3.1181 \(\int \frac{\sqrt{d+e x^2} (a+b \tan ^{-1}(c x))}{x^5} \, dx\)

Optimal. Leaf size=97 \[ b \text{Unintegrable}\left (\frac{\tan ^{-1}(c x) \sqrt{d+e x^2}}{x^5},x\right )+\frac{a e^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{8 d^{3/2}}-\frac{a e \sqrt{d+e x^2}}{8 d x^2}-\frac{a \sqrt{d+e x^2}}{4 x^4} \]

[Out]

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

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Rubi [A]  time = 0.175813, 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 = {} \[ \int \frac{\sqrt{d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{x^5} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 50.3483, size = 0, normalized size = 0. \[ \int \frac{\sqrt{d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{x^5} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.785, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\arctan \left ( cx \right ) }{{x}^{5}}\sqrt{e{x}^{2}+d}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^(1/2)*(a+b*arctan(c*x))/x^5,x)

[Out]

int((e*x^2+d)^(1/2)*(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)^(1/2)*(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{\sqrt{e x^{2} + d}{\left (b \arctan \left (c x\right ) + a\right )}}{x^{5}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(1/2)*(a+b*arctan(c*x))/x^5,x, algorithm="fricas")

[Out]

integral(sqrt(e*x^2 + d)*(b*arctan(c*x) + a)/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 ) \sqrt{d + e x^{2}}}{x^{5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**(1/2)*(a+b*atan(c*x))/x**5,x)

[Out]

Integral((a + b*atan(c*x))*sqrt(d + e*x**2)/x**5, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e x^{2} + d}{\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)^(1/2)*(a+b*arctan(c*x))/x^5,x, algorithm="giac")

[Out]

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