\(\int \frac {1}{(a+b \sqrt [3]{x})^2 x^3} \, dx\) [2371]

Optimal result

Integrand size = 15, antiderivative size = 125 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^3} \, dx=\frac {3 b^6}{a^7 \left (a+b \sqrt [3]{x}\right )}-\frac {1}{2 a^2 x^2}+\frac {6 b}{5 a^3 x^{5/3}}-\frac {9 b^2}{4 a^4 x^{4/3}}+\frac {4 b^3}{a^5 x}-\frac {15 b^4}{2 a^6 x^{2/3}}+\frac {18 b^5}{a^7 \sqrt [3]{x}}-\frac {21 b^6 \log \left (a+b \sqrt [3]{x}\right )}{a^8}+\frac {7 b^6 \log (x)}{a^8} \]

[Out]

3*b^6/a^7/(a+b*x^(1/3))-1/2/a^2/x^2+6/5*b/a^3/x^(5/3)-9/4*b^2/a^4/x^(4/3)+4*b^3/a^5/x-15/2*b^4/a^6/x^(2/3)+18*
b^5/a^7/x^(1/3)-21*b^6*ln(a+b*x^(1/3))/a^8+7*b^6*ln(x)/a^8

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 46} \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^3} \, dx=-\frac {21 b^6 \log \left (a+b \sqrt [3]{x}\right )}{a^8}+\frac {7 b^6 \log (x)}{a^8}+\frac {3 b^6}{a^7 \left (a+b \sqrt [3]{x}\right )}+\frac {18 b^5}{a^7 \sqrt [3]{x}}-\frac {15 b^4}{2 a^6 x^{2/3}}+\frac {4 b^3}{a^5 x}-\frac {9 b^2}{4 a^4 x^{4/3}}+\frac {6 b}{5 a^3 x^{5/3}}-\frac {1}{2 a^2 x^2} \]

[In]

Int[1/((a + b*x^(1/3))^2*x^3),x]

[Out]

(3*b^6)/(a^7*(a + b*x^(1/3))) - 1/(2*a^2*x^2) + (6*b)/(5*a^3*x^(5/3)) - (9*b^2)/(4*a^4*x^(4/3)) + (4*b^3)/(a^5
*x) - (15*b^4)/(2*a^6*x^(2/3)) + (18*b^5)/(a^7*x^(1/3)) - (21*b^6*Log[a + b*x^(1/3)])/a^8 + (7*b^6*Log[x])/a^8

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[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*} \text {integral}& = 3 \text {Subst}\left (\int \frac {1}{x^7 (a+b x)^2} \, dx,x,\sqrt [3]{x}\right ) \\ & = 3 \text {Subst}\left (\int \left (\frac {1}{a^2 x^7}-\frac {2 b}{a^3 x^6}+\frac {3 b^2}{a^4 x^5}-\frac {4 b^3}{a^5 x^4}+\frac {5 b^4}{a^6 x^3}-\frac {6 b^5}{a^7 x^2}+\frac {7 b^6}{a^8 x}-\frac {b^7}{a^7 (a+b x)^2}-\frac {7 b^7}{a^8 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {3 b^6}{a^7 \left (a+b \sqrt [3]{x}\right )}-\frac {1}{2 a^2 x^2}+\frac {6 b}{5 a^3 x^{5/3}}-\frac {9 b^2}{4 a^4 x^{4/3}}+\frac {4 b^3}{a^5 x}-\frac {15 b^4}{2 a^6 x^{2/3}}+\frac {18 b^5}{a^7 \sqrt [3]{x}}-\frac {21 b^6 \log \left (a+b \sqrt [3]{x}\right )}{a^8}+\frac {7 b^6 \log (x)}{a^8} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^3} \, dx=\frac {\frac {a \left (-10 a^6+14 a^5 b \sqrt [3]{x}-21 a^4 b^2 x^{2/3}+35 a^3 b^3 x-70 a^2 b^4 x^{4/3}+210 a b^5 x^{5/3}+420 b^6 x^2\right )}{\left (a+b \sqrt [3]{x}\right ) x^2}-420 b^6 \log \left (a+b \sqrt [3]{x}\right )+140 b^6 \log (x)}{20 a^8} \]

[In]

Integrate[1/((a + b*x^(1/3))^2*x^3),x]

[Out]

((a*(-10*a^6 + 14*a^5*b*x^(1/3) - 21*a^4*b^2*x^(2/3) + 35*a^3*b^3*x - 70*a^2*b^4*x^(4/3) + 210*a*b^5*x^(5/3) +
420*b^6*x^2))/((a + b*x^(1/3))*x^2) - 420*b^6*Log[a + b*x^(1/3)] + 140*b^6*Log[x])/(20*a^8)

