Optimal. Leaf size=602 \[ -\frac {4 b d^2 \left (1-c^2 x^2\right )}{3 c e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}+\frac {8 b d^2 \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 e^3 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \left (2 c^2 d^2-e^2\right ) \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 e^3 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {32 b d \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} F\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {64 b d^2 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \Pi \left (2;\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 2.15, antiderivative size = 602, normalized size of antiderivative = 1.00, number of steps
used = 31, number of rules used = 18, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {45,
5355, 12, 6853, 6874, 759, 21, 733, 435, 972, 946, 174, 552, 551, 849, 858, 430, 1665}
\begin {gather*} \frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}+\frac {64 b d^2 \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c e^4 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}+\frac {8 b d^2 \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 e^3 x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \sqrt {1-c^2 x^2} \left (2 c^2 d^2-e^2\right ) \sqrt {d+e x} E\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 e^3 x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {32 b d \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 e^3 x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}-\frac {4 b d^2 \left (1-c^2 x^2\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 21
Rule 45
Rule 174
Rule 430
Rule 435
Rule 551
Rule 552
Rule 733
Rule 759
Rule 849
Rule 858
Rule 946
Rule 972
Rule 1665
Rule 5355
Rule 6853
Rule 6874
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx &=\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}+\frac {b \int \frac {2 \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )}{3 e^4 \sqrt {1-\frac {1}{c^2 x^2}} x^2 (d+e x)^{3/2}} \, dx}{c}\\ &=\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}+\frac {(2 b) \int \frac {-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3}{\sqrt {1-\frac {1}{c^2 x^2}} x^2 (d+e x)^{3/2}} \, dx}{3 c e^4}\\ &=\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3}{x (d+e x)^{3/2} \sqrt {1-c^2 x^2}} \, dx}{3 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \left (-\frac {24 d^2 e}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}-\frac {16 d^3}{x (d+e x)^{3/2} \sqrt {1-c^2 x^2}}-\frac {6 d e^2 x}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}+\frac {e^3 x^2}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}\right ) \, dx}{3 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}-\frac {\left (32 b d^3 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x (d+e x)^{3/2} \sqrt {1-c^2 x^2}} \, dx}{3 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (16 b d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}} \, dx}{c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (4 b d \sqrt {1-c^2 x^2}\right ) \int \frac {x}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}} \, dx}{c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {x^2}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}} \, dx}{3 c e \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=-\frac {68 b d^2 \left (1-c^2 x^2\right )}{3 c e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}-\frac {\left (32 b d^3 \sqrt {1-c^2 x^2}\right ) \int \left (-\frac {e}{d (d+e x)^{3/2} \sqrt {1-c^2 x^2}}+\frac {1}{d x \sqrt {d+e x} \sqrt {1-c^2 x^2}}\right ) \, dx}{3 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (32 b c d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {-\frac {d}{2}-\frac {e x}{2}}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{e^3 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (8 b d \sqrt {1-c^2 x^2}\right ) \int \frac {-\frac {e}{2}-\frac {1}{2} c^2 d x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{c e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (4 b \sqrt {1-c^2 x^2}\right ) \int \frac {\frac {d}{2}+\frac {1}{2} \left (\frac {2 c^2 d^2}{e}-e\right ) x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{3 c e \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=-\frac {68 b d^2 \left (1-c^2 x^2\right )}{3 c e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}-\frac {\left (32 b d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{3 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (32 b d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}} \, dx}{3 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (4 b c d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{e^3 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (16 b c d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{e^3 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (4 b d \left (1-\frac {c^2 d^2}{e^2}\right ) \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{3 c e \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b \left (-1+\frac {2 c^2 d^2}{e^2}\right ) \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{3 c e \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (4 b d (c d-e) (c d+e) \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{c e^3 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=-\frac {4 b d^2 \left (1-c^2 x^2\right )}{3 c e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}-\frac {\left (32 b d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x \sqrt {1-c x} \sqrt {1+c x} \sqrt {d+e x}} \, dx}{3 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (64 b c d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {-\frac {d}{2}-\frac {e x}{2}}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{3 