Optimal. Leaf size=129 \[ -\frac {c^2 d x^{m+3} \left (a+b \sin ^{-1}(c x)\right )}{m+3}+\frac {d x^{m+1} \left (a+b \sin ^{-1}(c x)\right )}{m+1}-\frac {b c d (3 m+7) x^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{(m+1) (m+2) (m+3)^2}-\frac {b c d \sqrt {1-c^2 x^2} x^{m+2}}{(m+3)^2} \]
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Rubi [A] time = 0.14, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {14, 4687, 12, 459, 364} \[ -\frac {c^2 d x^{m+3} \left (a+b \sin ^{-1}(c x)\right )}{m+3}+\frac {d x^{m+1} \left (a+b \sin ^{-1}(c x)\right )}{m+1}-\frac {b c d (3 m+7) x^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};c^2 x^2\right )}{(m+1) (m+2) (m+3)^2}-\frac {b c d \sqrt {1-c^2 x^2} x^{m+2}}{(m+3)^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 364
Rule 459
Rule 4687
Rubi steps
\begin {align*} \int x^m \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {d x^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{1+m}-\frac {c^2 d x^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{3+m}-(b c) \int \frac {d x^{1+m} \left (\frac {1}{1+m}-\frac {c^2 x^2}{3+m}\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {d x^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{1+m}-\frac {c^2 d x^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{3+m}-(b c d) \int \frac {x^{1+m} \left (\frac {1}{1+m}-\frac {c^2 x^2}{3+m}\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {b c d x^{2+m} \sqrt {1-c^2 x^2}}{(3+m)^2}+\frac {d x^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{1+m}-\frac {c^2 d x^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{3+m}-\frac {(b c d (7+3 m)) \int \frac {x^{1+m}}{\sqrt {1-c^2 x^2}} \, dx}{(1+m) (3+m)^2}\\ &=-\frac {b c d x^{2+m} \sqrt {1-c^2 x^2}}{(3+m)^2}+\frac {d x^{1+m} \left (a+b \sin ^{-1}(c x)\right )}{1+m}-\frac {c^2 d x^{3+m} \left (a+b \sin ^{-1}(c x)\right )}{3+m}-\frac {b c d (7+3 m) x^{2+m} \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};c^2 x^2\right )}{(1+m) (2+m) (3+m)^2}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 118, normalized size = 0.91 \[ -\frac {d x^{m+1} \left ((m+2) \left (m \left (c^2 x^2-1\right )+c^2 x^2-3\right ) \left (a+b \sin ^{-1}(c x)\right )+b c (m+1) x \, _2F_1\left (-\frac {1}{2},\frac {m}{2}+1;\frac {m}{2}+2;c^2 x^2\right )+2 b c x \, _2F_1\left (\frac {1}{2},\frac {m}{2}+1;\frac {m}{2}+2;c^2 x^2\right )\right )}{(m+1) (m+2) (m+3)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a c^{2} d x^{2} - a d + {\left (b c^{2} d x^{2} - b d\right )} \arcsin \left (c x\right )\right )} x^{m}, x\right ) \]
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 \[ \int -{\left (c^{2} d x^{2} - d\right )} {\left (b \arcsin \left (c x\right ) + a\right )} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.43, size = 0, normalized size = 0.00 \[ \int x^{m} \left (-c^{2} d \,x^{2}+d \right ) \left (a +b \arcsin \left (c x \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int x^m\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (d-c^2\,d\,x^2\right ) \,d x \]
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 \[ - d \left (\int \left (- a x^{m}\right )\, dx + \int \left (- b x^{m} \operatorname {asin}{\left (c x \right )}\right )\, dx + \int a c^{2} x^{2} x^{m}\, dx + \int b c^{2} x^{2} x^{m} \operatorname {asin}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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