Optimal. Leaf size=393 \[ -\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}+\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}+\frac{4 b \left (c^2 x^2+1\right )}{3 c x \sqrt{\frac{1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt{d+e x}}-\frac{4 b \sqrt{-c^2} \sqrt{c^2 x^2+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{3 c e x \sqrt{\frac{1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}+\frac{8 b \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right )}{3 c e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}} \]
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Rubi [A] time = 2.19119, antiderivative size = 393, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 14, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.737, Rules used = {43, 6310, 12, 6721, 6742, 745, 21, 719, 424, 958, 932, 168, 538, 537} \[ -\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}+\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}+\frac{4 b \left (c^2 x^2+1\right )}{3 c x \sqrt{\frac{1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt{d+e x}}-\frac{4 b \sqrt{-c^2} \sqrt{c^2 x^2+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{3 c e x \sqrt{\frac{1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}+\frac{8 b \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right )}{3 c e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6310
Rule 12
Rule 6721
Rule 6742
Rule 745
Rule 21
Rule 719
Rule 424
Rule 958
Rule 932
Rule 168
Rule 538
Rule 537
Rubi steps
\begin{align*} \int \frac{x \left (a+b \text{csch}^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx &=\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}+\frac{b \int \frac{2 (-2 d-3 e x)}{3 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x^2 (d+e x)^{3/2}} \, dx}{c}\\ &=\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}+\frac{(2 b) \int \frac{-2 d-3 e x}{\sqrt{1+\frac{1}{c^2 x^2}} x^2 (d+e x)^{3/2}} \, dx}{3 c e^2}\\ &=\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}+\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \int \frac{-2 d-3 e x}{x (d+e x)^{3/2} \sqrt{1+c^2 x^2}} \, dx}{3 c e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}+\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \int \left (-\frac{3 e}{(d+e x)^{3/2} \sqrt{1+c^2 x^2}}-\frac{2 d}{x (d+e x)^{3/2} \sqrt{1+c^2 x^2}}\right ) \, dx}{3 c e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}-\frac{\left (4 b d \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^2 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \int \frac{1}{(d+e x)^{3/2} \sqrt{1+c^2 x^2}} \, dx}{c e \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=\frac{4 b \left (1+c^2 x^2\right )}{c \left (c^2 d^2+e^2\right ) \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}-\frac{\left (4 b d \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^2 \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (4 b c \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 \left (c^2 d^2+e^2\right ) \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=\frac{4 b \left (1+c^2 x^2\right )}{c \left (c^2 d^2+e^2\right ) \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}-\frac{\left (4 b \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^2 \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (4 b \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 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (2 b c \sqrt{1+c^2 x^2}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{1+c^2 x^2}} \, dx}{e \left (c^2 d^2+e^2\right ) \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=\frac{4 b \left (1+c^2 x^2\right )}{3 c \left (c^2 d^2+e^2\right ) \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}-\frac{\left (4 b \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^2 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{\left (8 b c \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 \left (c^2 d^2+e^2\right ) \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 )}{c 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-\sqrt{-c^2} e}}}\\ &=\frac{4 b \left (1+c^2 x^2\right )}{3 c \left (c^2 d^2+e^2\right ) \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}-\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 )}{c 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-\sqrt{-c^2} e}}}+\frac{\left (8 b \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^2 \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (4 b c \sqrt{1+c^2 x^2}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{1+c^2 x^2}} \, dx}{3 e \left (c^2 d^2+e^2\right ) \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=\frac{4 b \left (1+c^2 x^2\right )}{3 c \left (c^2 d^2+e^2\right ) \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}-\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 )}{c 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-\sqrt{-c^2} e}}}+\frac{\left (8 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 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-\sqrt{-c^2} e}}}+\frac{\left (8 b \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^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=\frac{4 b \left (1+c^2 x^2\right )}{3 c \left (c^2 d^2+e^2\right ) \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}-\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 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-\sqrt{-c^2} e}}}+\frac{8 b \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^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ \end{align*}
Mathematica [C] time = 2.41961, size = 390, normalized size = 0.99 \[ \frac{2}{3} \left (\frac{2 i b \sqrt{-\frac{c}{c d-i e}} \sqrt{-\frac{e (c x-i)}{c d+i e}} \sqrt{-\frac{e (c x+i)}{c d-i e}} \left (-c d \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right ),\frac{c d-i e}{c d+i e}\right )+c d E\left (i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right )|\frac{c d-i e}{c d+i e}\right )+2 (c d-i e) \Pi \left (1-\frac{i e}{c d};i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right )|\frac{c d-i e}{c d+i e}\right )\right )}{c^2 d e^2 x \sqrt{\frac{1}{c^2 x^2}+1}}-\frac{a (2 d+3 e x)}{e^2 (d+e x)^{3/2}}+\frac{2 b c x \sqrt{\frac{1}{c^2 x^2}+1}}{\left (c^2 d^2+e^2\right ) \sqrt{d+e x}}-\frac{b \text{csch}^{-1}(c x) (2 d+3 e x)}{e^2 (d+e x)^{3/2}}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.308, size = 2107, normalized size = 5.4 \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 \operatorname{arcsch}\left (c x\right ) + a x\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{arcsch}\left (c x\right ) + a\right )} x}{{\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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