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

Optimal. Leaf size=67 \[ b \text {Int}\left (\frac {\tan ^{-1}(c x)}{x \left (d+e x^2\right )^{3/2}},x\right )-\frac {a \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{d^{3/2}}+\frac {a}{d \sqrt {d+e x^2}} \]

[Out]

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

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Rubi [A]  time = 0.18, 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 = {} \[ \int \frac {a+b \tan ^{-1}(c x)}{x \left (d+e x^2\right )^{3/2}} \, dx \]

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*}

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Mathematica [A]  time = 56.73, size = 0, normalized size = 0.00 \[ \int \frac {a+b \tan ^{-1}(c x)}{x \left (d+e x^2\right )^{3/2}} \, dx \]

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

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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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/(e*x^2+d)^(3/2),x, algorithm="giac")

[Out]

sage0*x

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maple [A]  time = 1.11, size = 0, normalized size = 0.00 \[ \int \frac {a +b \arctan \left (c x \right )}{x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx \]

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

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]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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]

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

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