Optimal. Leaf size=180 \[ \frac {8 \sqrt {\pi } \sin \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{3 b^{5/2} c^2}-\frac {8 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{3 b^{5/2} c^2}-\frac {4}{3 b^2 c^2 \sqrt {a+b \sin ^{-1}(c x)}}+\frac {8 x^2}{3 b^2 \sqrt {a+b \sin ^{-1}(c x)}}-\frac {2 x \sqrt {1-c^2 x^2}}{3 b c \left (a+b \sin ^{-1}(c x)\right )^{3/2}} \]
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Rubi [A] time = 0.51, antiderivative size = 180, 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 = {4633, 4719, 4635, 4406, 12, 3306, 3305, 3351, 3304, 3352, 4641} \[ \frac {8 \sqrt {\pi } \sin \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {\pi } \sqrt {b}}\right )}{3 b^{5/2} c^2}-\frac {8 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{3 b^{5/2} c^2}-\frac {4}{3 b^2 c^2 \sqrt {a+b \sin ^{-1}(c x)}}+\frac {8 x^2}{3 b^2 \sqrt {a+b \sin ^{-1}(c x)}}-\frac {2 x \sqrt {1-c^2 x^2}}{3 b c \left (a+b \sin ^{-1}(c x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3304
Rule 3305
Rule 3306
Rule 3351
Rule 3352
Rule 4406
Rule 4633
Rule 4635
Rule 4641
Rule 4719
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b \sin ^{-1}(c x)\right )^{5/2}} \, dx &=-\frac {2 x \sqrt {1-c^2 x^2}}{3 b c \left (a+b \sin ^{-1}(c x)\right )^{3/2}}+\frac {2 \int \frac {1}{\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}} \, dx}{3 b c}-\frac {(4 c) \int \frac {x^2}{\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^{3/2}} \, dx}{3 b}\\ &=-\frac {2 x \sqrt {1-c^2 x^2}}{3 b c \left (a+b \sin ^{-1}(c x)\right )^{3/2}}-\frac {4}{3 b^2 c^2 \sqrt {a+b \sin ^{-1}(c x)}}+\frac {8 x^2}{3 b^2 \sqrt {a+b \sin ^{-1}(c x)}}-\frac {16 \int \frac {x}{\sqrt {a+b \sin ^{-1}(c x)}} \, dx}{3 b^2}\\ &=-\frac {2 x \sqrt {1-c^2 x^2}}{3 b c \left (a+b \sin ^{-1}(c x)\right )^{3/2}}-\frac {4}{3 b^2 c^2 \sqrt {a+b \sin ^{-1}(c x)}}+\frac {8 x^2}{3 b^2 \sqrt {a+b \sin ^{-1}(c x)}}-\frac {16 \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{3 b^2 c^2}\\ &=-\frac {2 x \sqrt {1-c^2 x^2}}{3 b c \left (a+b \sin ^{-1}(c x)\right )^{3/2}}-\frac {4}{3 b^2 c^2 \sqrt {a+b \sin ^{-1}(c x)}}+\frac {8 x^2}{3 b^2 \sqrt {a+b \sin ^{-1}(c x)}}-\frac {16 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{3 b^2 c^2}\\ &=-\frac {2 x \sqrt {1-c^2 x^2}}{3 b c \left (a+b \sin ^{-1}(c x)\right )^{3/2}}-\frac {4}{3 b^2 c^2 \sqrt {a+b \sin ^{-1}(c x)}}+\frac {8 x^2}{3 b^2 \sqrt {a+b \sin ^{-1}(c x)}}-\frac {8 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{3 b^2 c^2}\\ &=-\frac {2 x \sqrt {1-c^2 x^2}}{3 b c \left (a+b \sin ^{-1}(c x)\right )^{3/2}}-\frac {4}{3 b^2 c^2 \sqrt {a+b \sin ^{-1}(c x)}}+\frac {8 x^2}{3 b^2 \sqrt {a+b \sin ^{-1}(c x)}}-\frac {\left (8 \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,\sin ^{-1}(c x)\right )}{3 b^2 c^2}+\frac {\left (8 \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,\sin ^{-1}(c x)\right )}{3 b^2 c^2}\\ &=-\frac {2 x \sqrt {1-c^2 x^2}}{3 b c \left (a+b \sin ^{-1}(c x)\right )^{3/2}}-\frac {4}{3 b^2 c^2 \sqrt {a+b \sin ^{-1}(c x)}}+\frac {8 x^2}{3 b^2 \sqrt {a+b \sin ^{-1}(c x)}}-\frac {\left (16 \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 \sin ^{-1}(c x)}\right )}{3 b^3 c^2}+\frac {\left (16 \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 \sin ^{-1}(c x)}\right )}{3 b^3 c^2}\\ &=-\frac {2 x \sqrt {1-c^2 x^2}}{3 b c \left (a+b \sin ^{-1}(c x)\right )^{3/2}}-\frac {4}{3 b^2 c^2 \sqrt {a+b \sin ^{-1}(c x)}}+\frac {8 x^2}{3 b^2 \sqrt {a+b \sin ^{-1}(c x)}}-\frac {8 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{3 b^{5/2} c^2}+\frac {8 \sqrt {\pi } C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{3 b^{5/2} c^2}\\ \end {align*}
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Mathematica [C] time = 1.36, size = 173, normalized size = 0.96 \[ -\frac {b \sin \left (2 \sin ^{-1}(c x)\right )+2 \left (a+b \sin ^{-1}(c x)\right ) \left (-\sqrt {2} e^{-\frac {2 i a}{b}} \sqrt {-\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {1}{2},-\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )-\sqrt {2} e^{\frac {2 i a}{b}} \sqrt {\frac {i \left (a+b \sin ^{-1}(c x)\right )}{b}} \Gamma \left (\frac {1}{2},\frac {2 i \left (a+b \sin ^{-1}(c x)\right )}{b}\right )+e^{-2 i \sin ^{-1}(c x)}+e^{2 i \sin ^{-1}(c x)}\right )}{3 b^2 c^2 \left (a+b \sin ^{-1}(c x)\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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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 [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 311, normalized size = 1.73 \[ -\frac {8 \arcsin \left (c x \right ) \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {2 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) b -8 \arcsin \left (c x \right ) \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {2 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) b +8 \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {2 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) a -8 \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {2 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) a +4 \arcsin \left (c x \right ) \cos \left (\frac {2 a +2 b \arcsin \left (c x \right )}{b}-\frac {2 a}{b}\right ) b +\sin \left (\frac {2 a +2 b \arcsin \left (c x \right )}{b}-\frac {2 a}{b}\right ) b +4 \cos \left (\frac {2 a +2 b \arcsin \left (c x \right )}{b}-\frac {2 a}{b}\right ) a}{3 c^{2} b^{2} \left (a +b \arcsin \left (c x \right )\right )^{\frac {3}{2}}} \]
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 \[ \int \frac {x}{{\left (b \arcsin \left (c x\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {x}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{5/2}} \,d x \]
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 \[ \int \frac {x}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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