Optimal. Leaf size=142 \[ \frac {x^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d}-\frac {2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 d}-\frac {b x^3 \sqrt {c^2 x^2+1}}{9 c \sqrt {c^2 d x^2+d}}+\frac {2 b x \sqrt {c^2 x^2+1}}{3 c^3 \sqrt {c^2 d x^2+d}} \]
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Rubi [A] time = 0.16, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5758, 5717, 8, 30} \[ \frac {x^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d}-\frac {2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 d}-\frac {b x^3 \sqrt {c^2 x^2+1}}{9 c \sqrt {c^2 d x^2+d}}+\frac {2 b x \sqrt {c^2 x^2+1}}{3 c^3 \sqrt {c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 5717
Rule 5758
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {d+c^2 d x^2}} \, dx &=\frac {x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d}-\frac {2 \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {d+c^2 d x^2}} \, dx}{3 c^2}-\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int x^2 \, dx}{3 c \sqrt {d+c^2 d x^2}}\\ &=-\frac {b x^3 \sqrt {1+c^2 x^2}}{9 c \sqrt {d+c^2 d x^2}}-\frac {2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 d}+\frac {x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int 1 \, dx}{3 c^3 \sqrt {d+c^2 d x^2}}\\ &=\frac {2 b x \sqrt {1+c^2 x^2}}{3 c^3 \sqrt {d+c^2 d x^2}}-\frac {b x^3 \sqrt {1+c^2 x^2}}{9 c \sqrt {d+c^2 d x^2}}-\frac {2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^4 d}+\frac {x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^2 d}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 93, normalized size = 0.65 \[ \frac {3 a \left (c^4 x^4-c^2 x^2-2\right )+b c x \sqrt {c^2 x^2+1} \left (6-c^2 x^2\right )+3 b \left (c^4 x^4-c^2 x^2-2\right ) \sinh ^{-1}(c x)}{9 c^4 \sqrt {c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 132, normalized size = 0.93 \[ \frac {3 \, {\left (b c^{4} x^{4} - b c^{2} x^{2} - 2 \, b\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (3 \, a c^{4} x^{4} - 3 \, a c^{2} x^{2} - {\left (b c^{3} x^{3} - 6 \, b c x\right )} \sqrt {c^{2} x^{2} + 1} - 6 \, a\right )} \sqrt {c^{2} d x^{2} + d}}{9 \, {\left (c^{6} d x^{2} + c^{4} d\right )}} \]
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 [B] time = 0.23, size = 358, normalized size = 2.52 \[ a \left (\frac {x^{2} \sqrt {c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 \sqrt {c^{2} d \,x^{2}+d}}{3 d \,c^{4}}\right )+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}+3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+3 \arcsinh \left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+\arcsinh \left (c x \right )\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (1+\arcsinh \left (c x \right )\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}-3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (1+3 \arcsinh \left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}+1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 117, normalized size = 0.82 \[ \frac {1}{3} \, b {\left (\frac {\sqrt {c^{2} d x^{2} + d} x^{2}}{c^{2} d} - \frac {2 \, \sqrt {c^{2} d x^{2} + d}}{c^{4} d}\right )} \operatorname {arsinh}\left (c x\right ) + \frac {1}{3} \, a {\left (\frac {\sqrt {c^{2} d x^{2} + d} x^{2}}{c^{2} d} - \frac {2 \, \sqrt {c^{2} d x^{2} + d}}{c^{4} d}\right )} - \frac {{\left (c^{2} x^{3} - 6 \, x\right )} b}{9 \, c^{3} \sqrt {d}} \]
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 \[ \int \frac {x^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{\sqrt {d\,c^2\,x^2+d}} \,d x \]
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 \[ \int \frac {x^{3} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\sqrt {d \left (c^{2} x^{2} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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