Optimal. Leaf size=246 \[ \frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{\frac{e x^2}{d}+1} \text{EllipticF}\left (\sin ^{-1}(c x),-\frac{e}{c^2 d}\right )}{3 c d e \sqrt{d+e x^2}}+\frac{x^3 \left (a+b \text{sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac{b x \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt{d+e x^2}}-\frac{b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3 d e \left (c^2 d+e\right ) \sqrt{\frac{e x^2}{d}+1}} \]
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Rubi [A] time = 0.246232, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {264, 6301, 12, 471, 423, 426, 424, 421, 419} \[ \frac{x^3 \left (a+b \text{sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac{b x \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt{d+e x^2}}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{\frac{e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3 c d e \sqrt{d+e x^2}}-\frac{b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3 d e \left (c^2 d+e\right ) \sqrt{\frac{e x^2}{d}+1}} \]
Antiderivative was successfully verified.
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Rule 264
Rule 6301
Rule 12
Rule 471
Rule 423
Rule 426
Rule 424
Rule 421
Rule 419
Rubi steps
\begin{align*} \int \frac{x^2 \left (a+b \text{sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=\frac{x^3 \left (a+b \text{sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x^2}{3 d \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx\\ &=\frac{x^3 \left (a+b \text{sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x^2}{\sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 d}\\ &=-\frac{b x \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt{d+e x^2}}+\frac{x^3 \left (a+b \text{sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{\sqrt{1-c^2 x^2}}{\sqrt{d+e x^2}} \, dx}{3 d \left (c^2 d+e\right )}\\ &=-\frac{b x \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt{d+e x^2}}+\frac{x^3 \left (a+b \text{sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{\sqrt{1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{3 d e}-\frac{\left (b c^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{\sqrt{d+e x^2}}{\sqrt{1-c^2 x^2}} \, dx}{3 d e \left (c^2 d+e\right )}\\ &=-\frac{b x \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt{d+e x^2}}+\frac{x^3 \left (a+b \text{sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac{\left (b c^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{d+e x^2}\right ) \int \frac{\sqrt{1+\frac{e x^2}{d}}}{\sqrt{1-c^2 x^2}} \, dx}{3 d e \left (c^2 d+e\right ) \sqrt{1+\frac{e x^2}{d}}}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1+\frac{e x^2}{d}}\right ) \int \frac{1}{\sqrt{1-c^2 x^2} \sqrt{1+\frac{e x^2}{d}}} \, dx}{3 d e \sqrt{d+e x^2}}\\ &=-\frac{b x \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{3 d \left (c^2 d+e\right ) \sqrt{d+e x^2}}+\frac{x^3 \left (a+b \text{sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac{b c \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3 d e \left (c^2 d+e\right ) \sqrt{1+\frac{e x^2}{d}}}+\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1+\frac{e x^2}{d}} F\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3 c d e \sqrt{d+e x^2}}\\ \end{align*}
Mathematica [C] time = 2.47446, size = 488, normalized size = 1.98 \[ \frac{\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (d+e x^2\right ) \sqrt{\frac{c \left (\sqrt{d}+i \sqrt{e} x\right )}{(c x+1) \left (c \sqrt{d}+i \sqrt{e}\right )}} \sqrt{\frac{c \left (\sqrt{e} x+i \sqrt{d}\right )}{(c x+1) \left (\sqrt{e}+i c \sqrt{d}\right )}} \left (\left (\sqrt{e}+i c \sqrt{d}\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{\left (d c^2+e\right ) (1-c x)}{\left (\sqrt{d} c+i \sqrt{e}\right )^2 (c x+1)}}\right )|\frac{\left (\sqrt{d} c+i \sqrt{e}\right )^2}{\left (c \sqrt{d}-i \sqrt{e}\right )^2}\right )-2 \sqrt{e} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{(1-c x) \left (c^2 d+e\right )}{(c x+1) \left (c \sqrt{d}+i \sqrt{e}\right )^2}}\right ),\frac{\left (c \sqrt{d}+i \sqrt{e}\right )^2}{\left (c \sqrt{d}-i \sqrt{e}\right )^2}\right )\right )}{c e \left (c \sqrt{d}+i \sqrt{e}\right ) \sqrt{\frac{(c x-1) \left (\sqrt{e}+i c \sqrt{d}\right )}{(c x+1) \left (\sqrt{e}-i c \sqrt{d}\right )}}}+a x^3-\frac{b \sqrt{\frac{1-c x}{c x+1}} \left (d+e x^2\right ) (e x-c d)}{e \left (c^2 d+e\right )}+b x^3 \text{sech}^{-1}(c x)}{3 d \left (d+e x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.457, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ( a+b{\rm arcsech} \left (cx\right ) \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{x}{{\left (e x^{2} + d\right )}^{\frac{3}{2}} e} - \frac{x}{\sqrt{e x^{2} + d} d e}\right )} + b \int \frac{x^{2} \log \left (\sqrt{\frac{1}{c x} + 1} \sqrt{\frac{1}{c x} - 1} + \frac{1}{c x}\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{2} \operatorname{arsech}\left (c x\right ) + a x^{2}\right )} \sqrt{e x^{2} + d}}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + 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{arsech}\left (c x\right ) + a\right )} x^{2}}{{\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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