Integrand size = 21, antiderivative size = 555 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{a+c x^4} \, dx=\frac {13 d e^2 x \sqrt {d+e x^2}}{8 c}+\frac {e^3 x^3 \sqrt {d+e x^2}}{4 c}+\frac {\sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (4 c d^2 e \left (c d^2-a e^2\right )-\left (c^2 d^4-6 a c d^2 e^2+a^2 e^4\right ) \left (e-\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{2 \sqrt {2} \sqrt [4]{a} c^{5/2} d \sqrt {c d^2+a e^2}}+\frac {e^{3/2} \left (35 c d^2-8 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{8 c^2}+\frac {\sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (4 c d^2 e \left (c d^2-a e^2\right )-\left (c^2 d^4-6 a c d^2 e^2+a^2 e^4\right ) \left (e+\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e-\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{2 \sqrt {2} \sqrt [4]{a} c^{5/2} d \sqrt {c d^2+a e^2}} \] Output:
13/8*d*e^2*x*(e*x^2+d)^(1/2)/c+1/4*e^3*x^3*(e*x^2+d)^(1/2)/c+1/4*(a^(1/2)* e+(a*e^2+c*d^2)^(1/2))^(1/2)*(4*c*d^2*e*(-a*e^2+c*d^2)-(a^2*e^4-6*a*c*d^2* e^2+c^2*d^4)*(e-(a*e^2+c*d^2)^(1/2)/a^(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^(1/4)/c^(5/2)/d/(a*e^2+c*d^2 )^(1/2)+1/8*e^(3/2)*(-8*a*e^2+35*c*d^2)*arctanh(e^(1/2)*x/(e*x^2+d)^(1/2)) /c^2+1/4*(-a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*(4*c*d^2*e*(-a*e^2+c*d^2)- (a^2*e^4-6*a*c*d^2*e^2+c^2*d^4)*(e+(a*e^2+c*d^2)^(1/2)/a^(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^(1/4)/c ^(5/2)/d/(a*e^2+c*d^2)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.34 (sec) , antiderivative size = 466, normalized size of antiderivative = 0.84 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{a+c x^4} \, dx=\frac {e^{3/2} \left (c \sqrt {e} x \sqrt {d+e x^2} \left (13 d+2 e x^2\right )+\left (-35 c d^2+8 a e^2\right ) \log \left (-\sqrt {e} x+\sqrt {d+e x^2}\right )+16 \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2+16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {c^2 d^5 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )-a c d^3 e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )-c^2 d^4 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}-4 a c d^2 e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+a^2 e^4 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+c^2 d^3 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-a c d e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}-8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]\right )}{8 c^2} \] Input:
Integrate[(d + e*x^2)^(7/2)/(a + c*x^4),x]
Output:
(e^(3/2)*(c*Sqrt[e]*x*Sqrt[d + e*x^2]*(13*d + 2*e*x^2) + (-35*c*d^2 + 8*a* e^2)*Log[-(Sqrt[e]*x) + Sqrt[d + e*x^2]] + 16*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 & , (c^2*d^5*Log[d + 2 *e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - a*c*d^3*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - c^2*d^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x *Sqrt[d + e*x^2] - #1]*#1 - 4*a*c*d^2*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sq rt[d + e*x^2] - #1]*#1 + a^2*e^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e* x^2] - #1]*#1 + c^2*d^3*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1 ]*#1^2 - a*c*d*e^2*Log[d + 2*e*x^2 - 2*Sqrt[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) & ]))/(8*c^2)
