Optimal. Leaf size=157 \[ \frac {1}{7} c^4 d^2 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{5} c^2 d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} d^2 x^3 \left (a+b \sinh ^{-1}(c x)\right )-\frac {b d^2 \left (c^2 x^2+1\right )^{7/2}}{49 c^3}+\frac {b d^2 \left (c^2 x^2+1\right )^{5/2}}{175 c^3}+\frac {4 b d^2 \left (c^2 x^2+1\right )^{3/2}}{315 c^3}+\frac {8 b d^2 \sqrt {c^2 x^2+1}}{105 c^3} \]
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Rubi [A] time = 0.17, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {270, 5730, 12, 1251, 771} \[ \frac {1}{7} c^4 d^2 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{5} c^2 d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} d^2 x^3 \left (a+b \sinh ^{-1}(c x)\right )-\frac {b d^2 \left (c^2 x^2+1\right )^{7/2}}{49 c^3}+\frac {b d^2 \left (c^2 x^2+1\right )^{5/2}}{175 c^3}+\frac {4 b d^2 \left (c^2 x^2+1\right )^{3/2}}{315 c^3}+\frac {8 b d^2 \sqrt {c^2 x^2+1}}{105 c^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 270
Rule 771
Rule 1251
Rule 5730
Rubi steps
\begin {align*} \int x^2 \left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac {1}{3} d^2 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{5} c^2 d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{7} c^4 d^2 x^7 \left (a+b \sinh ^{-1}(c x)\right )-(b c) \int \frac {d^2 x^3 \left (35+42 c^2 x^2+15 c^4 x^4\right )}{105 \sqrt {1+c^2 x^2}} \, dx\\ &=\frac {1}{3} d^2 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{5} c^2 d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{7} c^4 d^2 x^7 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{105} \left (b c d^2\right ) \int \frac {x^3 \left (35+42 c^2 x^2+15 c^4 x^4\right )}{\sqrt {1+c^2 x^2}} \, dx\\ &=\frac {1}{3} d^2 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{5} c^2 d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{7} c^4 d^2 x^7 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{210} \left (b c d^2\right ) \operatorname {Subst}\left (\int \frac {x \left (35+42 c^2 x+15 c^4 x^2\right )}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )\\ &=\frac {1}{3} d^2 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{5} c^2 d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{7} c^4 d^2 x^7 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{210} \left (b c d^2\right ) \operatorname {Subst}\left (\int \left (-\frac {8}{c^2 \sqrt {1+c^2 x}}-\frac {4 \sqrt {1+c^2 x}}{c^2}-\frac {3 \left (1+c^2 x\right )^{3/2}}{c^2}+\frac {15 \left (1+c^2 x\right )^{5/2}}{c^2}\right ) \, dx,x,x^2\right )\\ &=\frac {8 b d^2 \sqrt {1+c^2 x^2}}{105 c^3}+\frac {4 b d^2 \left (1+c^2 x^2\right )^{3/2}}{315 c^3}+\frac {b d^2 \left (1+c^2 x^2\right )^{5/2}}{175 c^3}-\frac {b d^2 \left (1+c^2 x^2\right )^{7/2}}{49 c^3}+\frac {1}{3} d^2 x^3 \left (a+b \sinh ^{-1}(c x)\right )+\frac {2}{5} c^2 d^2 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{7} c^4 d^2 x^7 \left (a+b \sinh ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.08, size = 111, normalized size = 0.71 \[ \frac {d^2 \left (105 a c^3 x^3 \left (15 c^4 x^4+42 c^2 x^2+35\right )-b \sqrt {c^2 x^2+1} \left (225 c^6 x^6+612 c^4 x^4+409 c^2 x^2-818\right )+105 b c^3 x^3 \left (15 c^4 x^4+42 c^2 x^2+35\right ) \sinh ^{-1}(c x)\right )}{11025 c^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 153, normalized size = 0.97 \[ \frac {1575 \, a c^{7} d^{2} x^{7} + 4410 \, a c^{5} d^{2} x^{5} + 3675 \, a c^{3} d^{2} x^{3} + 105 \, {\left (15 \, b c^{7} d^{2} x^{7} + 42 \, b c^{5} d^{2} x^{5} + 35 \, b c^{3} d^{2} x^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (225 \, b c^{6} d^{2} x^{6} + 612 \, b c^{4} d^{2} x^{4} + 409 \, b c^{2} d^{2} x^{2} - 818 \, b d^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{11025 \, c^{3}} \]
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 = 148, normalized size = 0.94 \[ \frac {d^{2} a \left (\frac {1}{7} c^{7} x^{7}+\frac {2}{5} c^{5} x^{5}+\frac {1}{3} c^{3} x^{3}\right )+d^{2} b \left (\frac {\arcsinh \left (c x \right ) c^{7} x^{7}}{7}+\frac {2 \arcsinh \left (c x \right ) c^{5} x^{5}}{5}+\frac {\arcsinh \left (c x \right ) c^{3} x^{3}}{3}-\frac {c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{49}-\frac {68 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{1225}-\frac {409 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{11025}+\frac {818 \sqrt {c^{2} x^{2}+1}}{11025}\right )}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 261, normalized size = 1.66 \[ \frac {1}{7} \, a c^{4} d^{2} x^{7} + \frac {2}{5} \, a c^{2} d^{2} x^{5} + \frac {1}{245} \, {\left (35 \, x^{7} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac {6 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac {16 \, \sqrt {c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b c^{4} d^{2} + \frac {2}{75} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b c^{2} d^{2} + \frac {1}{3} \, a d^{2} x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d^{2} \]
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 \[ \int x^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.14, size = 202, normalized size = 1.29 \[ \begin {cases} \frac {a c^{4} d^{2} x^{7}}{7} + \frac {2 a c^{2} d^{2} x^{5}}{5} + \frac {a d^{2} x^{3}}{3} + \frac {b c^{4} d^{2} x^{7} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {b c^{3} d^{2} x^{6} \sqrt {c^{2} x^{2} + 1}}{49} + \frac {2 b c^{2} d^{2} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {68 b c d^{2} x^{4} \sqrt {c^{2} x^{2} + 1}}{1225} + \frac {b d^{2} x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {409 b d^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{11025 c} + \frac {818 b d^{2} \sqrt {c^{2} x^{2} + 1}}{11025 c^{3}} & \text {for}\: c \neq 0 \\\frac {a d^{2} x^{3}}{3} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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