Optimal. Leaf size=89 \[ \frac{(e (c+d x))^{m+1} \left (a+b \sin ^{-1}(c+d x)\right )}{d e (m+1)}-\frac{b (e (c+d x))^{m+2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},(c+d x)^2\right )}{d e^2 (m+1) (m+2)} \]
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Rubi [A] time = 0.0621221, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4805, 4627, 364} \[ \frac{(e (c+d x))^{m+1} \left (a+b \sin ^{-1}(c+d x)\right )}{d e (m+1)}-\frac{b (e (c+d x))^{m+2} \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};(c+d x)^2\right )}{d e^2 (m+1) (m+2)} \]
Antiderivative was successfully verified.
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Rule 4805
Rule 4627
Rule 364
Rubi steps
\begin{align*} \int (c e+d e x)^m \left (a+b \sin ^{-1}(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int (e x)^m \left (a+b \sin ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{(e (c+d x))^{1+m} \left (a+b \sin ^{-1}(c+d x)\right )}{d e (1+m)}-\frac{b \operatorname{Subst}\left (\int \frac{(e x)^{1+m}}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{d e (1+m)}\\ &=\frac{(e (c+d x))^{1+m} \left (a+b \sin ^{-1}(c+d x)\right )}{d e (1+m)}-\frac{b (e (c+d x))^{2+m} \, _2F_1\left (\frac{1}{2},\frac{2+m}{2};\frac{4+m}{2};(c+d x)^2\right )}{d e^2 (1+m) (2+m)}\\ \end{align*}
Mathematica [A] time = 0.0418228, size = 77, normalized size = 0.87 \[ -\frac{(c+d x) (e (c+d x))^m \left (b (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{m+2}{2},\frac{m+4}{2},(c+d x)^2\right )-(m+2) \left (a+b \sin ^{-1}(c+d x)\right )\right )}{d (m+1) (m+2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.263, size = 0, normalized size = 0. \begin{align*} \int \left ( dex+ce \right ) ^{m} \left ( a+b\arcsin \left ( dx+c \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \arcsin \left (d x + c\right ) + a\right )}{\left (d e x + c e\right )}^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e \left (c + d x\right )\right )^{m} \left (a + b \operatorname{asin}{\left (c + d x \right )}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arcsin \left (d x + c\right ) + a\right )}{\left (d e x + c e\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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