3.1202 \(\int \frac{x^2 (a+b \tan ^{-1}(c x))}{\sqrt{d+e x^2}} \, dx\)

Optimal. Leaf size=74 \[ b \text{Unintegrable}\left (\frac{x^2 \tan ^{-1}(c x)}{\sqrt{d+e x^2}},x\right )-\frac{a d \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{2 e^{3/2}}+\frac{a x \sqrt{d+e x^2}}{2 e} \]

[Out]

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

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Rubi [A]  time = 0.155581, 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 = {} \[ \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{\sqrt{d+e x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 9.11565, size = 0, normalized size = 0. \[ \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{\sqrt{d+e x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.785, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ( a+b\arctan \left ( cx \right ) \right ){\frac{1}{\sqrt{e{x}^{2}+d}}}}\, 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)^(1/2),x)

[Out]

int(x^2*(a+b*arctan(c*x))/(e*x^2+d)^(1/2),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)^(1/2),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{b x^{2} \arctan \left (c x\right ) + a x^{2}}{\sqrt{e x^{2} + d}}, 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)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \left (a + b \operatorname{atan}{\left (c x \right )}\right )}{\sqrt{d + e x^{2}}}\, dx \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)**(1/2),x)

[Out]

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

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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}}{\sqrt{e x^{2} + d}}\,{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)^(1/2),x, algorithm="giac")

[Out]

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