Optimal. Leaf size=139 \[ \frac{20 b \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right ),-1\right )}{189 d e^{11/2}}-\frac{2 \left (a+b \sin ^{-1}(c+d x)\right )}{9 d e (e (c+d x))^{9/2}}-\frac{20 b \sqrt{1-(c+d x)^2}}{189 d e^4 (e (c+d x))^{3/2}}-\frac{4 b \sqrt{1-(c+d x)^2}}{63 d e^2 (e (c+d x))^{7/2}} \]
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Rubi [A] time = 0.113851, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {4805, 4627, 325, 329, 221} \[ -\frac{2 \left (a+b \sin ^{-1}(c+d x)\right )}{9 d e (e (c+d x))^{9/2}}-\frac{20 b \sqrt{1-(c+d x)^2}}{189 d e^4 (e (c+d x))^{3/2}}-\frac{4 b \sqrt{1-(c+d x)^2}}{63 d e^2 (e (c+d x))^{7/2}}+\frac{20 b F\left (\left .\sin ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right )\right |-1\right )}{189 d e^{11/2}} \]
Antiderivative was successfully verified.
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Rule 4805
Rule 4627
Rule 325
Rule 329
Rule 221
Rubi steps
\begin{align*} \int \frac{a+b \sin ^{-1}(c+d x)}{(c e+d e x)^{11/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b \sin ^{-1}(x)}{(e x)^{11/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac{2 \left (a+b \sin ^{-1}(c+d x)\right )}{9 d e (e (c+d x))^{9/2}}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{(e x)^{9/2} \sqrt{1-x^2}} \, dx,x,c+d x\right )}{9 d e}\\ &=-\frac{4 b \sqrt{1-(c+d x)^2}}{63 d e^2 (e (c+d x))^{7/2}}-\frac{2 \left (a+b \sin ^{-1}(c+d x)\right )}{9 d e (e (c+d x))^{9/2}}+\frac{(10 b) \operatorname{Subst}\left (\int \frac{1}{(e x)^{5/2} \sqrt{1-x^2}} \, dx,x,c+d x\right )}{63 d e^3}\\ &=-\frac{4 b \sqrt{1-(c+d x)^2}}{63 d e^2 (e (c+d x))^{7/2}}-\frac{20 b \sqrt{1-(c+d x)^2}}{189 d e^4 (e (c+d x))^{3/2}}-\frac{2 \left (a+b \sin ^{-1}(c+d x)\right )}{9 d e (e (c+d x))^{9/2}}+\frac{(10 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{e x} \sqrt{1-x^2}} \, dx,x,c+d x\right )}{189 d e^5}\\ &=-\frac{4 b \sqrt{1-(c+d x)^2}}{63 d e^2 (e (c+d x))^{7/2}}-\frac{20 b \sqrt{1-(c+d x)^2}}{189 d e^4 (e (c+d x))^{3/2}}-\frac{2 \left (a+b \sin ^{-1}(c+d x)\right )}{9 d e (e (c+d x))^{9/2}}+\frac{(20 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^4}{e^2}}} \, dx,x,\sqrt{e (c+d x)}\right )}{189 d e^6}\\ &=-\frac{4 b \sqrt{1-(c+d x)^2}}{63 d e^2 (e (c+d x))^{7/2}}-\frac{20 b \sqrt{1-(c+d x)^2}}{189 d e^4 (e (c+d x))^{3/2}}-\frac{2 \left (a+b \sin ^{-1}(c+d x)\right )}{9 d e (e (c+d x))^{9/2}}+\frac{20 b F\left (\left .\sin ^{-1}\left (\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right )\right |-1\right )}{189 d e^{11/2}}\\ \end{align*}
Mathematica [C] time = 0.0446952, size = 66, normalized size = 0.47 \[ -\frac{2 \sqrt{e (c+d x)} \left (2 b (c+d x) \text{Hypergeometric2F1}\left (-\frac{7}{4},\frac{1}{2},-\frac{3}{4},(c+d x)^2\right )+7 \left (a+b \sin ^{-1}(c+d x)\right )\right )}{63 d e^6 (c+d x)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 203, normalized size = 1.5 \begin{align*} 2\,{\frac{1}{de} \left ( -1/9\,{\frac{a}{ \left ( dex+ce \right ) ^{9/2}}}+b \left ( -1/9\,{\frac{1}{ \left ( dex+ce \right ) ^{9/2}}\arcsin \left ({\frac{dex+ce}{e}} \right ) }+2/9\,{\frac{1}{e} \left ( -1/7\,{\frac{1}{ \left ( dex+ce \right ) ^{7/2}}\sqrt{-{\frac{ \left ( dex+ce \right ) ^{2}}{{e}^{2}}}+1}}-{\frac{5}{21\,{e}^{2} \left ( dex+ce \right ) ^{3/2}}\sqrt{-{\frac{ \left ( dex+ce \right ) ^{2}}{{e}^{2}}}+1}}+{\frac{5\,{\it EllipticF} \left ( \sqrt{dex+ce}\sqrt{{e}^{-1}},i \right ) }{21\,{e}^{4}\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 ) \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{\sqrt{d e x + c e}{\left (b \arcsin \left (d x + c\right ) + a\right )}}{d^{6} e^{6} x^{6} + 6 \, c d^{5} e^{6} x^{5} + 15 \, c^{2} d^{4} e^{6} x^{4} + 20 \, c^{3} d^{3} e^{6} x^{3} + 15 \, c^{4} d^{2} e^{6} x^{2} + 6 \, c^{5} d e^{6} x + c^{6} e^{6}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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