Optimal. Leaf size=107 \[ \frac {16 b^2 c^2 (d x)^{5/2} \, _3F_2\left (1,\frac {5}{4},\frac {5}{4};\frac {7}{4},\frac {9}{4};c^2 x^2\right )}{15 d^3}+\frac {8 b c (d x)^{3/2} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};c^2 x^2\right ) \left (a+b \cos ^{-1}(c x)\right )}{3 d^2}+\frac {2 \sqrt {d x} \left (a+b \cos ^{-1}(c x)\right )^2}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 107, normalized size of antiderivative = 1.00, 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 = 1.49, size = 142, normalized size = 1.33 \[ \frac {3 \sqrt {2} \pi b^2 c^2 x^3 \, _3F_2\left (1,\frac {5}{4},\frac {5}{4};\frac {7}{4},\frac {9}{4};c^2 x^2\right )+8 x \Gamma \left (\frac {7}{4}\right ) \Gamma \left (\frac {9}{4}\right ) \left (4 a b c x \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};c^2 x^2\right )+3 \left (a+b \cos ^{-1}(c x)\right )^2+2 b^2 \, _2F_1\left (1,\frac {5}{4};\frac {7}{4};c^2 x^2\right ) \cos ^{-1}(c x) \sin \left (2 \cos ^{-1}(c x)\right )\right )}{12 \Gamma \left (\frac {7}{4}\right ) \Gamma \left (\frac {9}{4}\right ) \sqrt {d x}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} \arccos \left (c x\right )^{2} + 2 \, a b \arccos \left (c x\right ) + a^{2}\right )} \sqrt {d x}}{d x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{\sqrt {d x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arccos \left (c x \right )\right )^{2}}{\sqrt {d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2}{\sqrt {d\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________