Optimal. Leaf size=130 \[ \frac {b c x}{8 \left (c^2 d-e\right ) e \left (d+e x^2\right )}-\frac {b \text {ArcTan}(c x)}{4 d \left (c^2 d-e\right )^2}+\frac {x^4 (a+b \text {ArcTan}(c x))}{4 d \left (d+e x^2\right )^2}-\frac {b c \left (c^2 d-3 e\right ) \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 \sqrt {d} \left (c^2 d-e\right )^2 e^{3/2}} \]
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Rubi [A]
time = 0.14, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {270, 5096, 12,
481, 536, 211} \begin {gather*} \frac {x^4 (a+b \text {ArcTan}(c x))}{4 d \left (d+e x^2\right )^2}-\frac {b c \left (c^2 d-3 e\right ) \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 \sqrt {d} e^{3/2} \left (c^2 d-e\right )^2}-\frac {b \text {ArcTan}(c x)}{4 d \left (c^2 d-e\right )^2}+\frac {b c x}{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 270
Rule 481
Rule 536
Rule 5096
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx &=\frac {x^4 \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}-(b c) \int \frac {x^4}{4 \left (d+c^2 d x^2\right ) \left (d+e x^2\right )^2} \, dx\\ &=\frac {x^4 \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}-\frac {1}{4} (b c) \int \frac {x^4}{\left (d+c^2 d x^2\right ) \left (d+e x^2\right )^2} \, dx\\ &=\frac {b c x}{8 \left (c^2 d-e\right ) e \left (d+e x^2\right )}+\frac {x^4 \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}-\frac {(b c) \int \frac {d^2+d \left (c^2 d-2 e\right ) x^2}{\left (d+c^2 d x^2\right ) \left (d+e x^2\right )} \, dx}{8 d \left (c^2 d-e\right ) e}\\ &=\frac {b c x}{8 \left (c^2 d-e\right ) e \left (d+e x^2\right )}+\frac {x^4 \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}-\frac {(b c) \int \frac {1}{d+c^2 d x^2} \, dx}{4 \left (c^2 d-e\right )^2}-\frac {\left (b c \left (c^2 d-3 e\right )\right ) \int \frac {1}{d+e x^2} \, dx}{8 \left (c^2 d-e\right )^2 e}\\ &=\frac {b c x}{8 \left (c^2 d-e\right ) e \left (d+e x^2\right )}-\frac {b \tan ^{-1}(c x)}{4 d \left (c^2 d-e\right )^2}+\frac {x^4 \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}-\frac {b c \left (c^2 d-3 e\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 \sqrt {d} \left (c^2 d-e\right )^2 e^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 2.33, size = 158, normalized size = 1.22 \begin {gather*} \frac {\frac {2 a d}{\left (d+e x^2\right )^2}+\frac {-4 a c^2 d+4 a e+b c e x}{\left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac {2 b c^2 \left (c^2 d-2 e\right ) \text {ArcTan}(c x)}{\left (-c^2 d+e\right )^2}-\frac {2 b \left (d+2 e x^2\right ) \text {ArcTan}(c x)}{\left (d+e x^2\right )^2}-\frac {b c \left (c^2 d-3 e\right ) \sqrt {e} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \left (-c^2 d+e\right )^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. \(303\) vs.
\(2(114)=228\).
time = 0.38, size = 304, normalized size = 2.34 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 207, normalized size = 1.59 \begin {gather*} -\frac {1}{8} \, {\left ({\left (\frac {{\left (c^{2} d - 3 \, e\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )}}{{\left (c^{4} d^{2} e - 2 \, c^{2} d e^{2} + e^{3}\right )} \sqrt {d}} - \frac {x}{c^{2} d^{2} e + {\left (c^{2} d e^{2} - e^{3}\right )} x^{2} - d e^{2}} - \frac {2 \, {\left (c^{4} d - 2 \, c^{2} e\right )} \arctan \left (c x\right )}{{\left (c^{4} d^{2} e^{2} - 2 \, c^{2} d e^{3} + e^{4}\right )} c}\right )} c + \frac {2 \, {\left (2 \, x^{2} e + d\right )} \arctan \left (c x\right )}{x^{4} e^{4} + 2 \, d x^{2} e^{3} + d^{2} e^{2}}\right )} b - \frac {{\left (2 \, x^{2} e + d\right )} a}{4 \, {\left (x^{4} e^{4} + 2 \, d x^{2} e^{3} + d^{2} e^{2}\right )}} \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. 325 vs.
