3.458 \(\int x^m (c+a^2 c x^2)^2 \tan ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=25 \[ \text {Int}\left (x^m \left (a^2 c x^2+c\right )^2 \tan ^{-1}(a x)^3,x\right ) \]

[Out]

Unintegrable(x^m*(a^2*c*x^2+c)^2*arctan(a*x)^3,x)

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Rubi [A]  time = 0.05, 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 = {} \[ \int x^m \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3 \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^m*(c + a^2*c*x^2)^2*ArcTan[a*x]^3,x]

[Out]

Defer[Int][x^m*(c + a^2*c*x^2)^2*ArcTan[a*x]^3, x]

Rubi steps

\begin {align*} \int x^m \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3 \, dx &=\int x^m \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3 \, dx\\ \end {align*}

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Mathematica [A]  time = 2.12, size = 0, normalized size = 0.00 \[ \int x^m \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3 \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x^m*(c + a^2*c*x^2)^2*ArcTan[a*x]^3,x]

[Out]

Integrate[x^m*(c + a^2*c*x^2)^2*ArcTan[a*x]^3, x]

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fricas [A]  time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} x^{m} \arctan \left (a x\right )^{3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(a^2*c*x^2+c)^2*arctan(a*x)^3,x, algorithm="fricas")

[Out]

integral((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)*x^m*arctan(a*x)^3, x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(a^2*c*x^2+c)^2*arctan(a*x)^3,x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 1.54, size = 0, normalized size = 0.00 \[ \int x^{m} \left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(a^2*c*x^2+c)^2*arctan(a*x)^3,x)

[Out]

int(x^m*(a^2*c*x^2+c)^2*arctan(a*x)^3,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\frac {15}{2} \, {\left ({\left (a^{4} c^{2} m^{2} + 4 \, a^{4} c^{2} m + 3 \, a^{4} c^{2}\right )} x^{5} + 2 \, {\left (a^{2} c^{2} m^{2} + 6 \, a^{2} c^{2} m + 5 \, a^{2} c^{2}\right )} x^{3} + {\left (c^{2} m^{2} + 8 \, c^{2} m + 15 \, c^{2}\right )} x\right )} x^{m} \arctan \left (a x\right )^{3} - \frac {21}{8} \, {\left ({\left (a^{4} c^{2} m^{2} + 4 \, a^{4} c^{2} m + 3 \, a^{4} c^{2}\right )} x^{5} + 2 \, {\left (a^{2} c^{2} m^{2} + 6 \, a^{2} c^{2} m + 5 \, a^{2} c^{2}\right )} x^{3} + {\left (c^{2} m^{2} + 8 \, c^{2} m + 15 \, c^{2}\right )} x\right )} x^{m} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right )^{2} + {\left (m^{3} + 9 \, m^{2} + 23 \, m + 15\right )} \int \frac {196 \, {\left ({\left (a^{6} c^{2} m^{3} + 9 \, a^{6} c^{2} m^{2} + 23 \, a^{6} c^{2} m + 15 \, a^{6} c^{2}\right )} x^{6} + c^{2} m^{3} + 3 \, {\left (a^{4} c^{2} m^{3} + 9 \, a^{4} c^{2} m^{2} + 23 \, a^{4} c^{2} m + 15 \, a^{4} c^{2}\right )} x^{4} + 9 \, c^{2} m^{2} + 23 \, c^{2} m + 3 \, {\left (a^{2} c^{2} m^{3} + 9 \, a^{2} c^{2} m^{2} + 23 \, a^{2} c^{2} m + 15 \, a^{2} c^{2}\right )} x^{2} + 15 \, c^{2}\right )} x^{m} \arctan \left (a x\right )^{3} - 180 \, {\left ({\left (a^{5} c^{2} m^{2} + 4 \, a^{5} c^{2} m + 3 \, a^{5} c^{2}\right )} x^{5} + 2 \, {\left (a^{3} c^{2} m^{2} + 6 \, a^{3} c^{2} m + 5 \, a^{3} c^{2}\right )} x^{3} + {\left (a c^{2} m^{2} + 8 \, a c^{2} m + 15 \, a c^{2}\right )} x\right )} x^{m} \arctan \left (a x\right )^{2} + 84 \, {\left ({\left (a^{6} c^{2} m^{2} + 4 \, a^{6} c^{2} m + 3 \, a^{6} c^{2}\right )} x^{6} + 2 \, {\left (a^{4} c^{2} m^{2} + 6 \, a^{4} c^{2} m + 5 \, a^{4} c^{2}\right )} x^{4} + {\left (a^{2} c^{2} m^{2} + 8 \, a^{2} c^{2} m + 15 \, a^{2} c^{2}\right )} x^{2}\right )} x^{m} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right ) + 21 \, {\left ({\left ({\left (a^{6} c^{2} m^{3} + 9 \, a^{6} c^{2} m^{2} + 23 \, a^{6} c^{2} m + 15 \, a^{6} c^{2}\right )} x^{6} + c^{2} m^{3} + 3 \, {\left (a^{4} c^{2} m^{3} + 9 \, a^{4} c^{2} m^{2} + 23 \, a^{4} c^{2} m + 15 \, a^{4} c^{2}\right )} x^{4} + 9 \, c^{2} m^{2} + 23 \, c^{2} m + 3 \, {\left (a^{2} c^{2} m^{3} + 9 \, a^{2} c^{2} m^{2} + 23 \, a^{2} c^{2} m + 15 \, a^{2} c^{2}\right )} x^{2} + 15 \, c^{2}\right )} x^{m} \arctan \left (a x\right ) + {\left ({\left (a^{5} c^{2} m^{2} + 4 \, a^{5} c^{2} m + 3 \, a^{5} c^{2}\right )} x^{5} + 2 \, {\left (a^{3} c^{2} m^{2} + 6 \, a^{3} c^{2} m + 5 \, a^{3} c^{2}\right )} x^{3} + {\left (a c^{2} m^{2} + 8 \, a c^{2} m + 15 \, a c^{2}\right )} x\right )} x^{m}\right )} \log \left (a^{2} x^{2} + 1\right )^{2}}{8 \, {\left (m^{3} + {\left (a^{2} m^{3} + 9 \, a^{2} m^{2} + 23 \, a^{2} m + 15 \, a^{2}\right )} x^{2} + 9 \, m^{2} + 23 \, m + 15\right )}}\,{d x}}{32 \, {\left (m^{3} + 9 \, m^{2} + 23 \, m + 15\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(a^2*c*x^2+c)^2*arctan(a*x)^3,x, algorithm="maxima")

