Optimal. Leaf size=266 \[ \frac{2 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{\frac{e x^2}{d}+1} \text{EllipticF}\left (\sin ^{-1}(c x),-\frac{e}{c^2 d}\right )}{3 c d^2 \sqrt{d+e x^2}}+\frac{2 x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}+\frac{x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{b e x \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{3 d^2 \left (c^2 d+e\right ) \sqrt{d+e x^2}}+\frac{b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3 d^2 \left (c^2 d+e\right ) \sqrt{\frac{e x^2}{d}+1}} \]
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Rubi [A] time = 0.20608, antiderivative size = 266, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {192, 191, 6291, 12, 527, 524, 426, 424, 421, 419} \[ \frac{2 x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}+\frac{x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{b e x \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{3 d^2 \left (c^2 d+e\right ) \sqrt{d+e x^2}}+\frac{2 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{\frac{e x^2}{d}+1} F\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3 c d^2 \sqrt{d+e x^2}}+\frac{b c \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{d+e x^2} E\left (\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right )}{3 d^2 \left (c^2 d+e\right ) \sqrt{\frac{e x^2}{d}+1}} \]
Antiderivative was successfully verified.
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Rule 192
Rule 191
Rule 6291
Rule 12
Rule 527
Rule 524
Rule 426
Rule 424
Rule 421
Rule 419
Rubi steps
\begin{align*} \int \frac{a+b \text{sech}^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx &=\frac{x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}+\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{3 d+2 e x^2}{3 d^2 \sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx\\ &=\frac{x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{3 d+2 e x^2}{\sqrt{1-c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 d^2}\\ &=\frac{b e x \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{3 d^2 \left (c^2 d+e\right ) \sqrt{d+e x^2}}+\frac{x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{-d \left (3 c^2 d+2 e\right )-c^2 d e x^2}{\sqrt{1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{3 d^3 \left (c^2 d+e\right )}\\ &=\frac{b e x \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{3 d^2 \left (c^2 d+e\right ) \sqrt{d+e x^2}}+\frac{x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^2}}+\frac{\left (2 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^2}+\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^2 \left (c^2 d+e\right )}\\ &=\frac{b e x \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{3 d^2 \left (c^2 d+e\right ) \sqrt{d+e x^2}}+\frac{x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^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^2 \left (c^2 d+e\right ) \sqrt{1+\frac{e x^2}{d}}}+\frac{\left (2 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^2 \sqrt{d+e x^2}}\\ &=\frac{b e x \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{3 d^2 \left (c^2 d+e\right ) \sqrt{d+e x^2}}+\frac{x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac{2 x \left (a+b \text{sech}^{-1}(c x)\right )}{3 d^2 \sqrt{d+e x^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^2 \left (c^2 d+e\right ) \sqrt{1+\frac{e x^2}{d}}}+\frac{2 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^2 \sqrt{d+e x^2}}\\ \end{align*}
Mathematica [C] time = 5.08504, size = 517, normalized size = 1.94 \[ \frac{-\frac{i b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (d+e x^2\right ) \sqrt{\frac{c \left (\sqrt{d}-i \sqrt{e} x\right )}{(c x+1) \left (c \sqrt{d}-i \sqrt{e}\right )}} \sqrt{\frac{c \left (\sqrt{d}+i \sqrt{e} x\right )}{(c x+1) \left (c \sqrt{d}+i \sqrt{e}\right )}} \left (\left (c \sqrt{d}-i \sqrt{e}\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{\left (d c^2+e\right ) (1-c x)}{\left (\sqrt{d} c+i \sqrt{e}\right )^2 (c x+1)}}\right )|\frac{\left (\sqrt{d} c+i \sqrt{e}\right )^2}{\left (c \sqrt{d}-i \sqrt{e}\right )^2}\right )-2 \left (3 c \sqrt{d}+2 i \sqrt{e}\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{(1-c x) \left (c^2 d+e\right )}{(c x+1) \left (c \sqrt{d}+i \sqrt{e}\right )^2}}\right ),\frac{\left (c \sqrt{d}+i \sqrt{e}\right )^2}{\left (c \sqrt{d}-i \sqrt{e}\right )^2}\right )\right )}{c \left (c \sqrt{d}+i \sqrt{e}\right ) \sqrt{-\frac{(c x-1) \left (c \sqrt{d}-i \sqrt{e}\right )}{(c x+1) \left (c \sqrt{d}+i \sqrt{e}\right )}}}+a x \left (3 d+2 e x^2\right )+\frac{b \sqrt{\frac{1-c x}{c x+1}} \left (d+e x^2\right ) (e x-c d)}{c^2 d+e}+b x \text{sech}^{-1}(c x) \left (3 d+2 e x^2\right )}{3 d^2 \left (d+e x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.012, size = 0, normalized size = 0. \begin{align*} \int{(a+b{\rm arcsech} \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 (\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{\sqrt{e x^{2} + d}{\left (b \operatorname{arsech}\left (c x\right ) + a\right )}}{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{b \operatorname{arsech}\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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