Optimal. Leaf size=130 \[ -\frac{16 b^2 \text{HypergeometricPFQ}\left (\left \{-\frac{3}{4},-\frac{3}{4},1\right \},\left \{-\frac{1}{4},\frac{1}{4}\right \},(c+d x)^2\right )}{105 d e^3 (e (c+d x))^{3/2}}-\frac{8 b \text{Hypergeometric2F1}\left (-\frac{5}{4},\frac{1}{2},-\frac{1}{4},(c+d x)^2\right ) \left (a+b \sin ^{-1}(c+d x)\right )}{35 d e^2 (e (c+d x))^{5/2}}-\frac{2 \left (a+b \sin ^{-1}(c+d x)\right )^2}{7 d e (e (c+d x))^{7/2}} \]
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Rubi [A] time = 0.215817, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {4805, 4627, 4711} \[ -\frac{16 b^2 \, _3F_2\left (-\frac{3}{4},-\frac{3}{4},1;-\frac{1}{4},\frac{1}{4};(c+d x)^2\right )}{105 d e^3 (e (c+d x))^{3/2}}-\frac{8 b \, _2F_1\left (-\frac{5}{4},\frac{1}{2};-\frac{1}{4};(c+d x)^2\right ) \left (a+b \sin ^{-1}(c+d x)\right )}{35 d e^2 (e (c+d x))^{5/2}}-\frac{2 \left (a+b \sin ^{-1}(c+d x)\right )^2}{7 d e (e (c+d x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 4805
Rule 4627
Rule 4711
Rubi steps
\begin{align*} \int \frac{\left (a+b \sin ^{-1}(c+d x)\right )^2}{(c e+d e x)^{9/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \sin ^{-1}(x)\right )^2}{(e x)^{9/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac{2 \left (a+b \sin ^{-1}(c+d x)\right )^2}{7 d e (e (c+d x))^{7/2}}+\frac{(4 b) \operatorname{Subst}\left (\int \frac{a+b \sin ^{-1}(x)}{(e x)^{7/2} \sqrt{1-x^2}} \, dx,x,c+d x\right )}{7 d e}\\ &=-\frac{2 \left (a+b \sin ^{-1}(c+d x)\right )^2}{7 d e (e (c+d x))^{7/2}}-\frac{8 b \left (a+b \sin ^{-1}(c+d x)\right ) \, _2F_1\left (-\frac{5}{4},\frac{1}{2};-\frac{1}{4};(c+d x)^2\right )}{35 d e^2 (e (c+d x))^{5/2}}-\frac{16 b^2 \, _3F_2\left (-\frac{3}{4},-\frac{3}{4},1;-\frac{1}{4},\frac{1}{4};(c+d x)^2\right )}{105 d e^3 (e (c+d x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0882999, size = 114, normalized size = 0.88 \[ -\frac{2 \sqrt{e (c+d x)} \left (8 b^2 (c+d x)^2 \text{HypergeometricPFQ}\left (\left \{-\frac{3}{4},-\frac{3}{4},1\right \},\left \{-\frac{1}{4},\frac{1}{4}\right \},(c+d x)^2\right )+3 \left (a+b \sin ^{-1}(c+d x)\right ) \left (4 b (c+d x) \text{Hypergeometric2F1}\left (-\frac{5}{4},\frac{1}{2},-\frac{1}{4},(c+d x)^2\right )+5 \left (a+b \sin ^{-1}(c+d x)\right )\right )\right )}{105 d e^5 (c+d x)^4} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.299, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\arcsin \left ( dx+c \right ) \right ) ^{2} \left ( dex+ce \right ) ^{-{\frac{9}{2}}}}\, 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 (\frac{{\left (b^{2} \arcsin \left (d x + c\right )^{2} + 2 \, a b \arcsin \left (d x + c\right ) + a^{2}\right )} \sqrt{d e x + c e}}{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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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