Optimal. Leaf size=371 \[ \frac {4 b^2 c^2 d x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};c^2 x^2\right )}{(m+3)^2 \left (m^2+3 m+2\right )}+\frac {2 b^2 c^2 d x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};c^2 x^2\right )}{(m+2) (m+3)^3}-\frac {4 b c d x^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{m^3+6 m^2+11 m+6}-\frac {2 b c d x^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{(m+2) (m+3)^2}+\frac {d \left (1-c^2 x^2\right ) x^{m+1} \left (a+b \sin ^{-1}(c x)\right )^2}{m+3}-\frac {2 b c d \sqrt {1-c^2 x^2} x^{m+2} \left (a+b \sin ^{-1}(c x)\right )}{(m+3)^2}+\frac {2 d x^{m+1} \left (a+b \sin ^{-1}(c x)\right )^2}{m^2+4 m+3}+\frac {2 b^2 c^2 d x^{m+3}}{(m+3)^3} \]
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Rubi [F] time = 0.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x^m \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int x^m \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\int x^m \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx\\ \end {align*}
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Mathematica [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int x^m \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{2} c^{2} d x^{2} - a^{2} d + {\left (b^{2} c^{2} d x^{2} - b^{2} d\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{2} d x^{2} - a b d\right )} \arcsin \left (c x\right )\right )} x^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -{\left (c^{2} d x^{2} - d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.48, size = 0, normalized size = 0.00 \[ \int x^{m} \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arcsin \left (c x \right )\right )^{2}\, dx \]
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 \[ -\frac {a^{2} c^{2} d x^{m + 3}}{m + 3} + \frac {a^{2} d x^{m + 1}}{m + 1} - \frac {{\left ({\left (b^{2} c^{2} d m + b^{2} c^{2} d\right )} x^{3} - {\left (b^{2} d m + 3 \, b^{2} d\right )} x\right )} x^{m} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (m^{2} + 4 \, m + 3\right )} \int \frac {{\left ({\left (b^{2} c^{3} d m + b^{2} c^{3} d\right )} x^{3} - {\left (b^{2} c d m + 3 \, b^{2} c d\right )} x\right )} \sqrt {c x + 1} \sqrt {-c x + 1} x^{m} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + {\left (a b d m^{2} + {\left (a b c^{4} d m^{2} + 4 \, a b c^{4} d m + 3 \, a b c^{4} d\right )} x^{4} + 4 \, a b d m + 3 \, a b d - 2 \, {\left (a b c^{2} d m^{2} + 4 \, a b c^{2} d m + 3 \, a b c^{2} d\right )} x^{2}\right )} x^{m} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{{\left (c^{2} m^{2} + 4 \, c^{2} m + 3 \, c^{2}\right )} x^{2} - m^{2} - 4 \, m - 3}\,{d x}}{m^{2} + 4 \, m + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^m\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\left (d-c^2\,d\,x^2\right ) \,d x \]
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 \[ - d \left (\int \left (- a^{2} x^{m}\right )\, dx + \int \left (- b^{2} x^{m} \operatorname {asin}^{2}{\left (c x \right )}\right )\, dx + \int \left (- 2 a b x^{m} \operatorname {asin}{\left (c x \right )}\right )\, dx + \int a^{2} c^{2} x^{2} x^{m}\, dx + \int b^{2} c^{2} x^{2} x^{m} \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int 2 a b c^{2} x^{2} x^{m} \operatorname {asin}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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