3.4.40 \(\int \frac {x^{11}}{(a+b x^3)^3} \, dx\) [340]

Optimal. Leaf size=61 \[ \frac {x^3}{3 b^3}+\frac {a^3}{6 b^4 \left (a+b x^3\right )^2}-\frac {a^2}{b^4 \left (a+b x^3\right )}-\frac {a \log \left (a+b x^3\right )}{b^4} \]

[Out]

1/3*x^3/b^3+1/6*a^3/b^4/(b*x^3+a)^2-a^2/b^4/(b*x^3+a)-a*ln(b*x^3+a)/b^4

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \begin {gather*} \frac {a^3}{6 b^4 \left (a+b x^3\right )^2}-\frac {a^2}{b^4 \left (a+b x^3\right )}-\frac {a \log \left (a+b x^3\right )}{b^4}+\frac {x^3}{3 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^11/(a + b*x^3)^3,x]

[Out]

x^3/(3*b^3) + a^3/(6*b^4*(a + b*x^3)^2) - a^2/(b^4*(a + b*x^3)) - (a*Log[a + b*x^3])/b^4

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {x^{11}}{\left (a+b x^3\right )^3} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^3}{(a+b x)^3} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (\frac {1}{b^3}-\frac {a^3}{b^3 (a+b x)^3}+\frac {3 a^2}{b^3 (a+b x)^2}-\frac {3 a}{b^3 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac {x^3}{3 b^3}+\frac {a^3}{6 b^4 \left (a+b x^3\right )^2}-\frac {a^2}{b^4 \left (a+b x^3\right )}-\frac {a \log \left (a+b x^3\right )}{b^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.05, size = 48, normalized size = 0.79 \begin {gather*} -\frac {-2 b x^3+\frac {a^2 \left (5 a+6 b x^3\right )}{\left (a+b x^3\right )^2}+6 a \log \left (a+b x^3\right )}{6 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^11/(a + b*x^3)^3,x]

[Out]

-1/6*(-2*b*x^3 + (a^2*(5*a + 6*b*x^3))/(a + b*x^3)^2 + 6*a*Log[a + b*x^3])/b^4

________________________________________________________________________________________

Maple [A]
time = 0.13, size = 62, normalized size = 1.02

method	result	size
norman	\(\frac {\frac {x^{9}}{3 b}-\frac {3 a^{3}}{2 b^{4}}-\frac {2 a^{2} x^{3}}{b^{3}}}{\left (b \,x^{3}+a \right )^{2}}-\frac {a \ln \left (b \,x^{3}+a \right )}{b^{4}}\)	\(54\)
risch	\(\frac {x^{3}}{3 b^{3}}+\frac {-a^{2} x^{3}-\frac {5 a^{3}}{6 b}}{b^{3} \left (b \,x^{3}+a \right )^{2}}-\frac {a \ln \left (b \,x^{3}+a \right )}{b^{4}}\)	\(54\)
default	\(\frac {x^{3}}{3 b^{3}}-\frac {a \left (\frac {3 a}{b \left (b \,x^{3}+a \right )}-\frac {a^{2}}{2 b \left (b \,x^{3}+a \right )^{2}}+\frac {3 \ln \left (b \,x^{3}+a \right )}{b}\right )}{3 b^{3}}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(b*x^3+a)^3,x,method=_RETURNVERBOSE)

[Out]

1/3*x^3/b^3-1/3*a/b^3*(3*a/b/(b*x^3+a)-1/2/b*a^2/(b*x^3+a)^2+3*ln(b*x^3+a)/b)

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 66, normalized size = 1.08 \begin {gather*} -\frac {6 \, a^{2} b x^{3} + 5 \, a^{3}}{6 \, {\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} + \frac {x^{3}}{3 \, b^{3}} - \frac {a \log \left (b x^{3} + a\right )}{b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

-1/6*(6*a^2*b*x^3 + 5*a^3)/(b^6*x^6 + 2*a*b^5*x^3 + a^2*b^4) + 1/3*x^3/b^3 - a*log(b*x^3 + a)/b^4

________________________________________________________________________________________

Fricas [A]
time = 0.33, size = 91, normalized size = 1.49 \begin {gather*} \frac {2 \, b^{3} x^{9} + 4 \, a b^{2} x^{6} - 4 \, a^{2} b x^{3} - 5 \, a^{3} - 6 \, {\left (a b^{2} x^{6} + 2 \, a^{2} b x^{3} + a^{3}\right )} \log \left (b x^{3} + a\right )}{6 \, {\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

1/6*(2*b^3*x^9 + 4*a*b^2*x^6 - 4*a^2*b*x^3 - 5*a^3 - 6*(a*b^2*x^6 + 2*a^2*b*x^3 + a^3)*log(b*x^3 + a))/(b^6*x^
6 + 2*a*b^5*x^3 + a^2*b^4)

________________________________________________________________________________________

Sympy [A]
time = 0.22, size = 65, normalized size = 1.07 \begin {gather*} - \frac {a \log {\left (a + b x^{3} \right )}}{b^{4}} + \frac {- 5 a^{3} - 6 a^{2} b x^{3}}{6 a^{2} b^{4} + 12 a b^{5} x^{3} + 6 b^{6} x^{6}} + \frac {x^{3}}{3 b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11/(b*x**3+a)**3,x)

[Out]

-a*log(a + b*x**3)/b**4 + (-5*a**3 - 6*a**2*b*x**3)/(6*a**2*b**4 + 12*a*b**5*x**3 + 6*b**6*x**6) + x**3/(3*b**
3)

________________________________________________________________________________________

Giac [A]
time = 2.14, size = 62, normalized size = 1.02 \begin {gather*} \frac {x^{3}}{3 \, b^{3}} - \frac {a \log \left ({\left | b x^{3} + a \right |}\right )}{b^{4}} + \frac {9 \, a b^{2} x^{6} + 12 \, a^{2} b x^{3} + 4 \, a^{3}}{6 \, {\left (b x^{3} + a\right )}^{2} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x^3+a)^3,x, algorithm="giac")

[Out]

1/3*x^3/b^3 - a*log(abs(b*x^3 + a))/b^4 + 1/6*(9*a*b^2*x^6 + 12*a^2*b*x^3 + 4*a^3)/((b*x^3 + a)^2*b^4)

________________________________________________________________________________________

Mupad [B]
time = 1.02, size = 67, normalized size = 1.10 \begin {gather*} \frac {x^3}{3\,b^3}-\frac {\frac {5\,a^3}{6\,b}+a^2\,x^3}{a^2\,b^3+2\,a\,b^4\,x^3+b^5\,x^6}-\frac {a\,\ln \left (b\,x^3+a\right )}{b^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(a + b*x^3)^3,x)

[Out]

x^3/(3*b^3) - ((5*a^3)/(6*b) + a^2*x^3)/(a^2*b^3 + b^5*x^6 + 2*a*b^4*x^3) - (a*log(a + b*x^3))/b^4

________________________________________________________________________________________