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

Optimal. Leaf size=69 \[ b \text {Int}\left (\frac {x^2 \tan ^{-1}(c x)}{\left (d+e x^2\right )^{3/2}},x\right )+\frac {a \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{e^{3/2}}-\frac {a x}{e \sqrt {d+e x^2}} \]

[Out]

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

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Rubi [A]  time = 0.17, 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 = {} \[ \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx \]

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*}

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Mathematica [A]  time = 20.12, size = 0, normalized size = 0.00 \[ \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx \]

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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fricas [A]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} \arctan \left (c x\right ) + a x^{2}\right )} \sqrt {e x^{2} + d}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \]

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

[Out]

sage0*x

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maple [A]  time = 1.13, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \left (a +b \arctan \left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arctan(c*x))/(e*x^2+d)^(3/2),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 \[ \int \frac {x^2\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \left (a + b \operatorname {atan}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]

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]

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

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