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

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

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 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])

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 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]

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*}

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Mathematica [A]  time = 0.00, size = 64, normalized size = 1.08 \[ \frac {a x^6}{6}+\frac {b \tan ^{-1}(c x)}{6 c^6}-\frac {b x}{6 c^5}+\frac {b x^3}{18 c^3}+\frac {1}{6} b x^6 \tan ^{-1}(c x)-\frac {b x^5}{30 c} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*ArcTan[c*x]),x]

[Out]

-1/6*(b*x)/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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fricas [A]  time = 0.46, size = 54, normalized size = 0.92 \[ \frac {15 \, a c^{6} x^{6} - 3 \, b c^{5} x^{5} + 5 \, b c^{3} x^{3} - 15 \, b c x + 15 \, {\left (b c^{6} x^{6} + b\right )} \arctan \left (c x\right )}{90 \, c^{6}} \]

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

[Out]

sage0*x

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maple [A]  time = 0.01, size = 53, normalized size = 0.90 \[ \frac {x^{6} a}{6}+\frac {b \,x^{6} \arctan \left (c x \right )}{6}-\frac {b \,x^{5}}{30 c}+\frac {b \,x^{3}}{18 c^{3}}-\frac {b x}{6 c^{5}}+\frac {b \arctan \left (c x \right )}{6 c^{6}} \]

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 = 0.43, size = 57, normalized size = 0.97 \[ \frac {1}{6} \, a x^{6} + \frac {1}{90} \, {\left (15 \, x^{6} \arctan \left (c x\right ) - c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} b \]

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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mupad [B]  time = 0.44, size = 52, normalized size = 0.88 \[ \frac {\frac {b\,\mathrm {atan}\left (c\,x\right )}{6}+\frac {b\,c^3\,x^3}{18}-\frac {b\,c^5\,x^5}{30}-\frac {b\,c\,x}{6}}{c^6}+\frac {a\,x^6}{6}+\frac {b\,x^6\,\mathrm {atan}\left (c\,x\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a + b*atan(c*x)),x)

[Out]

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

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sympy [A]  time = 1.42, size = 63, normalized size = 1.07 \[ \begin {cases} \frac {a x^{6}}{6} + \frac {b x^{6} \operatorname {atan}{\left (c x \right )}}{6} - \frac {b x^{5}}{30 c} + \frac {b x^{3}}{18 c^{3}} - \frac {b x}{6 c^{5}} + \frac {b \operatorname {atan}{\left (c x \right )}}{6 c^{6}} & \text {for}\: c \neq 0 \\\frac {a x^{6}}{6} & \text {otherwise} \end {cases} \]

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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