3.1210 \(\int \frac{x^2 (a+b \tan ^{-1}(c x))}{(d+e x^2)^{3/2}} \, dx\)

Optimal. Leaf size=68 \[ b \text{Unintegrable}\left (\frac{x^2 \tan ^{-1}(c x)}{\left (d+e x^2\right )^{3/2}},x\right )+\frac{a \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{e^{3/2}}-\frac{a x}{e \sqrt{d+e x^2}} \]

[Out]

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

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Rubi [A]  time = 0.167533, 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 = {} \[ \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 18.2179, size = 0, normalized size = 0. \[ \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.598, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ( a+b\arctan \left ( cx \right ) \right ) \left ( e{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arctan \left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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