Optimal. Leaf size=153 \[ -\frac {b c x \sqrt {1-c^2 x^2}}{8 e \left (c^2 d+e\right ) \left (d+e x^2\right )}-\frac {b \text {ArcSin}(c x)}{4 d e^2}+\frac {x^4 (a+b \text {ArcSin}(c x))}{4 d \left (d+e x^2\right )^2}+\frac {b c \left (2 c^2 d+3 e\right ) \text {ArcTan}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{8 \sqrt {d} e^2 \left (c^2 d+e\right )^{3/2}} \]
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Rubi [A]
time = 0.14, 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 = {270, 4815, 12,
481, 537, 222, 385, 211} \begin {gather*} \frac {x^4 (a+b \text {ArcSin}(c x))}{4 d \left (d+e x^2\right )^2}-\frac {b \text {ArcSin}(c x)}{4 d e^2}+\frac {b c \left (2 c^2 d+3 e\right ) \text {ArcTan}\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 )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 211
Rule 222
Rule 270
Rule 385
Rule 481
Rule 537
Rule 4815
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 ) \text {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 = 0.36, size = 152, normalized size = 0.99 \begin {gather*} \frac {-\frac {\frac {b c e x \sqrt {1-c^2 x^2} \left (d+e x^2\right )}{c^2 d+e}+2 a \left (d+2 e x^2\right )}{\left (d+e x^2\right )^2}-\frac {2 b \left (d+2 e x^2\right ) \text {ArcSin}(c x)}{\left (d+e x^2\right )^2}+\frac {b c \left (2 c^2 d+3 e\right ) \text {ArcTan}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{\sqrt {d} \left (c^2 d+e\right )^{3/2}}}{8 e^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1059\) vs.
\(2(133)=266\).
time = 0.11, size = 1060, normalized size = 6.93 Too large to display
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 449 vs.
\(2 (136) = 272\).
time = 2.46, size = 939, normalized size = 6.14 \begin {gather*} \left [-\frac {8 \, a c^{4} d^{4} + 16 \, a d x^{2} e^{3} + {\left (2 \, b c^{3} d^{3} + 3 \, b c x^{4} e^{3} + 2 \, {\left (b c^{3} d x^{4} + 3 \, b c d x^{2}\right )} e^{2} + {\left (4 \, b c^{3} d^{2} x^{2} + 3 \, b c d^{2}\right )} e\right )} \sqrt {-c^{2} d^{2} - d e} \log \left (\frac {8 \, c^{4} d^{2} x^{4} - 8 \, c^{2} d^{2} x^{2} + x^{4} e^{2} - 4 \, {\left (2 \, c^{2} d x^{3} + x^{3} e - d x\right )} \sqrt {-c^{2} d^{2} - d e} \sqrt {-c^{2} x^{2} + 1} + d^{2} + 2 \, {\left (4 \, c^{2} d x^{4} - 3 \, d x^{2}\right )} e}{x^{4} e^{2} + 2 \, d x^{2} e + d^{2}}\right ) + 8 \, {\left (b c^{4} d^{4} + 2 \, b d x^{2} e^{3} + {\left (4 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} e^{2} + 2 \, {\left (b c^{4} d^{3} x^{2} + b c^{2} d^{3}\right )} e\right )} \arcsin \left (c x\right ) + 8 \, {\left (4 \, a c^{2} d^{2} x^{2} + a d^{2}\right )} e^{2} + 16 \, {\left (a c^{4} d^{3} x^{2} + a c^{2} d^{3}\right )} e + 4 \, {\left (b c^{3} d^{3} x e + b c d x^{3} e^{3} + {\left (b c^{3} d^{2} x^{3} + b c d^{2} x\right )} e^{2}\right )} \sqrt {-c^{2} x^{2} + 1}}{32 \, {\left (c^{4} d^{5} e^{2} + d x^{4} e^{6} + 2 \, {\left (c^{2} d^{2} x^{4} + d^{2} x^{2}\right )} e^{5} + {\left (c^{4} d^{3} x^{4} + 4 \, c^{2} d^{3} x^{2} + d^{3}\right )} e^{4} + 2 \, {\left (c^{4} d^{4} x^{2} + c^{2} d^{4}\right )} e^{3}\right )}}, -\frac {4 \, a c^{4} d^{4} + 8 \, a d x^{2} e^{3} + {\left (2 \, b c^{3} d^{3} + 3 \, b c x^{4} e^{3} + 2 \, {\left (b c^{3} d x^{4} + 3 \, b c d x^{2}\right )} e^{2} + {\left (4 \, b c^{3} d^{2} x^{2} + 3 \, b c d^{2}\right )} e\right )} \sqrt {c^{2} d^{2} + d e} \arctan \left (\frac {{\left (2 \, c^{2} d x^{2} + x^{2} e - d\right )} \sqrt {c^{2} d^{2} + d e} \sqrt {-c^{2} x^{2} + 1}}{2 \, {\left (c^{4} d^{2} x^{3} - c^{2} d^{2} x + {\left (c^{2} d x^{3} - d x\right )} e\right )}}\right ) + 4 \, {\left (b c^{4} d^{4} + 2 \, b d x^{2} e^{3} + {\left (4 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} e^{2} + 2 \, {\left (b c^{4} d^{3} x^{2} + b c^{2} d^{3}\right )} e\right )} \arcsin \left (c x\right ) + 4 \, {\left (4 \, a c^{2} d^{2} x^{2} + a d^{2}\right )} e^{2} + 8 \, {\left (a c^{4} d^{3} x^{2} + a c^{2} d^{3}\right )} e + 2 \, {\left (b c^{3} d^{3} x e + b c d x^{3} e^{3} + {\left (b c^{3} d^{2} x^{3} + b c d^{2} x\right )} e^{2}\right )} \sqrt {-c^{2} x^{2} + 1}}{16 \, {\left (c^{4} d^{5} e^{2} + d x^{4} e^{6} + 2 \, {\left (c^{2} d^{2} x^{4} + d^{2} x^{2}\right )} e^{5} + {\left (c^{4} d^{3} x^{4} + 4 \, c^{2} d^{3} x^{2} + d^{3}\right )} e^{4} + 2 \, {\left (c^{4} d^{4} x^{2} + c^{2} d^{4}\right )} e^{3}\right )}}\right ] \end {gather*}
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 \begin {gather*} \int \frac {x^{3} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{3}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^3\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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