3.1198 \(\int \frac{(d+e x^2)^{5/2} (a+b \tan ^{-1}(c x))}{x^2} \, dx\)

Optimal. Leaf size=110 \[ b \text{Unintegrable}\left (\frac{\tan ^{-1}(c x) \left (d+e x^2\right )^{5/2}}{x^2},x\right )+\frac{15}{8} a d^2 \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )-\frac{a \left (d+e x^2\right )^{5/2}}{x}+\frac{5}{4} a e x \left (d+e x^2\right )^{3/2}+\frac{15}{8} a d e x \sqrt{d+e x^2} \]

[Out]

(15*a*d*e*x*Sqrt[d + e*x^2])/8 + (5*a*e*x*(d + e*x^2)^(3/2))/4 - (a*(d + e*x^2)^(5/2))/x + (15*a*d^2*Sqrt[e]*A
rcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/8 + b*Unintegrable[((d + e*x^2)^(5/2)*ArcTan[c*x])/x^2, x]

________________________________________________________________________________________

Rubi [A]  time = 0.186994, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(15*a*d*e*x*Sqrt[d + e*x^2])/8 + (5*a*e*x*(d + e*x^2)^(3/2))/4 - (a*(d + e*x^2)^(5/2))/x + (15*a*d^2*Sqrt[e]*A
rcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/8 + b*Defer[Int][((d + e*x^2)^(5/2)*ArcTan[c*x])/x^2, x]

Rubi steps

\begin{align*} \int \frac{\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{x^2} \, dx &=a \int \frac{\left (d+e x^2\right )^{5/2}}{x^2} \, dx+b \int \frac{\left (d+e x^2\right )^{5/2} \tan ^{-1}(c x)}{x^2} \, dx\\ &=-\frac{a \left (d+e x^2\right )^{5/2}}{x}+b \int \frac{\left (d+e x^2\right )^{5/2} \tan ^{-1}(c x)}{x^2} \, dx+(5 a e) \int \left (d+e x^2\right )^{3/2} \, dx\\ &=\frac{5}{4} a e x \left (d+e x^2\right )^{3/2}-\frac{a \left (d+e x^2\right )^{5/2}}{x}+b \int \frac{\left (d+e x^2\right )^{5/2} \tan ^{-1}(c x)}{x^2} \, dx+\frac{1}{4} (15 a d e) \int \sqrt{d+e x^2} \, dx\\ &=\frac{15}{8} a d e x \sqrt{d+e x^2}+\frac{5}{4} a e x \left (d+e x^2\right )^{3/2}-\frac{a \left (d+e x^2\right )^{5/2}}{x}+b \int \frac{\left (d+e x^2\right )^{5/2} \tan ^{-1}(c x)}{x^2} \, dx+\frac{1}{8} \left (15 a d^2 e\right ) \int \frac{1}{\sqrt{d+e x^2}} \, dx\\ &=\frac{15}{8} a d e x \sqrt{d+e x^2}+\frac{5}{4} a e x \left (d+e x^2\right )^{3/2}-\frac{a \left (d+e x^2\right )^{5/2}}{x}+b \int \frac{\left (d+e x^2\right )^{5/2} \tan ^{-1}(c x)}{x^2} \, dx+\frac{1}{8} \left (15 a d^2 e\right ) \operatorname{Subst}\left (\int \frac{1}{1-e x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )\\ &=\frac{15}{8} a d e x \sqrt{d+e x^2}+\frac{5}{4} a e x \left (d+e x^2\right )^{3/2}-\frac{a \left (d+e x^2\right )^{5/2}}{x}+\frac{15}{8} a d^2 \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )+b \int \frac{\left (d+e x^2\right )^{5/2} \tan ^{-1}(c x)}{x^2} \, dx\\ \end{align*}

Mathematica [A]  time = 9.01508, size = 0, normalized size = 0. \[ \int \frac{\left (d+e x^2\right )^{5/2} \left (a+b \tan ^{-1}(c x)\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.585, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\arctan \left ( cx \right ) }{{x}^{2}} \left ( e{x}^{2}+d \right ) ^{{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a e^{2} x^{4} + 2 \, a d e x^{2} + a d^{2} +{\left (b e^{2} x^{4} + 2 \, b d e x^{2} + b d^{2}\right )} \arctan \left (c x\right )\right )} \sqrt{e x^{2} + d}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a*e^2*x^4 + 2*a*d*e*x^2 + a*d^2 + (b*e^2*x^4 + 2*b*d*e*x^2 + b*d^2)*arctan(c*x))*sqrt(e*x^2 + d)/x^2
, x)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{2} + d\right )}^{\frac{5}{2}}{\left (b \arctan \left (c x\right ) + a\right )}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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