Optimal. Leaf size=109 \[ \frac {16 b^2 c^2 (d x)^{7/2} \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};c^2 x^2\right )}{105 d^3}+\frac {8 b c (d x)^{5/2} \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {9}{4};c^2 x^2\right ) \left (a+b \cos ^{-1}(c x)\right )}{15 d^2}+\frac {2 (d x)^{3/2} \left (a+b \cos ^{-1}(c x)\right )^2}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, 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} \[ \frac {16 b^2 c^2 (d x)^{7/2} \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};c^2 x^2\right )}{105 d^3}+\frac {8 b c (d x)^{5/2} \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {9}{4};c^2 x^2\right ) \left (a+b \cos ^{-1}(c x)\right )}{15 d^2}+\frac {2 (d x)^{3/2} \left (a+b \cos ^{-1}(c x)\right )^2}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4628
Rule 4712
Rubi steps
\begin {align*} \int \sqrt {d x} \left (a+b \cos ^{-1}(c x)\right )^2 \, dx &=\frac {2 (d x)^{3/2} \left (a+b \cos ^{-1}(c x)\right )^2}{3 d}+\frac {(4 b c) \int \frac {(d x)^{3/2} \left (a+b \cos ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{3 d}\\ &=\frac {2 (d x)^{3/2} \left (a+b \cos ^{-1}(c x)\right )^2}{3 d}+\frac {8 b c (d x)^{5/2} \left (a+b \cos ^{-1}(c x)\right ) \, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {9}{4};c^2 x^2\right )}{15 d^2}+\frac {16 b^2 c^2 (d x)^{7/2} \, _3F_2\left (1,\frac {7}{4},\frac {7}{4};\frac {9}{4},\frac {11}{4};c^2 x^2\right )}{105 d^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.62, size = 202, normalized size = 1.85 \[ \frac {1}{27} \sqrt {d x} \left (\frac {3 \sqrt {2} \pi b^2 x \, _3F_2\left (\frac {3}{4},\frac {3}{4},1;\frac {5}{4},\frac {7}{4};c^2 x^2\right )}{\Gamma \left (\frac {5}{4}\right ) \Gamma \left (\frac {7}{4}\right )}+\frac {2 \left (9 a^2 c x+12 a b \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};c^2 x^2\right )-12 a b \sqrt {1-c^2 x^2}+18 a b c x \cos ^{-1}(c x)+12 b^2 \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {3}{4},1;\frac {5}{4};c^2 x^2\right ) \cos ^{-1}(c x)-12 b^2 \sqrt {1-c^2 x^2} \cos ^{-1}(c x)-8 b^2 c x+9 b^2 c x \cos ^{-1}(c x)^2\right )}{c}\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} \arccos \left (c x\right )^{2} + 2 \, a b \arccos \left (c x\right ) + a^{2}\right )} \sqrt {d x}, x\right ) \]
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 = 0.35, size = 0, normalized size = 0.00 \[ \int \left (a +b \arccos \left (c x \right )\right )^{2} \sqrt {d x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2}{3} \, b^{2} \sqrt {d} x^{\frac {3}{2}} \arctan \left (\sqrt {c x + 1} \sqrt {-c x + 1}, c x\right )^{2} + \frac {1}{6} \, a^{2} c^{2} \sqrt {d} {\left (\frac {4 \, x^{\frac {3}{2}}}{c^{2}} + \frac {6 \, \arctan \left (\sqrt {c} \sqrt {x}\right )}{c^{\frac {7}{2}}} + \frac {3 \, \log \left (\frac {c \sqrt {x} - \sqrt {c}}{c \sqrt {x} + \sqrt {c}}\right )}{c^{\frac {7}{2}}}\right )} + 6 \, a b c^{2} \sqrt {d} \int \frac {x^{\frac {5}{2}} \arctan \left (\frac {\sqrt {c x + 1} \sqrt {-c x + 1}}{c x}\right )}{3 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x} - 4 \, b^{2} c \sqrt {d} \int \frac {\sqrt {c x + 1} \sqrt {-c x + 1} x^{\frac {3}{2}} \arctan \left (\frac {\sqrt {c x + 1} \sqrt {-c x + 1}}{c x}\right )}{3 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x} - \frac {1}{2} \, a^{2} \sqrt {d} {\left (\frac {2 \, \arctan \left (\sqrt {c} \sqrt {x}\right )}{c^{\frac {3}{2}}} + \frac {\log \left (\frac {c \sqrt {x} - \sqrt {c}}{c \sqrt {x} + \sqrt {c}}\right )}{c^{\frac {3}{2}}}\right )} - 6 \, a b \sqrt {d} \int \frac {\sqrt {x} \arctan \left (\frac {\sqrt {c x + 1} \sqrt {-c x + 1}}{c x}\right )}{3 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2\,\sqrt {d\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {d x} \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________