Optimal. Leaf size=105 \[ -\frac {16 b^2 c^2 (d x)^{3/2} \, _3F_2\left (\frac {3}{4},\frac {3}{4},1;\frac {5}{4},\frac {7}{4};c^2 x^2\right )}{3 d^3}-\frac {8 b c \sqrt {d x} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};c^2 x^2\right ) \left (a+b \cos ^{-1}(c x)\right )}{d^2}-\frac {2 \left (a+b \cos ^{-1}(c x)\right )^2}{d \sqrt {d x}} \]
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Rubi [A] time = 0.13, antiderivative size = 105, 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} \[ -\frac {16 b^2 c^2 (d x)^{3/2} \, _3F_2\left (\frac {3}{4},\frac {3}{4},1;\frac {5}{4},\frac {7}{4};c^2 x^2\right )}{3 d^3}-\frac {8 b c \sqrt {d x} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};c^2 x^2\right ) \left (a+b \cos ^{-1}(c x)\right )}{d^2}-\frac {2 \left (a+b \cos ^{-1}(c x)\right )^2}{d \sqrt {d x}} \]
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)^{3/2}} \, dx &=-\frac {2 \left (a+b \cos ^{-1}(c x)\right )^2}{d \sqrt {d x}}-\frac {(4 b c) \int \frac {a+b \cos ^{-1}(c x)}{\sqrt {d x} \sqrt {1-c^2 x^2}} \, dx}{d}\\ &=-\frac {2 \left (a+b \cos ^{-1}(c x)\right )^2}{d \sqrt {d x}}-\frac {8 b c \sqrt {d x} \left (a+b \cos ^{-1}(c x)\right ) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};c^2 x^2\right )}{d^2}-\frac {16 b^2 c^2 (d x)^{3/2} \, _3F_2\left (\frac {3}{4},\frac {3}{4},1;\frac {5}{4},\frac {7}{4};c^2 x^2\right )}{3 d^3}\\ \end {align*}
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Mathematica [A] time = 0.52, size = 129, normalized size = 1.23 \[ \frac {x \left (-\frac {\sqrt {2} \pi b^2 c^2 x^2 \, _3F_2\left (\frac {3}{4},\frac {3}{4},1;\frac {5}{4},\frac {7}{4};c^2 x^2\right )}{\Gamma \left (\frac {5}{4}\right ) \Gamma \left (\frac {7}{4}\right )}-2 \left (4 a b c x \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};c^2 x^2\right )+\left (a+b \cos ^{-1}(c x)\right )^2+2 b^2 \, _2F_1\left (\frac {3}{4},1;\frac {5}{4};c^2 x^2\right ) \cos ^{-1}(c x) \sin \left (2 \cos ^{-1}(c x)\right )\right )\right )}{(d x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} \arccos \left (c x\right )^{2} + 2 \, a b \arccos \left (c x\right ) + a^{2}\right )} \sqrt {d x}}{d^{2} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{\left (d x\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.36, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arccos \left (c x \right )\right )^{2}}{\left (d x \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2}{{\left (d\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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