Time = 2.04 (sec) , antiderivative size = 742, normalized size of antiderivative = 1.34, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {1489, 27, 318, 25, 403, 27, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^{7/2}}{a+c x^4} \, dx\) |
| \(\Big \downarrow \) 1489 |
| \(\displaystyle -\frac {\sqrt {c} \int \frac {\left (e x^2+d\right )^{7/2}}{\sqrt {c} \left (\sqrt {-a}-\sqrt {c} x^2\right )}dx}{2 \sqrt {-a}}-\frac {\sqrt {c} \int \frac {\left (e x^2+d\right )^{7/2}}{\sqrt {c} \left (\sqrt {c} x^2+\sqrt {-a}\right )}dx}{2 \sqrt {-a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\left (e x^2+d\right )^{7/2}}{\sqrt {-a}-\sqrt {c} x^2}dx}{2 \sqrt {-a}}-\frac {\int \frac {\left (e x^2+d\right )^{7/2}}{\sqrt {c} x^2+\sqrt {-a}}dx}{2 \sqrt {-a}}\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle -\frac {\frac {\int \frac {\left (e x^2+d\right )^{3/2} \left (e \left (11 \sqrt {c} d-6 \sqrt {-a} e\right ) x^2+d \left (6 \sqrt {c} d-\sqrt {-a} e\right )\right )}{\sqrt {c} x^2+\sqrt {-a}}dx}{6 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {-a}}-\frac {-\frac {\int -\frac {\left (e x^2+d\right )^{3/2} \left (e \left (11 \sqrt {c} d+6 \sqrt {-a} e\right ) x^2+d \left (6 \sqrt {c} d+\sqrt {-a} e\right )\right )}{\sqrt {-a}-\sqrt {c} x^2}dx}{6 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {-a}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int \frac {\left (e x^2+d\right )^{3/2} \left (e \left (11 \sqrt {c} d-6 \sqrt {-a} e\right ) x^2+d \left (6 \sqrt {c} d-\sqrt {-a} e\right )\right )}{\sqrt {c} x^2+\sqrt {-a}}dx}{6 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {-a}}-\frac {\frac {\int \frac {\left (e x^2+d\right )^{3/2} \left (e \left (11 \sqrt {c} d+6 \sqrt {-a} e\right ) x^2+d \left (6 \sqrt {c} d+\sqrt {-a} e\right )\right )}{\sqrt {-a}-\sqrt {c} x^2}dx}{6 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {-a}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {3 \sqrt {e x^2+d} \left (e \left (19 c d^2-22 \sqrt {-a} \sqrt {c} e d-8 a e^2\right ) x^2+d \left (8 c d^2-5 \sqrt {-a} \sqrt {c} e d-2 a e^2\right )\right )}{\sqrt {c} x^2+\sqrt {-a}}dx}{4 \sqrt {c}}+\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (11 d-\frac {6 \sqrt {-a} e}{\sqrt {c}}\right )}{6 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {-a}}-\frac {\frac {-\frac {\int -\frac {3 \sqrt {e x^2+d} \left (e \left (19 c d^2+22 \sqrt {-a} \sqrt {c} e d-8 a e^2\right ) x^2+d \left (8 c d^2+5 \sqrt {-a} \sqrt {c} e d-2 a e^2\right )\right )}{\sqrt {-a}-\sqrt {c} x^2}dx}{4 \sqrt {c}}-\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (\frac {6 \sqrt {-a} e}{\sqrt {c}}+11 d\right )}{6 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {-a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {3 \int \frac {\sqrt {e x^2+d} \left (e \left (19 c d^2-22 \sqrt {-a} \sqrt {c} e d-8 a e^2\right ) x^2+d \left (8 c d^2-5 \sqrt {-a} \sqrt {c} e d-2 a e^2\right )\right )}{\sqrt {c} x^2+\sqrt {-a}}dx}{4 \sqrt {c}}+\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (11 d-\frac {6 \sqrt {-a} e}{\sqrt {c}}\right )}{6 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {-a}}-\frac {\frac {\frac {3 \int \frac {\sqrt {e x^2+d} \left (e \left (19 c d^2+22 \sqrt {-a} \sqrt {c} e d-8 a e^2\right ) x^2+d \left (8 c d^2+5 \sqrt {-a} \sqrt {c} e d-2 a e^2\right )\right )}{\sqrt {-a}-\sqrt {c} x^2}dx}{4 \sqrt {c}}-\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (\frac {6 \sqrt {-a} e}{\sqrt {c}}+11 d\right )}{6 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {-a}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle -\frac {\frac {\frac {3 \left (-\frac {\int -\frac {e \left (35 c^{3/2} d^3+70 \sqrt {-a} c e d^2-56 a \sqrt {c} e^2 d+16 (-a)^{3/2} e^3\right ) x^2+d \left (16 c^{3/2} d^3+29 \sqrt {-a} c e d^2-26 a \sqrt {c} e^2 d-8 \sqrt {-a} a e^3\right )}{\left (\sqrt {-a}-\sqrt {c} x^2\right ) \sqrt {e x^2+d}}dx}{2 \sqrt {c}}-\frac {e x \sqrt {d+e x^2} \left (22 \sqrt {-a} \sqrt {c} d e-8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (\frac {6 \sqrt {-a} e}{\sqrt {c}}+11 d\right )}{6 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {-a}}-\frac {\frac {\frac {3 \left (\frac {\int \frac {e \left (35 c^{3/2} d^3-70 \sqrt {-a} c e d^2-56 a \sqrt {c} e^2 d+16 \sqrt {-a} a e^3\right ) x^2+d \left (16 c^{3/2} d^3-29 \sqrt {-a} c e d^2-26 a \sqrt {c} e^2 d+8 \sqrt {-a} a e^3\right )}{\left (\sqrt {c} x^2+\sqrt {-a}\right ) \sqrt {e x^2+d}}dx}{2 \sqrt {c}}+\frac {e x \sqrt {d+e x^2} \left (-22 \sqrt {-a} \sqrt {c} d e-8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}+\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (11 d-\frac {6 \sqrt {-a} e}{\sqrt {c}}\right )}{6 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {-a}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\frac {3 \left (\frac {\int \frac {e \left (35 c^{3/2} d^3+70 \sqrt {-a} c e d^2-56 a \sqrt {c} e^2 d+16 (-a)^{3/2} e^3\right ) x^2+d \left (16 c^{3/2} d^3+29 \sqrt {-a} c e d^2-26 a \sqrt {c} e^2 d+8 (-a)^{3/2} e^3\right )}{\left (\sqrt {-a}-\sqrt {c} x^2\right ) \sqrt {e x^2+d}}dx}{2 \sqrt {c}}-\frac {e x \sqrt {d+e x^2} \left (22 \sqrt {-a} \sqrt {c} d e-8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (\frac {6 \sqrt {-a} e}{\sqrt {c}}+11 d\right )}{6 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {-a}}-\frac {\frac {\frac {3 \left (\frac {\int \frac {e \left (35 c^{3/2} d^3-70 \sqrt {-a} c e d^2-56 a \sqrt {c} e^2 d+16 \sqrt {-a} a e^3\right ) x^2+d \left (16 c^{3/2} d^3-29 \sqrt {-a} c e d^2-26 a \sqrt {c} e^2 d+8 \sqrt {-a} a e^3\right )}{\left (\sqrt {c} x^2+\sqrt {-a}\right ) \sqrt {e x^2+d}}dx}{2 \sqrt {c}}+\frac {e x \sqrt {d+e x^2} \left (-22 \sqrt {-a} \sqrt {c} d e-8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}+\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (11 d-\frac {6 \sqrt {-a} e}{\sqrt {c}}\right )}{6 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {-a}}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle -\frac {\frac {\frac {3 \left (\frac {\frac {16 \left (a^2 e^4+4 \sqrt {-a} c^{3/2} d^3 e-6 a c d^2 e^2+4 (-a)^{3/2} \sqrt {c} d e^3+c^2 d^4\right ) \int \frac {1}{\left (\sqrt {-a}-\sqrt {c} x^2\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}-\frac {e \left (70 \sqrt {-a} c d^2 e-56 a \sqrt {c} d e^2+16 (-a)^{3/2} e^3+35 c^{3/2} d^3\right ) \int \frac {1}{\sqrt {e x^2+d}}dx}{\sqrt {c}}}{2 \sqrt {c}}-\frac {e x \sqrt {d+e x^2} \left (22 \sqrt {-a} \sqrt {c} d e-8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (\frac {6 \sqrt {-a} e}{\sqrt {c}}+11 d\right )}{6 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {-a}}-\frac {\frac {\frac {3 \left (\frac {\frac {16 \left (a^2 e^4-4 \sqrt {-a} c^{3/2} d^3 e-6 a c d^2 e^2+4 \sqrt {-a} a \sqrt {c} d e^3+c^2 d^4\right ) \int \frac {1}{\left (\sqrt {c} x^2+\sqrt {-a}\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}+\frac {e \left (-70 \sqrt {-a} c d^2 e-56 a \sqrt {c} d e^2+16 \sqrt {-a} a e^3+35 c^{3/2} d^3\right ) \int \frac {1}{\sqrt {e x^2+d}}dx}{\sqrt {c}}}{2 \sqrt {c}}+\frac {e x \sqrt {d+e x^2} \left (-22 \sqrt {-a} \sqrt {c} d e-8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}+\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (11 d-\frac {6 \sqrt {-a} e}{\sqrt {c}}\right )}{6 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {-a}}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle -\frac {\frac {\frac {3 \left (\frac {\frac {16 \left (a^2 e^4+4 \sqrt {-a} c^{3/2} d^3 e-6 a c d^2 e^2+4 (-a)^{3/2} \sqrt {c} d e^3+c^2 d^4\right ) \int \frac {1}{\left (\sqrt {-a}-\sqrt {c} x^2\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}-\frac {e \left (70 \sqrt {-a} c d^2 e-56 a \sqrt {c} d e^2+16 (-a)^{3/2} e^3+35 c^{3/2} d^3\right ) \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{\sqrt {c}}}{2 \sqrt {c}}-\frac {e x \sqrt {d+e x^2} \left (22 \sqrt {-a} \sqrt {c} d e-8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (\frac {6 \sqrt {-a} e}{\sqrt {c}}+11 d\right )}{6 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {-a}}-\frac {\frac {\frac {3 \left (\frac {\frac {16 \left (a^2 e^4-4 \sqrt {-a} c^{3/2} d^3 e-6 a c d^2 e^2+4 \sqrt {-a} a \sqrt {c} d e^3+c^2 d^4\right ) \int \frac {1}{\left (\sqrt {c} x^2+\sqrt {-a}\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}+\frac {e \left (-70 \sqrt {-a} c d^2 e-56 a \sqrt {c} d e^2+16 \sqrt {-a} a e^3+35 c^{3/2} d^3\right ) \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{\sqrt {c}}}{2 \sqrt {c}}+\frac {e x \sqrt {d+e x^2} \left (-22 \sqrt {-a} \sqrt {c} d e-8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}+\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (11 d-\frac {6 \sqrt {-a} e}{\sqrt {c}}\right )}{6 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {-a}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {\frac {3 \left (\frac {\frac {16 \left (a^2 e^4+4 \sqrt {-a} c^{3/2} d^3 e-6 a c d^2 e^2+4 (-a)^{3/2} \sqrt {c} d e^3+c^2 d^4\right ) \int \frac {1}{\left (\sqrt {-a}-\sqrt {c} x^2\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}-\frac {\sqrt {e} \left (70 \sqrt {-a} c d^2 e-56 a \sqrt {c} d e^2+16 (-a)^{3/2} e^3+35 c^{3/2} d^3\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}-\frac {e x \sqrt {d+e x^2} \left (22 \sqrt {-a} \sqrt {c} d e-8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (\frac {6 \sqrt {-a} e}{\sqrt {c}}+11 d\right )}{6 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {-a}}-\frac {\frac {\frac {3 \left (\frac {\frac {16 \left (a^2 e^4-4 \sqrt {-a} c^{3/2} d^3 e-6 a c d^2 e^2+4 \sqrt {-a} a \sqrt {c} d e^3+c^2 d^4\right ) \int \frac {1}{\left (\sqrt {c} x^2+\sqrt {-a}\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}+\frac {\sqrt {e} \left (-70 \sqrt {-a} c d^2 e-56 a \sqrt {c} d e^2+16 \sqrt {-a} a e^3+35 c^{3/2} d^3\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}+\frac {e x \sqrt {d+e x^2} \left (-22 \sqrt {-a} \sqrt {c} d e-8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}+\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (11 d-\frac {6 \sqrt {-a} e}{\sqrt {c}}\right )}{6 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {-a}}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle -\frac {\frac {\frac {3 \left (\frac {\frac {16 \left (a^2 e^4-4 \sqrt {-a} c^{3/2} d^3 e-6 a c d^2 e^2+4 \sqrt {-a} a \sqrt {c} d e^3+c^2 d^4\right ) \int \frac {1}{\sqrt {-a}-\frac {\left (\sqrt {-a} e-\sqrt {c} d\right ) x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{\sqrt {c}}+\frac {\sqrt {e} \left (-70 \sqrt {-a} c d^2 e-56 a \sqrt {c} d e^2+16 \sqrt {-a} a e^3+35 c^{3/2} d^3\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}+\frac {e x \sqrt {d+e x^2} \left (-22 \sqrt {-a} \sqrt {c} d e-8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}+\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (11 d-\frac {6 \sqrt {-a} e}{\sqrt {c}}\right )}{6 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {-a}}-\frac {\frac {\frac {3 \left (\frac {\frac {16 \left (a^2 e^4+4 \sqrt {-a} c^{3/2} d^3 e-6 a c d^2 e^2+4 (-a)^{3/2} \sqrt {c} d e^3+c^2 d^4\right ) \int \frac {1}{\sqrt {-a}-\frac {\left (\sqrt {c} d+\sqrt {-a} e\right ) x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{\sqrt {c}}-\frac {\sqrt {e} \left (70 \sqrt {-a} c d^2 e-56 a \sqrt {c} d e^2+16 (-a)^{3/2} e^3+35 c^{3/2} d^3\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}-\frac {e x \sqrt {d+e x^2} \left (22 \sqrt {-a} \sqrt {c} d e-8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (\frac {6 \sqrt {-a} e}{\sqrt {c}}+11 d\right )}{6 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {-a}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\frac {\frac {3 \left (\frac {\frac {16 \left (a^2 e^4+4 \sqrt {-a} c^{3/2} d^3 e-6 a c d^2 e^2+4 (-a)^{3/2} \sqrt {c} d e^3+c^2 d^4\right ) \int \frac {1}{\sqrt {-a}-\frac {\left (\sqrt {c} d+\sqrt {-a} e\right ) x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{\sqrt {c}}-\frac {\sqrt {e} \left (70 \sqrt {-a} c d^2 e-56 a \sqrt {c} d e^2+16 (-a)^{3/2} e^3+35 c^{3/2} d^3\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}-\frac {e x \sqrt {d+e x^2} \left (22 \sqrt {-a} \sqrt {c} d e-8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (\frac {6 \sqrt {-a} e}{\sqrt {c}}+11 d\right )}{6 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {-a}}-\frac {\frac {\frac {3 \left (\frac {\frac {16 \left (a^2 e^4-4 \sqrt {-a} c^{3/2} d^3 e-6 a c d^2 e^2+4 \sqrt {-a} a \sqrt {c} d e^3+c^2 d^4\right ) \arctan \left (\frac {x \sqrt {\sqrt {c} d-\sqrt {-a} e}}{\sqrt [4]{-a} \sqrt {d+e x^2}}\right )}{\sqrt [4]{-a} \sqrt {c} \sqrt {\sqrt {c} d-\sqrt {-a} e}}+\frac {\sqrt {e} \left (-70 \sqrt {-a} c d^2 e-56 a \sqrt {c} d e^2+16 \sqrt {-a} a e^3+35 c^{3/2} d^3\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}+\frac {e x \sqrt {d+e x^2} \left (-22 \sqrt {-a} \sqrt {c} d e-8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}+\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (11 d-\frac {6 \sqrt {-a} e}{\sqrt {c}}\right )}{6 