Optimal. Leaf size=369 \[ \frac{20 b d \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right ),\frac{2 e}{c d+e}\right )}{3 c^2 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac{32 b d^2 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 c e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 c^2 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{c (d+e x)}{c d+e}}} \]
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Rubi [A] time = 1.81351, antiderivative size = 369, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.619, Rules used = {43, 5247, 12, 6721, 6742, 719, 419, 932, 168, 538, 537, 844, 424} \[ -\frac{2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac{32 b d^2 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 c e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{20 b d \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 c^2 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{3 c^2 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{c (d+e x)}{c d+e}}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 5247
Rule 12
Rule 6721
Rule 6742
Rule 719
Rule 419
Rule 932
Rule 168
Rule 538
Rule 537
Rule 844
Rule 424
Rubi steps
\begin{align*} \int \frac{x^2 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx &=-\frac{2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac{b \int \frac{2 \left (-8 d^2-4 d e x+e^2 x^2\right )}{3 e^3 \sqrt{1-\frac{1}{c^2 x^2}} x^2 \sqrt{d+e x}} \, dx}{c}\\ &=-\frac{2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac{(2 b) \int \frac{-8 d^2-4 d e x+e^2 x^2}{\sqrt{1-\frac{1}{c^2 x^2}} x^2 \sqrt{d+e x}} \, dx}{3 c e^3}\\ &=-\frac{2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \int \frac{-8 d^2-4 d e x+e^2 x^2}{x \sqrt{d+e x} \sqrt{1-c^2 x^2}} \, dx}{3 c e^3 \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \int \left (-\frac{4 d e}{\sqrt{d+e x} \sqrt{1-c^2 x^2}}-\frac{8 d^2}{x \sqrt{d+e x} \sqrt{1-c^2 x^2}}+\frac{e^2 x}{\sqrt{d+e x} \sqrt{1-c^2 x^2}}\right ) \, dx}{3 c e^3 \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac{\left (16 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^3 \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\left (8 b d \sqrt{1-c^2 x^2}\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{1-c^2 x^2}} \, dx}{3 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}{\sqrt{d+e x} \sqrt{1-c^2 x^2}} \, dx}{3 c e \sqrt{1-\frac{1}{c^2 x^2}} x}\\ &=-\frac{2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac{\left (16 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^3 \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{1-c^2 x^2}} \, dx}{3 c e^2 \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\left (2 b d \sqrt{1-c^2 x^2}\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{1-c^2 x^2}} \, dx}{3 c e^2 \sqrt{1-\frac{1}{c^2 x^2}} x}+\frac{\left (16 b d \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^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=-\frac{2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac{16 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^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{\left (32 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^3 \sqrt{1-\frac{1}{c^2 x^2}} x}-\frac{\left (4 b \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^2 \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 d \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^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=-\frac{2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac{4 b \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^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{\frac{c (d+e x)}{c d+e}}}+\frac{20 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^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{\left (32 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^3 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=-\frac{2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac{4 b \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^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{\frac{c (d+e x)}{c d+e}}}+\frac{20 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^2 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{32 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^3 \sqrt{1-\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ \end{align*}
