Optimal. Leaf size=81 \[ \frac{2 \sqrt{e (c+d x)} \left (a+b \sin ^{-1}(c+d x)\right )}{d e}+\frac{4 b \sqrt{e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{-c-d x+1}}{\sqrt{2}}\right )\right |2\right )}{d e \sqrt{c+d x}} \]
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Rubi [A] time = 0.0759981, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {4805, 4627, 320, 318, 424} \[ \frac{2 \sqrt{e (c+d x)} \left (a+b \sin ^{-1}(c+d x)\right )}{d e}+\frac{4 b \sqrt{e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{-c-d x+1}}{\sqrt{2}}\right )\right |2\right )}{d e \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Rule 4805
Rule 4627
Rule 320
Rule 318
Rule 424
Rubi steps
\begin{align*} \int \frac{a+b \sin ^{-1}(c+d x)}{\sqrt{c e+d e x}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b \sin ^{-1}(x)}{\sqrt{e x}} \, dx,x,c+d x\right )}{d}\\ &=\frac{2 \sqrt{e (c+d x)} \left (a+b \sin ^{-1}(c+d x)\right )}{d e}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{\sqrt{e x}}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{d e}\\ &=\frac{2 \sqrt{e (c+d x)} \left (a+b \sin ^{-1}(c+d x)\right )}{d e}-\frac{\left (2 b \sqrt{e (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{d e \sqrt{c+d x}}\\ &=\frac{2 \sqrt{e (c+d x)} \left (a+b \sin ^{-1}(c+d x)\right )}{d e}+\frac{\left (4 b \sqrt{e (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-2 x^2}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-c-d x}}{\sqrt{2}}\right )}{d e \sqrt{c+d x}}\\ &=\frac{2 \sqrt{e (c+d x)} \left (a+b \sin ^{-1}(c+d x)\right )}{d e}+\frac{4 b \sqrt{e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{1-c-d x}}{\sqrt{2}}\right )\right |2\right )}{d e \sqrt{c+d x}}\\ \end{align*}
Mathematica [C] time = 0.0311126, size = 59, normalized size = 0.73 \[ -\frac{2 \sqrt{e (c+d x)} \left (2 b (c+d x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},(c+d x)^2\right )-3 \left (a+b \sin ^{-1}(c+d x)\right )\right )}{3 d e} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.008, size = 149, normalized size = 1.8 \begin{align*} 2\,{\frac{1}{de} \left ( a\sqrt{dex+ce}+b \left ( \sqrt{dex+ce}\arcsin \left ({\frac{dex+ce}{e}} \right ) +2\,{\frac{{\it EllipticF} \left ( \sqrt{dex+ce}\sqrt{{e}^{-1}},i \right ) -{\it EllipticE} \left ( \sqrt{dex+ce}\sqrt{{e}^{-1}},i \right ) }{\sqrt{{e}^{-1}}}\sqrt{1-{\frac{dex+ce}{e}}}\sqrt{{\frac{dex+ce}{e}}+1}{\frac{1}{\sqrt{-{\frac{ \left ( dex+ce \right ) ^{2}}{{e}^{2}}}+1}}}} \right ) \right ) } \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 (\frac{b \arcsin \left (d x + c\right ) + a}{\sqrt{d e x + c e}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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 \frac{b \arcsin \left (d x + c\right ) + a}{\sqrt{d e x + c e}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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