Maple [A] (verified)

Time = 6.01 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.85

method	result	size
derivativedivides	\(\frac {3 b^{6}}{a^{7} \left (a +b \,x^{\frac {1}{3}}\right )}-\frac {1}{2 a^{2} x^{2}}+\frac {6 b}{5 a^{3} x^{\frac {5}{3}}}-\frac {9 b^{2}}{4 a^{4} x^{\frac {4}{3}}}+\frac {4 b^{3}}{a^{5} x}-\frac {15 b^{4}}{2 a^{6} x^{\frac {2}{3}}}+\frac {18 b^{5}}{a^{7} x^{\frac {1}{3}}}-\frac {21 b^{6} \ln \left (a +b \,x^{\frac {1}{3}}\right )}{a^{8}}+\frac {7 b^{6} \ln \left (x \right )}{a^{8}}\)	\(106\)
default	\(\frac {3 b^{6}}{a^{7} \left (a +b \,x^{\frac {1}{3}}\right )}-\frac {1}{2 a^{2} x^{2}}+\frac {6 b}{5 a^{3} x^{\frac {5}{3}}}-\frac {9 b^{2}}{4 a^{4} x^{\frac {4}{3}}}+\frac {4 b^{3}}{a^{5} x}-\frac {15 b^{4}}{2 a^{6} x^{\frac {2}{3}}}+\frac {18 b^{5}}{a^{7} x^{\frac {1}{3}}}-\frac {21 b^{6} \ln \left (a +b \,x^{\frac {1}{3}}\right )}{a^{8}}+\frac {7 b^{6} \ln \left (x \right )}{a^{8}}\)	\(106\)

[In]

int(1/(a+b*x^(1/3))^2/x^3,x,method=_RETURNVERBOSE)

[Out]

3*b^6/a^7/(a+b*x^(1/3))-1/2/a^2/x^2+6/5*b/a^3/x^(5/3)-9/4*b^2/a^4/x^(4/3)+4*b^3/a^5/x-15/2*b^4/a^6/x^(2/3)+18*
b^5/a^7/x^(1/3)-21*b^6*ln(a+b*x^(1/3))/a^8+7*b^6*ln(x)/a^8

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.31 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^3} \, dx=\frac {140 \, a^{3} b^{6} x^{2} + 70 \, a^{6} b^{3} x - 10 \, a^{9} - 420 \, {\left (b^{9} x^{3} + a^{3} b^{6} x^{2}\right )} \log \left (b x^{\frac {1}{3}} + a\right ) + 420 \, {\left (b^{9} x^{3} + a^{3} b^{6} x^{2}\right )} \log \left (x^{\frac {1}{3}}\right ) + 15 \, {\left (28 \, a b^{8} x^{2} + 21 \, a^{4} b^{5} x - 3 \, a^{7} b^{2}\right )} x^{\frac {2}{3}} - 6 \, {\left (35 \, a^{2} b^{7} x^{2} + 21 \, a^{5} b^{4} x - 4 \, a^{8} b\right )} x^{\frac {1}{3}}}{20 \, {\left (a^{8} b^{3} x^{3} + a^{11} x^{2}\right )}} \]

[In]

integrate(1/(a+b*x^(1/3))^2/x^3,x, algorithm="fricas")

[Out]

1/20*(140*a^3*b^6*x^2 + 70*a^6*b^3*x - 10*a^9 - 420*(b^9*x^3 + a^3*b^6*x^2)*log(b*x^(1/3) + a) + 420*(b^9*x^3
+ a^3*b^6*x^2)*log(x^(1/3)) + 15*(28*a*b^8*x^2 + 21*a^4*b^5*x - 3*a^7*b^2)*x^(2/3) - 6*(35*a^2*b^7*x^2 + 21*a^
5*b^4*x - 4*a^8*b)*x^(1/3))/(a^8*b^3*x^3 + a^11*x^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 405 vs. \(2 (124) = 248\).

