Optimal. Leaf size=97 \[ \frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (c^2 x^2+1\right )^2}-\frac {b \sinh ^{-1}(c x)}{4 c^4 d^3}+\frac {b x^3}{12 c d^3 \left (c^2 x^2+1\right )^{3/2}}+\frac {b x}{4 c^3 d^3 \sqrt {c^2 x^2+1}} \]
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Rubi [A] time = 0.09, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5723, 288, 215} \[ \frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (c^2 x^2+1\right )^2}+\frac {b x^3}{12 c d^3 \left (c^2 x^2+1\right )^{3/2}}+\frac {b x}{4 c^3 d^3 \sqrt {c^2 x^2+1}}-\frac {b \sinh ^{-1}(c x)}{4 c^4 d^3} \]
Antiderivative was successfully verified.
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Rule 215
Rule 288
Rule 5723
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} \, dx &=\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {(b c) \int \frac {x^4}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{4 d^3}\\ &=\frac {b x^3}{12 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {b \int \frac {x^2}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{4 c d^3}\\ &=\frac {b x^3}{12 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b x}{4 c^3 d^3 \sqrt {1+c^2 x^2}}+\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {b \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{4 c^3 d^3}\\ &=\frac {b x^3}{12 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b x}{4 c^3 d^3 \sqrt {1+c^2 x^2}}-\frac {b \sinh ^{-1}(c x)}{4 c^4 d^3}+\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (1+c^2 x^2\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 79, normalized size = 0.81 \[ \frac {-3 a \left (2 c^2 x^2+1\right )+b c x \sqrt {c^2 x^2+1} \left (4 c^2 x^2+3\right )-3 \left (2 b c^2 x^2+b\right ) \sinh ^{-1}(c x)}{12 c^4 d^3 \left (c^2 x^2+1\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 99, normalized size = 1.02 \[ \frac {3 \, a c^{4} x^{4} - 3 \, {\left (2 \, b c^{2} x^{2} + b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (4 \, b c^{3} x^{3} + 3 \, b c x\right )} \sqrt {c^{2} x^{2} + 1}}{12 \, {\left (c^{8} d^{3} x^{4} + 2 \, c^{6} d^{3} x^{2} + c^{4} d^{3}\right )}} \]
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 [A] time = 0.02, size = 108, normalized size = 1.11 \[ \frac {\frac {a \left (-\frac {1}{2 \left (c^{2} x^{2}+1\right )}+\frac {1}{4 \left (c^{2} x^{2}+1\right )^{2}}\right )}{d^{3}}+\frac {b \left (-\frac {\arcsinh \left (c x \right )}{2 \left (c^{2} x^{2}+1\right )}+\frac {\arcsinh \left (c x \right )}{4 \left (c^{2} x^{2}+1\right )^{2}}-\frac {c x}{12 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {c x}{3 \sqrt {c^{2} x^{2}+1}}\right )}{d^{3}}}{c^{4}} \]
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 \[ -\frac {1}{16} \, b {\left (\frac {4 \, c^{2} x^{2} + 4 \, {\left (2 \, c^{2} x^{2} + 1\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 3}{c^{8} d^{3} x^{4} + 2 \, c^{6} d^{3} x^{2} + c^{4} d^{3}} - 16 \, \int \frac {2 \, c^{2} x^{2} + 1}{4 \, {\left (c^{10} d^{3} x^{7} + 3 \, c^{8} d^{3} x^{5} + 3 \, c^{6} d^{3} x^{3} + c^{4} d^{3} x + {\left (c^{9} d^{3} x^{6} + 3 \, c^{7} d^{3} x^{4} + 3 \, c^{5} d^{3} x^{2} + c^{3} d^{3}\right )} \sqrt {c^{2} x^{2} + 1}\right )}}\,{d x}\right )} - \frac {{\left (2 \, c^{2} x^{2} + 1\right )} a}{4 \, {\left (c^{8} d^{3} x^{4} + 2 \, c^{6} d^{3} x^{2} + c^{4} d^{3}\right )}} \]
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 \[ \int \frac {x^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (d\,c^2\,x^2+d\right )}^3} \,d x \]
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 \[ \frac {\int \frac {a x^{3}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac {b x^{3} \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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