Integrand size = 22, antiderivative size = 232 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{a-c x^4} \, dx=-\frac {13 d e^2 x \sqrt {d+e x^2}}{8 c}-\frac {e^3 x^3 \sqrt {d+e x^2}}{4 c}+\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{7/2} \arctan \left (\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{2 a^{3/4} c^2}-\frac {e^{3/2} \left (35 c d^2+8 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{8 c^2}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right )^{7/2} \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{2 a^{3/4} c^2} \] 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+1/2*(c^(1/2) *d-a^(1/2)*e)^(7/2)*arctan((c^(1/2)*d-a^(1/2)*e)^(1/2)*x/a^(1/4)/(e*x^2+d) ^(1/2))/a^(3/4)/c^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+1/2*(c^(1/2)*d+a^(1/2)*e)^(7/2)*arctanh((c^(1/2)*d+a^(1/2) *e)^(1/2)*x/a^(1/4)/(e*x^2+d)^(1/2))/a^(3/4)/c^2
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.33 (sec) , antiderivative size = 465, normalized size of antiderivative = 2.00 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{a-c x^4} \, dx=\frac {e^{3/2} \left (-c \sqrt {e} x \sqrt {d+e x^2} \left (13 d+2 e x^2\right )+\left (35 c d^2+8 a e^2\right ) \log \left (-\sqrt {e} x+\sqrt {d+e x^2}\right )-16 \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2-16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {c^2 d^5 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+a c d^3 e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )-c^2 d^4 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+4 a c d^2 e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+a^2 e^4 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+c^2 d^3 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+a c d e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}+8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]\right )}{8 c^2} \] 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* Sqrt[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)
Leaf count is larger than twice the leaf count of optimal. \(582\) vs. \(2(232)=464\).
Time = 1.59 (sec) , antiderivative size = 582, normalized size of antiderivative = 2.51, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, 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 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\int \frac {e \left (35 c^{3/2} d^3-70 \sqrt {a} c e d^2+56 a \sqrt {c} e^2 d-16 a^{3/2} e^3\right ) x^2+d \left (16 c^{3/2} d^3-29 \sqrt {a} c e d^2+26 a \sqrt {c} e^2 d-8 a^{3/2} e^3\right )}{\left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {e x^2+d}}dx}{2 \sqrt {c}}+\frac {e x \sqrt {d+e x^2} \left (-22 \sqrt {a} \sqrt {c} d e+8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}+\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (11 d-\frac {6 \sqrt {a} e}{\sqrt {c}}\right )}{6 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {a}}+\frac {\frac {\frac {3 \left (-\frac {\int -\frac {e \left (35 c^{3/2} d^3+70 \sqrt {a} c e d^2+56 a \sqrt {c} e^2 d+16 a^{3/2} e^3\right ) x^2+d \left (16 c^{3/2} d^3+29 \sqrt {a} c e d^2+26 a \sqrt {c} e^2 d+8 a^{3/2} e^3\right )}{\left (\sqrt {a}-\sqrt {c} x^2\right ) \sqrt {e x^2+d}}dx}{2 \sqrt {c}}-\frac {e x \sqrt {d+e x^2} \left (22 \sqrt {a} \sqrt {c} d e+8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (\frac {6 \sqrt {a} e}{\sqrt {c}}+11 d\right )}{6 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {a}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\int \frac {e \left (35 c^{3/2} d^3-70 \sqrt {a} c e d^2+56 a \sqrt {c} e^2 d-16 a^{3/2} e^3\right ) x^2+d \left (16 c^{3/2} d^3-29 \sqrt {a} c e d^2+26 a \sqrt {c} e^2 d-8 a^{3/2} e^3\right )}{\left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {e x^2+d}}dx}{2 \sqrt {c}}+\frac {e x \sqrt {d+e x^2} \left (-22 \sqrt {a} \sqrt {c} d e+8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}+\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (11 d-\frac {6 \sqrt {a} e}{\sqrt {c}}\right )}{6 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {a}}+\frac {\frac {\frac {3 \left (\frac {\int \frac {e \left (35 c^{3/2} d^3+70 \sqrt {a} c e d^2+56 a \sqrt {c} e^2 d+16 a^{3/2} e^3\right ) x^2+d \left (16 c^{3/2} d^3+29 \sqrt {a} c e d^2+26 a \sqrt {c} e^2 d+8 a^{3/2} e^3\right )}{\left (\sqrt {a}-\sqrt {c} x^2\right ) \sqrt {e x^2+d}}dx}{2 \sqrt {c}}-\frac {e x \sqrt {d+e x^2} \left (22 \sqrt {a} \sqrt {c} d e+8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (\frac {6 \sqrt {a} e}{\sqrt {c}}+11 d\right )}{6 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {a}}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {16 \left (\sqrt {a} e+\sqrt {c} d\right )^4 \int \frac {1}{\left (\sqrt {a}-\sqrt {c} x^2\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}-\frac {e \left (16 a^{3/2} e^3+70 \sqrt {a} c d^2 e+56 a \sqrt {c} d e^2+35 c^{3/2} d^3\right ) \int \frac {1}{\sqrt {e x^2+d}}dx}{\sqrt {c}}}{2 \sqrt {c}}-\frac {e x \sqrt {d+e x^2} \left (22 \sqrt {a} \sqrt {c} d e+8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (\frac {6 \sqrt {a} e}{\sqrt {c}}+11 d\right )}{6 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {a}}+\frac {\frac {\frac {3 \left (\frac {\frac {e \left (-16 a^{3/2} e^3-70 \sqrt {a} c d^2 e+56 a \sqrt {c} d e^2+35 c^{3/2} d^3\right ) \int \frac {1}{\sqrt {e x^2+d}}dx}{\sqrt {c}}+\frac {16 \left (\sqrt {c} d-\sqrt {a} e\right )^4 \int \frac {1}{\left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}}{2 \sqrt {c}}+\frac {e x \sqrt {d+e x^2} \left (-22 \sqrt {a} \sqrt {c} d e+8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}+\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (11 d-\frac {6 \sqrt {a} e}{\sqrt {c}}\right )}{6 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {a}}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {e \left (-16 a^{3/2} e^3-70 \sqrt {a} c d^2 e+56 a \sqrt {c} d e^2+35 c^{3/2} d^3\right ) \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{\sqrt {c}}+\frac {16 \left (\sqrt {c} d-\sqrt {a} e\right )^4 \int \frac {1}{\left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}}{2 \sqrt {c}}+\frac {e x \sqrt {d+e x^2} \left (-22 \sqrt {a} \sqrt {c} d e+8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}+\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (11 d-\frac {6 \sqrt {a} e}{\sqrt {c}}\right )}{6 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {a}}+\frac {\frac {\frac {3 \left (\frac {\frac {16 \left (\sqrt {a} e+\sqrt {c} d\right )^4 \int \frac {1}{\left (\sqrt {a}-\sqrt {c} x^2\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}-\frac {e \left (16 a^{3/2} e^3+70 \sqrt {a} c d^2 e+56 a \sqrt {c} d e^2+35 c^{3/2} d^3\right ) \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{\sqrt {c}}}{2 \sqrt {c}}-\frac {e x \sqrt {d+e x^2} \left (22 \sqrt {a} \sqrt {c} d e+8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (\frac {6 \sqrt {a} e}{\sqrt {c}}+11 d\right )}{6 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {a}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {16 \left (\sqrt {a} e+\sqrt {c} d\right )^4 \int \frac {1}{\left (\sqrt {a}-\sqrt {c} x^2\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}-\frac {\sqrt {e} \left (16 a^{3/2} e^3+70 \sqrt {a} c d^2 e+56 a \sqrt {c} d e^2+35 c^{3/2} d^3\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}-\frac {e x \sqrt {d+e x^2} \left (22 \sqrt {a} \sqrt {c} d e+8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (\frac {6 \sqrt {a} e}{\sqrt {c}}+11 d\right )}{6 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {a}}+\frac {\frac {\frac {3 \left (\frac {\frac {16 \left (\sqrt {c} d-\sqrt {a} e\right )^4 \int \frac {1}{\left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}+\frac {\sqrt {e} \left (-16 a^{3/2} e^3-70 \sqrt {a} c d^2 e+56 a \sqrt {c} d e^2+35 c^{3/2} d^3\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}+\frac {e x \sqrt {d+e x^2} \left (-22 \sqrt {a} \sqrt {c} d e+8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}+\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (11 d-\frac {6 \sqrt {a} e}{\sqrt {c}}\right )}{6 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {a}}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {16 \left (\sqrt {c} d-\sqrt {a} e\right )^4 \int \frac {1}{\sqrt {a}-\frac {\left (\sqrt {a} e-\sqrt {c} d\right ) x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{\sqrt {c}}+\frac {\sqrt {e} \left (-16 a^{3/2} e^3-70 \sqrt {a} c d^2 e+56 a \sqrt {c} d e^2+35 c^{3/2} d^3\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}+\frac {e x \sqrt {d+e x^2} \left (-22 \sqrt {a} \sqrt {c} d