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

Optimal. Leaf size=76 \[ b \text {Int}\left (\frac {\tan ^{-1}(c x)}{x^3 \sqrt {d+e x^2}},x\right )+\frac {a e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{2 d^{3/2}}-\frac {a \sqrt {d+e x^2}}{2 d x^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.17, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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 = 57.96, size = 0, normalized size = 0.00 \[ \int \frac {a+b \tan ^{-1}(c x)}{x^3 \sqrt {d+e x^2}} \, dx \]

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]

________________________________________________________________________________________

fricas [A]  time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \arctan \left (c x\right ) + a\right )}}{e x^{5} + d x^{3}}, x\right ) \]

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)

________________________________________________________________________________________

giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

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]

sage0*x

________________________________________________________________________________________

maple [A]  time = 1.15, size = 0, normalized size = 0.00 \[ \int \frac {a +b \arctan \left (c x \right )}{x^{3} \sqrt {e \,x^{2}+d}}\, dx \]

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 [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, a {\left (\frac {e \operatorname {arsinh}\left (\frac {d}{\sqrt {d e} {\left | x \right |}}\right )}{d^{\frac {3}{2}}} - \frac {\sqrt {e x^{2} + d}}{d x^{2}}\right )} + b \int \frac {\arctan \left (c x\right )}{\sqrt {e x^{2} + d} x^{3}}\,{d x} \]

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]

1/2*a*(e*arcsinh(d/(sqrt(d*e)*abs(x)))/d^(3/2) - sqrt(e*x^2 + d)/(d*x^2)) + b*integrate(arctan(c*x)/(sqrt(e*x^
2 + d)*x^3), x)

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x^3\,\sqrt {e\,x^2+d}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {atan}{\left (c x \right )}}{x^{3} \sqrt {d + e x^{2}}}\, dx \]

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)

________________________________________________________________________________________