3.1184 \(\int x^2 (d+e x^2)^{3/2} (a+b \tan ^{-1}(c x)) \, dx\)

Optimal. Leaf size=118 \[ b \text{Unintegrable}\left (x^2 \tan ^{-1}(c x) \left (d+e x^2\right )^{3/2},x\right )-\frac{a d^3 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{16 e^{3/2}}+\frac{a d^2 x \sqrt{d+e x^2}}{16 e}+\frac{1}{8} a d x^3 \sqrt{d+e x^2}+\frac{1}{6} a x^3 \left (d+e x^2\right )^{3/2} \]

[Out]

(a*d^2*x*Sqrt[d + e*x^2])/(16*e) + (a*d*x^3*Sqrt[d + e*x^2])/8 + (a*x^3*(d + e*x^2)^(3/2))/6 - (a*d^3*ArcTanh[
(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(16*e^(3/2)) + b*Unintegrable[x^2*(d + e*x^2)^(3/2)*ArcTan[c*x], x]

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Rubi [A]  time = 0.19486, 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 = {} \[ \int x^2 \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(a*d^2*x*Sqrt[d + e*x^2])/(16*e) + (a*d*x^3*Sqrt[d + e*x^2])/8 + (a*x^3*(d + e*x^2)^(3/2))/6 - (a*d^3*ArcTanh[
(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(16*e^(3/2)) + b*Defer[Int][x^2*(d + e*x^2)^(3/2)*ArcTan[c*x], x]

Rubi steps

\begin{align*} \int x^2 \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right ) \, dx &=a \int x^2 \left (d+e x^2\right )^{3/2} \, dx+b \int x^2 \left (d+e x^2\right )^{3/2} \tan ^{-1}(c x) \, dx\\ &=\frac{1}{6} a x^3 \left (d+e x^2\right )^{3/2}+b \int x^2 \left (d+e x^2\right )^{3/2} \tan ^{-1}(c x) \, dx+\frac{1}{2} (a d) \int x^2 \sqrt{d+e x^2} \, dx\\ &=\frac{1}{8} a d x^3 \sqrt{d+e x^2}+\frac{1}{6} a x^3 \left (d+e x^2\right )^{3/2}+b \int x^2 \left (d+e x^2\right )^{3/2} \tan ^{-1}(c x) \, dx+\frac{1}{8} \left (a d^2\right ) \int \frac{x^2}{\sqrt{d+e x^2}} \, dx\\ &=\frac{a d^2 x \sqrt{d+e x^2}}{16 e}+\frac{1}{8} a d x^3 \sqrt{d+e x^2}+\frac{1}{6} a x^3 \left (d+e x^2\right )^{3/2}+b \int x^2 \left (d+e x^2\right )^{3/2} \tan ^{-1}(c x) \, dx-\frac{\left (a d^3\right ) \int \frac{1}{\sqrt{d+e x^2}} \, dx}{16 e}\\ &=\frac{a d^2 x \sqrt{d+e x^2}}{16 e}+\frac{1}{8} a d x^3 \sqrt{d+e x^2}+\frac{1}{6} a x^3 \left (d+e x^2\right )^{3/2}+b \int x^2 \left (d+e x^2\right )^{3/2} \tan ^{-1}(c x) \, dx-\frac{\left (a d^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-e x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )}{16 e}\\ &=\frac{a d^2 x \sqrt{d+e x^2}}{16 e}+\frac{1}{8} a d x^3 \sqrt{d+e x^2}+\frac{1}{6} a x^3 \left (d+e x^2\right )^{3/2}-\frac{a d^3 \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d+e x^2}}\right )}{16 e^{3/2}}+b \int x^2 \left (d+e x^2\right )^{3/2} \tan ^{-1}(c x) \, dx\\ \end{align*}

Mathematica [A]  time = 10.7402, size = 0, normalized size = 0. \[ \int x^2 \left (d+e x^2\right )^{3/2} \left (a+b \tan ^{-1}(c x)\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.593, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}} \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)^(3/2)*(a+b*arctan(c*x)),x)

[Out]

int(x^2*(e*x^2+d)^(3/2)*(a+b*arctan(c*x)),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*(e*x^2+d)^(3/2)*(a+b*arctan(c*x)),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 ({\left (a e x^{4} + a d x^{2} +{\left (b e x^{4} + b d x^{2}\right )} \arctan \left (c x\right )\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)^(3/2)*(a+b*arctan(c*x)),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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