Optimal. Leaf size=71 \[ \frac {b c \tan ^{-1}\left (\frac {x \sqrt {c^2 d-e}}{\sqrt {d+e x^2}}\right )}{e \sqrt {c^2 d-e}}-\frac {a+b \tan ^{-1}(c x)}{e \sqrt {d+e x^2}} \]
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Rubi [A] time = 0.07, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4974, 377, 203} \[ \frac {b c \tan ^{-1}\left (\frac {x \sqrt {c^2 d-e}}{\sqrt {d+e x^2}}\right )}{e \sqrt {c^2 d-e}}-\frac {a+b \tan ^{-1}(c x)}{e \sqrt {d+e x^2}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 377
Rule 4974
Rubi steps
\begin {align*} \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx &=-\frac {a+b \tan ^{-1}(c x)}{e \sqrt {d+e x^2}}+\frac {(b c) \int \frac {1}{\left (1+c^2 x^2\right ) \sqrt {d+e x^2}} \, dx}{e}\\ &=-\frac {a+b \tan ^{-1}(c x)}{e \sqrt {d+e x^2}}+\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{1-\left (-c^2 d+e\right ) x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{e}\\ &=-\frac {a+b \tan ^{-1}(c x)}{e \sqrt {d+e x^2}}+\frac {b c \tan ^{-1}\left (\frac {\sqrt {c^2 d-e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {c^2 d-e} e}\\ \end {align*}
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Mathematica [C] time = 0.40, size = 210, normalized size = 2.96 \[ -\frac {\frac {2 a}{\sqrt {d+e x^2}}+\frac {i b c \log \left (-\frac {4 i e \left (\sqrt {c^2 d-e} \sqrt {d+e x^2}+c d-i e x\right )}{b (c x+i) \sqrt {c^2 d-e}}\right )}{\sqrt {c^2 d-e}}-\frac {i b c \log \left (\frac {4 i e \left (\sqrt {c^2 d-e} \sqrt {d+e x^2}+c d+i e x\right )}{b (c x-i) \sqrt {c^2 d-e}}\right )}{\sqrt {c^2 d-e}}+\frac {2 b \tan ^{-1}(c x)}{\sqrt {d+e x^2}}}{2 e} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 379, normalized size = 5.34 \[ \left [-\frac {{\left (b c e x^{2} + b c d\right )} \sqrt {-c^{2} d + e} \log \left (\frac {{\left (c^{4} d^{2} - 8 \, c^{2} d e + 8 \, e^{2}\right )} x^{4} - 2 \, {\left (3 \, c^{2} d^{2} - 4 \, d e\right )} x^{2} - 4 \, {\left ({\left (c^{2} d - 2 \, e\right )} x^{3} - d x\right )} \sqrt {-c^{2} d + e} \sqrt {e x^{2} + d} + d^{2}}{c^{4} x^{4} + 2 \, c^{2} x^{2} + 1}\right ) + 4 \, {\left (a c^{2} d - a e + {\left (b c^{2} d - b e\right )} \arctan \left (c x\right )\right )} \sqrt {e x^{2} + d}}{4 \, {\left (c^{2} d^{2} e - d e^{2} + {\left (c^{2} d e^{2} - e^{3}\right )} x^{2}\right )}}, \frac {{\left (b c e x^{2} + b c d\right )} \sqrt {c^{2} d - e} \arctan \left (\frac {\sqrt {c^{2} d - e} {\left ({\left (c^{2} d - 2 \, e\right )} x^{2} - d\right )} \sqrt {e x^{2} + d}}{2 \, {\left ({\left (c^{2} d e - e^{2}\right )} x^{3} + {\left (c^{2} d^{2} - d e\right )} x\right )}}\right ) - 2 \, {\left (a c^{2} d - a e + {\left (b c^{2} d - b e\right )} \arctan \left (c x\right )\right )} \sqrt {e x^{2} + d}}{2 \, {\left (c^{2} d^{2} e - d e^{2} + {\left (c^{2} d e^{2} - e^{3}\right )} x^{2}\right )}}\right ] \]
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 [F] time = 0.99, size = 0, normalized size = 0.00 \[ \int \frac {x \left (a +b \arctan \left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {x\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x \left (a + b \operatorname {atan}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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