Optimal. Leaf size=297 \[ -\frac {b \left (7 c^2 d+3 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{40 c^4}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}-\frac {b \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {ArcTan}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{40 c^5 \sqrt {e}}-\frac {b d^{5/2} \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{5 e} \]
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Rubi [A]
time = 0.29, antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 11, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {6434, 531,
457, 104, 159, 163, 65, 223, 209, 95, 213} \begin {gather*} \frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) \text {ArcTan}\left (\frac {\sqrt {e} \sqrt {1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{40 c^5 \sqrt {e}}-\frac {b d^{5/2} \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 )}{5 e}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c^2}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (7 c^2 d+3 e\right ) \sqrt {d+e x^2}}{40 c^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 95
Rule 104
Rule 159
Rule 163
Rule 209
Rule 213
Rule 223
Rule 457
Rule 531
Rule 6434
Rubi steps
\begin {align*} \int x \left (d+e x^2\right )^{3/2} \left (a+b \text {sech}^{-1}(c x)\right ) \, dx &=\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\left (d+e x^2\right )^{5/2}}{x \sqrt {1-c x} \sqrt {1+c x}} \, dx}{5 e}\\ &=\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {\left (d+e x^2\right )^{5/2}}{x \sqrt {1-c^2 x^2}} \, dx}{5 e}\\ &=\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {(d+e x)^{5/2}}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{10 e}\\ &=-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\sqrt {d+e x} \left (-2 c^2 d^2-\frac {1}{2} e \left (7 c^2 d+3 e\right ) x\right )}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{20 c^2 e}\\ &=-\frac {b \left (7 c^2 d+3 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{40 c^4}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}+\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {2 c^4 d^3+\frac {1}{4} e \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) x}{x \sqrt {1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{20 c^4 e}\\ &=-\frac {b \left (7 c^2 d+3 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{40 c^4}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}+\frac {\left (b d^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{10 e}+\frac {\left (b \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{80 c^4}\\ &=-\frac {b \left (7 c^2 d+3 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{40 c^4}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}+\frac {\left (b d^3 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{-d+x^2} \, dx,x,\frac {\sqrt {d+e x^2}}{\sqrt {1-c^2 x^2}}\right )}{5 e}-\frac {\left (b \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {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 )}{40 c^6}\\ &=-\frac {b \left (7 c^2 d+3 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{40 c^4}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}-\frac {b d^{5/2} \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 )}{5 e}-\frac {\left (b \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {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 )}{40 c^6}\\ &=-\frac {b \left (7 c^2 d+3 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{40 c^4}-\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {sech}^{-1}(c x)\right )}{5 e}-\frac {b \left (15 c^4 d^2+10 c^2 d e+3 e^2\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 )}{40 c^5 \sqrt {e}}-\frac {b d^{5/2} \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 )}{5 e}\\ \end {align*}
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Mathematica [A]
time = 21.02, size = 342, normalized size = 1.15 \begin {gather*} \frac {\sqrt {d+e x^2} \left (8 a c^4 \left (d+e x^2\right )^2-b e \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (3 e+c^2 \left (9 d+2 e x^2\right )\right )+8 b c^4 \left (d+e x^2\right )^2 \text {sech}^{-1}(c x)\right )}{40 c^4 e}+\frac {b \sqrt {\frac {1-c x}{1+c x}} \sqrt {1-c^2 x^2} \left (\sqrt {-c^2} \sqrt {-c^2 d-e} \sqrt {e} \left (15 c^4 d^2+10 c^2 d e+3 e^2\right ) \sqrt {\frac {c^2 \left (d+e x^2\right )}{c^2 d+e}} \text {ArcSin}\left (\frac {c \sqrt {e} \sqrt {1-c^2 x^2}}{\sqrt {-c^2} \sqrt {-c^2 d-e}}\right )+8 c^7 d^{5/2} \sqrt {-d-e x^2} \text {ArcTan}\left (\frac {\sqrt {d} \sqrt {1-c^2 x^2}}{\sqrt {-d-e x^2}}\right )\right )}{40 c^7 e (-1+c x) \sqrt {d+e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.75, size = 0, normalized size = 0.00 \[\int x \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 703 vs.
\(2 (183) = 366\).
time = 0.92, size = 1441, normalized size = 4.85 \begin {gather*} \text {Too large to display} \end {gather*}
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 \begin {gather*} \int x \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x\,{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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