Optimal. Leaf size=109 \[ \frac{16 b^2 c^2 \sqrt{d x} \text{HypergeometricPFQ}\left (\left \{\frac{1}{4},\frac{1}{4},1\right \},\left \{\frac{3}{4},\frac{5}{4}\right \},c^2 x^2\right )}{3 d^3}+\frac{8 b c \text{Hypergeometric2F1}\left (-\frac{1}{4},\frac{1}{2},\frac{3}{4},c^2 x^2\right ) \left (a+b \cos ^{-1}(c x)\right )}{3 d^2 \sqrt{d x}}-\frac{2 \left (a+b \cos ^{-1}(c x)\right )^2}{3 d (d x)^{3/2}} \]
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Rubi [A] time = 0.141738, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {4628, 4712} \[ \frac{16 b^2 c^2 \sqrt{d x} \, _3F_2\left (\frac{1}{4},\frac{1}{4},1;\frac{3}{4},\frac{5}{4};c^2 x^2\right )}{3 d^3}+\frac{8 b c \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};c^2 x^2\right ) \left (a+b \cos ^{-1}(c x)\right )}{3 d^2 \sqrt{d x}}-\frac{2 \left (a+b \cos ^{-1}(c x)\right )^2}{3 d (d x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4628
Rule 4712
Rubi steps
\begin{align*} \int \frac{\left (a+b \cos ^{-1}(c x)\right )^2}{(d x)^{5/2}} \, dx &=-\frac{2 \left (a+b \cos ^{-1}(c x)\right )^2}{3 d (d x)^{3/2}}-\frac{(4 b c) \int \frac{a+b \cos ^{-1}(c x)}{(d x)^{3/2} \sqrt{1-c^2 x^2}} \, dx}{3 d}\\ &=-\frac{2 \left (a+b \cos ^{-1}(c x)\right )^2}{3 d (d x)^{3/2}}+\frac{8 b c \left (a+b \cos ^{-1}(c x)\right ) \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};c^2 x^2\right )}{3 d^2 \sqrt{d x}}+\frac{16 b^2 c^2 \sqrt{d x} \, _3F_2\left (\frac{1}{4},\frac{1}{4},1;\frac{3}{4},\frac{5}{4};c^2 x^2\right )}{3 d^3}\\ \end{align*}
Mathematica [A] time = 0.786154, size = 198, normalized size = 1.82 \[ \frac{x \left (3 \sqrt{2} \pi b^2 c^4 x^4 \text{HypergeometricPFQ}\left (\left \{1,\frac{5}{4},\frac{5}{4}\right \},\left \{\frac{7}{4},\frac{9}{4}\right \},c^2 x^2\right )-8 \text{Gamma}\left (\frac{7}{4}\right ) \text{Gamma}\left (\frac{9}{4}\right ) \left (-12 a b c x \text{Hypergeometric2F1}\left (-\frac{1}{4},\frac{1}{2},\frac{3}{4},c^2 x^2\right )-4 b^2 c^3 x^3 \sqrt{1-c^2 x^2} \cos ^{-1}(c x) \text{Hypergeometric2F1}\left (1,\frac{5}{4},\frac{7}{4},c^2 x^2\right )+3 \left (a^2+2 b \cos ^{-1}(c x) \left (a-2 b c x \sqrt{1-c^2 x^2}\right )-8 b^2 c^2 x^2+b^2 \cos ^{-1}(c x)^2\right )\right )\right )}{36 \text{Gamma}\left (\frac{7}{4}\right ) \text{Gamma}\left (\frac{9}{4}\right ) (d x)^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.342, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\arccos \left ( cx \right ) \right ) ^{2} \left ( dx \right ) ^{-{\frac{5}{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} \arccos \left (c x\right )^{2} + 2 \, a b \arccos \left (c x\right ) + a^{2}\right )} \sqrt{d x}}{d^{3} x^{3}}, 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{{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{\left (d x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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