3.1188 \(\int \frac{(d+e x^2)^{3/2} (a+b \tan ^{-1}(c x))}{x^2} \, dx\)

Optimal. Leaf size=89 \[ b \text{Unintegrable}\left (\frac{\tan ^{-1}(c x) \left (d+e x^2\right )^{3/2}}{x^2},x\right )-\frac{a \left (d+e x^2\right )^{3/2}}{x}+\frac{3}{2} a e x \sqrt{d+e x^2}+\frac{3}{2} a d \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.174019, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{x^2} \, dx &=a \int \frac{\left (d+e x^2\right )^{3/2}}{x^2} \, dx+b \int \frac{\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^2} \, dx\\ &=-\frac{a \left (d+e x^2\right )^{3/2}}{x}+b \int \frac{\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^2} \, dx+(3 a e) \int \sqrt{d+e x^2} \, dx\\ &=\frac{3}{2} a e x \sqrt{d+e x^2}-\frac{a \left (d+e x^2\right )^{3/2}}{x}+b \int \frac{\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^2} \, dx+\frac{1}{2} (3 a d e) \int \frac{1}{\sqrt{d+e x^2}} \, dx\\ &=\frac{3}{2} a e x \sqrt{d+e x^2}-\frac{a \left (d+e x^2\right )^{3/2}}{x}+b \int \frac{\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^2} \, dx+\frac{1}{2} (3 a d e) \operatorname{Subst}\left (\int \frac{1}{1-e x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )\\ &=\frac{3}{2} a e x \sqrt{d+e x^2}-\frac{a \left (d+e x^2\right )^{3/2}}{x}+\frac{3}{2} a d \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )+b \int \frac{\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^2} \, dx\\ \end{align*}

Mathematica [A]  time = 8.68745, size = 0, normalized size = 0. \[ \int \frac{\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.574, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\arctan \left ( cx \right ) }{{x}^{2}} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a e x^{2} + a d +{\left (b e x^{2} + b d\right )} \arctan \left (c x\right )\right )} \sqrt{e x^{2} + d}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac{3}{2}}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{2} + d\right )}^{\frac{3}{2}}{\left (b \arctan \left (c x\right ) + a\right )}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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