Optimal. Leaf size=146 \[ \frac{2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac{2 b \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{c^2 x^2+1}}{c \sqrt{d+e x^2}}\right )}{3 d^2 \sqrt{e}}-\frac{b c \sqrt{c^2 x^2+1}}{3 d \left (c^2 d-e\right ) \sqrt{d+e x^2}} \]
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Rubi [A] time = 0.168782, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {192, 191, 5704, 12, 571, 78, 63, 217, 206} \[ \frac{2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac{2 b \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{c^2 x^2+1}}{c \sqrt{d+e x^2}}\right )}{3 d^2 \sqrt{e}}-\frac{b c \sqrt{c^2 x^2+1}}{3 d \left (c^2 d-e\right ) \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Rule 192
Rule 191
Rule 5704
Rule 12
Rule 571
Rule 78
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{a+b \sinh ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx &=\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}-(b c) \int \frac{x \left (3 d+2 e x^2\right )}{3 d^2 \sqrt{1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx\\ &=\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}-\frac{(b c) \int \frac{x \left (3 d+2 e x^2\right )}{\sqrt{1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 d^2}\\ &=\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}-\frac{(b c) \operatorname{Subst}\left (\int \frac{3 d+2 e x}{\sqrt{1+c^2 x} (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 d^2}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{3 d \left (c^2 d-e\right ) \sqrt{d+e x^2}}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \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 )}{3 d^2}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{3 d \left (c^2 d-e\right ) \sqrt{d+e x^2}}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}-\frac{(2 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 )}{3 c d^2}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{3 d \left (c^2 d-e\right ) \sqrt{d+e x^2}}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}-\frac{(2 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 )}{3 c d^2}\\ &=-\frac{b c \sqrt{1+c^2 x^2}}{3 d \left (c^2 d-e\right ) \sqrt{d+e x^2}}+\frac{x \left (a+b \sinh ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}-\frac{2 b \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{1+c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{3 d^2 \sqrt{e}}\\ \end{align*}
Mathematica [C] time = 0.335032, size = 139, normalized size = 0.95 \[ \frac{a x \left (3 d+2 e x^2\right )-b c x^2 \left (d+e x^2\right ) \sqrt{\frac{e x^2}{d}+1} F_1\left (1;\frac{1}{2},\frac{1}{2};2;-c^2 x^2,-\frac{e x^2}{d}\right )-\frac{b c d \sqrt{c^2 x^2+1} \left (d+e x^2\right )}{c^2 d-e}+b x \sinh ^{-1}(c x) \left (3 d+2 e x^2\right )}{3 d^2 \left (d+e x^2\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.325, size = 0, normalized size = 0. \begin{align*} \int{(a+b{\it Arcsinh} \left ( cx \right ) ) \left ( e{x}^{2}+d \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \, a{\left (\frac{2 \, x}{\sqrt{e x^{2} + d} d^{2}} + \frac{x}{{\left (e x^{2} + d\right )}^{\frac{3}{2}} d}\right )} + b \int \frac{\log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{{\left (e x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.29745, size = 1503, normalized size = 10.29 \begin{align*} \left [\frac{{\left (b c^{2} d^{3} +{\left (b c^{2} d e^{2} - b e^{3}\right )} x^{4} - b d^{2} e + 2 \,{\left (b c^{2} d^{2} e - b d e^{2}\right )} x^{2}\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 ) + 2 \,{\left (2 \,{\left (b c^{2} d e^{2} - b e^{3}\right )} x^{3} + 3 \,{\left (b c^{2} d^{2} e - b d e^{2}\right )} x\right )} \sqrt{e x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + 2 \,{\left (2 \,{\left (a c^{2} d e^{2} - a e^{3}\right )} x^{3} + 3 \,{\left (a c^{2} d^{2} e - a d e^{2}\right )} x -{\left (b c d e^{2} x^{2} + b c d^{2} e\right )} \sqrt{c^{2} x^{2} + 1}\right )} \sqrt{e x^{2} + d}}{6 \,{\left (c^{2} d^{5} e - d^{4} e^{2} +{\left (c^{2} d^{3} e^{3} - d^{2} e^{4}\right )} x^{4} + 2 \,{\left (c^{2} d^{4} e^{2} - d^{3} e^{3}\right )} x^{2}\right )}}, \frac{{\left (b c^{2} d^{3} +{\left (b c^{2} d e^{2} - b e^{3}\right )} x^{4} - b d^{2} e + 2 \,{\left (b c^{2} d^{2} e - b d e^{2}\right )} x^{2}\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 ) +{\left (2 \,{\left (b c^{2} d e^{2} - b e^{3}\right )} x^{3} + 3 \,{\left (b c^{2} d^{2} e - b d e^{2}\right )} x\right )} \sqrt{e x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) +{\left (2 \,{\left (a c^{2} d e^{2} - a e^{3}\right )} x^{3} + 3 \,{\left (a c^{2} d^{2} e - a d e^{2}\right )} x -{\left (b c d e^{2} x^{2} + b c d^{2} e\right )} \sqrt{c^{2} x^{2} + 1}\right )} \sqrt{e x^{2} + d}}{3 \,{\left (c^{2} d^{5} e - d^{4} e^{2} +{\left (c^{2} d^{3} e^{3} - d^{2} e^{4}\right )} x^{4} + 2 \,{\left (c^{2} d^{4} e^{2} - d^{3} e^{3}\right )} x^{2}\right )}}\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{b \operatorname{arsinh}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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