3.1191 \(\int \frac {(d+e x^2)^{3/2} (a+b \tan ^{-1}(c x))}{x^5} \, dx\)

Optimal. Leaf size=95 \[ b \text {Int}\left (\frac {\tan ^{-1}(c x) \left (d+e x^2\right )^{3/2}}{x^5},x\right )-\frac {3 a e^2 \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{8 \sqrt {d}}-\frac {3 a e \sqrt {d+e x^2}}{8 x^2}-\frac {a \left (d+e x^2\right )^{3/2}}{4 x^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.20, 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 = {} \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{x^5} \, dx \]

Verification is Not applicable to the result.

[In]

Int[((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/x^5,x]

[Out]

(-3*a*e*Sqrt[d + e*x^2])/(8*x^2) - (a*(d + e*x^2)^(3/2))/(4*x^4) - (3*a*e^2*ArcTanh[Sqrt[d + e*x^2]/Sqrt[d]])/
(8*Sqrt[d]) + b*Defer[Int][((d + e*x^2)^(3/2)*ArcTan[c*x])/x^5, x]

Rubi steps

\begin {align*} \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{x^5} \, dx &=a \int \frac {\left (d+e x^2\right )^{3/2}}{x^5} \, dx+b \int \frac {\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^5} \, dx\\ &=\frac {1}{2} a \operatorname {Subst}\left (\int \frac {(d+e x)^{3/2}}{x^3} \, dx,x,x^2\right )+b \int \frac {\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^5} \, dx\\ &=-\frac {a \left (d+e x^2\right )^{3/2}}{4 x^4}+b \int \frac {\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^5} \, dx+\frac {1}{8} (3 a e) \operatorname {Subst}\left (\int \frac {\sqrt {d+e x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {3 a e \sqrt {d+e x^2}}{8 x^2}-\frac {a \left (d+e x^2\right )^{3/2}}{4 x^4}+b \int \frac {\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^5} \, dx+\frac {1}{16} \left (3 a e^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^2\right )\\ &=-\frac {3 a e \sqrt {d+e x^2}}{8 x^2}-\frac {a \left (d+e x^2\right )^{3/2}}{4 x^4}+b \int \frac {\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^5} \, dx+\frac {1}{8} (3 a e) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^2}\right )\\ &=-\frac {3 a e \sqrt {d+e x^2}}{8 x^2}-\frac {a \left (d+e x^2\right )^{3/2}}{4 x^4}-\frac {3 a e^2 \tanh ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{8 \sqrt {d}}+b \int \frac {\left (d+e x^2\right )^{3/2} \tan ^{-1}(c x)}{x^5} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 57.17, size = 0, normalized size = 0.00 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{x^5} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/x^5,x]

[Out]

Integrate[((d + e*x^2)^(3/2)*(a + b*ArcTan[c*x]))/x^5, x]

________________________________________________________________________________________

fricas [A]  time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a e x^{2} + a d + {\left (b e x^{2} + b d\right )} \arctan \left (c x\right )\right )} \sqrt {e x^{2} + d}}{x^{5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(3/2)*(a+b*arctan(c*x))/x^5,x, algorithm="fricas")

[Out]

integral((a*e*x^2 + a*d + (b*e*x^2 + b*d)*arctan(c*x))*sqrt(e*x^2 + d)/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)^(3/2)*(a+b*arctan(c*x))/x^5,x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [A]  time = 1.06, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arctan \left (c x \right )\right )}{x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^(3/2)*(a+b*arctan(c*x))/x^5,x)

[Out]

int((e*x^2+d)^(3/2)*(a+b*arctan(c*x))/x^5,x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^(3/2)*(a+b*arctan(c*x))/x^5,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^{3/2}}{x^5} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*atan(c*x))*(d + e*x^2)^(3/2))/x^5,x)

[Out]

int(((a + b*atan(c*x))*(d + e*x^2)^(3/2))/x^5, x)

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {atan}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{\frac {3}{2}}}{x^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**(3/2)*(a+b*atan(c*x))/x**5,x)

[Out]

Integral((a + b*atan(c*x))*(d + e*x**2)**(3/2)/x**5, x)

________________________________________________________________________________________