Optimal. Leaf size=114 \[ -\frac {a+b \sinh ^{-1}(c x)}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}+\frac {b x}{6 c d^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}}+\frac {b \sqrt {c^2 x^2+1} \tan ^{-1}(c x)}{6 c^2 d^2 \sqrt {c^2 d x^2+d}} \]
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Rubi [A] time = 0.08, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5717, 199, 203} \[ -\frac {a+b \sinh ^{-1}(c x)}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}+\frac {b x}{6 c d^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}}+\frac {b \sqrt {c^2 x^2+1} \tan ^{-1}(c x)}{6 c^2 d^2 \sqrt {c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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Rule 199
Rule 203
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 )^{5/2}} \, dx &=-\frac {a+b \sinh ^{-1}(c x)}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\left (1+c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d+c^2 d x^2}}\\ &=\frac {b x}{6 c d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {a+b \sinh ^{-1}(c x)}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int \frac {1}{1+c^2 x^2} \, dx}{6 c d^2 \sqrt {d+c^2 d x^2}}\\ &=\frac {b x}{6 c d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {a+b \sinh ^{-1}(c x)}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {b \sqrt {1+c^2 x^2} \tan ^{-1}(c x)}{6 c^2 d^2 \sqrt {d+c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 130, normalized size = 1.14 \[ \frac {\sqrt {c^2 d x^2+d} \left (-2 a \sqrt {c^2 x^2+1}+b c^3 x^3-2 b \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)+b c x\right )}{6 c^2 d^3 \left (c^2 x^2+1\right )^{5/2}}+\frac {b \sqrt {d \left (c^2 x^2+1\right )} \tan ^{-1}(c x)}{6 c^2 d^3 \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 166, normalized size = 1.46 \[ -\frac {{\left (b c^{4} x^{4} + 2 \, b c^{2} x^{2} + b\right )} \sqrt {d} \arctan \left (\frac {2 \, \sqrt {c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} + 1} c \sqrt {d} x}{c^{4} d x^{4} - d}\right ) + 4 \, \sqrt {c^{2} d x^{2} + d} b \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 2 \, \sqrt {c^{2} d x^{2} + d} {\left (\sqrt {c^{2} x^{2} + 1} b c x - 2 \, a\right )}}{12 \, {\left (c^{6} d^{3} x^{4} + 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}\right )}} \]
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 \[ \int \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.14, size = 198, normalized size = 1.74 \[ -\frac {a}{3 c^{2} d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x}{6 d^{3} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} c}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )}{3 d^{3} \left (c^{2} x^{2}+1\right )^{2} c^{2}}+\frac {i b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (c x +\sqrt {c^{2} x^{2}+1}+i\right )}{6 \sqrt {c^{2} x^{2}+1}\, c^{2} d^{3}}-\frac {i b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (c x +\sqrt {c^{2} x^{2}+1}-i\right )}{6 \sqrt {c^{2} x^{2}+1}\, c^{2} d^{3}} \]
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 \[ b \int \frac {x \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}\,{d x} - \frac {a}{3 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d} \]
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 \[ \int \frac {x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (d\,c^2\,x^2+d\right )}^{5/2}} \,d x \]
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 \[ \int \frac {x \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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