Optimal. Leaf size=70 \[ \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt{d+e x^2}}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{c^2 x^2+1}}{c \sqrt{d+e x^2}}\right )}{d \sqrt{e}} \]
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Rubi [A] time = 0.101097, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {191, 5704, 12, 444, 63, 217, 206} \[ \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt{d+e x^2}}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{c^2 x^2+1}}{c \sqrt{d+e x^2}}\right )}{d \sqrt{e}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 5704
Rule 12
Rule 444
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{a+b \sinh ^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx &=\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt{d+e x^2}}-(b c) \int \frac{x}{d \sqrt{1+c^2 x^2} \sqrt{d+e x^2}} \, dx\\ &=\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt{d+e x^2}}-\frac{(b c) \int \frac{x}{\sqrt{1+c^2 x^2} \sqrt{d+e x^2}} \, dx}{d}\\ &=\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt{d+e x^2}}-\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{2 d}\\ &=\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt{d+e x^2}}-\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{d-\frac{e}{c^2}+\frac{e x^2}{c^2}}} \, dx,x,\sqrt{1+c^2 x^2}\right )}{c d}\\ &=\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt{d+e x^2}}-\frac{b \operatorname{Subst}\left (\int \frac{1}{1-\frac{e x^2}{c^2}} \, dx,x,\frac{\sqrt{1+c^2 x^2}}{\sqrt{d+e x^2}}\right )}{c d}\\ &=\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{d \sqrt{d+e x^2}}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{1+c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{d \sqrt{e}}\\ \end{align*}
Mathematica [C] time = 0.111725, size = 75, normalized size = 1.07 \[ \frac{x \left (2 \left (a+b \sinh ^{-1}(c x)\right )-b c x \sqrt{\frac{e x^2}{d}+1} F_1\left (1;\frac{1}{2},\frac{1}{2};2;-c^2 x^2,-\frac{e x^2}{d}\right )\right )}{2 d \sqrt{d+e x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.295, size = 0, normalized size = 0. \begin{align*} \int{(a+b{\it Arcsinh} \left ( cx \right ) ) \left ( e{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}\, dx \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 [B] time = 2.75295, size = 736, normalized size = 10.51 \begin{align*} \left [\frac{4 \, \sqrt{e x^{2} + d} b e x \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + 4 \, \sqrt{e x^{2} + d} a e x +{\left (b e x^{2} + b d\right )} \sqrt{e} \log \left (8 \, c^{4} e^{2} x^{4} + c^{4} d^{2} + 6 \, c^{2} d e + 8 \,{\left (c^{4} d e + c^{2} e^{2}\right )} x^{2} - 4 \,{\left (2 \, c^{3} e x^{2} + c^{3} d + c e\right )} \sqrt{c^{2} x^{2} + 1} \sqrt{e x^{2} + d} \sqrt{e} + e^{2}\right )}{4 \,{\left (d e^{2} x^{2} + d^{2} e\right )}}, \frac{2 \, \sqrt{e x^{2} + d} b e x \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + 2 \, \sqrt{e x^{2} + d} a e x +{\left (b e x^{2} + b d\right )} \sqrt{-e} \arctan \left (\frac{{\left (2 \, c^{2} e x^{2} + c^{2} d + e\right )} \sqrt{c^{2} x^{2} + 1} \sqrt{e x^{2} + d} \sqrt{-e}}{2 \,{\left (c^{3} e^{2} x^{4} + c d e +{\left (c^{3} d e + c e^{2}\right )} x^{2}\right )}}\right )}{2 \,{\left (d e^{2} x^{2} + d^{2} e\right )}}\right ] \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{asinh}{\left (c x \right )}}{\left (d + e x^{2}\right )^{\frac{3}{2}}}\, 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{arsinh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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