Optimal. Leaf size=153 \[ \frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {b c \left (2 c^2 d+3 e\right ) \tan ^{-1}\left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 \sqrt {d} e^2 \left (c^2 d+e\right )^{3/2}}-\frac {b c x \sqrt {1-c^2 x^2}}{8 e \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {b \sin ^{-1}(c x)}{4 d e^2} \]
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Rubi [A] time = 0.19, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {264, 4731, 12, 470, 523, 216, 377, 205} \[ \frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {b c \left (2 c^2 d+3 e\right ) \tan ^{-1}\left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 \sqrt {d} e^2 \left (c^2 d+e\right )^{3/2}}-\frac {b c x \sqrt {1-c^2 x^2}}{8 e \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {b \sin ^{-1}(c x)}{4 d e^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 216
Rule 264
Rule 377
Rule 470
Rule 523
Rule 4731
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx &=\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}-(b c) \int \frac {x^4}{4 d \sqrt {1-c^2 x^2} \left (d+e x^2\right )^2} \, dx\\ &=\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}-\frac {(b c) \int \frac {x^4}{\sqrt {1-c^2 x^2} \left (d+e x^2\right )^2} \, dx}{4 d}\\ &=-\frac {b c x \sqrt {1-c^2 x^2}}{8 e \left (c^2 d+e\right ) \left (d+e x^2\right )}+\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {(b c) \int \frac {d-2 \left (c^2 d+e\right ) x^2}{\sqrt {1-c^2 x^2} \left (d+e x^2\right )} \, dx}{8 d e \left (c^2 d+e\right )}\\ &=-\frac {b c x \sqrt {1-c^2 x^2}}{8 e \left (c^2 d+e\right ) \left (d+e x^2\right )}+\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}-\frac {(b c) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{4 d e^2}+\frac {\left (b c \left (2 c^2 d+3 e\right )\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \left (d+e x^2\right )} \, dx}{8 e^2 \left (c^2 d+e\right )}\\ &=-\frac {b c x \sqrt {1-c^2 x^2}}{8 e \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {b \sin ^{-1}(c x)}{4 d e^2}+\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {\left (b c \left (2 c^2 d+3 e\right )\right ) \operatorname {Subst}\left (\int \frac {1}{d-\left (-c^2 d-e\right ) x^2} \, dx,x,\frac {x}{\sqrt {1-c^2 x^2}}\right )}{8 e^2 \left (c^2 d+e\right )}\\ &=-\frac {b c x \sqrt {1-c^2 x^2}}{8 e \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {b \sin ^{-1}(c x)}{4 d e^2}+\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}+\frac {b c \left (2 c^2 d+3 e\right ) \tan ^{-1}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 \sqrt {d} e^2 \left (c^2 d+e\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.54, size = 152, normalized size = 0.99 \[ \frac {-\frac {2 a \left (d+2 e x^2\right )+\frac {b c e x \sqrt {1-c^2 x^2} \left (d+e x^2\right )}{c^2 d+e}}{\left (d+e x^2\right )^2}+\frac {b c \left (2 c^2 d+3 e\right ) \tan ^{-1}\left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{\sqrt {d} \left (c^2 d+e\right )^{3/2}}-\frac {2 b \sin ^{-1}(c x) \left (d+2 e x^2\right )}{\left (d+e x^2\right )^2}}{8 e^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.17, size = 921, normalized size = 6.02 \[ \left [-\frac {8 \, a c^{4} d^{4} + 16 \, a