Optimal. Leaf size=136 \[ \frac {\sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{c^4 d^2}+\frac {a+b \sinh ^{-1}(c x)}{c^4 d \sqrt {c^2 d x^2+d}}-\frac {b \sqrt {c^2 d x^2+d} \tan ^{-1}(c x)}{c^4 d^2 \sqrt {c^2 x^2+1}}-\frac {b x \sqrt {c^2 d x^2+d}}{c^3 d^2 \sqrt {c^2 x^2+1}} \]
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Rubi [A] time = 0.18, antiderivative size = 141, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {5751, 5717, 8, 321, 203} \[ \frac {2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{c^4 d^2}-\frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{c^2 d \sqrt {c^2 d x^2+d}}-\frac {b x \sqrt {c^2 x^2+1}}{c^3 d \sqrt {c^2 d x^2+d}}-\frac {b \sqrt {c^2 x^2+1} \tan ^{-1}(c x)}{c^4 d \sqrt {c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 203
Rule 321
Rule 5717
Rule 5751
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\left (d+c^2 d x^2\right )^{3/2}} \, dx &=-\frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {2 \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {d+c^2 d x^2}} \, dx}{c^2 d}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int \frac {x^2}{1+c^2 x^2} \, dx}{c d \sqrt {d+c^2 d x^2}}\\ &=\frac {b x \sqrt {1+c^2 x^2}}{c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^4 d^2}-\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int \frac {1}{1+c^2 x^2} \, dx}{c^3 d \sqrt {d+c^2 d x^2}}-\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int 1 \, dx}{c^3 d \sqrt {d+c^2 d x^2}}\\ &=-\frac {b x \sqrt {1+c^2 x^2}}{c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^4 d^2}-\frac {b \sqrt {1+c^2 x^2} \tan ^{-1}(c x)}{c^4 d \sqrt {d+c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 143, normalized size = 1.05 \[ \frac {\sqrt {c^2 d x^2+d} \left (a \sqrt {c^2 x^2+1} \left (c^2 x^2+2\right )-b \left (c^3 x^3+c x\right )+b \sqrt {c^2 x^2+1} \left (c^2 x^2+2\right ) \sinh ^{-1}(c x)\right )}{c^4 d^2 \left (c^2 x^2+1\right )^{3/2}}-\frac {b \sqrt {d \left (c^2 x^2+1\right )} \tan ^{-1}(c x)}{c^4 d^2 \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 166, normalized size = 1.22 \[ \frac {{\left (b c^{2} x^{2} + b\right )} \sqrt {d} \arctan \left (\frac {2 \, \sqrt {c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} + 1} c \sqrt {d} x}{c^{4} d x^{4} - d}\right ) + 2 \, {\left (b c^{2} x^{2} + 2 \, b\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 2 \, {\left (a c^{2} x^{2} - \sqrt {c^{2} x^{2} + 1} b c x + 2 \, a\right )} \sqrt {c^{2} d x^{2} + d}}{2 \, {\left (c^{6} d^{2} x^{2} + c^{4} d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.22, size = 260, normalized size = 1.91 \[ \frac {a \,x^{2}}{c^{2} d \sqrt {c^{2} d \,x^{2}+d}}+\frac {2 a}{d \,c^{4} \sqrt {c^{2} d \,x^{2}+d}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x^{2}}{c^{2} d^{2} \left (c^{2} x^{2}+1\right )}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x}{c^{3} d^{2} \sqrt {c^{2} x^{2}+1}}+\frac {2 b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )}{c^{4} d^{2} \left (c^{2} x^{2}+1\right )}+\frac {i b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (c x +\sqrt {c^{2} x^{2}+1}-i\right )}{\sqrt {c^{2} x^{2}+1}\, c^{4} d^{2}}-\frac {i b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (c x +\sqrt {c^{2} x^{2}+1}+i\right )}{\sqrt {c^{2} x^{2}+1}\, c^{4} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 119, normalized size = 0.88 \[ -b c {\left (\frac {x}{c^{4} d^{\frac {3}{2}}} + \frac {\arctan \left (c x\right )}{c^{5} d^{\frac {3}{2}}}\right )} + b {\left (\frac {x^{2}}{\sqrt {c^{2} d x^{2} + d} c^{2} d} + \frac {2}{\sqrt {c^{2} d x^{2} + d} c^{4} d}\right )} \operatorname {arsinh}\left (c x\right ) + a {\left (\frac {x^{2}}{\sqrt {c^{2} d x^{2} + d} c^{2} d} + \frac {2}{\sqrt {c^{2} d x^{2} + d} c^{4} d}\right )} \]
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 \[ \int \frac {x^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (d\,c^2\,x^2+d\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{3} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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