Optimal. Leaf size=172 \[ -\frac {3 \sqrt {\pi } b^{3/2} \sin \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{32 c^2}+\frac {3 \sqrt {\pi } b^{3/2} \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{32 c^2}-\frac {3 b x \sqrt {1-c^2 x^2} \sqrt {a+b \cos ^{-1}(c x)}}{8 c}-\frac {\left (a+b \cos ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^{3/2} \]
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Rubi [A] time = 0.46, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {4630, 4708, 4642, 4636, 4406, 12, 3306, 3305, 3351, 3304, 3352} \[ -\frac {3 \sqrt {\pi } b^{3/2} \sin \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {\pi } \sqrt {b}}\right )}{32 c^2}+\frac {3 \sqrt {\pi } b^{3/2} \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{32 c^2}-\frac {3 b x \sqrt {1-c^2 x^2} \sqrt {a+b \cos ^{-1}(c x)}}{8 c}-\frac {\left (a+b \cos ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3304
Rule 3305
Rule 3306
Rule 3351
Rule 3352
Rule 4406
Rule 4630
Rule 4636
Rule 4642
Rule 4708
Rubi steps
\begin {align*} \int x \left (a+b \cos ^{-1}(c x)\right )^{3/2} \, dx &=\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^{3/2}+\frac {1}{4} (3 b c) \int \frac {x^2 \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {3 b x \sqrt {1-c^2 x^2} \sqrt {a+b \cos ^{-1}(c x)}}{8 c}+\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^{3/2}-\frac {1}{16} \left (3 b^2\right ) \int \frac {x}{\sqrt {a+b \cos ^{-1}(c x)}} \, dx+\frac {(3 b) \int \frac {\sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {1-c^2 x^2}} \, dx}{8 c}\\ &=-\frac {3 b x \sqrt {1-c^2 x^2} \sqrt {a+b \cos ^{-1}(c x)}}{8 c}-\frac {\left (a+b \cos ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^{3/2}+\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{16 c^2}\\ &=-\frac {3 b x \sqrt {1-c^2 x^2} \sqrt {a+b \cos ^{-1}(c x)}}{8 c}-\frac {\left (a+b \cos ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^{3/2}+\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{16 c^2}\\ &=-\frac {3 b x \sqrt {1-c^2 x^2} \sqrt {a+b \cos ^{-1}(c x)}}{8 c}-\frac {\left (a+b \cos ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^{3/2}+\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{32 c^2}\\ &=-\frac {3 b x \sqrt {1-c^2 x^2} \sqrt {a+b \cos ^{-1}(c x)}}{8 c}-\frac {\left (a+b \cos ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^{3/2}+\frac {\left (3 b^2 \cos \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{32 c^2}-\frac {\left (3 b^2 \sin \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{32 c^2}\\ &=-\frac {3 b x \sqrt {1-c^2 x^2} \sqrt {a+b \cos ^{-1}(c x)}}{8 c}-\frac {\left (a+b \cos ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^{3/2}+\frac {\left (3 b \cos \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \cos ^{-1}(c x)}\right )}{16 c^2}-\frac {\left (3 b \sin \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \cos ^{-1}(c x)}\right )}{16 c^2}\\ &=-\frac {3 b x \sqrt {1-c^2 x^2} \sqrt {a+b \cos ^{-1}(c x)}}{8 c}-\frac {\left (a+b \cos ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^{3/2}+\frac {3 b^{3/2} \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{32 c^2}-\frac {3 b^{3/2} \sqrt {\pi } C\left (\frac {2 \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{32 c^2}\\ \end {align*}
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Mathematica [A] time = 0.95, size = 155, normalized size = 0.90 \[ \frac {-3 \sqrt {\pi } b \sin \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {\pi }}\right )+3 \sqrt {\pi } b \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {\pi }}\right )+2 \sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}(c x)} \left (4 a \cos \left (2 \cos ^{-1}(c x)\right )+4 b \cos ^{-1}(c x) \cos \left (2 \cos ^{-1}(c x)\right )-3 b \sin \left (2 \cos ^{-1}(c x)\right )\right )}{32 \sqrt {\frac {1}{b}} c^2} \]
Antiderivative was successfully verified.
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fricas [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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giac [B] time = 5.42, size = 884, normalized size = 5.14 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 267, normalized size = 1.55 \[ \frac {3 \sqrt {\pi }\, \sqrt {\frac {1}{b}}\, \sqrt {a +b \arccos \left (c x \right )}\, \cos \left (\frac {2 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) b^{2}-3 \sqrt {\pi }\, \sqrt {\frac {1}{b}}\, \sqrt {a +b \arccos \left (c x \right )}\, \sin \left (\frac {2 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) b^{2}+8 \arccos \left (c x \right )^{2} \cos \left (\frac {2 a +2 b \arccos \left (c x \right )}{b}-\frac {2 a}{b}\right ) b^{2}+16 \arccos \left (c x \right ) \cos \left (\frac {2 a +2 b \arccos \left (c x \right )}{b}-\frac {2 a}{b}\right ) a b -6 \arccos \left (c x \right ) \sin \left (\frac {2 a +2 b \arccos \left (c x \right )}{b}-\frac {2 a}{b}\right ) b^{2}+8 \cos \left (\frac {2 a +2 b \arccos \left (c x \right )}{b}-\frac {2 a}{b}\right ) a^{2}-6 \sin \left (\frac {2 a +2 b \arccos \left (c x \right )}{b}-\frac {2 a}{b}\right ) a b}{32 c^{2} \sqrt {a +b \arccos \left (c x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \arccos \left (c x\right ) + a\right )}^{\frac {3}{2}} x\,{d x} \]
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 \[ \int x\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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