3.12.87 \(\int \frac {(d+e x^2)^{3/2} (a+b \text {ArcTan}(c x))}{x} \, dx\) [1187]

Optimal. Leaf size=81 \[ a d \sqrt {d+e x^2}+\frac {1}{3} a \left (d+e x^2\right )^{3/2}-a d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )+b \text {Int}\left (\frac {\left (d+e x^2\right )^{3/2} \text {ArcTan}(c x)}{x},x\right ) \]

[Out]

1/3*a*(e*x^2+d)^(3/2)-a*d^(3/2)*arctanh((e*x^2+d)^(1/2)/d^(1/2))+a*d*(e*x^2+d)^(1/2)+b*Unintegrable((e*x^2+d)^
(3/2)*arctan(c*x)/x,x)

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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.

[In]

Int[((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/x,x]

[Out]

a*d*Sqrt[d + e*x^2] + (a*(d + e*x^2)^(3/2))/3 - a*d^(3/2)*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]] + b*Defer[Int][((d
+ e*x^2)^(3/2)*ArcTan[c*x])/x, x]

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.

[In]

Integrate[((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/x,x]

[Out]

Integrate[((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/x, x]

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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.

[In]

int((e*x^2+d)^(3/2)*(a+b*arctan(c*x))/x,x)

[Out]

int((e*x^2+d)^(3/2)*(a+b*arctan(c*x))/x,x)

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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.

[In]

integrate((e*x^2+d)^(3/2)*(a+b*arctan(c*x))/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c^2*d-%e>0)', see `assume?` fo
r more detai

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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.

[In]

integrate((e*x^2+d)^(3/2)*(a+b*arctan(c*x))/x,x, algorithm="fricas")

[Out]

integral((a*x^2*e + a*d + (b*x^2*e + b*d)*arctan(c*x))*sqrt(x^2*e + d)/x, x)

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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.

[In]

integrate((e*x**2+d)**(3/2)*(a+b*atan(c*x))/x,x)

[Out]

Integral((a + b*atan(c*x))*(d + e*x**2)**(3/2)/x, x)

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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.

[In]

integrate((e*x^2+d)^(3/2)*(a+b*arctan(c*x))/x,x, algorithm="giac")

[Out]

Timed out

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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.

[In]

int(((a + b*atan(c*x))*(d + e*x^2)^(3/2))/x,x)

[Out]

int(((a + b*atan(c*x))*(d + e*x^2)^(3/2))/x, x)

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