3.1191 \(\int \frac{(d+e x^2)^{3/2} (a+b \tan ^{-1}(c x))}{x^5} \, dx\)

Optimal. Leaf size=94 \[ b \text{Unintegrable}\left (\frac{\tan ^{-1}(c x) \left (d+e x^2\right )^{3/2}}{x^5},x\right )-\frac{3 a e^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{8 \sqrt{d}}-\frac{3 a e \sqrt{d+e x^2}}{8 x^2}-\frac{a \left (d+e x^2\right )^{3/2}}{4 x^4} \]

[Out]

(-3*a*e*Sqrt[d + e*x^2])/(8*x^2) - (a*(d + e*x^2)^(3/2))/(4*x^4) - (3*a*e^2*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/
(8*Sqrt[d]) + b*Unintegrable[((d + e*x^2)^(3/2)*ArcTan[c*x])/x^5, x]

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Rubi [A]  time = 0.196105, 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 = {} \[ \int \frac{\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{x^5} \, dx \]

Verification is Not applicable to the result.

[In]

Int[((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/x^5,x]

[Out]

(-3*a*e*Sqrt[d + e*x^2])/(8*x^2) - (a*(d + e*x^2)^(3/2))/(4*x^4) - (3*a*e^2*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/
(8*Sqrt[d]) + b*Defer[Int][((d + e*x^2)^(3/2)*ArcTan[c*x])/x^5, x]

Rubi steps

\begin{align*} \int \frac{\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{x^5} \, dx &=a \int \frac{\left (d+e x^2\right )^{3/2}}{x^5} \, dx+b \int \frac{\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^5} \, dx\\ &=\frac{1}{2} a \operatorname{Subst}\left (\int \frac{(d+e x)^{3/2}}{x^3} \, dx,x,x^2\right )+b \int \frac{\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^5} \, dx\\ &=-\frac{a \left (d+e x^2\right )^{3/2}}{4 x^4}+b \int \frac{\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^5} \, dx+\frac{1}{8} (3 a e) \operatorname{Subst}\left (\int \frac{\sqrt{d+e x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac{3 a e \sqrt{d+e x^2}}{8 x^2}-\frac{a \left (d+e x^2\right )^{3/2}}{4 x^4}+b \int \frac{\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^5} \, dx+\frac{1}{16} \left (3 a e^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{d+e x}} \, dx,x,x^2\right )\\ &=-\frac{3 a e \sqrt{d+e x^2}}{8 x^2}-\frac{a \left (d+e x^2\right )^{3/2}}{4 x^4}+b \int \frac{\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^5} \, dx+\frac{1}{8} (3 a e) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x^2}{e}} \, dx,x,\sqrt{d+e x^2}\right )\\ &=-\frac{3 a e \sqrt{d+e x^2}}{8 x^2}-\frac{a \left (d+e x^2\right )^{3/2}}{4 x^4}-\frac{3 a e^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{8 \sqrt{d}}+b \int \frac{\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^5} \, dx\\ \end{align*}

Mathematica [A]  time = 49.6443, size = 0, normalized size = 0. \[ \int \frac{\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{x^5} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/x^5,x]

[Out]

Integrate[((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/x^5, x]

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Maple [A]  time = 0.594, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\arctan \left ( cx \right ) }{{x}^{5}} \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^5,x)

[Out]

int((e*x^2+d)^(3/2)*(a+b*arctan(c*x))/x^5,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^5,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^{5}}, 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^5,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^5, 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^{5}}\, 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**5,x)

[Out]

Integral((a + b*atan(c*x))*(d + e*x**2)**(3/2)/x**5, 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^{5}}\,{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^5,x, algorithm="giac")

[Out]

integrate((e*x^2 + d)^(3/2)*(b*arctan(c*x) + a)/x^5, x)