3.1188 \(\int \frac {(d+e x^2)^{3/2} (a+b \tan ^{-1}(c x))}{x^2} \, dx\)

Optimal. Leaf size=90 \[ b \text {Int}\left (\frac {\tan ^{-1}(c x) \left (d+e x^2\right )^{3/2}}{x^2},x\right )-\frac {a \left (d+e x^2\right )^{3/2}}{x}+\frac {3}{2} a e x \sqrt {d+e x^2}+\frac {3}{2} a d \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \]

[Out]

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

________________________________________________________________________________________

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

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 9.25, size = 0, normalized size = 0.00 \[ \int \frac {\left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a e x^{2} + a d + {\left (b e x^{2} + b d\right )} \arctan \left (c x\right )\right )} \sqrt {e x^{2} + d}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

maple [A]  time = 1.14, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arctan \left (c x \right )\right )}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e-c^2*d>0)', see `assume?` for
more details)Is e-c^2*d positive or negative?

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________