Optimal. Leaf size=251 \[ -\frac {d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}+\frac {2 b d^{3/2} \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{3 e^2}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{6 c^2 e}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (3 c^2 d-e\right ) \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{6 c^3 e^{3/2}} \]
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Rubi [A] time = 0.33, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {266, 43, 6301, 12, 573, 154, 157, 63, 217, 203, 93, 207} \[ -\frac {d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}+\frac {2 b d^{3/2} \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{3 e^2}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (3 c^2 d-e\right ) \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{6 c^3 e^{3/2}}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{6 c^2 e} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 63
Rule 93
Rule 154
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 )}{\sqrt {d+e x^2}} \, dx &=-\frac {d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}+\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\left (-2 d+e x^2\right ) \sqrt {d+e x^2}}{3 e^2 x \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\left (-2 d+e x^2\right ) \sqrt {d+e x^2}}{x \sqrt {1-c^2 x^2}} \, dx}{3 e^2}\\ &=-\frac {d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 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) \sqrt {d+e x}}{x \sqrt {1-c^2 x}} \, 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} \sqrt {d+e x^2}}{6 c^2 e}-\frac {d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {2 c^2 d^2+\frac {1}{2} \left (3 c^2 d-e\right ) e x}{x \sqrt {1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{6 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}-\frac {d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}-\frac {\left (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^2}-\frac {\left (b \left (3 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}\\ &=-\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}-\frac {d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}-\frac {\left (2 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^2}+\frac {\left (b \left (3 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}\\ &=-\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}-\frac {d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}+\frac {2 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^2}+\frac {\left (b \left (3 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}\\ &=-\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}-\frac {d \sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right )}{3 e^2}+\frac {b \left (3 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^{3/2}}+\frac {2 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^2}\\ \end {align*}
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Mathematica [A] time = 1.32, size = 406, normalized size = 1.62 \[ -\frac {\sqrt {d+e x^2} \left (2 a c^2 \left (2 d-e x^2\right )+2 b c^2 \text {sech}^{-1}(c x) \left (2 d-e x^2\right )+b e \sqrt {\frac {1-c x}{c x+1}} (c x+1)\right )}{6 c^2 e^2}-\frac {b \sqrt {\frac {1-c x}{c x+1}} \sqrt {1-c^2 x^2} \left (\sqrt {-c^2} e^{3/2} \sqrt {c^2 (-d)-e} \sqrt {\frac {c^2 \left (d+e x^2\right )}{c^2 d+e}} \sin ^{-1}\left (\frac {\sqrt {-c^2} \sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {c^2 (-d)-e}}\right )-3 \left (-c^2\right )^{3/2} d \sqrt {e} \sqrt {c^2 (-d)-e} \sqrt {\frac {c^2 \left (d+e x^2\right )}{c^2 d+e}} \sin ^{-1}\left (\frac {c \sqrt {e} \sqrt {1-c^2 x^2}}{\sqrt {-c^2} \sqrt {c^2 (-d)-e}}\right )+4 c^5 d^{3/2} \sqrt {-d-e x^2} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {1-c^2 x^2}}{\sqrt {-d-e x^2}}\right )\right )}{6 c^5 e^2 (c x-1) \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.01, size = 1389, normalized size = 5.53 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{3}}{\sqrt {e x^{2} + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.61, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}{\sqrt {e \,x^{2}+d}}\, dx \]
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 \[ \frac {1}{3} \, {\left (\frac {\sqrt {e x^{2} + d} x^{2}}{e} - \frac {2 \, \sqrt {e x^{2} + d} d}{e^{2}}\right )} a + \frac {1}{3} \, b {\left (\frac {{\left (e^{2} x^{4} - d e x^{2} - 2 \, d^{2}\right )} \log \left (\sqrt {c x + 1} \sqrt {-c x + 1} + 1\right )}{\sqrt {e x^{2} + d} e^{2}} - 3 \, \int \frac {6 \, {\left (c^{2} e^{2} x^{4} - e^{2} x^{2}\right )} x^{3} \log \left (\sqrt {x}\right ) + 3 \, {\left (c^{2} e^{2} x^{4} \log \relax (c) - e^{2} x^{2} \log \relax (c)\right )} x^{3} + {\left (6 \, {\left (c^{2} e^{2} x^{4} - e^{2} x^{2}\right )} x^{3} \log \left (\sqrt {x}\right ) + {\left ({\left (3 \, e^{2} \log \relax (c) + e^{2}\right )} c^{2} x^{4} - 2 \, c^{2} d^{2} - {\left (c^{2} d e + 3 \, e^{2} \log \relax (c)\right )} x^{2}\right )} x^{3}\right )} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (-c x + 1\right )\right )}}{3 \, {\left (c^{2} e^{2} x^{4} - e^{2} x^{2} + {\left (c^{2} e^{2} x^{4} - e^{2} x^{2}\right )} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (-c x + 1\right )\right )}\right )} \sqrt {e x^{2} + d}}\,{d x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {e\,x^2+d}} \,d x \]
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 \[ \int \frac {x^{3} \left (a + b \operatorname {asech}{\left (c x \right )}\right )}{\sqrt {d + e x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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