Optimal. Leaf size=499 \[ -\frac{20 b c d \sqrt{c^2 x^2+1} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right ),-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}-\frac{2 d^2 \left (a+b \text{csch}^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^3}+\frac{32 b d^2 \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right )}{3 c e^3 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{4 b c \sqrt{c^2 x^2+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}} \]
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Rubi [A] time = 2.00958, antiderivative size = 499, 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, 6310, 12, 6721, 6742, 719, 419, 932, 168, 538, 537, 844, 424} \[ -\frac{2 d^2 \left (a+b \text{csch}^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^3}+\frac{32 b d^2 \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right )}{3 c e^3 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}-\frac{20 b c d \sqrt{c^2 x^2+1} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{4 b c \sqrt{c^2 x^2+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6310
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 \text{csch}^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx &=-\frac{2 d^2 \left (a+b \text{csch}^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \text{csch}^{-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 \text{csch}^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \text{csch}^{-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 \text{csch}^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \text{csch}^{-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 \text{csch}^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \text{csch}^{-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 \text{csch}^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \text{csch}^{-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 \text{csch}^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \text{csch}^{-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-\sqrt{-c^2} x} \sqrt{1+\sqrt{-c^2} 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 \sqrt{-c^2} d \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 \sqrt{-c^2} e x^2}{c^2 d-\sqrt{-c^2} e}}} \, dx,x,\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )}{3 c^3 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=-\frac{2 d^2 \left (a+b \text{csch}^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^3}-\frac{16 b \sqrt{-c^2} d \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{3 c^3 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}{\sqrt{-c^2}}-\frac{e x^2}{\sqrt{-c^2}}}} \, dx,x,\sqrt{1-\sqrt{-c^2} x}\right )}{3 c e^3 \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (4 b \sqrt{-c^2} \sqrt{d+e x} \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 \sqrt{-c^2} e x^2}{c^2 d-\sqrt{-c^2} e}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )}{3 c^3 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}-\frac{\left (4 b \sqrt{-c^2} d \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 \sqrt{-c^2} e x^2}{c^2 d-\sqrt{-c^2} e}}} \, dx,x,\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )}{3 c^3 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=-\frac{2 d^2 \left (a+b \text{csch}^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^3}+\frac{4 b \sqrt{-c^2} \sqrt{d+e x} \sqrt{1+c^2 x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{3 c^3 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}-\frac{20 b \sqrt{-c^2} d \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{3 c^3 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} \sqrt{1+\frac{e \left (-1+\sqrt{-c^2} x\right )}{\sqrt{-c^2} d+e}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{2-x^2} \sqrt{1-\frac{e x^2}{\sqrt{-c^2} \left (d+\frac{e}{\sqrt{-c^2}}\right )}}} \, dx,x,\sqrt{1-\sqrt{-c^2} 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 \text{csch}^{-1}(c x)\right )}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^3}+\frac{2 (d+e x)^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^3}+\frac{4 b \sqrt{-c^2} \sqrt{d+e x} \sqrt{1+c^2 x^2} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{3 c^3 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}-\frac{20 b \sqrt{-c^2} d \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{3 c^3 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{32 b d^2 \sqrt{1+c^2 x^2} \sqrt{1-\frac{e \left (1-\sqrt{-c^2} x\right )}{\sqrt{-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} 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 = 14.3665, size = 979, normalized size = 1.96 \[ \frac{b \left (\frac{2 \left (\frac{d}{x}+e\right )^{3/2} (c x)^{3/2} \left (-\frac{5 \sqrt{2} c d e \sqrt{i c x+1} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{-\frac{e (c x+i)}{c d-i e}}\right ),\frac{i c d+e}{2 e}\right )}{\sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2} \sqrt{\frac{e (1-i c x)}{i c d+e}}}+\frac{i \sqrt{2} (c d-i e) \left (8 c^2 d^2-e^2\right ) \sqrt{i c x+1} \sqrt{\frac{e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left (\frac{i c d}{e}+1;\sin ^{-1}\left (\sqrt{-\frac{e (c x+i)}{c d-i e}}\right )|\frac{i c d+e}{2 e}\right )}{e \sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}+\frac{2 e \cosh \left (2 \text{csch}^{-1}(c x)\right ) \left (\frac{c x \left (c d \sqrt{2 i c x+2} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{-\frac{e (c x+i)}{c d-i e}}\right ),\frac{i c d+e}{2 e}\right )+2 \sqrt{-\frac{e (c x-i)}{c d+i e}} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} \left ((c d+i e) E\left (\sin ^{-1}\left (\sqrt{\frac{c d+c e x}{c d-i e}}\right )|\frac{c d-i e}{c d+i e}\right )-i e \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{c d+c e x}{c d-i e}}\right ),\frac{c d-i e}{c d+i e}\right )\right )+(i c d+e) \sqrt{2 i c x+2} \sqrt{-\frac{e (c x+i)}{c d-i e}} \sqrt{\frac{e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left (\frac{i c d}{e}+1;\sin ^{-1}\left (\sqrt{-\frac{e (c x+i)}{c d-i e}}\right )|\frac{i c d+e}{2 e}\right )\right )}{2 \sqrt{-\frac{e (c x+i)}{c d-i e}}}-(c d+c e x) \left (c^2 x^2+1\right )\right )}{\sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} \sqrt{c x} \left (c^2 x^2+2\right )}\right )}{3 e^3 (d+e x)^{3/2}}-\frac{c^2 \left (\frac{d}{x}+e\right )^2 x^2 \left (-\frac{2 c x \text{csch}^{-1}(c x)}{3 e^2}-\frac{2 c d \text{csch}^{-1}(c x)}{e^2 \left (\frac{d}{x}+e\right )}+\frac{16 c d \text{csch}^{-1}(c x)}{3 e^3}-\frac{4 \sqrt{1+\frac{1}{c^2 x^2}}}{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 [C] time = 0.299, size = 896, 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(-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{arcsch}\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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