Optimal. Leaf size=278 \[ -\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left (a+b \text{sech}^{-1}(c x)\right )}{e^3}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^3}+\frac{8 b d^{3/2} \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{3 e^3}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{6 c^2 e^2}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (9 c^2 d-e\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{6 c^3 e^{5/2}} \]
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Rubi [A] time = 1.11681, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {266, 43, 6301, 12, 1615, 157, 63, 217, 203, 93, 207} \[ -\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left (a+b \text{sech}^{-1}(c x)\right )}{e^3}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^3}+\frac{8 b d^{3/2} \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{3 e^3}-\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{6 c^2 e^2}+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (9 c^2 d-e\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{6 c^3 e^{5/2}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rule 6301
Rule 12
Rule 1615
Rule 157
Rule 63
Rule 217
Rule 203
Rule 93
Rule 207
Rubi steps
\begin{align*} \int \frac{x^5 \left (a+b \text{sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx &=-\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left (a+b \text{sech}^{-1}(c x)\right )}{e^3}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^3}+\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{-8 d^2-4 d e x^2+e^2 x^4}{3 e^3 x \sqrt{1-c^2 x^2} \sqrt{d+e x^2}} \, dx\\ &=-\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left (a+b \text{sech}^{-1}(c x)\right )}{e^3}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^3}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{-8 d^2-4 d e x^2+e^2 x^4}{x \sqrt{1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{3 e^3}\\ &=-\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left (a+b \text{sech}^{-1}(c x)\right )}{e^3}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^3}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{-8 d^2-4 d e x+e^2 x^2}{x \sqrt{1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{6 e^3}\\ &=-\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{6 c^2 e^2}-\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left (a+b \text{sech}^{-1}(c x)\right )}{e^3}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^3}-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{8 c^2 d^2 e+\frac{1}{2} \left (9 c^2 d-e\right ) e^2 x}{x \sqrt{1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{6 c^2 e^4}\\ &=-\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{6 c^2 e^2}-\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left (a+b \text{sech}^{-1}(c x)\right )}{e^3}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^3}-\frac{\left (4 b d^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{3 e^3}-\frac{\left (b \left (9 c^2 d-e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{12 c^2 e^2}\\ &=-\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{6 c^2 e^2}-\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left (a+b \text{sech}^{-1}(c x)\right )}{e^3}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^3}-\frac{\left (8 b d^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{1}{-d+x^2} \, dx,x,\frac{\sqrt{d+e x^2}}{\sqrt{1-c^2 x^2}}\right )}{3 e^3}+\frac{\left (b \left (9 c^2 d-e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \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 )}{6 c^4 e^2}\\ &=-\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{6 c^2 e^2}-\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left (a+b \text{sech}^{-1}(c x)\right )}{e^3}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^3}+\frac{8 b d^{3/2} \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{3 e^3}+\frac{\left (b \left (9 c^2 d-e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \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 )}{6 c^4 e^2}\\ &=-\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}{6 c^2 e^2}-\frac{d^2 \left (a+b \text{sech}^{-1}(c x)\right )}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left (a+b \text{sech}^{-1}(c x)\right )}{e^3}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{3 e^3}+\frac{b \left (9 c^2 d-e\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{6 c^3 e^{5/2}}+\frac{8 b d^{3/2} \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{3 e^3}\\ \end{align*}
Mathematica [A] time = 1.42311, size = 436, normalized size = 1.57 \[ \frac{-2 a c^2 \left (8 d^2+4 d e x^2-e^2 x^4\right )-2 b c^2 \text{sech}^{-1}(c x) \left (8 d^2+4 d e x^2-e^2 x^4\right )-b e \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (d+e x^2\right )}{6 c^2 e^3 \sqrt{d+e x^2}}-\frac{b \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^2 x^2} \left (16 c^5 d^{3/2} \sqrt{-d-e x^2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{1-c^2 x^2}}{\sqrt{-d-e x^2}}\right )+\sqrt{-c^2} e^{3/2} \sqrt{c^2 (-d)-e} \sqrt{\frac{c^2 \left (d+e x^2\right )}{c^2 d+e}} \sin ^{-1}\left (\frac{\sqrt{-c^2} \sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{c^2 (-d)-e}}\right )-9 \left (-c^2\right )^{3/2} d \sqrt{e} \sqrt{c^2 (-d)-e} \sqrt{\frac{c^2 \left (d+e x^2\right )}{c^2 d+e}} \sin ^{-1}\left (\frac{c \sqrt{e} \sqrt{1-c^2 x^2}}{\sqrt{-c^2} \sqrt{c^2 (-d)-e}}\right )\right )}{6 c^5 e^3 (c x-1) \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.541, size = 0, normalized size = 0. \begin{align*} \int{{x}^{5} \left ( a+b{\rm arcsech} \left (cx\right ) \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 = 4.14966, size = 3760, normalized size = 13.53 \begin{align*} \text{result too large to display} \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^{5}}{{\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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