Optimal. Leaf size=107 \[ \frac{16 b^2 c^2 (d x)^{5/2} \text{HypergeometricPFQ}\left (\left \{1,\frac{5}{4},\frac{5}{4}\right \},\left \{\frac{7}{4},\frac{9}{4}\right \},c^2 x^2\right )}{15 d^3}+\frac{8 b c (d x)^{3/2} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},c^2 x^2\right ) \left (a+b \cos ^{-1}(c x)\right )}{3 d^2}+\frac{2 \sqrt{d x} \left (a+b \cos ^{-1}(c x)\right )^2}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.124588, antiderivative size = 107, 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} \[ \frac{16 b^2 c^2 (d x)^{5/2} \, _3F_2\left (1,\frac{5}{4},\frac{5}{4};\frac{7}{4},\frac{9}{4};c^2 x^2\right )}{15 d^3}+\frac{8 b c (d x)^{3/2} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};c^2 x^2\right ) \left (a+b \cos ^{-1}(c x)\right )}{3 d^2}+\frac{2 \sqrt{d x} \left (a+b \cos ^{-1}(c x)\right )^2}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4628
Rule 4712
Rubi steps
\begin{align*} \int \frac{\left (a+b \cos ^{-1}(c x)\right )^2}{\sqrt{d x}} \, dx &=\frac{2 \sqrt{d x} \left (a+b \cos ^{-1}(c x)\right )^2}{d}+\frac{(4 b c) \int \frac{\sqrt{d x} \left (a+b \cos ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{d}\\ &=\frac{2 \sqrt{d x} \left (a+b \cos ^{-1}(c x)\right )^2}{d}+\frac{8 b c (d x)^{3/2} \left (a+b \cos ^{-1}(c x)\right ) \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};c^2 x^2\right )}{3 d^2}+\frac{16 b^2 c^2 (d x)^{5/2} \, _3F_2\left (1,\frac{5}{4},\frac{5}{4};\frac{7}{4},\frac{9}{4};c^2 x^2\right )}{15 d^3}\\ \end{align*}
Mathematica [A] time = 1.27198, size = 142, normalized size = 1.33 \[ \frac{3 \sqrt{2} \pi b^2 c^2 x^3 \text{HypergeometricPFQ}\left (\left \{1,\frac{5}{4},\frac{5}{4}\right \},\left \{\frac{7}{4},\frac{9}{4}\right \},c^2 x^2\right )+8 x \text{Gamma}\left (\frac{7}{4}\right ) \text{Gamma}\left (\frac{9}{4}\right ) \left (4 a b c x \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},c^2 x^2\right )+2 b^2 \cos ^{-1}(c x) \text{Hypergeometric2F1}\left (1,\frac{5}{4},\frac{7}{4},c^2 x^2\right ) \sin \left (2 \cos ^{-1}(c x)\right )+3 \left (a+b \cos ^{-1}(c x)\right )^2\right )}{12 \text{Gamma}\left (\frac{7}{4}\right ) \text{Gamma}\left (\frac{9}{4}\right ) \sqrt{d x}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.518, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\arccos \left ( cx \right ) \right ) ^{2}{\frac{1}{\sqrt{dx}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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 x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{\sqrt{d x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]