e+8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}+\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (11 d-\frac {6 \sqrt {a} e}{\sqrt {c}}\right )}{6 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {a}}+\frac {\frac {\frac {3 \left (\frac {\frac {16 \left (\sqrt {a} e+\sqrt {c} d\right )^4 \int \frac {1}{\sqrt {a}-\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{\sqrt {c}}-\frac {\sqrt {e} \left (16 a^{3/2} e^3+70 \sqrt {a} c d^2 e+56 a \sqrt {c} d e^2+35 c^{3/2} d^3\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}-\frac {e x \sqrt {d+e x^2} \left (22 \sqrt {a} \sqrt {c} d e+8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (\frac {6 \sqrt {a} e}{\sqrt {c}}+11 d\right )}{6 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {a}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {16 \left (\sqrt {a} e+\sqrt {c} d\right )^4 \int \frac {1}{\sqrt {a}-\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{\sqrt {c}}-\frac {\sqrt {e} \left (16 a^{3/2} e^3+70 \sqrt {a} c d^2 e+56 a \sqrt {c} d e^2+35 c^{3/2} d^3\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}-\frac {e x \sqrt {d+e x^2} \left (22 \sqrt {a} \sqrt {c} d e+8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (\frac {6 \sqrt {a} e}{\sqrt {c}}+11 d\right )}{6 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {a}}+\frac {\frac {\frac {3 \left (\frac {\frac {\sqrt {e} \left (-16 a^{3/2} e^3-70 \sqrt {a} c d^2 e+56 a \sqrt {c} d e^2+35 c^{3/2} d^3\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}+\frac {16 \left (\sqrt {c} d-\sqrt {a} e\right )^{7/2} \arctan \left (\frac {x \sqrt {\sqrt {c} d-\sqrt {a} e}}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{\sqrt [4]{a} \sqrt {c}}}{2 \sqrt {c}}+\frac {e x \sqrt {d+e x^2} \left (-22 \sqrt {a} \sqrt {c} d e+8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}+\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (11 d-\frac {6 \sqrt {a} e}{\sqrt {c}}\right )}{6 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {a}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {\sqrt {e} \left (-16 a^{3/2} e^3-70 \sqrt {a} c d^2 e+56 a \sqrt {c} d e^2+35 c^{3/2} d^3\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}+\frac {16 \left (\sqrt {c} d-\sqrt {a} e\right )^{7/2} \arctan \left (\frac {x \sqrt {\sqrt {c} d-\sqrt {a} e}}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{\sqrt [4]{a} \sqrt {c}}}{2 \sqrt {c}}+\frac {e x \sqrt {d+e x^2} \left (-22 \sqrt {a} \sqrt {c} d e+8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}+\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (11 d-\frac {6 \sqrt {a} e}{\sqrt {c}}\right )}{6 \sqrt {c}}+\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {a}}+\frac {\frac {\frac {3 \left (\frac {\frac {16 \left (\sqrt {a} e+\sqrt {c} d\right )^{7/2} \text {arctanh}\left (\frac {x \sqrt {\sqrt {a} e+\sqrt {c} d}}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{\sqrt [4]{a} \sqrt {c}}-\frac {\sqrt {e} \left (16 a^{3/2} e^3+70 \sqrt {a} c d^2 e+56 a \sqrt {c} d e^2+35 c^{3/2} d^3\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c}}}{2 \sqrt {c}}-\frac {e x \sqrt {d+e x^2} \left (22 \sqrt {a} \sqrt {c} d e+8 a e^2+19 c d^2\right )}{2 \sqrt {c}}\right )}{4 \sqrt {c}}-\frac {1}{4} e x \left (d+e x^2\right )^{3/2} \left (\frac {6 \sqrt {a} e}{\sqrt {c}}+11 d\right )}{6 \sqrt {c}}-\frac {e x \left (d+e x^2\right )^{5/2}}{6 \sqrt {c}}}{2 \sqrt {a}}\) |
Input:
Int[(d + e*x^2)^(7/2)/(a - c*x^4),x]
Output:
((e*x*(d + e*x^2)^(5/2))/(6*Sqrt[c]) + ((e*(11*d - (6*Sqrt[a]*e)/Sqrt[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*(Sqrt[c]*d - Sqrt[a]*e)^(7/2)*Ar cTan[(Sqrt[Sqrt[c]*d - Sqrt[a]*e]*x)/(a^(1/4)*Sqrt[d + e*x^2])])/(a^(1/4)* 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)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/Sqrt[c])/(2*Sq rt[c])))/(4*Sqrt[c]))/(6*Sqrt[c]))/(2*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]*Sqrt[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*S qrt[c]*d*e^2 + 16*a^(3/2)*e^3)*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/Sqrt[ c]) + (16*(Sqrt[c]*d + Sqrt[a]*e)^(7/2)*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[a]* e]*x)/(a^(1/4)*Sqrt[d + e*x^2])])/(a^(1/4)*Sqrt[c]))/(2*Sqrt[c])))/(4*Sqrt [c]))/(6*Sqrt[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. \(356\) vs. \(2(176)=352\).
