Optimal. Leaf size=109 \[ \frac {16 b^2 c^2 (d x)^{9/2} \, _3F_2\left (1,\frac {9}{4},\frac {9}{4};\frac {11}{4},\frac {13}{4};c^2 x^2\right )}{315 d^3}+\frac {8 b c (d x)^{7/2} \, _2F_1\left (\frac {1}{2},\frac {7}{4};\frac {11}{4};c^2 x^2\right ) \left (a+b \cos ^{-1}(c x)\right )}{35 d^2}+\frac {2 (d x)^{5/2} \left (a+b \cos ^{-1}(c x)\right )^2}{5 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 109, 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} \[ \frac {16 b^2 c^2 (d x)^{9/2} \, _3F_2\left (1,\frac {9}{4},\frac {9}{4};\frac {11}{4},\frac {13}{4};c^2 x^2\right )}{315 d^3}+\frac {8 b c (d x)^{7/2} \, _2F_1\left (\frac {1}{2},\frac {7}{4};\frac {11}{4};c^2 x^2\right ) \left (a+b \cos ^{-1}(c x)\right )}{35 d^2}+\frac {2 (d x)^{5/2} \left (a+b \cos ^{-1}(c x)\right )^2}{5 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4628
Rule 4712
Rubi steps
\begin {align*} \int (d x)^{3/2} \left (a+b \cos ^{-1}(c x)\right )^2 \, dx &=\frac {2 (d x)^{5/2} \left (a+b \cos ^{-1}(c x)\right )^2}{5 d}+\frac {(4 b c) \int \frac {(d x)^{5/2} \left (a+b \cos ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{5 d}\\ &=\frac {2 (d x)^{5/2} \left (a+b \cos ^{-1}(c x)\right )^2}{5 d}+\frac {8 b c (d x)^{7/2} \left (a+b \cos ^{-1}(c x)\right ) \, _2F_1\left (\frac {1}{2},\frac {7}{4};\frac {11}{4};c^2 x^2\right )}{35 d^2}+\frac {16 b^2 c^2 (d x)^{9/2} \, _3F_2\left (1,\frac {9}{4},\frac {9}{4};\frac {11}{4},\frac {13}{4};c^2 x^2\right )}{315 d^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 8.62, size = 176, normalized size = 1.61 \[ \frac {(d x)^{3/2} \left (\frac {525 \sqrt {2} \pi b^2 c^2 x^3 \, _3F_2\left (1,\frac {9}{4},\frac {9}{4};\frac {11}{4},\frac {13}{4};c^2 x^2\right )}{\Gamma \left (\frac {11}{4}\right ) \Gamma \left (\frac {13}{4}\right )}+4480 a^2 x+\frac {128 b \left (28 a \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};c^2 x^2\right )-28 a \sqrt {1-c^2 x^2}+70 a c x \cos ^{-1}(c x)+20 b c^2 x^2 \sqrt {1-c^2 x^2} \, _2F_1\left (1,\frac {9}{4};\frac {11}{4};c^2 x^2\right ) \cos ^{-1}(c x)+35 b c x \cos ^{-1}(c x)^2\right )}{c}\right )}{11200} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} d x \arccos \left (c x\right )^{2} + 2 \, a b d x \arccos \left (c x\right ) + a^{2} d x\right )} \sqrt {d x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int \left (d x \right )^{\frac {3}{2}} \left (a +b \arccos \left (c x \right )\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2}{5} \, b^{2} d^{\frac {3}{2}} x^{\frac {5}{2}} \arctan \left (\sqrt {c x + 1} \sqrt {-c x + 1}, c x\right )^{2} + \frac {1}{10} \, a^{2} c^{2} d^{\frac {3}{2}} {\left (\frac {4 \, {\left (c^{2} x^{\frac {5}{2}} + 5 \, \sqrt {x}\right )}}{c^{4}} - \frac {10 \, \arctan \left (\sqrt {c} \sqrt {x}\right )}{c^{\frac {9}{2}}} + \frac {5 \, \log \left (\frac {c \sqrt {x} - \sqrt {c}}{c \sqrt {x} + \sqrt {c}}\right )}{c^{\frac {9}{2}}}\right )} + 10 \, a b c^{2} d^{\frac {3}{2}} \int \frac {x^{\frac {7}{2}} \arctan \left (\frac {\sqrt {c x + 1} \sqrt {-c x + 1}}{c x}\right )}{5 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x} - 4 \, b^{2} c d^{\frac {3}{2}} \int \frac {\sqrt {c x + 1} \sqrt {-c x + 1} x^{\frac {5}{2}} \arctan \left (\frac {\sqrt {c x + 1} \sqrt {-c x + 1}}{c x}\right )}{5 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x} - \frac {1}{2} \, a^{2} d^{\frac {3}{2}} {\left (\frac {4 \, \sqrt {x}}{c^{2}} - \frac {2 \, \arctan \left (\sqrt {c} \sqrt {x}\right )}{c^{\frac {5}{2}}} + \frac {\log \left (\frac {c \sqrt {x} - \sqrt {c}}{c \sqrt {x} + \sqrt {c}}\right )}{c^{\frac {5}{2}}}\right )} - 10 \, a b d^{\frac {3}{2}} \int \frac {x^{\frac {3}{2}} \arctan \left (\frac {\sqrt {c x + 1} \sqrt {-c x + 1}}{c x}\right )}{5 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2\,{\left (d\,x\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d x\right )^{\frac {3}{2}} \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________