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {-a}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {\frac {3 \left (\frac {\frac {16 \left (a^2 e^4-4 \sqrt {-a} c^{3/2} d^3 e-6 a c d^2 e^2+4 \sqrt {-a} a \sqrt {c} d e^3+c^2 d^4\right ) \arctan \left (\frac {x \sqrt {\sqrt {c} d-\sqrt {-a} e}}{\sqrt [4]{-a} \sqrt {d+e x^2}}\right )}{\sqrt [4]{-a} \sqrt {c} \sqrt {\sqrt {c} d-\sqrt {-a} e}}+\frac {\sqrt {e} \left (-70 \sqrt {-a} c d^2 e-56 a \sqrt {c} d e^2+16 \sqrt {-a} a e^3+35 c^{3/2} d^3\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}+\frac {e x \sqrt {d+e x^2} \left (-22 \sqrt {-a} \sqrt {c} d e-8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}+\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (11 d-\frac {6 \sqrt {-a} e}{\sqrt {c}}\right )}{6 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {-a}}-\frac {\frac {\frac {3 \left (\frac {\frac {16 \left (a^2 e^4+4 \sqrt {-a} c^{3/2} d^3 e-6 a c d^2 e^2+4 (-a)^{3/2} \sqrt {c} d e^3+c^2 d^4\right ) \text {arctanh}\left (\frac {x \sqrt {\sqrt {-a} e+\sqrt {c} d}}{\sqrt [4]{-a} \sqrt {d+e x^2}}\right )}{\sqrt [4]{-a} \sqrt {c} \sqrt {\sqrt {-a} e+\sqrt {c} d}}-\frac {\sqrt {e} \left (70 \sqrt {-a} c d^2 e-56 a \sqrt {c} d e^2+16 (-a)^{3/2} e^3+35 c^{3/2} d^3\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}-\frac {e x \sqrt {d+e x^2} \left (22 \sqrt {-a} \sqrt {c} d e-8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (\frac {6 \sqrt {-a} e}{\sqrt {c}}+11 d\right )}{6 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {-a}}\) |
Input:
Int[(d + e*x^2)^(7/2)/(a + c*x^4),x]
Output:
-1/2*((e*x*(d + e*x^2)^(5/2))/(6*Sqrt[c]) + ((e*(11*d - (6*Sqrt[-a]*e)/Sqr t[c])*x*(d + e*x^2)^(3/2))/4 + (3*((e*(19*c*d^2 - 22*Sqrt[-a]*Sqrt[c]*d*e - 8*a*e^2)*x*Sqrt[d + e*x^2])/(2*Sqrt[c]) + ((16*(c^2*d^4 - 4*Sqrt[-a]*c^( 3/2)*d^3*e - 6*a*c*d^2*e^2 + 4*Sqrt[-a]*a*Sqrt[c]*d*e^3 + a^2*e^4)*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]*(35*c^(3/2)*d^3 - 70*Sq rt[-a]*c*d^2*e - 56*a*Sqrt[c]*d*e^2 + 16*Sqrt[-a]*a*e^3)*ArcTanh[(Sqrt[e]* x)/Sqrt[d + e*x^2]])/Sqrt[c])/(2*Sqrt[c])))/(4*Sqrt[c]))/(6*Sqrt[c]))/Sqrt [-a] - (-1/6*(e*x*(d + e*x^2)^(5/2))/Sqrt[c] + (-1/4*(e*(11*d + (6*Sqrt[-a ]*e)/Sqrt[c])*x*(d + e*x^2)^(3/2)) + (3*(-1/2*(e*(19*c*d^2 + 22*Sqrt[-a]*S qrt[c]*d*e - 8*a*e^2)*x*Sqrt[d + e*x^2])/Sqrt[c] + (-((Sqrt[e]*(35*c^(3/2) *d^3 + 70*Sqrt[-a]*c*d^2*e - 56*a*Sqrt[c]*d*e^2 + 16*(-a)^(3/2)*e^3)*ArcTa nh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/Sqrt[c]) + (16*(c^2*d^4 + 4*Sqrt[-a]*c^(3 /2)*d^3*e - 6*a*c*d^2*e^2 + 4*(-a)^(3/2)*Sqrt[c]*d*e^3 + a^2*e^4)*ArcTanh[ (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]))/(2*Sqrt[c])))/(4*Sqrt[c]))/(6*Sqr t[c]))/(2*Sqrt[-a])
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[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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[((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]
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/(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]
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]
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]
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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(989\) vs. \(2(459)=918\).