Time = 0.97 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.54
| method | result | size |
| pseudoelliptic | \(-\frac {2 \sqrt {e}\, \left (\left (a^{2} e^{4}+6 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \sqrt {d^{2} a c}-4 a^{2} c \,d^{2} e^{3}-4 c^{2} d^{4} e a \right ) \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \arctan \left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}}\right )+\sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}\, \left (-2 \left (\left (a^{2} e^{4}+6 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \sqrt {d^{2} a c}+4 a^{2} c \,d^{2} e^{3}+4 c^{2} d^{4} e a \right ) \sqrt {e}\, \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}}\right )+\sqrt {d^{2} a c}\, \left (\left (4 e^{4} a +\frac {35}{2} c \,d^{2} e^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right )+x \left (e \,x^{2}+\frac {13 d}{2}\right ) e^{\frac {5}{2}} c \sqrt {e \,x^{2}+d}\right ) \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\right )}{4 \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}\, \sqrt {e}\, \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \sqrt {d^{2} a c}\, c^{2}}\) | \(357\) |
| risch | \(-\frac {e^{2} x \left (2 e \,x^{2}+13 d \right ) \sqrt {e \,x^{2}+d}}{8 c}-\frac {\frac {e^{\frac {3}{2}} \left (8 a \,e^{2}+35 c \,d^{2}\right ) \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{c}+\frac {4 \left (4 a^{\frac {3}{2}} \sqrt {c}\, d \,e^{3}+4 \sqrt {a}\, c^{\frac {3}{2}} d^{3} e -a^{2} e^{4}-6 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \left (-\frac {\ln \left (\frac {\frac {-2 \sqrt {a}\, \sqrt {c}\, e +2 c d}{c}+\frac {2 e \sqrt {-\sqrt {a}\, \sqrt {c}}\, \left (x -\frac {\sqrt {-\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+2 \sqrt {\frac {-\sqrt {a}\, \sqrt {c}\, e +c d}{c}}\, \sqrt {\left (x -\frac {\sqrt {-\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )^{2} e +\frac {2 e \sqrt {-\sqrt {a}\, \sqrt {c}}\, \left (x -\frac {\sqrt {-\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {-\sqrt {a}\, \sqrt {c}\, e +c d}{c}}}{x -\frac {\sqrt {-\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}}\right )}{2 \sqrt {-\sqrt {a}\, \sqrt {c}}\, \sqrt {\frac {-\sqrt {a}\, \sqrt {c}\, e +c d}{c}}}+\frac {\ln \left (\frac {\frac {-2 \sqrt {a}\, \sqrt {c}\, e +2 c d}{c}-\frac {2 e \sqrt {-\sqrt {a}\, \sqrt {c}}\, \left (x +\frac {\sqrt {-\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+2 \sqrt {\frac {-\sqrt {a}\, \sqrt {c}\, e +c d}{c}}\, \sqrt {\left (x +\frac {\sqrt {-\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )^{2} e -\frac {2 e \sqrt {-\sqrt {a}\, \sqrt {c}}\, \left (x +\frac {\sqrt {-\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {-\sqrt {a}\, \sqrt {c}\, e +c d}{c}}}{x +\frac {\sqrt {-\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}}\right )}{2 \sqrt {-\sqrt {a}\, \sqrt {c}}\, \sqrt {\frac {-\sqrt {a}\, \sqrt {c}\, e +c d}{c}}}\right )}{c \sqrt {a}}-\frac {4 \left (4 a^{\frac {3}{2}} \sqrt {c}\, d \,e^{3}+4 \sqrt {a}\, c^{\frac {3}{2}} d^{3} e +a^{2} e^{4}+6 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (\frac {\ln \left (\frac {\frac {2 \sqrt {a}\, \sqrt {c}\, e +2 c d}{c}+\frac {2 e \sqrt {\sqrt {a}\, \sqrt {c}}\, \left (x -\frac {\sqrt {\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+2 \sqrt {\frac {\sqrt {a}\, \sqrt {c}\, e +c d}{c}}\, \sqrt {\left (x -\frac {\sqrt {\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )^{2} e +\frac {2 e \sqrt {\sqrt {a}\, \sqrt {c}}\, \left (x -\frac {\sqrt {\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {\sqrt {a}\, \sqrt {c}\, e +c d}{c}}}{x -\frac {\sqrt {\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}}\right )}{2 \sqrt {\sqrt {a}\, \sqrt {c}}\, \sqrt {\frac {\sqrt {a}\, \sqrt {c}\, e +c d}{c}}}-\frac {\ln \left (\frac {\frac {2 \sqrt {a}\, \sqrt {c}\, e +2 c d}{c}-\frac {2 e \sqrt {\sqrt {a}\, \sqrt {c}}\, \left (x +\frac {\sqrt {\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+2 \sqrt {\frac {\sqrt {a}\, \sqrt {c}\, e +c d}{c}}\, \sqrt {\left (x +\frac {\sqrt {\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )^{2} e -\frac {2 e \sqrt {\sqrt {a}\, \sqrt {c}}\, \left (x +\frac {\sqrt {\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {\sqrt {a}\, \sqrt {c}\, e +c d}{c}}}{x +\frac {\sqrt {\sqrt {a}\, \sqrt {c}}}{\sqrt {c}}}\right )}{2 \sqrt {\sqrt {a}\, \sqrt {c}}\, \sqrt {\frac {\sqrt {a}\, \sqrt {c}\, e +c d}{c}}}\right )}{c \sqrt {a}}}{8 c}\) | \(956\) |
| default | \(\text {Expression too large to display}\) | \(7092\) |
Input:
int((e*x^2+d)^(7/2)/(-c*x^4+a),x,method=_RETURNVERBOSE)
Output:
-1/4/((-a*e+(d^2*a*c)^(1/2))*a)^(1/2)/e^(1/2)/((a*e+(d^2*a*c)^(1/2))*a)^(1 /2)*(2*e^(1/2)*((a^2*e^4+6*a*c*d^2*e^2+c^2*d^4)*(d^2*a*c)^(1/2)-4*a^2*c*d^ 2*e^3-4*c^2*d^4*e*a)*((a*e+(d^2*a*c)^(1/2))*a)^(1/2)*arctan((e*x^2+d)^(1/2 )/x*a/((-a*e+(d^2*a*c)^(1/2))*a)^(1/2))+((-a*e+(d^2*a*c)^(1/2))*a)^(1/2)*( -2*((a^2*e^4+6*a*c*d^2*e^2+c^2*d^4)*(d^2*a*c)^(1/2)+4*a^2*c*d^2*e^3+4*c^2* d^4*e*a)*e^(1/2)*arctanh((e*x^2+d)^(1/2)/x*a/((a*e+(d^2*a*c)^(1/2))*a)^(1/ 2))+(d^2*a*c)^(1/2)*((4*e^4*a+35/2*c*d^2*e^2)*arctanh((e*x^2+d)^(1/2)/x/e^ (1/2))+x*(e*x^2+13/2*d)*e^(5/2)*c*(e*x^2+d)^(1/2))*((a*e+(d^2*a*c)^(1/2))* a)^(1/2)))/(d^2*a*c)^(1/2)/c^2
Leaf count of result is larger than twice the leaf count of optimal. 3366 vs. \(2 (176) = 352\).
Time = 52.55 (sec) , antiderivative size = 6742, normalized size of antiderivative = 29.06 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{a-c x^4} \, dx=\text {Too large to display} \] Input:
integrate((e*x^2+d)^(7/2)/(-c*x^4+a),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\left (d+e x^2\right )^{7/2}}{a-c x^4} \, dx=- \int \frac {d^{3} \sqrt {d + e x^{2}}}{- a + c x^{4}}\, dx - \int \frac {e^{3} x^{6} \sqrt {d + e x^{2}}}{- a + c x^{4}}\, dx - \int \frac {3 d e^{2} x^{4} \sqrt {d + e x^{2}}}{- a + c x^{4}}\, dx - \int \frac {3 d^{2} e x^{2} \sqrt {d + e x^{2}}}{- a + c x^{4}}\, dx \] Input:
integrate((e*x**2+d)**(7/2)/(-c*x**4+a),x)
Output:
-Integral(d**3*sqrt(d + e*x**2)/(-a + c*x**4), x) - Integral(e**3*x**6*sqr t(d + e*x**2)/(-a + c*x**4), x) - Integral(3*d*e**2*x**4*sqrt(d + e*x**2)/ (-a + c*x**4), x) - Integral(3*d**2*e*x**2*sqrt(d + e*x**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. \[ \int \frac {\left (d+e x^2\right )^{7/2}}{a-c x^4} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{7/2}}{a-c\,x^4} \,d x \] 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