3.1 \(\int x^5 (a+b \tan ^{-1}(c x)) \, dx\)

Optimal. Leaf size=59 \[ \frac{1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac{b x^3}{18 c^3}-\frac{b x}{6 c^5}+\frac{b \tan ^{-1}(c x)}{6 c^6}-\frac{b x^5}{30 c} \]

[Out]

-(b*x)/(6*c^5) + (b*x^3)/(18*c^3) - (b*x^5)/(30*c) + (b*ArcTan[c*x])/(6*c^6) + (x^6*(a + b*ArcTan[c*x]))/6

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Rubi [A]  time = 0.0316717, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4852, 302, 203} \[ \frac{1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac{b x^3}{18 c^3}-\frac{b x}{6 c^5}+\frac{b \tan ^{-1}(c x)}{6 c^6}-\frac{b x^5}{30 c} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*ArcTan[c*x]),x]

[Out]

-(b*x)/(6*c^5) + (b*x^3)/(18*c^3) - (b*x^5)/(30*c) + (b*ArcTan[c*x])/(6*c^6) + (x^6*(a + b*ArcTan[c*x]))/6

Rule 4852

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTa
n[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcTan[c*x])^(p - 1))/(1 + c^
2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int x^5 \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac{1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{6} (b c) \int \frac{x^6}{1+c^2 x^2} \, dx\\ &=\frac{1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{6} (b c) \int \left (\frac{1}{c^6}-\frac{x^2}{c^4}+\frac{x^4}{c^2}-\frac{1}{c^6 \left (1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac{b x}{6 c^5}+\frac{b x^3}{18 c^3}-\frac{b x^5}{30 c}+\frac{1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )+\frac{b \int \frac{1}{1+c^2 x^2} \, dx}{6 c^5}\\ &=-\frac{b x}{6 c^5}+\frac{b x^3}{18 c^3}-\frac{b x^5}{30 c}+\frac{b \tan ^{-1}(c x)}{6 c^6}+\frac{1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.0031498, size = 64, normalized size = 1.08 \[ \frac{a x^6}{6}+\frac{b x^3}{18 c^3}-\frac{b x}{6 c^5}+\frac{b \tan ^{-1}(c x)}{6 c^6}-\frac{b x^5}{30 c}+\frac{1}{6} b x^6 \tan ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*ArcTan[c*x]),x]

[Out]

-(b*x)/(6*c^5) + (b*x^3)/(18*c^3) - (b*x^5)/(30*c) + (a*x^6)/6 + (b*ArcTan[c*x])/(6*c^6) + (b*x^6*ArcTan[c*x])
/6

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Maple [A]  time = 0.006, size = 53, normalized size = 0.9 \begin{align*}{\frac{{x}^{6}a}{6}}+{\frac{b{x}^{6}\arctan \left ( cx \right ) }{6}}-{\frac{b{x}^{5}}{30\,c}}+{\frac{b{x}^{3}}{18\,{c}^{3}}}-{\frac{bx}{6\,{c}^{5}}}+{\frac{b\arctan \left ( cx \right ) }{6\,{c}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a+b*arctan(c*x)),x)

[Out]

1/6*x^6*a+1/6*b*x^6*arctan(c*x)-1/30*b*x^5/c+1/18*b*x^3/c^3-1/6*b*x/c^5+1/6*b*arctan(c*x)/c^6

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Maxima [A]  time = 1.46044, size = 77, normalized size = 1.31 \begin{align*} \frac{1}{6} \, a x^{6} + \frac{1}{90} \,{\left (15 \, x^{6} \arctan \left (c x\right ) - c{\left (\frac{3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac{15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arctan(c*x)),x, algorithm="maxima")

[Out]

1/6*a*x^6 + 1/90*(15*x^6*arctan(c*x) - c*((3*c^4*x^5 - 5*c^2*x^3 + 15*x)/c^6 - 15*arctan(c*x)/c^7))*b

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Fricas [A]  time = 2.36432, size = 131, normalized size = 2.22 \begin{align*} \frac{15 \, a c^{6} x^{6} - 3 \, b c^{5} x^{5} + 5 \, b c^{3} x^{3} - 15 \, b c x + 15 \,{\left (b c^{6} x^{6} + b\right )} \arctan \left (c x\right )}{90 \, c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arctan(c*x)),x, algorithm="fricas")

[Out]

1/90*(15*a*c^6*x^6 - 3*b*c^5*x^5 + 5*b*c^3*x^3 - 15*b*c*x + 15*(b*c^6*x^6 + b)*arctan(c*x))/c^6

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Sympy [A]  time = 2.02291, size = 63, normalized size = 1.07 \begin{align*} \begin{cases} \frac{a x^{6}}{6} + \frac{b x^{6} \operatorname{atan}{\left (c x \right )}}{6} - \frac{b x^{5}}{30 c} + \frac{b x^{3}}{18 c^{3}} - \frac{b x}{6 c^{5}} + \frac{b \operatorname{atan}{\left (c x \right )}}{6 c^{6}} & \text{for}\: c \neq 0 \\\frac{a x^{6}}{6} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(a+b*atan(c*x)),x)

[Out]

Piecewise((a*x**6/6 + b*x**6*atan(c*x)/6 - b*x**5/(30*c) + b*x**3/(18*c**3) - b*x/(6*c**5) + b*atan(c*x)/(6*c*
*6), Ne(c, 0)), (a*x**6/6, True))

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Giac [A]  time = 1.22262, size = 89, normalized size = 1.51 \begin{align*} \frac{15 \, b c^{6} x^{6} \arctan \left (c x\right ) + 15 \, a c^{6} x^{6} - 3 \, b c^{5} x^{5} + 5 \, b c^{3} x^{3} - 15 \, \pi b \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (x\right ) - 15 \, b c x + 15 \, b \arctan \left (c x\right )}{90 \, c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arctan(c*x)),x, algorithm="giac")

[Out]

1/90*(15*b*c^6*x^6*arctan(c*x) + 15*a*c^6*x^6 - 3*b*c^5*x^5 + 5*b*c^3*x^3 - 15*pi*b*sgn(c)*sgn(x) - 15*b*c*x +
15*b*arctan(c*x))/c^6