Optimal. Leaf size=131 \[ -\frac {a+b \tan ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}-\frac {b c \left (3 c^2 d-e\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{3/2} \sqrt {e} \left (c^2 d-e\right )^2}-\frac {b c x}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac {b c^4 \tan ^{-1}(c x)}{4 e \left (c^2 d-e\right )^2} \]
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Rubi [A] time = 0.11, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {4974, 414, 522, 203, 205} \[ -\frac {a+b \tan ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}-\frac {b c \left (3 c^2 d-e\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{3/2} \sqrt {e} \left (c^2 d-e\right )^2}-\frac {b c x}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac {b c^4 \tan ^{-1}(c x)}{4 e \left (c^2 d-e\right )^2} \]
Antiderivative was successfully verified.
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Rule 203
Rule 205
Rule 414
Rule 522
Rule 4974
Rubi steps
\begin {align*} \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx &=-\frac {a+b \tan ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {(b c) \int \frac {1}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )^2} \, dx}{4 e}\\ &=-\frac {b c x}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {a+b \tan ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}+\frac {(b c) \int \frac {2 c^2 d-e-c^2 e x^2}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )} \, dx}{8 d \left (c^2 d-e\right ) e}\\ &=-\frac {b c x}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {a+b \tan ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}-\frac {\left (b c \left (3 c^2 d-e\right )\right ) \int \frac {1}{d+e x^2} \, dx}{8 d \left (c^2 d-e\right )^2}+\frac {\left (b c^5\right ) \int \frac {1}{1+c^2 x^2} \, dx}{4 \left (c^2 d-e\right )^2 e}\\ &=-\frac {b c x}{8 d \left (c^2 d-e\right ) \left (d+e x^2\right )}+\frac {b c^4 \tan ^{-1}(c x)}{4 \left (c^2 d-e\right )^2 e}-\frac {a+b \tan ^{-1}(c x)}{4 e \left (d+e x^2\right )^2}-\frac {b c \left (3 c^2 d-e\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{3/2} \left (c^2 d-e\right )^2 \sqrt {e}}\\ \end {align*}
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Mathematica [A] time = 1.19, size = 131, normalized size = 1.00 \[ \frac {1}{8} \left (-\frac {\frac {2 a}{e}+\frac {b c x \left (d+e x^2\right )}{d \left (c^2 d-e\right )}}{\left (d+e x^2\right )^2}-\frac {b c \left (3 c^2 d-e\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2} \sqrt {e} \left (e-c^2 d\right )^2}+\frac {2 b \tan ^{-1}(c x) \left (\frac {c^4}{\left (e-c^2 d\right )^2}-\frac {1}{\left (d+e x^2\right )^2}\right )}{e}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.68, size = 637, normalized size = 4.86 \[ \left [-\frac {4 \, a c^{4} d^{4} - 8 \, a c^{2} d^{3} e + 4 \, a d^{2} e^{2} + 2 \, {\left (b c^{3} d^{2} e^{2} - b c d e^{3}\right )} x^{3} - {\left (3 \, b c^{3} d^{3} - b c d^{2} e + {\left (3 \, b c^{3} d e^{2} - b c e^{3}\right )} x^{4} + 2 \, {\left (3 \, b c^{3} d^{2} e - b c d e^{2}\right )} x^{2}\right )} \sqrt {-d e} \log \left (\frac {e x^{2} - 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right ) + 2 \, {\left (b c^{3} d^{3} e - b c d^{2} e^{2}\right )} x - 4 \, {\left (b c^{4} d^{2} e^{2} x^{4} + 2 \, b c^{4} d^{3} e x^{2} + 2 \, b c^{2} d^{3} e - b d^{2} e^{2}\right )} \arctan \left (c x\right )}{16 \, {\left (c^{4} d^{6} e - 2 \, c^{2} d^{5} e^{2} + d^{4} e^{3} + {\left (c^{4} d^{4} e^{3} - 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\right )} x^{4} + 2 \, {\left (c^{4} d^{5} e^{2} - 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4}\right )} x^{2}\right )}}, -\frac {2 \, a c^{4} d^{4} - 4 \, a c^{2} d^{3} e + 2 \, a d^{2} e^{2} + {\left (b c^{3} d^{2} e^{2} - b c d e^{3}\right )} x^{3} + {\left (3 \, b c^{3} d^{3} - b c d^{2} e + {\left (3 \, b c^{3} d e^{2} - b c e^{3}\right )} x^{4} + 2 \, {\left (3 \, b c^{3} d^{2} e - b c d e^{2}\right )} x^{2}\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right ) + {\left (b c^{3} d^{3} e - b c d^{2} e^{2}\right )} x - 2 \, {\left (b c^{4} d^{2} e^{2} x^{4} + 2 \, b c^{4} d^{3} e x^{2} + 2 \, b c^{2} d^{3} e - b d^{2} e^{2}\right )} \arctan \left (c x\right )}{8 \, {\left (c^{4} d^{6} e - 2 \, c^{2} d^{5} e^{2} + d^{4} e^{3} + {\left (c^{4} d^{4} e^{3} - 2 \, c^{2} d^{3} e^{4} + d^{2} e^{5}\right )} x^{4} + 2 \, {\left (c^{4} d^{5} e^{2} - 2 \, c^{2} d^{4} e^{3} + d^{3} e^{4}\right )} x^{2}\right )}}\right ] \]
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 [A] time = 0.05, size = 216, normalized size = 1.65 \[ -\frac {c^{4} a}{4 e \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}-\frac {c^{4} b \arctan \left (c x \right )}{4 e \left (c^{2} e \,x^{2}+c^{2} d \right )^{2}}-\frac {c^{5} b x}{8 \left (c^{2} d -e \right )^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {c^{3} b e x}{8 \left (c^{2} d -e \right )^{2} d \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {3 c^{3} b \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 \left (c^{2} d -e \right )^{2} \sqrt {d e}}+\frac {c b e \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 \left (c^{2} d -e \right )^{2} d \sqrt {d e}}+\frac {b \,c^{4} \arctan \left (c x \right )}{4 \left (c^{2} d -e \right )^{2} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 185, normalized size = 1.41 \[ \frac {1}{8} \, {\left ({\left (\frac {2 \, c^{3} \arctan \left (c x\right )}{c^{4} d^{2} e - 2 \, c^{2} d e^{2} + e^{3}} - \frac {{\left (3 \, c^{2} d - e\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{{\left (c^{4} d^{3} - 2 \, c^{2} d^{2} e + d e^{2}\right )} \sqrt {d e}} - \frac {x}{c^{2} d^{3} - d^{2} e + {\left (c^{2} d^{2} e - d e^{2}\right )} x^{2}}\right )} c - \frac {2 \, \arctan \left (c x\right )}{e^{3} x^{4} + 2 \, d e^{2} x^{2} + d^{2} e}\right )} b - \frac {a}{4 \, {\left (e^{3} x^{4} + 2 \, d e^{2} x^{2} + d^{2} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.61, size = 201, normalized size = 1.53 \[ \frac {b\,c\,x}{8\,\left (e-c^2\,d\right )\,{\left (e\,x^2+d\right )}^2}-\frac {b\,\mathrm {atan}\left (c\,x\right )}{4\,e\,{\left (e\,x^2+d\right )}^2}-\frac {a}{4\,e\,{\left (e\,x^2+d\right )}^2}+\frac {b\,c^4\,\mathrm {atan}\left (c\,x\right )}{4\,e\,{\left (e-c^2\,d\right )}^2}+\frac {b\,c\,e\,x^3}{8\,d\,\left (e-c^2\,d\right )\,{\left (e\,x^2+d\right )}^2}+\frac {b\,c\,\mathrm {atan}\left (\frac {x\,\sqrt {-d^3\,e}\,1{}\mathrm {i}}{d^2}\right )\,\sqrt {-d^3\,e}\,1{}\mathrm {i}}{8\,d^3\,{\left (e-c^2\,d\right )}^2}-\frac {b\,c^3\,\mathrm {atan}\left (\frac {x\,\sqrt {-d^3\,e}\,1{}\mathrm {i}}{d^2}\right )\,\sqrt {-d^3\,e}\,3{}\mathrm {i}}{8\,d^2\,e\,{\left (e-c^2\,d\right )}^2} \]
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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