\(2 (117) = 234\).
time = 2.34, size = 680, normalized size = 5.23 \begin {gather*} \left [-\frac {4 \, a c^{4} d^{4} - {\left (b c^{3} d^{3} - 3 \, b c x^{4} e^{3} + {\left (b c^{3} d x^{4} - 6 \, b c d x^{2}\right )} e^{2} + {\left (2 \, b c^{3} d^{2} x^{2} - 3 \, b c d^{2}\right )} e\right )} \sqrt {-d e} \log \left (\frac {x^{2} e - 2 \, \sqrt {-d e} x - d}{x^{2} e + d}\right ) + 4 \, {\left (2 \, {\left (b c^{2} d x^{4} + b d x^{2}\right )} e^{3} - {\left (b c^{4} d^{2} x^{4} - b d^{2}\right )} e^{2}\right )} \arctan \left (c x\right ) + 2 \, {\left (b c d x^{3} + 4 \, a d x^{2}\right )} e^{3} - 2 \, {\left (b c^{3} d^{2} x^{3} + 8 \, a c^{2} d^{2} x^{2} - b c d^{2} x - 2 \, a d^{2}\right )} e^{2} + 2 \, {\left (4 \, a c^{4} d^{3} x^{2} - b c^{3} d^{3} x - 4 \, a c^{2} d^{3}\right )} e}{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 )}}, -\frac {2 \, a c^{4} d^{4} + {\left (b c^{3} d^{3} - 3 \, b c x^{4} e^{3} + {\left (b c^{3} d x^{4} - 6 \, b c d x^{2}\right )} e^{2} + {\left (2 \, b c^{3} d^{2} x^{2} - 3 \, b c d^{2}\right )} e\right )} \sqrt {d} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\frac {1}{2}} + 2 \, {\left (2 \, {\left (b c^{2} d x^{4} + b d x^{2}\right )} e^{3} - {\left (b c^{4} d^{2} x^{4} - b d^{2}\right )} e^{2}\right )} \arctan \left (c x\right ) + {\left (b c d x^{3} + 4 \, a d x^{2}\right )} e^{3} - {\left (b c^{3} d^{2} x^{3} + 8 \, a c^{2} d^{2} x^{2} - b c d^{2} x - 2 \, a d^{2}\right )} e^{2} + {\left (4 \, a c^{4} d^{3} x^{2} - b c^{3} d^{3} x - 4 \, a c^{2} d^{3}\right )} e}{8 \, {\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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 [B]
time = 3.30, size = 273, normalized size = 2.10 \begin {gather*} \frac {b\,c^4\,d\,\mathrm {atan}\left (c\,x\right )}{4\,e^2\,{\left (e-c^2\,d\right )}^2}-\frac {a\,d}{4\,e^2\,{\left (e\,x^2+d\right )}^2}-\frac {b\,d\,\mathrm {atan}\left (c\,x\right )}{4\,e^2\,{\left (e\,x^2+d\right )}^2}-\frac {b\,c\,x^3}{8\,\left (e-c^2\,d\right )\,{\left (e\,x^2+d\right )}^2}-\frac {b\,c^2\,\mathrm {atan}\left (c\,x\right )}{2\,e\,{\left (e-c^2\,d\right )}^2}-\frac {b\,x^2\,\mathrm {atan}\left (c\,x\right )}{2\,e\,{\left (e\,x^2+d\right )}^2}-\frac {b\,c^3\,\mathrm {atan}\left (\frac {x\,\sqrt {-d\,e^3}\,1{}\mathrm {i}}{d\,e}\right )\,\sqrt {-d\,e^3}\,1{}\mathrm {i}}{8\,e^3\,{\left (e-c^2\,d\right )}^2}-\frac {a\,x^2}{2\,e\,{\left (e\,x^2+d\right )}^2}-\frac {b\,c\,d\,x}{8\,e\,\left (e-c^2\,d\right )\,{\left (e\,x^2+d\right )}^2}+\frac {b\,c\,\mathrm {atan}\left (\frac {x\,\sqrt {-d\,e^3}\,1{}\mathrm {i}}{d\,e}\right )\,\sqrt {-d\,e^3}\,3{}\mathrm {i}}{8\,d\,e^2\,{\left (e-c^2\,d\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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