Optimal. Leaf size=179 \[ -\frac {a+b \text {sech}^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}+\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{3 \sqrt {d} e^2}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{3 e \left (c^2 d+e\right ) \sqrt {d+e x^2}} \]
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Rubi [A] time = 0.26, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {266, 43, 6301, 12, 573, 152, 93, 207} \[ -\frac {a+b \text {sech}^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}+\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{3 \sqrt {d} e^2}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{3 e \left (c^2 d+e\right ) \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 93
Rule 152
Rule 207
Rule 266
Rule 573
Rule 6301
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{5/2}} \, dx &=\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \text {sech}^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-2 d-3 e x^2}{3 e^2 x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx\\ &=\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \text {sech}^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-2 d-3 e x^2}{x \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 e^2}\\ &=\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \text {sech}^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {-2 d-3 e x}{x \sqrt {1-c^2 x} (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 e^2}\\ &=\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{3 e \left (c^2 d+e\right ) \sqrt {d+e x^2}}+\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \text {sech}^{-1}(c x)}{e^2 \sqrt {d+e x^2}}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {d \left (c^2 d+e\right )}{x \sqrt {1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{3 d e^2 \left (c^2 d+e\right )}\\ &=\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{3 e \left (c^2 d+e\right ) \sqrt {d+e x^2}}+\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \text {sech}^{-1}(c x)}{e^2 \sqrt {d+e x^2}}-\frac {\left (b \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^2}\\ &=\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{3 e \left (c^2 d+e\right ) \sqrt {d+e x^2}}+\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \text {sech}^{-1}(c x)}{e^2 \sqrt {d+e x^2}}-\frac {\left (2 b \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^2}\\ &=\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{3 e \left (c^2 d+e\right ) \sqrt {d+e x^2}}+\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2 \left (d+e x^2\right )^{3/2}}-\frac {a+b \text {sech}^{-1}(c x)}{e^2 \sqrt {d+e x^2}}+\frac {2 b \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 \sqrt {d} e^2}\\ \end {align*}
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Mathematica [A] time = 0.38, size = 218, normalized size = 1.22 \[ \frac {-a \left (c^2 d+e\right ) \left (2 d+3 e x^2\right )-b \left (c^2 d+e\right ) \text {sech}^{-1}(c x) \left (2 d+3 e x^2\right )+b e \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (d+e x^2\right )}{3 e^2 \left (c^2 d+e\right ) \left (d+e x^2\right )^{3/2}}-\frac {2 b \sqrt {\frac {1-c x}{c x+1}} \sqrt {1-c^2 x^2} \sqrt {-d-e x^2} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {1-c^2 x^2}}{\sqrt {-d-e x^2}}\right )}{3 \sqrt {d} e^2 (c x-1) \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.78, size = 786, normalized size = 4.39 \[ \left [-\frac {2 \, {\left (2 \, b c^{2} d^{3} + 2 \, b d^{2} e + 3 \, {\left (b c^{2} d^{2} e + b d e^{2}\right )} x^{2}\right )} \sqrt {e x^{2} + d} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\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 {d} \log \left (\frac {{\left (c^{4} d^{2} - 6 \, c^{2} d e + e^{2}\right )} x^{4} - 8 \, {\left (c^{2} d^{2} - d e\right )} x^{2} - 4 \, {\left ({\left (c^{3} d - c e\right )} x^{3} - 2 \, c d x\right )} \sqrt {e x^{2} + d} \sqrt {d} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 8 \, d^{2}}{x^{4}}\right ) + 2 \, {\left (2 \, a c^{2} d^{3} + 2 \, a d^{2} e + 3 \, {\left (a c^{2} d^{2} e + a d e^{2}\right )} x^{2} - {\left (b c d e^{2} x^{3} + b c d^{2} e x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}\right )} \sqrt {e x^{2} + d}}{6 \, {\left (c^{2} d^{4} e^{2} + d^{3} e^{3} + {\left (c^{2} d^{2} e^{4} + d e^{5}\right )} x^{4} + 2 \, {\left (c^{2} d^{3} e^{3} + d^{2} e^{4}\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 {-d} \arctan \left (-\frac {{\left ({\left (c^{3} d - c e\right )} x^{3} - 2 \, c d x\right )} \sqrt {e x^{2} + d} \sqrt {-d} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{2 \, {\left (c^{2} d e x^{4} + {\left (c^{2} d^{2} - d e\right )} x^{2} - d^{2}\right )}}\right ) - {\left (2 \, b c^{2} d^{3} + 2 \, b d^{2} e + 3 \, {\left (b c^{2} d^{2} e + b d e^{2}\right )} x^{2}\right )} \sqrt {e x^{2} + d} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (2 \, a c^{2} d^{3} + 2 \, a d^{2} e + 3 \, {\left (a c^{2} d^{2} e + a d e^{2}\right )} x^{2} - {\left (b c d e^{2} x^{3} + b c d^{2} e x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}\right )} \sqrt {e x^{2} + d}}{3 \, {\left (c^{2} d^{4} e^{2} + d^{3} e^{3} + {\left (c^{2} d^{2} e^{4} + d e^{5}\right )} x^{4} + 2 \, {\left (c^{2} d^{3} e^{3} + d^{2} e^{4}\right )} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.48, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{3} \, a {\left (\frac {3 \, x^{2}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e} + \frac {2 \, d}{{\left (e x^{2} + d\right )}^{\frac {3}{2}} e^{2}}\right )} + b \int \frac {x^{3} \log \left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )}{{\left (e x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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