Optimal. Leaf size=89 \[ \frac {(e (c+d x))^{m+1} \left (a+b \sin ^{-1}(c+d x)\right )}{d e (m+1)}-\frac {b (e (c+d x))^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};(c+d x)^2\right )}{d e^2 (m+1) (m+2)} \]
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Rubi [A] time = 0.06, antiderivative size = 89, normalized size of antiderivative = 1.00, 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 364
Rule 4627
Rule 4805
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*}
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Mathematica [A] time = 0.05, size = 77, normalized size = 0.87 \[ -\frac {(c+d x) (e (c+d x))^m \left (b (c+d x) \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};(c+d x)^2\right )-(m+2) \left (a+b \sin ^{-1}(c+d x)\right )\right )}{d (m+1) (m+2)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \arcsin \left (d x + c\right ) + a\right )} {\left (d e x + c e\right )}^{m}, x\right ) \]
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 \[ \int {\left (b \arcsin \left (d x + c\right ) + a\right )} {\left (d e x + c e\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.80, size = 0, normalized size = 0.00 \[ \int \left (d e x +c e \right )^{m} \left (a +b \arcsin \left (d x +c \right )\right )\, dx \]
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 \[ \int {\left (c\,e+d\,e\,x\right )}^m\,\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right ) \,d x \]
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 \[ \int \left (e \left (c + d x\right )\right )^{m} \left (a + b \operatorname {asin}{\left (c + d x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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