c^{2} d^{3} e + 8 \, a d^{2} e^{2} + 16 \, {\left (a c^{4} d^{3} e + 2 \, a c^{2} d^{2} e^{2} + a d e^{3}\right )} x^{2} + {\left (2 \, b c^{3} d^{3} + 3 \, b c d^{2} e + {\left (2 \, b c^{3} d e^{2} + 3 \, b c e^{3}\right )} x^{4} + 2 \, {\left (2 \, b c^{3} d^{2} e + 3 \, b c d e^{2}\right )} x^{2}\right )} \sqrt {-c^{2} d^{2} - d e} \log \left (\frac {{\left (8 \, c^{4} d^{2} + 8 \, c^{2} d e + e^{2}\right )} x^{4} - 2 \, {\left (4 \, c^{2} d^{2} + 3 \, d e\right )} x^{2} - 4 \, \sqrt {-c^{2} d^{2} - d e} \sqrt {-c^{2} x^{2} + 1} {\left ({\left (2 \, c^{2} d + e\right )} x^{3} - d x\right )} + d^{2}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\right ) + 8 \, {\left (b c^{4} d^{4} + 2 \, b c^{2} d^{3} e + b d^{2} e^{2} + 2 \, {\left (b c^{4} d^{3} e + 2 \, b c^{2} d^{2} e^{2} + b d e^{3}\right )} x^{2}\right )} \arcsin \left (c x\right ) + 4 \, \sqrt {-c^{2} x^{2} + 1} {\left ({\left (b c^{3} d^{2} e^{2} + b c d e^{3}\right )} x^{3} + {\left (b c^{3} d^{3} e + b c d^{2} e^{2}\right )} x\right )}}{32 \, {\left (c^{4} d^{5} e^{2} + 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4} + {\left (c^{4} d^{3} e^{4} + 2 \, c^{2} d^{2} e^{5} + d e^{6}\right )} x^{4} + 2 \, {\left (c^{4} d^{4} e^{3} + 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\right )} x^{2}\right )}}, -\frac {4 \, a c^{4} d^{4} + 8 \, a c^{2} d^{3} e + 4 \, a d^{2} e^{2} + 8 \, {\left (a c^{4} d^{3} e + 2 \, a c^{2} d^{2} e^{2} + a d e^{3}\right )} x^{2} + {\left (2 \, b c^{3} d^{3} + 3 \, b c d^{2} e + {\left (2 \, b c^{3} d e^{2} + 3 \, b c e^{3}\right )} x^{4} + 2 \, {\left (2 \, b c^{3} d^{2} e + 3 \, b c d e^{2}\right )} x^{2}\right )} \sqrt {c^{2} d^{2} + d e} \arctan \left (\frac {\sqrt {c^{2} d^{2} + d e} \sqrt {-c^{2} x^{2} + 1} {\left ({\left (2 \, c^{2} d + e\right )} x^{2} - d\right )}}{2 \, {\left ({\left (c^{4} d^{2} + c^{2} d e\right )} x^{3} - {\left (c^{2} d^{2} + d e\right )} x\right )}}\right ) + 4 \, {\left (b c^{4} d^{4} + 2 \, b c^{2} d^{3} e + b d^{2} e^{2} + 2 \, {\left (b c^{4} d^{3} e + 2 \, b c^{2} d^{2} e^{2} + b d e^{3}\right )} x^{2}\right )} \arcsin \left (c x\right ) + 2 \, \sqrt {-c^{2} x^{2} + 1} {\left ({\left (b c^{3} d^{2} e^{2} + b c d e^{3}\right )} x^{3} + {\left (b c^{3} d^{3} e + b c d^{2} e^{2}\right )} x\right )}}{16 \, {\left (c^{4} d^{5} e^{2} + 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4} + {\left (c^{4} d^{3} e^{4} + 2 \, c^{2} d^{2} e^{5} + d e^{6}\right )} x^{4} + 2 \, {\left (c^{4} d^{4} e^{3} + 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\right )} x^{2}\right )}}\right ] \]
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 \[ \int \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 1055, normalized size = 6.90 \[ -\frac {c^{2} a}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {c^{4} a d}{4 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}-\frac {c^{2} b \arcsin \left (c x \right )}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {c^{4} b \arcsin \left (c x \right ) d}{4 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}+\frac {3 c^{2} b \ln \left (\frac {\frac {2 c^{2} d +2 e}{e}+\frac {2 \sqrt {-c^{2} e d}\, \left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-\left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )^{2}+\frac {2 \sqrt {-c^{2} e d}\, \left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{c x +\frac {\sqrt {-c^{2} e d}}{e}}\right )}{16 e^{2} \sqrt {-c^{2} e d}\, \sqrt {\frac {c^{2} d +e}{e}}}-\frac {3 c^{2} b \ln \left (\frac {\frac {2 c^{2} d +2 e}{e}-\frac {2 \sqrt {-c^{2} e d}\, \left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-\left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )^{2}-\frac {2 \sqrt {-c^{2} e d}\, \left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{c x -\frac {\sqrt {-c^{2} e d}}{e}}\right )}{16 e^{2} \sqrt {-c^{2} e d}\, \sqrt {\frac {c^{2} d +e}{e}}}-\frac {c^{2} b \sqrt {-\left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )^{2}-\frac {2 \sqrt {-c^{2} e d}\, \left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{16 e^{2} \left (c^{2} d +e \right ) \left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )}-\frac {c^{2} b \sqrt {-c^{2} e d}\, \ln \left (\frac {\frac {2 c^{2} d +2 e}{e}-\frac {2 \sqrt {-c^{2} e d}\, \left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-\left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )^{2}-\frac {2 \sqrt {-c^{2} e d}\, \left (c x -\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{c x -\frac {\sqrt {-c^{2} e d}}{e}}\right )}{16 e^{3} \left (c^{2} d +e \right ) \sqrt {\frac {c^{2} d +e}{e}}}-\frac {c^{2} b \sqrt {-\left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )^{2}+\frac {2 \sqrt {-c^{2} e d}\, \left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{16 e^{2} \left (c^{2} d +e \right ) \left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )}+\frac {c^{2} b \sqrt {-c^{2} e d}\, \ln \left (\frac {\frac {2 c^{2} d +2 e}{e}+\frac {2 \sqrt {-c^{2} e d}\, \left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d +e}{e}}\, \sqrt {-\left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )^{2}+\frac {2 \sqrt {-c^{2} e d}\, \left (c x +\frac {\sqrt {-c^{2} e d}}{e}\right )}{e}+\frac {c^{2} d +e}{e}}}{c x +\frac {\sqrt {-c^{2} e d}}{e}}\right )}{16 e^{3} \left (c^{2} d +e \right ) \sqrt {\frac {c^{2} d +e}{e}}} \]
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 \[ -\frac {{\left (2 \, e x^{2} + d\right )} a}{4 \, {\left (e^{4} x^{4} + 2 \, d e^{3} x^{2} + d^{2} e^{2}\right )}} - \frac {{\left ({\left (2 \, e x^{2} + d\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + {\left (e^{4} x^{4} + 2 \, d e^{3} x^{2} + d^{2} e^{2}\right )} \int \frac {{\left (2 \, c e x^{2} + c d\right )} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (-c x + 1\right )\right )}}{c^{4} e^{4} x^{8} - c^{2} d^{2} e^{2} x^{2} + {\left (2 \, c^{4} d e^{3} - c^{2} e^{4}\right )} x^{6} + {\left (c^{4} d^{2} e^{2} - 2 \, c^{2} d e^{3}\right )} x^{4} - {\left (c^{2} e^{4} x^{6} + {\left (2 \, c^{2} d e^{3} - e^{4}\right )} x^{4} - d^{2} e^{2} + {\left (c^{2} d^{2} e^{2} - 2 \, d e^{3}\right )} x^{2}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )}}\,{d x}\right )} b}{4 \, {\left (e^{4} x^{4} + 2 \, d e^{3} x^{2} + d^{2} e^{2}\right )}} \]
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 \[ \int \frac {x^3\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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