[Out]

1/32*(4*((a^4*c^2*m^2 + 4*a^4*c^2*m + 3*a^4*c^2)*x^5 + 2*(a^2*c^2*m^2 + 6*a^2*c^2*m + 5*a^2*c^2)*x^3 + (c^2*m^
2 + 8*c^2*m + 15*c^2)*x)*x^m*arctan(a*x)^3 - 3*((a^4*c^2*m^2 + 4*a^4*c^2*m + 3*a^4*c^2)*x^5 + 2*(a^2*c^2*m^2 +
6*a^2*c^2*m + 5*a^2*c^2)*x^3 + (c^2*m^2 + 8*c^2*m + 15*c^2)*x)*x^m*arctan(a*x)*log(a^2*x^2 + 1)^2 + 32*(m^3 +
9*m^2 + 23*m + 15)*integrate(1/32*(28*((a^6*c^2*m^3 + 9*a^6*c^2*m^2 + 23*a^6*c^2*m + 15*a^6*c^2)*x^6 + c^2*m^
3 + 3*(a^4*c^2*m^3 + 9*a^4*c^2*m^2 + 23*a^4*c^2*m + 15*a^4*c^2)*x^4 + 9*c^2*m^2 + 23*c^2*m + 3*(a^2*c^2*m^3 +
9*a^2*c^2*m^2 + 23*a^2*c^2*m + 15*a^2*c^2)*x^2 + 15*c^2)*x^m*arctan(a*x)^3 - 12*((a^5*c^2*m^2 + 4*a^5*c^2*m +
3*a^5*c^2)*x^5 + 2*(a^3*c^2*m^2 + 6*a^3*c^2*m + 5*a^3*c^2)*x^3 + (a*c^2*m^2 + 8*a*c^2*m + 15*a*c^2)*x)*x^m*arc
tan(a*x)^2 + 12*((a^6*c^2*m^2 + 4*a^6*c^2*m + 3*a^6*c^2)*x^6 + 2*(a^4*c^2*m^2 + 6*a^4*c^2*m + 5*a^4*c^2)*x^4 +
(a^2*c^2*m^2 + 8*a^2*c^2*m + 15*a^2*c^2)*x^2)*x^m*arctan(a*x)*log(a^2*x^2 + 1) + 3*(((a^6*c^2*m^3 + 9*a^6*c^2
*m^2 + 23*a^6*c^2*m + 15*a^6*c^2)*x^6 + c^2*m^3 + 3*(a^4*c^2*m^3 + 9*a^4*c^2*m^2 + 23*a^4*c^2*m + 15*a^4*c^2)*
x^4 + 9*c^2*m^2 + 23*c^2*m + 3*(a^2*c^2*m^3 + 9*a^2*c^2*m^2 + 23*a^2*c^2*m + 15*a^2*c^2)*x^2 + 15*c^2)*x^m*arc
tan(a*x) + ((a^5*c^2*m^2 + 4*a^5*c^2*m + 3*a^5*c^2)*x^5 + 2*(a^3*c^2*m^2 + 6*a^3*c^2*m + 5*a^3*c^2)*x^3 + (a*c
^2*m^2 + 8*a*c^2*m + 15*a*c^2)*x)*x^m)*log(a^2*x^2 + 1)^2)/(m^3 + (a^2*m^3 + 9*a^2*m^2 + 23*a^2*m + 15*a^2)*x^
2 + 9*m^2 + 23*m + 15), x))/(m^3 + 9*m^2 + 23*m + 15)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int x^m\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*atan(a*x)^3*(c + a^2*c*x^2)^2,x)

[Out]

int(x^m*atan(a*x)^3*(c + a^2*c*x^2)^2, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ c^{2} \left (\int x^{m} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int 2 a^{2} x^{2} x^{m} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{4} x^{4} x^{m} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(a**2*c*x**2+c)**2*atan(a*x)**3,x)

[Out]

c**2*(Integral(x**m*atan(a*x)**3, x) + Integral(2*a**2*x**2*x**m*atan(a*x)**3, x) + Integral(a**4*x**4*x**m*at
an(a*x)**3, x))

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