3.1213 \(\int \frac{a+b \tan ^{-1}(c x)}{x (d+e x^2)^{3/2}} \, dx\)

Optimal. Leaf size=66 \[ b \text{Unintegrable}\left (\frac{\tan ^{-1}(c x)}{x \left (d+e x^2\right )^{3/2}},x\right )-\frac{a \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{d^{3/2}}+\frac{a}{d \sqrt{d+e x^2}} \]

[Out]

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

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Rubi [A]  time = 0.176169, 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 = {} \[ \int \frac{a+b \tan ^{-1}(c x)}{x \left (d+e x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 49.9619, size = 0, normalized size = 0. \[ \int \frac{a+b \tan ^{-1}(c x)}{x \left (d+e x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.558, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\arctan \left ( cx \right ) }{x} \left ( e{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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