Optimal. Leaf size=70 \[ \frac {b \sqrt {c^2 x^2+1} \tan ^{-1}(c x)}{c^2 d \sqrt {c^2 d x^2+d}}-\frac {a+b \sinh ^{-1}(c x)}{c^2 d \sqrt {c^2 d x^2+d}} \]
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Rubi [A] time = 0.07, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {5717, 203} \[ \frac {b \sqrt {c^2 x^2+1} \tan ^{-1}(c x)}{c^2 d \sqrt {c^2 d x^2+d}}-\frac {a+b \sinh ^{-1}(c x)}{c^2 d \sqrt {c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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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 )^{3/2}} \, dx &=-\frac {a+b \sinh ^{-1}(c x)}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int \frac {1}{1+c^2 x^2} \, dx}{c d \sqrt {d+c^2 d x^2}}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {b \sqrt {1+c^2 x^2} \tan ^{-1}(c x)}{c^2 d \sqrt {d+c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 82, normalized size = 1.17 \[ \frac {b \sqrt {d \left (c^2 x^2+1\right )} \tan ^{-1}(c x)}{c^2 d^2 \sqrt {c^2 x^2+1}}-\frac {\sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{c^2 d^2 \left (c^2 x^2+1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 128, normalized size = 1.83 \[ -\frac {{\left (b c^{2} x^{2} + b\right )} \sqrt {d} \arctan \left (\frac {2 \, \sqrt {c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} + 1} c \sqrt {d} x}{c^{4} d x^{4} - d}\right ) + 2 \, \sqrt {c^{2} d x^{2} + d} b \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 2 \, \sqrt {c^{2} d x^{2} + d} a}{2 \, {\left (c^{4} d^{2} x^{2} + c^{2} d^{2}\right )}} \]
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 \[ \int \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.14, size = 164, normalized size = 2.34 \[ -\frac {a}{c^{2} d \sqrt {c^{2} d \,x^{2}+d}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )}{c^{2} d^{2} \left (c^{2} x^{2}+1\right )}+\frac {i b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (c x +\sqrt {c^{2} x^{2}+1}+i\right )}{\sqrt {c^{2} x^{2}+1}\, c^{2} d^{2}}-\frac {i b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (c x +\sqrt {c^{2} x^{2}+1}-i\right )}{\sqrt {c^{2} x^{2}+1}\, c^{2} d^{2}} \]
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 \[ b {\left (\frac {-\operatorname {arsinh}\left (\frac {1}{c {\left | x \right |}}\right )}{c^{2} d^{\frac {3}{2}}} - \frac {\log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{\sqrt {c^{2} x^{2} + 1} c^{2} d^{\frac {3}{2}}} - \int \frac {1}{c^{5} d^{\frac {3}{2}} x^{4} + c^{3} d^{\frac {3}{2}} x^{2} + {\left (c^{4} d^{\frac {3}{2}} x^{3} + c^{2} d^{\frac {3}{2}} x\right )} \sqrt {c^{2} x^{2} + 1}}\,{d x}\right )} - \frac {a}{\sqrt {c^{2} d x^{2} + d} c^{2} d} \]
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 \[ \int \frac {x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (d\,c^2\,x^2+d\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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