3.12.81 \(\int \frac {\sqrt {d+e x^2} (a+b \text {ArcTan}(c x))}{x^5} \, dx\) [1181]

Optimal. Leaf size=98 \[ -\frac {a \sqrt {d+e x^2}}{4 x^4}-\frac {a e \sqrt {d+e x^2}}{8 d x^2}+\frac {a e^2 \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{8 d^{3/2}}+b \text {Int}\left (\frac {\sqrt {d+e x^2} \text {ArcTan}(c x)}{x^5},x\right ) \]

[Out]

1/8*a*e^2*arctanh((e*x^2+d)^(1/2)/d^(1/2))/d^(3/2)-1/4*a*(e*x^2+d)^(1/2)/x^4-1/8*a*e*(e*x^2+d)^(1/2)/d/x^2+b*U
nintegrable(arctan(c*x)*(e*x^2+d)^(1/2)/x^5,x)

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Rubi [A]
time = 0.12, 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 = {} \begin {gather*} \int \frac {\sqrt {d+e x^2} (a+b \text {ArcTan}(c x))}{x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-1/4*(a*Sqrt[d + e*x^2])/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 \text {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) \text {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 ) \text {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) \text {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*}

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Mathematica [A]
time = 10.09, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {d+e x^2} (a+b \text {ArcTan}(c x))}{x^5} \, dx \end {gather*}

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 = 0.14, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {e \,x^{2}+d}\, \left (a +b \arctan \left (c x \right )\right )}{x^{5}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

1/8*a*(arcsinh(sqrt(d)*e^(-1/2)/abs(x))*e^2/d^(3/2) - sqrt(x^2*e + d)*e^2/d^2 + (x^2*e + d)^(3/2)*e/(d^2*x^2)
- 2*(x^2*e + d)^(3/2)/(d*x^4)) + b*integrate(sqrt(x^2*e + d)*arctan(c*x)/x^5, x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}}{x^{5}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]

sage0*x

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,\sqrt {e\,x^2+d}}{x^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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