Optimal. Leaf size=284 \[ \frac{4 b c \sqrt{c^2 x^2+1} \sqrt{\frac{d+e x}{\frac{e}{\sqrt{-c^2}}+d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right ),-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{\left (-c^2\right )^{3/2} x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e}-\frac{4 b d \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right )}{c e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}} \]
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Rubi [A] time = 0.413714, antiderivative size = 284, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6290, 1574, 944, 719, 419, 933, 168, 538, 537} \[ \frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e}+\frac{4 b c \sqrt{c^2 x^2+1} \sqrt{\frac{d+e x}{\frac{e}{\sqrt{-c^2}}+d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{\left (-c^2\right )^{3/2} x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}-\frac{4 b d \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right )}{c e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Rule 6290
Rule 1574
Rule 944
Rule 719
Rule 419
Rule 933
Rule 168
Rule 538
Rule 537
Rubi steps
\begin{align*} \int \frac{a+b \text{csch}^{-1}(c x)}{\sqrt{d+e x}} \, dx &=\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e}+\frac{(2 b) \int \frac{\sqrt{d+e x}}{\sqrt{1+\frac{1}{c^2 x^2}} x^2} \, dx}{c e}\\ &=\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e}+\frac{\left (2 b \sqrt{\frac{1}{c^2}+x^2}\right ) \int \frac{\sqrt{d+e x}}{x \sqrt{\frac{1}{c^2}+x^2}} \, dx}{c e \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e}+\frac{\left (2 b \sqrt{\frac{1}{c^2}+x^2}\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{\frac{1}{c^2}+x^2}} \, dx}{c \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (2 b d \sqrt{\frac{1}{c^2}+x^2}\right ) \int \frac{1}{x \sqrt{d+e x} \sqrt{\frac{1}{c^2}+x^2}} \, dx}{c e \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e}+\frac{\left (2 b d \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}{c e \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (4 b \sqrt{-c^2} \sqrt{\frac{d+e x}{d-\frac{\sqrt{-c^2} e}{c^2}}} \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 \left (d-\frac{\sqrt{-c^2} e}{c^2}\right )}}} \, dx,x,\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )}{c^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e}+\frac{4 b \sqrt{-c^2} \sqrt{\frac{d+e x}{d+\frac{e}{\sqrt{-c^2}}}} \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 )}{c^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{\left (4 b d \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 )}{c e \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e}+\frac{4 b \sqrt{-c^2} \sqrt{\frac{d+e x}{d+\frac{e}{\sqrt{-c^2}}}} \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 )}{c^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{\left (4 b d \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 )}{c e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e}+\frac{4 b \sqrt{-c^2} \sqrt{\frac{d+e x}{d+\frac{e}{\sqrt{-c^2}}}} \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 )}{c^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{4 b d \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 )}{c e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ \end{align*}
Mathematica [C] time = 4.95715, size = 307, normalized size = 1.08 \[ \frac{2 \left (a e (d+e x)-\frac{b \left (\frac{d}{x}+e\right ) \left (-c e x \text{csch}^{-1}(c x)+\frac{\sqrt{2} \sqrt{1+i c x} \left (c d (e+i c d) \sqrt{-\frac{e (c x+i)}{c d-i e}} \sqrt{\frac{c e (c x+i) (d+e x)}{(e+i c d)^2}} \Pi \left (\frac{i c d}{e}+1;\sin ^{-1}\left (\sqrt{-\frac{e (c x+i)}{c d-i e}}\right )|\frac{i c d+e}{2 e}\right )-e^2 (c x+i) \sqrt{\frac{c (d+e x)}{c d-i e}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{-\frac{e (c x+i)}{c d-i e}}\right ),\frac{e+i c d}{2 e}\right )\right )}{\sqrt{\frac{1}{c^2 x^2}+1} \sqrt{-\frac{e (c x+i)}{c d-i e}} (c d+c e x)}\right )}{c}\right )}{e^2 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.281, size = 395, normalized size = 1.4 \begin{align*} 2\,{\frac{1}{e} \left ( a\sqrt{ex+d}+b \left ( \sqrt{ex+d}{\rm arccsch} \left (cx\right )+2\,{\frac{1}{cx}\sqrt{-{\frac{i \left ( ex+d \right ) ce+ \left ( ex+d \right ){c}^{2}d-{c}^{2}{d}^{2}-{e}^{2}}{{c}^{2}{d}^{2}+{e}^{2}}}}\sqrt{{\frac{i \left ( ex+d \right ) ce- \left ( ex+d \right ){c}^{2}d+{c}^{2}{d}^{2}+{e}^{2}}{{c}^{2}{d}^{2}+{e}^{2}}}} \left ({\it EllipticF} \left ( \sqrt{ex+d}\sqrt{{\frac{ \left ( ie+cd \right ) c}{{c}^{2}{d}^{2}+{e}^{2}}}},\sqrt{-{\frac{2\,icde-{c}^{2}{d}^{2}+{e}^{2}}{{c}^{2}{d}^{2}+{e}^{2}}}} \right ) -{\it EllipticPi} \left ( \sqrt{ex+d}\sqrt{{\frac{ \left ( ie+cd \right ) c}{{c}^{2}{d}^{2}+{e}^{2}}}},{\frac{{c}^{2}{d}^{2}+{e}^{2}}{ \left ( ie+cd \right ) cd}},{\sqrt{-{\frac{ \left ( ie-cd \right ) c}{{c}^{2}{d}^{2}+{e}^{2}}}}{\frac{1}{\sqrt{{\frac{ \left ( ie+cd \right ) c}{{c}^{2}{d}^{2}+{e}^{2}}}}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{ \left ( ex+d \right ) ^{2}{c}^{2}-2\, \left ( ex+d \right ){c}^{2}d+{c}^{2}{d}^{2}+{e}^{2}}{{c}^{2}{x}^{2}{e}^{2}}}}}}{\frac{1}{\sqrt{{\frac{ \left ( ie+cd \right ) c}{{c}^{2}{d}^{2}+{e}^{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{acsch}{\left (c x \right )}}{\sqrt{d + e x}}\, dx \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{b \operatorname{arcsch}\left (c x\right ) + a}{\sqrt{e x + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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