3.1174 \(\int x^2 \sqrt{d+e x^2} (a+b \tan ^{-1}(c x)) \, dx\)

Optimal. Leaf size=96 \[ b \text{Unintegrable}\left (x^2 \tan ^{-1}(c x) \sqrt{d+e x^2},x\right )-\frac{a d^2 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{8 e^{3/2}}+\frac{1}{4} a x^3 \sqrt{d+e x^2}+\frac{a d x \sqrt{d+e x^2}}{8 e} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.158155, 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 = {} \[ \int x^2 \sqrt{d+e x^2} \left (a+b \tan ^{-1}(c x)\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 10.9487, size = 0, normalized size = 0. \[ \int x^2 \sqrt{d+e x^2} \left (a+b \tan ^{-1}(c x)\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.787, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}\sqrt{e{x}^{2}+d} \left ( a+b\arctan \left ( cx \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(e*x^2+d)^(1/2)*(a+b*arctan(c*x)),x)

[Out]

int(x^2*(e*x^2+d)^(1/2)*(a+b*arctan(c*x)),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(x^2*(e*x^2+d)^(1/2)*(a+b*arctan(c*x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(e*x^2+d)^(1/2)*(a+b*arctan(c*x)),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{e x^{2} + d}{\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(x^2*(e*x^2+d)^(1/2)*(a+b*arctan(c*x)),x, algorithm="giac")

[Out]

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