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

Optimal. Leaf size=97 \[ b \text {Int}\left (x^2 \tan ^{-1}(c x) \sqrt {d+e x^2},x\right )-\frac {a d^2 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{8 e^{3/2}}+\frac {a d x \sqrt {d+e x^2}}{8 e}+\frac {1}{4} a x^3 \sqrt {d+e x^2} \]

[Out]

-1/8*a*d^2*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))/e^(3/2)+1/8*a*d*x*(e*x^2+d)^(1/2)/e+1/4*a*x^3*(e*x^2+d)^(1/2)+b*
Unintegrable(x^2*arctan(c*x)*(e*x^2+d)^(1/2),x)

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Rubi [A]  time = 0.16, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 11.29, size = 0, normalized size = 0.00 \[ \int x^2 \sqrt {d+e x^2} \left (a+b \tan ^{-1}(c x)\right ) \, dx \]

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]

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fricas [A]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{2} \arctan \left (c x\right ) + a x^{2}\right )} \sqrt {e x^{2} + d}, x\right ) \]

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)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

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]

sage0*x

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maple [A]  time = 1.13, size = 0, normalized size = 0.00 \[ \int x^{2} \sqrt {e \,x^{2}+d}\, \left (a +b \arctan \left (c x \right )\right )\, dx \]

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)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e-c^2*d>0)', see `assume?` for
more details)Is e-c^2*d positive or negative?

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,\sqrt {e\,x^2+d} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \left (a + b \operatorname {atan}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}\, dx \]

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)

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