )-\arctan \left (\frac {2 \sqrt {a}\, \sqrt {e \,x^{2}+d}+\sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x}{x \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}}\right )\right ) \left (-\frac {\sqrt {e}\, a c \,d^{2}}{3}+a^{2} e^{\frac {5}{2}}-\frac {2 \sqrt {a \,e^{2}+c \,d^{2}}\, e^{\frac {3}{2}} a^{\frac {3}{2}}}{3}\right )\right ) d c \right )}{2 a^{\frac {3}{2}} \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}\, \sqrt {e}\, d \,c^{2}}\)	\(826\)
risch	\(\text {Expression too large to display}\)	\(1004\)
default	\(\text {Expression too large to display}\)	\(5338\)

Input:

int((e*x^2+d)^(5/2)/(c*x^4+a),x,method=_RETURNVERBOSE)
 

Output:

-3/2*(1/4*(2*(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*((1/3*e^(1/2)*c*d^2-2/3* 
(a*e^2+c*d^2)^(1/2)*e^(3/2)*a^(1/2)-e^(5/2)*a)*(a*(a*e^2+c*d^2))^(1/2)-1/3 
*e^(3/2)*a*c*d^2+a^2*e^(7/2)+2/3*(a*e^2+c*d^2)^(1/2)*e^(5/2)*a^(3/2))*(4*( 
a*e^2+c*d^2)^(1/2)*a^(1/2)-2*(a*(a*e^2+c*d^2))^(1/2)-2*a*e)^(1/2)*ln((a^(1 
/2)*(e*x^2+d)-(e*x^2+d)^(1/2)*(2*(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*x+(a 
*e^2+c*d^2)^(1/2)*x^2)/x^2)-1/4*(2*(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*(( 
1/3*e^(1/2)*c*d^2-2/3*(a*e^2+c*d^2)^(1/2)*e^(3/2)*a^(1/2)-e^(5/2)*a)*(a*(a 
*e^2+c*d^2))^(1/2)-1/3*e^(3/2)*a*c*d^2+a^2*e^(7/2)+2/3*(a*e^2+c*d^2)^(1/2) 
*e^(5/2)*a^(3/2))*(4*(a*e^2+c*d^2)^(1/2)*a^(1/2)-2*(a*(a*e^2+c*d^2))^(1/2) 
-2*a*e)^(1/2)*ln((a^(1/2)*(e*x^2+d)+(e*x^2+d)^(1/2)*(2*(a*(a*e^2+c*d^2))^( 
1/2)+2*a*e)^(1/2)*x+(a*e^2+c*d^2)^(1/2)*x^2)/x^2)+(-1/3*a^(3/2)*(5*arctanh 
((e*x^2+d)^(1/2)/x/e^(1/2))*d*e^2+(e*x^2+d)^(1/2)*e^(5/2)*x)*(4*(a*e^2+c*d 
^2)^(1/2)*a^(1/2)-2*(a*(a*e^2+c*d^2))^(1/2)-2*a*e)^(1/2)+d*(arctan(((2*(a* 
(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*x-2*a^(1/2)*(e*x^2+d)^(1/2))/x/(4*(a*e^2 
+c*d^2)^(1/2)*a^(1/2)-2*(a*(a*e^2+c*d^2))^(1/2)-2*a*e)^(1/2))-arctan((2*a^ 
(1/2)*(e*x^2+d)^(1/2)+(2*(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*x)/x/(4*(a*e 
^2+c*d^2)^(1/2)*a^(1/2)-2*(a*(a*e^2+c*d^2))^(1/2)-2*a*e)^(1/2)))*(-1/3*e^( 
1/2)*a*c*d^2+a^2*e^(5/2)-2/3*(a*e^2+c*d^2)^(1/2)*e^(3/2)*a^(3/2)))*d*c)/a^ 
(3/2)/(4*(a*e^2+c*d^2)^(1/2)*a^(1/2)-2*(a*(a*e^2+c*d^2))^(1/2)-2*a*e)^(1/2 
)/e^(1/2)/d/c^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2327 vs. \(2 (380) = 760\).

Time = 13.10 (sec) , antiderivative size = 4661, normalized size of antiderivative = 9.83 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{a+c x^4} \, dx=\text {Too large to display} \] Input:

integrate((e*x^2+d)^(5/2)/(c*x^4+a),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {\left (d+e x^2\right )^{5/2}}{a+c x^4} \, dx=\int \frac {\left (d + e x^{2}\right )^{\frac {5}{2}}}{a + c x^{4}}\, dx \] Input:

integrate((e*x**2+d)**(5/2)/(c*x**4+a),x)
 

Output:

Integral((d + e*x**2)**(5/2)/(a + c*x**4), x)
 

Maxima [F]

\[ \int \frac {\left (d+e x^2\right )^{5/2}}{a+c x^4} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {5}{2}}}{c x^{4} + a} \,d x } \] Input:

integrate((e*x^2+d)^(5/2)/(c*x^4+a),x, algorithm="maxima")
 

Output:

integrate((e*x^2 + d)^(5/2)/(c*x^4 + a), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d+e x^2\right )^{5/2}}{a+c x^4} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((e*x^2+d)^(5/2)/(c*x^4+a),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^{5/2}}{a+c x^4} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{5/2}}{c\,x^4+a} \,d x \] Input:

int((d + e*x^2)^(5/2)/(a + c*x^4),x)
 

Output:

int((d + e*x^2)^(5/2)/(a + c*x^4), x)
 

Reduce [F]

\[ \int \frac {\left (d+e x^2\right )^{5/2}}{a+c x^4} \, dx=\left (\int \frac {\sqrt {e \,x^{2}+d}}{c \,x^{4}+a}d x \right ) d^{2}+\left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{4}}{c \,x^{4}+a}d x \right ) e^{2}+2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{2}}{c \,x^{4}+a}d x \right ) d e \] Input:

int((e*x^2+d)^(5/2)/(c*x^4+a),x)
 

Output:

int(sqrt(d + e*x**2)/(a + c*x**4),x)*d**2 + int((sqrt(d + e*x**2)*x**4)/(a 
 + c*x**4),x)*e**2 + 2*int((sqrt(d + e*x**2)*x**2)/(a + c*x**4),x)*d*e