Mathematica [C] time = 13.6764, size = 750, normalized size = 2.03 \[ \frac{b \left (\frac{2 (c x)^{3/2} \left (\frac{d}{x}+e\right )^{3/2} \left (\frac{10 c d e \sqrt{1-c^2 x^2} \sqrt{\frac{c d+c e x}{c d+e}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right ),\frac{2 e}{c d+e}\right )}{\sqrt{1-\frac{1}{c^2 x^2}} (c x)^{3/2} \sqrt{\frac{d}{x}+e}}-\frac{2 e \cos \left (2 \csc ^{-1}(c x)\right ) \left (c^2 d x \sqrt{1-c^2 x^2} \sqrt{\frac{c d+c e x}{c d+e}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right ),\frac{2 e}{c d+e}\right )-\frac{c x (c x+1) \sqrt{\frac{e-c e x}{c d+e}} \sqrt{\frac{c d+c e x}{c d-e}} \left ((c d+e) E\left (\sin ^{-1}\left (\sqrt{\frac{c d+c e x}{c d-e}}\right )|\frac{c d-e}{c d+e}\right )-e \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{c d+c e x}{c d-e}}\right ),\frac{c d-e}{c d+e}\right )\right )}{\sqrt{\frac{e (c x+1)}{e-c d}}}+\left (c^2 x^2-1\right ) (c d+c e x)+c e x \sqrt{1-c^2 x^2} \sqrt{\frac{c d+c e x}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )\right )}{\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c x} \left (c^2 x^2-2\right ) \sqrt{\frac{d}{x}+e}}+\frac{2 \sqrt{1-c^2 x^2} \left (8 c^2 d^2+e^2\right ) \sqrt{\frac{c d+c e x}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{\sqrt{1-\frac{1}{c^2 x^2}} (c x)^{3/2} \sqrt{\frac{d}{x}+e}}\right )}{3 e^3 (d+e x)^{3/2}}-\frac{c^2 x^2 \left (\frac{d}{x}+e\right )^2 \left (-\frac{4 \sqrt{1-\frac{1}{c^2 x^2}}}{3 e^2}-\frac{2 c d \csc ^{-1}(c x)}{e^2 \left (\frac{d}{x}+e\right )}+\frac{16 c d \csc ^{-1}(c x)}{3 e^3}-\frac{2 c x \csc ^{-1}(c x)}{3 e^2}\right )}{(d+e x)^{3/2}}\right )}{c^3}-\frac{a d^3 \left (\frac{e x}{d}+1\right )^{3/2} B_{-\frac{e x}{d}}\left (3,-\frac{1}{2}\right )}{e^3 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.278, size = 441, normalized size = 1.2 \begin{align*} 2\,{\frac{1}{{e}^{3}} \left ( a \left ( 1/3\, \left ( ex+d \right ) ^{3/2}-2\,d\sqrt{ex+d}-{\frac{{d}^{2}}{\sqrt{ex+d}}} \right ) +b \left ( 1/3\, \left ( ex+d \right ) ^{3/2}{\rm arccsc} \left (cx\right )-2\,{\rm arccsc} \left (cx\right )d\sqrt{ex+d}-{\frac{{\rm arccsc} \left (cx\right ){d}^{2}}{\sqrt{ex+d}}}-2/3\,{\frac{1}{{c}^{2}x} \left ( 4\,d{\it EllipticF} \left ( \sqrt{ex+d}\sqrt{{\frac{c}{dc-e}}},\sqrt{{\frac{dc-e}{dc+e}}} \right ) c+{\it EllipticE} \left ( \sqrt{ex+d}\sqrt{{\frac{c}{dc-e}}},\sqrt{{\frac{dc-e}{dc+e}}} \right ) cd-8\,d{\it EllipticPi} \left ( \sqrt{ex+d}\sqrt{{\frac{c}{dc-e}}},{\frac{dc-e}{dc}},{\sqrt{{\frac{c}{dc+e}}}{\frac{1}{\sqrt{{\frac{c}{dc-e}}}}}} \right ) c-{\it EllipticF} \left ( \sqrt{ex+d}\sqrt{{\frac{c}{dc-e}}},\sqrt{{\frac{dc-e}{dc+e}}} \right ) e+{\it EllipticE} \left ( \sqrt{ex+d}\sqrt{{\frac{c}{dc-e}}},\sqrt{{\frac{dc-e}{dc+e}}} \right ) e \right ) \sqrt{-{\frac{ \left ( ex+d \right ) c-dc-e}{dc+e}}}\sqrt{-{\frac{ \left ( ex+d \right ) c-dc+e}{dc-e}}}{\frac{1}{\sqrt{{\frac{c}{dc-e}}}}}{\frac{1}{\sqrt{{\frac{{c}^{2} \left ( ex+d \right ) ^{2}-2\,d{c}^{2} \left ( ex+d \right ) +{c}^{2}{d}^{2}-{e}^{2}}{{c}^{2}{e}^{2}{x}^{2}}}}}}} \right ) \right ) } \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(-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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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^{2}}{{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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