Optimal. Leaf size=80 \[ -\frac {a+b \sinh ^{-1}(c x)}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}+\frac {b x}{6 c d^3 \sqrt {c^2 x^2+1}}+\frac {b x}{12 c d^3 \left (c^2 x^2+1\right )^{3/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {5717, 192, 191} \[ -\frac {a+b \sinh ^{-1}(c x)}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}+\frac {b x}{6 c d^3 \sqrt {c^2 x^2+1}}+\frac {b x}{12 c d^3 \left (c^2 x^2+1\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 5717
Rubi steps
\begin {align*} \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\left (d+c^2 d x^2\right )^3} \, dx &=-\frac {a+b \sinh ^{-1}(c x)}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {b \int \frac {1}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{4 c d^3}\\ &=\frac {b x}{12 c d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {a+b \sinh ^{-1}(c x)}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}+\frac {b \int \frac {1}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{6 c d^3}\\ &=\frac {b x}{12 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b x}{6 c d^3 \sqrt {1+c^2 x^2}}-\frac {a+b \sinh ^{-1}(c x)}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 56, normalized size = 0.70 \[ \frac {-3 a+b c x \sqrt {c^2 x^2+1} \left (2 c^2 x^2+3\right )-3 b \sinh ^{-1}(c x)}{12 d^3 \left (c^3 x^2+c\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 98, normalized size = 1.22 \[ \frac {3 \, a c^{4} x^{4} + 6 \, a c^{2} x^{2} - 3 \, b \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (2 \, b c^{3} x^{3} + 3 \, b c x\right )} \sqrt {c^{2} x^{2} + 1}}{12 \, {\left (c^{6} d^{3} x^{4} + 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}\right )}} \]
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 \[ \int \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x}{{\left (c^{2} d x^{2} + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 76, normalized size = 0.95 \[ \frac {-\frac {a}{4 d^{3} \left (c^{2} x^{2}+1\right )^{2}}+\frac {b \left (-\frac {\arcsinh \left (c x \right )}{4 \left (c^{2} x^{2}+1\right )^{2}}+\frac {c x}{12 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {c x}{6 \sqrt {c^{2} x^{2}+1}}\right )}{d^{3}}}{c^{2}} \]
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 \[ -\frac {1}{16} \, b {\left (\frac {4 \, \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 1}{c^{6} d^{3} x^{4} + 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}} - 16 \, \int \frac {1}{4 \, {\left (c^{8} d^{3} x^{7} + 3 \, c^{6} d^{3} x^{5} + 3 \, c^{4} d^{3} x^{3} + c^{2} d^{3} x + {\left (c^{7} d^{3} x^{6} + 3 \, c^{5} d^{3} x^{4} + 3 \, c^{3} d^{3} x^{2} + c d^{3}\right )} \sqrt {c^{2} x^{2} + 1}\right )}}\,{d x}\right )} - \frac {a}{4 \, {\left (c^{6} d^{3} x^{4} + 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (d\,c^2\,x^2+d\right )}^3} \,d x \]
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 \[ \frac {\int \frac {a x}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac {b x \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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