Integrand size = 29, antiderivative size = 412 \[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {4 a b x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \left (1-c^2 x^2\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {4 b^2 x \sqrt {1-c^2 x^2} \arcsin (c x)}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {2 b x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^4 d^2}+\frac {4 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}} \]
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Time = 0.30 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {4791, 4767, 4715, 267, 4795, 4749, 4266, 2317, 2438} \[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {4 i b \sqrt {1-c^2 x^2} \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^4 d^2}+\frac {2 b x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {4 a b x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {4 b^2 x \sqrt {1-c^2 x^2} \arcsin (c x)}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \left (1-c^2 x^2\right )}{c^4 d \sqrt {d-c^2 d x^2}} \]
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Rule 267
Rule 2317
Rule 2438
Rule 4266
Rule 4715
Rule 4749
Rule 4767
Rule 4791
Rule 4795
Rubi steps \begin{align*} \text {integral}& = \frac {x^2 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \int \frac {x (a+b \arcsin (c x))^2}{\sqrt {d-c^2 d x^2}} \, dx}{c^2 d}-\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {x^2 (a+b \arcsin (c x))}{1-c^2 x^2} \, dx}{c d \sqrt {d-c^2 d x^2}} \\ & = \frac {2 b x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^4 d^2}-\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \arcsin (c x)}{1-c^2 x^2} \, dx}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (4 b \sqrt {1-c^2 x^2}\right ) \int (a+b \arcsin (c x)) \, dx}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{c^2 d \sqrt {d-c^2 d x^2}} \\ & = -\frac {4 a b x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \left (1-c^2 x^2\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {2 b x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^4 d^2}-\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \text {Subst}(\int (a+b x) \sec (x) \, dx,x,\arcsin (c x))}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {\left (4 b^2 \sqrt {1-c^2 x^2}\right ) \int \arcsin (c x) \, dx}{c^3 d \sqrt {d-c^2 d x^2}} \\ & = -\frac {4 a b x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \left (1-c^2 x^2\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {4 b^2 x \sqrt {1-c^2 x^2} \arcsin (c x)}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {2 b x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^4 d^2}+\frac {4 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {\left (4 b^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{c^2 d \sqrt {d-c^2 d x^2}} \\ & = -\frac {4 a b x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \left (1-c^2 x^2\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {4 b^2 x \sqrt {1-c^2 x^2} \arcsin (c x)}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {2 b x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^4 d^2}+\frac {4 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {\left (2 i b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {\left (2 i b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}} \\ & = -\frac {4 a b x \sqrt {1-c^2 x^2}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \left (1-c^2 x^2\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {4 b^2 x \sqrt {1-c^2 x^2} \arcsin (c x)}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {2 b x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{c^4 d^2}+\frac {4 i b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \arctan \left (e^{i \arcsin (c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {2 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}}+\frac {2 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{c^4 d \sqrt {d-c^2 d x^2}} \\ \end{align*}
Time = 0.69 (sec) , antiderivative size = 369, normalized size of antiderivative = 0.90 \[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {4 a^2-2 b^2-2 a^2 c^2 x^2+6 a b \arcsin (c x)+3 b^2 \arcsin (c x)^2-2 b^2 \cos (2 \arcsin (c x))+2 a b \arcsin (c x) \cos (2 \arcsin (c x))+b^2 \arcsin (c x)^2 \cos (2 \arcsin (c x))-4 b^2 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )+4 b^2 \sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )+4 a b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-4 a b \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-4 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+4 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )-2 a b \sin (2 \arcsin (c x))-2 b^2 \arcsin (c x) \sin (2 \arcsin (c x))}{2 c^4 d \sqrt {d-c^2 d x^2}} \]
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Time = 0.40 (sec) , antiderivative size = 668, normalized size of antiderivative = 1.62
method | result | size |
default | \(a^{2} \left (-\frac {x^{2}}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2}{d \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}\right )+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (\arcsin \left (c x \right )^{2}-2+2 i \arcsin \left (c x \right )\right )}{2 c^{4} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )^{2}-2-2 i \arcsin \left (c x \right )\right )}{2 c^{4} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )^{2}}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}\right )+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, x}{c^{3} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x^{2}}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {4 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}\) | \(668\) |
parts | \(a^{2} \left (-\frac {x^{2}}{c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {2}{d \,c^{4} \sqrt {-c^{2} d \,x^{2}+d}}\right )+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i c x \sqrt {-c^{2} x^{2}+1}-1\right ) \left (\arcsin \left (c x \right )^{2}-2+2 i \arcsin \left (c x \right )\right )}{2 c^{4} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i c x \sqrt {-c^{2} x^{2}+1}+c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )^{2}-2-2 i \arcsin \left (c x \right )\right )}{2 c^{4} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )^{2}}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-\arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )\right )}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}\right )+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, x}{c^{3} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x^{2}}{c^{2} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {4 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{c^{4} d^{2} \left (c^{2} x^{2}-1\right )}\) | \(668\) |
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\[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{3}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^{3} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{3}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^3 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
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