Optimal. Leaf size=81 \[ a d \sqrt {d+e x^2}+\frac {1}{3} a \left (d+e x^2\right )^{3/2}-a d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )+b \text {Int}\left (\frac {\left (d+e x^2\right )^{3/2} \text {ArcTan}(c x)}{x},x\right ) \]
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Rubi [A]
time = 0.14, 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 {\left (d+e x^2\right )^{3/2} (a+b \text {ArcTan}(c x))}{x} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{x} \, dx &=a \int \frac {\left (d+e x^2\right )^{3/2}}{x} \, dx+b \int \frac {\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x} \, dx\\ &=\frac {1}{2} a \text {Subst}\left (\int \frac {(d+e x)^{3/2}}{x} \, dx,x,x^2\right )+b \int \frac {\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x} \, dx\\ &=\frac {1}{3} a \left (d+e x^2\right )^{3/2}+b \int \frac {\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x} \, dx+\frac {1}{2} (a d) \text {Subst}\left (\int \frac {\sqrt {d+e x}}{x} \, dx,x,x^2\right )\\ &=a d \sqrt {d+e x^2}+\frac {1}{3} a \left (d+e x^2\right )^{3/2}+b \int \frac {\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x} \, dx+\frac {1}{2} \left (a d^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )\\ &=a d \sqrt {d+e x^2}+\frac {1}{3} a \left (d+e x^2\right )^{3/2}+b \int \frac {\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x} \, dx+\frac {\left (a d^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )}{e}\\ &=a d \sqrt {d+e x^2}+\frac {1}{3} a \left (d+e x^2\right )^{3/2}-a d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )+b \int \frac {\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x} \, dx\\ \end {align*}
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Mathematica [A]
time = 7.06, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d+e x^2\right )^{3/2} (a+b \text {ArcTan}(c x))}{x} \, 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 \[\int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arctan \left (c x \right )\right )}{x}\, dx\]
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 \begin {gather*} \text {Exception raised: ValueError} \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 {\left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^{3/2}}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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