Optimal. Leaf size=67 \[ \frac {a}{d \sqrt {d+e x^2}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{d^{3/2}}+b \text {Int}\left (\frac {\text {ArcTan}(c x)}{x \left (d+e x^2\right )^{3/2}},x\right ) \]
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Rubi [A]
time = 0.12, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {a+b \text {ArcTan}(c x)}{x \left (d+e x^2\right )^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}(c x)}{x \left (d+e x^2\right )^{3/2}} \, dx &=a \int \frac {1}{x \left (d+e x^2\right )^{3/2}} \, dx+b \int \frac {\tan ^{-1}(c x)}{x \left (d+e x^2\right )^{3/2}} \, dx\\ &=\frac {1}{2} a \text {Subst}\left (\int \frac {1}{x (d+e x)^{3/2}} \, dx,x,x^2\right )+b \int \frac {\tan ^{-1}(c x)}{x \left (d+e x^2\right )^{3/2}} \, dx\\ &=\frac {a}{d \sqrt {d+e x^2}}+b \int \frac {\tan ^{-1}(c x)}{x \left (d+e x^2\right )^{3/2}} \, dx+\frac {a \text {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )}{2 d}\\ &=\frac {a}{d \sqrt {d+e x^2}}+b \int \frac {\tan ^{-1}(c x)}{x \left (d+e x^2\right )^{3/2}} \, dx+\frac {a \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{d e}\\ &=\frac {a}{d \sqrt {d+e x^2}}-\frac {a \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{d^{3/2}}+b \int \frac {\tan ^{-1}(c x)}{x \left (d+e x^2\right )^{3/2}} \, dx\\ \end {align*}
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Mathematica [A]
time = 8.42, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b \text {ArcTan}(c x)}{x \left (d+e x^2\right )^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.13, size = 0, normalized size = 0.00 \[\int \frac {a +b \arctan \left (c x \right )}{x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
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 [A]
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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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \operatorname {atan}{\left (c x \right )}}{x \left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
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 [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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