Optimal. Leaf size=318 \[ \frac{4 b c \sqrt{c^2 x^2+1} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right ),-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{\left (-c^2\right )^{3/2} e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}-\frac{8 b d \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right )}{c e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.69131, antiderivative size = 318, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.579, Rules used = {43, 6310, 12, 6721, 6742, 719, 419, 932, 168, 538, 537} \[ \frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}-\frac{8 b d \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right )}{c e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{4 b c \sqrt{c^2 x^2+1} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{\left (-c^2\right )^{3/2} e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 6310
Rule 12
Rule 6721
Rule 6742
Rule 719
Rule 419
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)^{3/2}} \, dx &=\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{b \int \frac{2 (2 d+e x)}{e^2 \sqrt{1+\frac{1}{c^2 x^2}} x^2 \sqrt{d+e x}} \, dx}{c}\\ &=\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{(2 b) \int \frac{2 d+e x}{\sqrt{1+\frac{1}{c^2 x^2}} x^2 \sqrt{d+e x}} \, dx}{c e^2}\\ &=\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \int \frac{2 d+e x}{x \sqrt{d+e x} \sqrt{1+c^2 x^2}} \, dx}{c e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \int \left (\frac{e}{\sqrt{d+e x} \sqrt{1+c^2 x^2}}+\frac{2 d}{x \sqrt{d+e x} \sqrt{1+c^2 x^2}}\right ) \, dx}{c e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{\left (4 b d \sqrt{1+c^2 x^2}\right ) \int \frac{1}{x \sqrt{d+e x} \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{1}{\sqrt{d+e x} \sqrt{1+c^2 x^2}} \, dx}{c e \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{\left (4 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^2 \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (4 b \sqrt{-c^2} \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 )}{c^3 e \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 )}{e^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{4 b \sqrt{-c^2} \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 )}{c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{\left (8 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^2 \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{4 b \sqrt{-c^2} \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 )}{c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{\left (8 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^2 \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 )}{e^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{4 b \sqrt{-c^2} \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 )}{c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{8 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^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ \end{align*}
Mathematica [C] time = 1.55801, size = 264, normalized size = 0.83 \[ \frac{2 \left (-\frac{2 i b \sqrt{-\frac{e (c x-i)}{c d+i e}} \sqrt{-\frac{e (c x+i)}{c d-i e}} \left (\text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right ),\frac{c d-i e}{c d+i e}\right )-2 \Pi \left (1-\frac{i e}{c d};i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right )|\frac{c d-i e}{c d+i e}\right )\right )}{c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{-\frac{c}{c d-i e}}}+\frac{a (2 d+e x)}{\sqrt{d+e x}}+\frac{b \text{csch}^{-1}(c x) (2 d+e x)}{\sqrt{d+e x}}\right )}{e^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.323, size = 418, normalized size = 1.3 \begin{align*} 2\,{\frac{1}{{e}^{2}} \left ( a \left ( \sqrt{ex+d}+{\frac{d}{\sqrt{ex+d}}} \right ) +b \left ( \sqrt{ex+d}{\rm arccsch} \left (cx\right )+{\frac{{\rm arccsch} \left (cx\right )d}{\sqrt{ex+d}}}+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 ) -2\,{\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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]