Optimal. Leaf size=177 \[ \frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{c e^{3/2}}-\frac {2 b \sqrt {d} \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{e^2} \]
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Rubi [A] time = 0.27, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {266, 43, 6301, 12, 573, 157, 63, 217, 203, 93, 207} \[ \frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{c e^{3/2}}-\frac {2 b \sqrt {d} \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{e^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 63
Rule 93
Rule 157
Rule 203
Rule 207
Rule 217
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 )^{3/2}} \, dx &=\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {2 d+e x^2}{e^2 x \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \, dx\\ &=\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {2 d+e x^2}{x \sqrt {1-c^2 x^2} \sqrt {d+e x^2}} \, dx}{e^2}\\ &=\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {2 d+e x}{x \sqrt {1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{2 e^2}\\ &=\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {\left (b d \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 )}{e^2}+\frac {\left (b \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 )}{2 e}\\ &=\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {\left (2 b d \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 )}{e^2}-\frac {\left (b \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 )}{c^2 e}\\ &=\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}-\frac {2 b \sqrt {d} \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 )}{e^2}-\frac {\left (b \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 )}{c^2 e}\\ &=\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}-\frac {b \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 )}{c e^{3/2}}-\frac {2 b \sqrt {d} \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 )}{e^2}\\ \end {align*}
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Mathematica [A] time = 1.78, size = 249, normalized size = 1.41 \[ \frac {\left (2 d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{e^2 \sqrt {d+e x^2}}+\frac {b \sqrt {\frac {1-c x}{c x+1}} \sqrt {1-c^2 x^2} \left (\sqrt {-c^2} \sqrt {e} \sqrt {c^2 (-d)-e} \sqrt {\frac {c^2 \left (d+e x^2\right )}{c^2 d+e}} \sin ^{-1}\left (\frac {c \sqrt {e} \sqrt {1-c^2 x^2}}{\sqrt {-c^2} \sqrt {c^2 (-d)-e}}\right )+2 c^3 \sqrt {d} \sqrt {-d-e x^2} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {1-c^2 x^2}}{\sqrt {-d-e x^2}}\right )\right )}{c^3 e^2 (c x-1) \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.80, size = 1311, normalized size = 7.41 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.33, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx \]
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 \[ a {\left (\frac {x^{2}}{\sqrt {e x^{2} + d} e} + \frac {2 \, d}{\sqrt {e x^{2} + d} e^{2}}\right )} + b \int \frac {x^{3} \log \left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^3\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \left (a + b \operatorname {asech}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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