e^3 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\left (8 b d^2 \sqrt {d+e x} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 c e x^2}{-c^2 d-c e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{e^3 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}}+\frac {\left (32 b d^2 \sqrt {d+e x} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 c e x^2}{-c^2 d-c e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{e^3 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}}-\frac {\left (4 b \left (-1+\frac {2 c^2 d^2}{e^2}\right ) \sqrt {d+e x} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 c e x^2}{-c^2 d-c e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{3 c^2 e \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}}-\frac {\left (8 b d \left (1-\frac {c^2 d^2}{e^2}\right ) \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 c e x^2}{-c^2 d-c e}}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{3 c^2 e \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {\left (8 b d (c d-e) (c d+e) \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 c e x^2}{-c^2 d-c e}}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{c^2 e^3 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=-\frac {4 b d^2 \left (1-c^2 x^2\right )}{3 c e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}+\frac {24 b d^2 \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{e^3 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \left (2 c^2 d^2-e^2\right ) \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 e^3 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {32 b d \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {\left (64 b d^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {d+\frac {e}{c}-\frac {e x^2}{c}}} \, dx,x,\sqrt {1-c x}\right )}{3 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (32 b c d^2 \sqrt {1-c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}} \, dx}{3 e^3 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x}\\ &=-\frac {4 b d^2 \left (1-c^2 x^2\right )}{3 c e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}+\frac {24 b d^2 \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{e^3 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \left (2 c^2 d^2-e^2\right ) \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 e^3 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {32 b d \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {\left (64 b d^2 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {1-\frac {e x^2}{c \left (d+\frac {e}{c}\right )}}} \, dx,x,\sqrt {1-c x}\right )}{3 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (64 b d^2 \sqrt {d+e x} \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 c e x^2}{-c^2 d-c e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{3 e^3 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}}\\ &=-\frac {4 b d^2 \left (1-c^2 x^2\right )}{3 c e^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac {6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt {d+e x}}-\frac {6 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}+\frac {8 b d^2 \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 e^3 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \left (2 c^2 d^2-e^2\right ) \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 e^3 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {32 b d \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {64 b d^2 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c e^4 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 4 in
optimal.
time = 30.67, size = 887, normalized size = 1.47 \begin {gather*} \frac {a d^4 \left (1+\frac {e x}{d}\right )^{5/2} B_{-\frac {e x}{d}}\left (4,-\frac {3}{2}\right )}{e^4 (d+e x)^{5/2}}+\frac {b \left (-\frac {c^3 \left (e+\frac {d}{x}\right )^3 x^3 \left (-\frac {4 \sqrt {1-\frac {1}{c^2 x^2}}}{3 e \left (-c^2 d^2+e^2\right )}+\frac {32 c d \csc ^{-1}(c x)}{3 e^4}-\frac {2 c d \csc ^{-1}(c x)}{3 e^2 \left (e+\frac {d}{x}\right )^2}-\frac {2 c x \csc ^{-1}(c x)}{3 e^3}-\frac {2 \left (-2 c^2 d^2 e \sqrt {1-\frac {1}{c^2 x^2}}-7 c^3 d^3 \csc ^{-1}(c x)+7 c d e^2 \csc ^{-1}(c x)\right )}{3 e^3 \left (-c^2 d^2+e^2\right ) \left (e+\frac {d}{x}\right )}\right )}{(d+e x)^{5/2}}+\frac {2 \left (e+\frac {d}{x}\right )^{5/2} (c x)^{5/2} \left (\frac {2 \left (8 c^3 d^3 e-8 c d e^3\right ) \sqrt {\frac {c d+c e x}{c d+e}} \sqrt {1-c^2 x^2} F\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} (c x)^{3/2}}+\frac {2 \left (16 c^4 d^4-16 c^2 d^2 e^2-e^4\right ) \sqrt {\frac {c d+c e x}{c d+e}} \sqrt {1-c^2 x^2} \Pi \left (2;\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} (c x)^{3/2}}+\frac {2 e^3 \cos \left (2 \csc ^{-1}(c x)\right ) \left ((c d+c e x) \left (-1+c^2 x^2\right )+c^2 d x \sqrt {\frac {c d+c e x}{c d+e}} \sqrt {1-c^2 x^2} F\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )-\frac {c x (1+c x) \sqrt {\frac {e-c e x}{c d+e}} \sqrt {\frac {c d+c e x}{c d-e}} \left ((c d+e) E\left (\text {ArcSin}\left (\sqrt {\frac {c d+c e x}{c d-e}}\right )|\frac {c d-e}{c d+e}\right )-e F\left (\text {ArcSin}\left (\sqrt {\frac {c d+c e x}{c d-e}}\right )|\frac {c d-e}{c d+e}\right )\right )}{\sqrt {\frac {e (1+c x)}{-c d+e}}}+c e x \sqrt {\frac {c d+c e x}{c d+e}} \sqrt {1-c^2 x^2} \Pi \left (2;\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )\right )}{\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} \sqrt {c x} \left (-2+c^2 x^2\right )}\right )}{3 (c d-e) e^4 (c d+e) (d+e x)^{5/2}}\right )}{c^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.96, size = 1067, normalized size = 1.77 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (d+e\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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