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

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

[Out]

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)

________________________________________________________________________________________

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

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 = 58.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \tan ^{-1}(c x)\right )}{x^5} \, dx \]

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]

________________________________________________________________________________________

fricas [A]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \arctan \left (c x\right ) + a\right )}}{x^{5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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^5,x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [A]  time = 1.14, 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)

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{8} \, {\left (\frac {e^{2} \operatorname {arsinh}\left (\frac {d}{\sqrt {d e} {\left | x \right |}}\right )}{d^{\frac {3}{2}}} - \frac {\sqrt {e x^{2} + d} e^{2}}{d^{2}} + \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}} e}{d^{2} x^{2}} - \frac {2 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}}}{d x^{4}}\right )} a + b \int \frac {\sqrt {e x^{2} + d} \arctan \left (c x\right )}{x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________