Optimal. Leaf size=102 \[ \frac {1}{5} c^2 d x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} d x^3 \left (a+b \sinh ^{-1}(c x)\right )-\frac {b d \left (c^2 x^2+1\right )^{5/2}}{25 c^3}+\frac {b d \left (c^2 x^2+1\right )^{3/2}}{45 c^3}+\frac {2 b d \sqrt {c^2 x^2+1}}{15 c^3} \]
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Rubi [A] time = 0.10, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {14, 5730, 12, 446, 77} \[ \frac {1}{5} c^2 d x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} d x^3 \left (a+b \sinh ^{-1}(c x)\right )-\frac {b d \left (c^2 x^2+1\right )^{5/2}}{25 c^3}+\frac {b d \left (c^2 x^2+1\right )^{3/2}}{45 c^3}+\frac {2 b d \sqrt {c^2 x^2+1}}{15 c^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 77
Rule 446
Rule 5730
Rubi steps
\begin {align*} \int x^2 \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac {1}{3} d x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} c^2 d x^5 \left (a+b \sinh ^{-1}(c x)\right )-(b c) \int \frac {d x^3 \left (5+3 c^2 x^2\right )}{15 \sqrt {1+c^2 x^2}} \, dx\\ &=\frac {1}{3} d x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} c^2 d x^5 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{15} (b c d) \int \frac {x^3 \left (5+3 c^2 x^2\right )}{\sqrt {1+c^2 x^2}} \, dx\\ &=\frac {1}{3} d x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} c^2 d x^5 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{30} (b c d) \operatorname {Subst}\left (\int \frac {x \left (5+3 c^2 x\right )}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )\\ &=\frac {1}{3} d x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} c^2 d x^5 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{30} (b c d) \operatorname {Subst}\left (\int \left (-\frac {2}{c^2 \sqrt {1+c^2 x}}-\frac {\sqrt {1+c^2 x}}{c^2}+\frac {3 \left (1+c^2 x\right )^{3/2}}{c^2}\right ) \, dx,x,x^2\right )\\ &=\frac {2 b d \sqrt {1+c^2 x^2}}{15 c^3}+\frac {b d \left (1+c^2 x^2\right )^{3/2}}{45 c^3}-\frac {b d \left (1+c^2 x^2\right )^{5/2}}{25 c^3}+\frac {1}{3} d x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} c^2 d x^5 \left (a+b \sinh ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.08, size = 78, normalized size = 0.76 \[ \frac {1}{225} d \left (15 a x^3 \left (3 c^2 x^2+5\right )+15 b x^3 \left (3 c^2 x^2+5\right ) \sinh ^{-1}(c x)+\frac {b \sqrt {c^2 x^2+1} \left (-9 c^4 x^4-13 c^2 x^2+26\right )}{c^3}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 103, normalized size = 1.01 \[ \frac {45 \, a c^{5} d x^{5} + 75 \, a c^{3} d x^{3} + 15 \, {\left (3 \, b c^{5} d x^{5} + 5 \, b c^{3} d x^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (9 \, b c^{4} d x^{4} + 13 \, b c^{2} d x^{2} - 26 \, b d\right )} \sqrt {c^{2} x^{2} + 1}}{225 \, c^{3}} \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 105, normalized size = 1.03 \[ \frac {d a \left (\frac {1}{5} c^{5} x^{5}+\frac {1}{3} c^{3} x^{3}\right )+d b \left (\frac {\arcsinh \left (c x \right ) c^{5} x^{5}}{5}+\frac {\arcsinh \left (c x \right ) c^{3} x^{3}}{3}-\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{25}-\frac {13 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{225}+\frac {26 \sqrt {c^{2} x^{2}+1}}{225}\right )}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 145, normalized size = 1.42 \[ \frac {1}{5} \, a c^{2} d x^{5} + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b c^{2} d + \frac {1}{3} \, a d x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d \]
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 \[ \int x^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\left (d\,c^2\,x^2+d\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.02, size = 126, normalized size = 1.24 \[ \begin {cases} \frac {a c^{2} d x^{5}}{5} + \frac {a d x^{3}}{3} + \frac {b c^{2} d x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {b c d x^{4} \sqrt {c^{2} x^{2} + 1}}{25} + \frac {b d x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {13 b d x^{2} \sqrt {c^{2} x^{2} + 1}}{225 c} + \frac {26 b d \sqrt {c^{2} x^{2} + 1}}{225 c^{3}} & \text {for}\: c \neq 0 \\\frac {a d x^{3}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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