Optimal. Leaf size=602 \[ \frac{32 b d \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right ),\frac{2 e}{c d+e}\right )}{3 c^2 e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac{6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt{d+e x}}-\frac{6 d \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac{2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}-\frac{4 b d^2 \left (1-c^2 x^2\right )}{3 c e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt{d+e x}}+\frac{8 b d^2 \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{4 b \sqrt{1-c^2 x^2} \left (2 c^2 d^2-e^2\right ) \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 c^2 e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt{\frac{c (d+e x)}{c d+e}}}+\frac{64 b d^2 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 c e^4 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}} \]
[Out]
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Rubi [A] time = 2.89572, antiderivative size = 602, normalized size of antiderivative = 1., 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 = {43, 5247, 12, 6721, 6742, 745, 21, 719, 424, 958, 932, 168, 538, 537, 835, 844, 419, 1651} \[ \frac{2 d^3 \left (a+b \csc ^{-1}(c x)\right )}{3 e^4 (d+e x)^{3/2}}-\frac{6 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^4 \sqrt{d+e x}}-\frac{6 d \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^4}+\frac{2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^4}-\frac{4 b d^2 \left (1-c^2 x^2\right )}{3 c e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt{d+e x}}+\frac{8 b d^2 \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{4 b \sqrt{1-c^2 x^2} \left (2 c^2 d^2-e^2\right ) \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 c^2 e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt{\frac{c (d+e x)}{c d+e}}}+\frac{64 b d^2 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 c e^4 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{32 b d \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 c^2 e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 5247
Rule 12
Rule 6721
Rule 6742
Rule 745
Rule 21
Rule 719
Rule 424
Rule 958
Rule 932
Rule 168
Rule 538
Rule 537
Rule 835
Rule 844
Rule 419
Rule 1651
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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [C] time = 13.935, size = 887, normalized size = 1.47 \[ \frac{a \left (\frac{e x}{d}+1\right )^{5/2} B_{-\frac{e x}{d}}\left (4,-\frac{3}{2}\right ) d^4}{e^4 (d+e x)^{5/2}}+\frac{b \left (\frac{2 \left (\frac{d}{x}+e\right )^{5/2} (c x)^{5/2} \left (\frac{2 \cos \left (2 \csc ^{-1}(c x)\right ) \left (d x \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right ),\frac{2 e}{c d+e}\right ) c^2-\frac{x (c x+1) \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 (\sin ^{-1}\left (\sqrt{\frac{c d+c e x}{c d-e}}\right )|\frac{c d-e}{c d+e}\right )-e \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{c d+c e x}{c d-e}}\right ),\frac{c d-e}{c d+e}\right )\right ) c}{\sqrt{\frac{e (c x+1)}{e-c d}}}+e x \sqrt{\frac{c d+c 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 ) c+(c d+c e x) \left (c^2 x^2-1\right )\right ) e^3}{\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} \sqrt{c x} \left (c^2 x^2-2\right )}+\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} \text{EllipticF}\left (\sin ^{-1}\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{\frac{d}{x}+e} (c x)^{3/2}}+\frac{2 \left (16 c^4 d^4-16 c^2 e^2 d^2-e^4\right ) \sqrt{\frac{c d+c 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 )}{\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}\right )}{3 (c d-e) e^4 (c d+e) (d+e x)^{5/2}}-\frac{c^3 \left (\frac{d}{x}+e\right )^3 x^3 \left (-\frac{2 c x \csc ^{-1}(c x)}{3 e^3}-\frac{2 c d \csc ^{-1}(c x)}{3 e^2 \left (\frac{d}{x}+e\right )^2}+\frac{32 c d \csc ^{-1}(c x)}{3 e^4}-\frac{2 \left (-7 c^3 \csc ^{-1}(c x) d^3-2 c^2 e \sqrt{1-\frac{1}{c^2 x^2}} d^2+7 c e^2 \csc ^{-1}(c x) d\right )}{3 e^3 \left (e^2-c^2 d^2\right ) \left (\frac{d}{x}+e\right )}-\frac{4 \sqrt{1-\frac{1}{c^2 x^2}}}{3 e \left (e^2-c^2 d^2\right )}\right )}{(d+e x)^{5/2}}\right )}{c^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.288, size = 1077, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{3} \operatorname{arccsc}\left (c x\right ) + a x^{3}\right )} \sqrt{e x + d}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arccsc}\left (c x\right ) + a\right )} x^{3}}{{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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