Optimal. Leaf size=159 \[ -\frac{2 \left (a+b \sin ^{-1}(c+d x)\right )}{7 d e (e (c+d x))^{7/2}}-\frac{12 b \sqrt{1-(c+d x)^2}}{35 d e^4 \sqrt{e (c+d x)}}-\frac{4 b \sqrt{1-(c+d x)^2}}{35 d e^2 (e (c+d x))^{5/2}}+\frac{12 b \sqrt{e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{-c-d x+1}}{\sqrt{2}}\right )\right |2\right )}{35 d e^5 \sqrt{c+d x}} \]
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Rubi [A] time = 0.127513, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {4805, 4627, 325, 320, 318, 424} \[ -\frac{2 \left (a+b \sin ^{-1}(c+d x)\right )}{7 d e (e (c+d x))^{7/2}}-\frac{12 b \sqrt{1-(c+d x)^2}}{35 d e^4 \sqrt{e (c+d x)}}-\frac{4 b \sqrt{1-(c+d x)^2}}{35 d e^2 (e (c+d x))^{5/2}}+\frac{12 b \sqrt{e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{-c-d x+1}}{\sqrt{2}}\right )\right |2\right )}{35 d e^5 \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Rule 4805
Rule 4627
Rule 325
Rule 320
Rule 318
Rule 424
Rubi steps
\begin{align*} \int \frac{a+b \sin ^{-1}(c+d x)}{(c e+d e x)^{9/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a+b \sin ^{-1}(x)}{(e x)^{9/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac{2 \left (a+b \sin ^{-1}(c+d x)\right )}{7 d e (e (c+d x))^{7/2}}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{(e x)^{7/2} \sqrt{1-x^2}} \, dx,x,c+d x\right )}{7 d e}\\ &=-\frac{4 b \sqrt{1-(c+d x)^2}}{35 d e^2 (e (c+d x))^{5/2}}-\frac{2 \left (a+b \sin ^{-1}(c+d x)\right )}{7 d e (e (c+d x))^{7/2}}+\frac{(6 b) \operatorname{Subst}\left (\int \frac{1}{(e x)^{3/2} \sqrt{1-x^2}} \, dx,x,c+d x\right )}{35 d e^3}\\ &=-\frac{4 b \sqrt{1-(c+d x)^2}}{35 d e^2 (e (c+d x))^{5/2}}-\frac{12 b \sqrt{1-(c+d x)^2}}{35 d e^4 \sqrt{e (c+d x)}}-\frac{2 \left (a+b \sin ^{-1}(c+d x)\right )}{7 d e (e (c+d x))^{7/2}}-\frac{(6 b) \operatorname{Subst}\left (\int \frac{\sqrt{e x}}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{35 d e^5}\\ &=-\frac{4 b \sqrt{1-(c+d x)^2}}{35 d e^2 (e (c+d x))^{5/2}}-\frac{12 b \sqrt{1-(c+d x)^2}}{35 d e^4 \sqrt{e (c+d x)}}-\frac{2 \left (a+b \sin ^{-1}(c+d x)\right )}{7 d e (e (c+d x))^{7/2}}-\frac{\left (6 b \sqrt{e (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{35 d e^5 \sqrt{c+d x}}\\ &=-\frac{4 b \sqrt{1-(c+d x)^2}}{35 d e^2 (e (c+d x))^{5/2}}-\frac{12 b \sqrt{1-(c+d x)^2}}{35 d e^4 \sqrt{e (c+d x)}}-\frac{2 \left (a+b \sin ^{-1}(c+d x)\right )}{7 d e (e (c+d x))^{7/2}}+\frac{\left (12 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 )}{35 d e^5 \sqrt{c+d x}}\\ &=-\frac{4 b \sqrt{1-(c+d x)^2}}{35 d e^2 (e (c+d x))^{5/2}}-\frac{12 b \sqrt{1-(c+d x)^2}}{35 d e^4 \sqrt{e (c+d x)}}-\frac{2 \left (a+b \sin ^{-1}(c+d x)\right )}{7 d e (e (c+d x))^{7/2}}+\frac{12 b \sqrt{e (c+d x)} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{1-c-d x}}{\sqrt{2}}\right )\right |2\right )}{35 d e^5 \sqrt{c+d x}}\\ \end{align*}
Mathematica [C] time = 0.0436403, size = 66, normalized size = 0.42 \[ -\frac{2 \sqrt{e (c+d x)} \left (2 b (c+d x) \text{Hypergeometric2F1}\left (-\frac{5}{4},\frac{1}{2},-\frac{1}{4},(c+d x)^2\right )+5 \left (a+b \sin ^{-1}(c+d x)\right )\right )}{35 d e^5 (c+d x)^4} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.015, size = 225, normalized size = 1.4 \begin{align*} 2\,{\frac{1}{de} \left ( -1/7\,{\frac{a}{ \left ( dex+ce \right ) ^{7/2}}}+b \left ( -1/7\,{\frac{1}{ \left ( dex+ce \right ) ^{7/2}}\arcsin \left ({\frac{dex+ce}{e}} \right ) }+2/7\,{\frac{1}{e} \left ( -1/5\,{\frac{1}{ \left ( dex+ce \right ) ^{5/2}}\sqrt{-{\frac{ \left ( dex+ce \right ) ^{2}}{{e}^{2}}}+1}}-3/5\,{\frac{1}{{e}^{2}\sqrt{dex+ce}}\sqrt{-{\frac{ \left ( dex+ce \right ) ^{2}}{{e}^{2}}}+1}}+3/5\,{\frac{{\it EllipticF} \left ( \sqrt{dex+ce}\sqrt{{e}^{-1}},i \right ) -{\it EllipticE} \left ( \sqrt{dex+ce}\sqrt{{e}^{-1}},i \right ) }{{e}^{3}\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^{5} e^{5} x^{5} + 5 \, c d^{4} e^{5} x^{4} + 10 \, c^{2} d^{3} e^{5} x^{3} + 10 \, c^{3} d^{2} e^{5} x^{2} + 5 \, c^{4} d e^{5} x + c^{5} e^{5}}, 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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