3.1207 \(\int \frac{a+b \tan ^{-1}(c x)}{x^3 \sqrt{d+e x^2}} \, dx\)

Optimal. Leaf size=75 \[ b \text{Unintegrable}\left (\frac{\tan ^{-1}(c x)}{x^3 \sqrt{d+e x^2}},x\right )+\frac{a e \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{2 d^{3/2}}-\frac{a \sqrt{d+e x^2}}{2 d x^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.172151, 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 = {} \[ \int \frac{a+b \tan ^{-1}(c x)}{x^3 \sqrt{d+e x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 50.9552, size = 0, normalized size = 0. \[ \int \frac{a+b \tan ^{-1}(c x)}{x^3 \sqrt{d+e x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.776, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\arctan \left ( cx \right ) }{{x}^{3}}{\frac{1}{\sqrt{e{x}^{2}+d}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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^3/(e*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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 x^{5} + d x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x))/x^3/(e*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atan(c*x))/x**3/(e*x**2+d)**(1/2),x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x))/x^3/(e*x^2+d)^(1/2),x, algorithm="giac")

[Out]

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