Time = 3.10 (sec) , antiderivative size = 990, normalized size of antiderivative = 1.78
| method | result | size |
| pseudoelliptic | \(-\frac {-\frac {\sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, \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}\, \left (\left (\left (3 e^{\frac {3}{2}} c \,d^{2} \sqrt {a}-e^{\frac {7}{2}} a^{\frac {3}{2}}\right ) \sqrt {a \,e^{2}+c \,d^{2}}-\sqrt {e}\, c^{2} d^{4}+6 e^{\frac {5}{2}} c \,d^{2} a -e^{\frac {9}{2}} a^{2}\right ) \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+\left (-3 e^{\frac {5}{2}} c \,d^{2} a^{\frac {3}{2}}+e^{\frac {9}{2}} a^{\frac {5}{2}}\right ) \sqrt {a \,e^{2}+c \,d^{2}}+a \left (e^{\frac {3}{2}} c^{2} d^{4}-6 a \,e^{\frac {7}{2}} c \,d^{2}+a^{2} e^{\frac {11}{2}}\right )\right ) \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}\, \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}\, \left (\left (\left (3 e^{\frac {3}{2}} c \,d^{2} \sqrt {a}-e^{\frac {7}{2}} a^{\frac {3}{2}}\right ) \sqrt {a \,e^{2}+c \,d^{2}}-\sqrt {e}\, c^{2} d^{4}+6 e^{\frac {5}{2}} c \,d^{2} a -e^{\frac {9}{2}} a^{2}\right ) \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+\left (-3 e^{\frac {5}{2}} c \,d^{2} a^{\frac {3}{2}}+e^{\frac {9}{2}} a^{\frac {5}{2}}\right ) \sqrt {a \,e^{2}+c \,d^{2}}+a \left (e^{\frac {3}{2}} c^{2} d^{4}-6 a \,e^{\frac {7}{2}} c \,d^{2}+a^{2} e^{\frac {11}{2}}\right )\right ) \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 (\left (\left (2 a^{\frac {5}{2}} e^{4}-\frac {35 a^{\frac {3}{2}} c \,d^{2} e^{2}}{4}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right )-\frac {13 x \,a^{\frac {3}{2}} \sqrt {e \,x^{2}+d}\, \left (\frac {2 e^{\frac {7}{2}} x^{2}}{13}+d \,e^{\frac {5}{2}}\right ) c}{4}\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}+\left (\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 )-\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 )\right ) \left (\left (3 e^{\frac {3}{2}} c \,d^{2} a^{\frac {3}{2}}-e^{\frac {7}{2}} a^{\frac {5}{2}}\right ) \sqrt {a \,e^{2}+c \,d^{2}}+a \left (\sqrt {e}\, c^{2} d^{4}-6 e^{\frac {5}{2}} c \,d^{2} a +e^{\frac {9}{2}} a^{2}\right )\right )\right ) d^{2} c}{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^{2} c^{3}}\) | \(990\) |
| risch | \(\text {Expression too large to display}\) | \(1058\) |
| default | \(\text {Expression too large to display}\) | \(7980\) |
Input:
int((e*x^2+d)^(7/2)/(c*x^4+a),x,method=_RETURNVERBOSE)
Output:
-1/2/a^(3/2)*(-1/4*(2*(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*(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)*(((3*e^(3/2)*c*d^2* a^(1/2)-e^(7/2)*a^(3/2))*(a*e^2+c*d^2)^(1/2)-e^(1/2)*c^2*d^4+6*e^(5/2)*c*d ^2*a-e^(9/2)*a^2)*(a*(a*e^2+c*d^2))^(1/2)+(-3*e^(5/2)*c*d^2*a^(3/2)+e^(9/2 )*a^(5/2))*(a*e^2+c*d^2)^(1/2)+a*(e^(3/2)*c^2*d^4-6*a*e^(7/2)*c*d^2+a^2*e^ (11/2)))*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)*(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)*(((3*e^(3/2)*c*d^2*a^(1/2)-e^(7/2)*a^(3/2))*(a*e^2+c*d^2)^(1/2) -e^(1/2)*c^2*d^4+6*e^(5/2)*c*d^2*a-e^(9/2)*a^2)*(a*(a*e^2+c*d^2))^(1/2)+(- 3*e^(5/2)*c*d^2*a^(3/2)+e^(9/2)*a^(5/2))*(a*e^2+c*d^2)^(1/2)+a*(e^(3/2)*c^ 2*d^4-6*a*e^(7/2)*c*d^2+a^2*e^(11/2)))*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) +(((2*a^(5/2)*e^4-35/4*a^(3/2)*c*d^2*e^2)*arctanh((e*x^2+d)^(1/2)/x/e^(1/2 ))-13/4*x*a^(3/2)*(e*x^2+d)^(1/2)*(2/13*e^(7/2)*x^2+d*e^(5/2))*c)*(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))-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)))*((3...
Leaf count of result is larger than twice the leaf count of optimal. 3394 vs. \(2 (461) = 922\).
Time = 53.72 (sec) , antiderivative size = 6799, normalized size of antiderivative = 12.25 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{a+c x^4} \, dx=\text {Too large to display} \] Input:
integrate((e*x^2+d)^(7/2)/(c*x^4+a),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\left (d+e x^2\right )^{7/2}}{a+c x^4} \, dx=\int \frac {\left (d + e x^{2}\right )^{\frac {7}{2}}}{a + c x^{4}}\, dx \] Input:
integrate((e*x**2+d)**(7/2)/(c*x**4+a),x)
Output:
Integral((d + e*x**2)**(7/2)/(a + c*x**4), x)
\[ \int \frac {\left (d+e x^2\right )^{7/2}}{a+c x^4} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {7}{2}}}{c x^{4} + a} \,d x } \] Input:
integrate((e*x^2+d)^(7/2)/(c*x^4+a),x, algorithm="maxima")
Output:
integrate((e*x^2 + d)^(7/2)/(c*x^4 + a), x)
Exception generated. \[ \int \frac {\left (d+e x^2\right )^{7/2}}{a+c x^4} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((e*x^2+d)^(7/2)/(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
Timed out. \[ \int \frac {\left (d+e x^2\right )^{7/2}}{a+c x^4} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{7/2}}{c\,x^4+a} \,d x \] Input:
int((d + e*x^2)^(7/2)/(a + c*x^4),x)
Output:
int((d + e*x^2)^(7/2)/(a + c*x^4), x)
\[ \int \frac {\left (d+e x^2\right )^{7/2}}{a+c x^4} \, dx=\left (\int \frac {\sqrt {e \,x^{2}+d}}{c \,x^{4}+a}d x \right ) d^{3}+\left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{6}}{c \,x^{4}+a}d x \right ) e^{3}+3 \left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{4}}{c \,x^{4}+a}d x \right ) d \,e^{2}+3 \left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{2}}{c \,x^{4}+a}d x \right ) d^{2} e \] Input:
int((e*x^2+d)^(7/2)/(c*x^4+a),x)
Output:
int(sqrt(d + e*x**2)/(a + c*x**4),x)*d**3 + int((sqrt(d + e*x**2)*x**6)/(a + c*x**4),x)*e**3 + 3*int((sqrt(d + e*x**2)*x**4)/(a + c*x**4),x)*d*e**2 + 3*int((sqrt(d + e*x**2)*x**2)/(a + c*x**4),x)*d**2*e