3.1189 \(\int \frac{(d+e x^2)^{3/2} (a+b \tan ^{-1}(c x))}{x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.204055, 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 = {} \[ \int \frac{\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{x^3} \, dx &=a \int \frac{\left (d+e x^2\right )^{3/2}}{x^3} \, dx+b \int \frac{\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^3} \, dx\\ &=\frac{1}{2} a \operatorname{Subst}\left (\int \frac{(d+e x)^{3/2}}{x^2} \, dx,x,x^2\right )+b \int \frac{\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^3} \, dx\\ &=-\frac{a \left (d+e x^2\right )^{3/2}}{2 x^2}+b \int \frac{\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^3} \, dx+\frac{1}{4} (3 a e) \operatorname{Subst}\left (\int \frac{\sqrt{d+e x}}{x} \, dx,x,x^2\right )\\ &=\frac{3}{2} a e \sqrt{d+e x^2}-\frac{a \left (d+e x^2\right )^{3/2}}{2 x^2}+b \int \frac{\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^3} \, dx+\frac{1}{4} (3 a d e) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d+e x}} \, dx,x,x^2\right )\\ &=\frac{3}{2} a e \sqrt{d+e x^2}-\frac{a \left (d+e x^2\right )^{3/2}}{2 x^2}+b \int \frac{\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^3} \, dx+\frac{1}{2} (3 a d) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x^2}\right )\\ &=\frac{3}{2} a e \sqrt{d+e x^2}-\frac{a \left (d+e x^2\right )^{3/2}}{2 x^2}-\frac{3}{2} a \sqrt{d} e \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^3} \, dx\\ \end{align*}

Mathematica [A]  time = 47.9191, size = 0, normalized size = 0. \[ \int \frac{\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.579, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\arctan \left ( cx \right ) }{{x}^{3}} \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^3,x)

[Out]

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

[Out]

Integral((a + b*atan(c*x))*(d + e*x**2)**(3/2)/x**3, 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^{3}}\,{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^3,x, algorithm="giac")

[Out]

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