3.1179 \(\int \frac {\sqrt {d+e x^2} (a+b \tan ^{-1}(c x))}{x^3} \, dx\)

Optimal. Leaf size=73 \[ b \text {Int}\left (\frac {\tan ^{-1}(c x) \sqrt {d+e x^2}}{x^3},x\right )-\frac {a \sqrt {d+e x^2}}{2 x^2}-\frac {a e \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{2 \sqrt {d}} \]

[Out]

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

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

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 54.93, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \arctan \left (c x\right ) + a\right )}}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [A]  time = 1.12, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e \,x^{2}+d}\, \left (a +b \arctan \left (c x \right )\right )}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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