Optimal. Leaf size=130 \[ \frac {x^4 \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}-\frac {b c \left (c^2 d-3 e\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 \sqrt {d} e^{3/2} \left (c^2 d-e\right )^2}+\frac {b c x}{8 e \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {b \tan ^{-1}(c x)}{4 d \left (c^2 d-e\right )^2} \]
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Rubi [A] time = 0.19, 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 = {264, 4976, 12, 470, 522, 205} \[ \frac {x^4 \left (a+b \tan ^{-1}(c x)\right )}{4 d \left (d+e x^2\right )^2}-\frac {b c \left (c^2 d-3 e\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 \sqrt {d} e^{3/2} \left (c^2 d-e\right )^2}+\frac {b c x}{8 e \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {b \tan ^{-1}(c x)}{4 d \left (c^2 d-e\right )^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 264
Rule 470
Rule 522
Rule 4976
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 = 3.76, size = 158, normalized size = 1.22 \[ \frac {\frac {-4 a c^2 d+4 a e+b c e x}{\left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac {2 a d}{\left (d+e x^2\right )^2}+\frac {2 b c^2 \left (c^2 d-2 e\right ) \tan ^{-1}(c x)}{\left (e-c^2 d\right )^2}-\frac {b c \sqrt {e} \left (c^2 d-3 e\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \left (e-c^2 d\right )^2}-\frac {2 b \tan ^{-1}(c x) \left (d+2 e x^2\right )}{\left (d+e x^2\right )^2}}{8 e^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 697, normalized size = 5.36 \[ \left [-\frac {4 \, a c^{4} d^{4} - 8 \, a c^{2} d^{3} e + 4 \, a d^{2} e^{2} - 2 \, {\left (b c^{3} d^{2} e^{2} - b c d e^{3}\right )} x^{3} + 8 \, {\left (a c^{4} d^{3} e - 2 \, a c^{2} d^{2} e^{2} + a d e^{3}\right )} x^{2} - {\left (b c^{3} d^{3} - 3 \, b c d^{2} e + {\left (b c^{3} d e^{2} - 3 \, b c e^{3}\right )} x^{4} + 2 \, {\left (b c^{3} d^{2} e - 3 \, b c d e^{2}\right )} x^{2}\right )} \sqrt {-d e} \log \left (\frac {e x^{2} - 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right ) - 2 \, {\left (b c^{3} d^{3} e - b c d^{2} e^{2}\right )} x + 4 \, {\left (2 \, b d e^{3} x^{2} + b d^{2} e^{2} - {\left (b c^{4} d^{2} e^{2} - 2 \, b c^{2} d e^{3}\right )} x^{4}\right )} \arctan \left (c x\right )}{16 \, {\left (c^{4} d^{5} e^{2} - 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4} + {\left (c^{4} d^{3} e^{4} - 2 \, c^{2} d^{2} e^{5} + d e^{6}\right )} x^{4} + 2 \, {\left (c^{4} d^{4} e^{3} - 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\right )} x^{2}\right )}}, -\frac {2 \, a c^{4} d^{4} - 4 \, a c^{2} d^{3} e + 2 \, a d^{2} e^{2} - {\left (b c^{3} d^{2} e^{2} - b c d e^{3}\right )} x^{3} + 4 \, {\left (a c^{4} d^{3} e - 2 \, a c^{2} d^{2} e^{2} + a d e^{3}\right )} x^{2} + {\left (b c^{3} d^{3} - 3 \, b c d^{2} e + {\left (b c^{3} d e^{2} - 3 \, b c e^{3}\right )} x^{4} + 2 \, {\left (b c^{3} d^{2} e - 3 \, b c d e^{2}\right )} x^{2}\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right ) - {\left (b c^{3} d^{3} e - b c d^{2} e^{2}\right )} x + 2 \, {\left (2 \, b d e^{3} x^{2} + b d^{2} e^{2} - {\left (b c^{4} d^{2} e^{2} - 2 \, b c^{2} d e^{3}\right )} x^{4}\right )} \arctan \left (c x\right )}{8 \, {\left (c^{4} d^{5} e^{2} - 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4} + {\left (c^{4} d^{3} e^{4} - 2 \, c^{2} d^{2} e^{5} + d e^{6}\right )} x^{4} + 2 \, {\left (c^{4} d^{4} e^{3} - 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\right )} x^{2}\right )}}\right ] \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 297, normalized size = 2.28 \[ \frac {c^{4} a d}{4 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}-\frac {c^{2} a}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {c^{4} b \arctan \left (c x \right ) d}{4 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}-\frac {c^{2} b \arctan \left (c x \right )}{2 e^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {c^{5} b d x}{8 e \left (c^{2} d -e \right )^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {c^{3} b x}{8 \left (c^{2} d -e \right )^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {c^{3} b d \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 e \left (c^{2} d -e \right )^{2} \sqrt {d e}}+\frac {3 c b \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 \left (c^{2} d -e \right )^{2} \sqrt {d e}}+\frac {c^{4} b d \arctan \left (c x \right )}{4 e^{2} \left (c^{2} d -e \right )^{2}}-\frac {c^{2} b \arctan \left (c x \right )}{2 e \left (c^{2} d -e \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 216, normalized size = 1.66 \[ -\frac {1}{8} \, {\left (c {\left (\frac {{\left (c^{2} d - 3 \, e\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{{\left (c^{4} d^{2} e - 2 \, c^{2} d e^{2} + e^{3}\right )} \sqrt {d e}} - \frac {x}{c^{2} d^{2} e - d e^{2} + {\left (c^{2} d e^{2} - e^{3}\right )} x^{2}} - \frac {2 \, {\left (c^{4} d - 2 \, c^{2} e\right )} \arctan \left (c x\right )}{{\left (c^{4} d^{2} e^{2} - 2 \, c^{2} d e^{3} + e^{4}\right )} c}\right )} + \frac {2 \, {\left (2 \, e x^{2} + d\right )} \arctan \left (c x\right )}{e^{4} x^{4} + 2 \, d e^{3} x^{2} + d^{2} e^{2}}\right )} b - \frac {{\left (2 \, e x^{2} + d\right )} a}{4 \, {\left (e^{4} x^{4} + 2 \, d e^{3} x^{2} + d^{2} e^{2}\right )}} \]
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 \[ \frac {b\,c^4\,d\,\mathrm {atan}\left (c\,x\right )}{4\,e^2\,{\left (e-c^2\,d\right )}^2}-\frac {a\,d}{4\,e^2\,{\left (e\,x^2+d\right )}^2}-\frac {b\,d\,\mathrm {atan}\left (c\,x\right )}{4\,e^2\,{\left (e\,x^2+d\right )}^2}-\frac {b\,c\,x^3}{8\,\left (e-c^2\,d\right )\,{\left (e\,x^2+d\right )}^2}-\frac {b\,c^2\,\mathrm {atan}\left (c\,x\right )}{2\,e\,{\left (e-c^2\,d\right )}^2}-\frac {b\,x^2\,\mathrm {atan}\left (c\,x\right )}{2\,e\,{\left (e\,x^2+d\right )}^2}-\frac {b\,c^3\,\mathrm {atan}\left (\frac {x\,\sqrt {-d\,e^3}\,1{}\mathrm {i}}{d\,e}\right )\,\sqrt {-d\,e^3}\,1{}\mathrm {i}}{8\,e^3\,{\left (e-c^2\,d\right )}^2}-\frac {a\,x^2}{2\,e\,{\left (e\,x^2+d\right )}^2}-\frac {b\,c\,d\,x}{8\,e\,\left (e-c^2\,d\right )\,{\left (e\,x^2+d\right )}^2}+\frac {b\,c\,\mathrm {atan}\left (\frac {x\,\sqrt {-d\,e^3}\,1{}\mathrm {i}}{d\,e}\right )\,\sqrt {-d\,e^3}\,3{}\mathrm {i}}{8\,d\,e^2\,{\left (e-c^2\,d\right )}^2} \]
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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