Time = 2.52 (sec) , antiderivative size = 405, normalized size of antiderivative = 3.24 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^3} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {8}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{2 a^{2} x^{2}} & \text {for}\: b = 0 \\- \frac {3}{8 b^{2} x^{\frac {8}{3}}} & \text {for}\: a = 0 \\- \frac {10 a^{7} x^{\frac {2}{3}}}{20 a^{9} x^{\frac {8}{3}} + 20 a^{8} b x^{3}} + \frac {14 a^{6} b x}{20 a^{9} x^{\frac {8}{3}} + 20 a^{8} b x^{3}} - \frac {21 a^{5} b^{2} x^{\frac {4}{3}}}{20 a^{9} x^{\frac {8}{3}} + 20 a^{8} b x^{3}} + \frac {35 a^{4} b^{3} x^{\frac {5}{3}}}{20 a^{9} x^{\frac {8}{3}} + 20 a^{8} b x^{3}} - \frac {70 a^{3} b^{4} x^{2}}{20 a^{9} x^{\frac {8}{3}} + 20 a^{8} b x^{3}} + \frac {210 a^{2} b^{5} x^{\frac {7}{3}}}{20 a^{9} x^{\frac {8}{3}} + 20 a^{8} b x^{3}} + \frac {140 a b^{6} x^{\frac {8}{3}} \log {\left (x \right )}}{20 a^{9} x^{\frac {8}{3}} + 20 a^{8} b x^{3}} - \frac {420 a b^{6} x^{\frac {8}{3}} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{20 a^{9} x^{\frac {8}{3}} + 20 a^{8} b x^{3}} + \frac {420 a b^{6} x^{\frac {8}{3}}}{20 a^{9} x^{\frac {8}{3}} + 20 a^{8} b x^{3}} + \frac {140 b^{7} x^{3} \log {\left (x \right )}}{20 a^{9} x^{\frac {8}{3}} + 20 a^{8} b x^{3}} - \frac {420 b^{7} x^{3} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{20 a^{9} x^{\frac {8}{3}} + 20 a^{8} b x^{3}} & \text {otherwise} \end {cases} \]

[In]

integrate(1/(a+b*x**(1/3))**2/x**3,x)

[Out]

Piecewise((zoo/x**(8/3), Eq(a, 0) & Eq(b, 0)), (-1/(2*a**2*x**2), Eq(b, 0)), (-3/(8*b**2*x**(8/3)), Eq(a, 0)),
(-10*a**7*x**(2/3)/(20*a**9*x**(8/3) + 20*a**8*b*x**3) + 14*a**6*b*x/(20*a**9*x**(8/3) + 20*a**8*b*x**3) - 21
*a**5*b**2*x**(4/3)/(20*a**9*x**(8/3) + 20*a**8*b*x**3) + 35*a**4*b**3*x**(5/3)/(20*a**9*x**(8/3) + 20*a**8*b*
x**3) - 70*a**3*b**4*x**2/(20*a**9*x**(8/3) + 20*a**8*b*x**3) + 210*a**2*b**5*x**(7/3)/(20*a**9*x**(8/3) + 20*
a**8*b*x**3) + 140*a*b**6*x**(8/3)*log(x)/(20*a**9*x**(8/3) + 20*a**8*b*x**3) - 420*a*b**6*x**(8/3)*log(a/b +
x**(1/3))/(20*a**9*x**(8/3) + 20*a**8*b*x**3) + 420*a*b**6*x**(8/3)/(20*a**9*x**(8/3) + 20*a**8*b*x**3) + 140*
b**7*x**3*log(x)/(20*a**9*x**(8/3) + 20*a**8*b*x**3) - 420*b**7*x**3*log(a/b + x**(1/3))/(20*a**9*x**(8/3) + 2
0*a**8*b*x**3), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^3} \, dx=\frac {420 \, b^{6} x^{2} + 210 \, a b^{5} x^{\frac {5}{3}} - 70 \, a^{2} b^{4} x^{\frac {4}{3}} + 35 \, a^{3} b^{3} x - 21 \, a^{4} b^{2} x^{\frac {2}{3}} + 14 \, a^{5} b x^{\frac {1}{3}} - 10 \, a^{6}}{20 \, {\left (a^{7} b x^{\frac {7}{3}} + a^{8} x^{2}\right )}} - \frac {21 \, b^{6} \log \left (b x^{\frac {1}{3}} + a\right )}{a^{8}} + \frac {7 \, b^{6} \log \left (x\right )}{a^{8}} \]

[In]

integrate(1/(a+b*x^(1/3))^2/x^3,x, algorithm="maxima")

[Out]

1/20*(420*b^6*x^2 + 210*a*b^5*x^(5/3) - 70*a^2*b^4*x^(4/3) + 35*a^3*b^3*x - 21*a^4*b^2*x^(2/3) + 14*a^5*b*x^(1
/3) - 10*a^6)/(a^7*b*x^(7/3) + a^8*x^2) - 21*b^6*log(b*x^(1/3) + a)/a^8 + 7*b^6*log(x)/a^8

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^3} \, dx=-\frac {21 \, b^{6} \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{a^{8}} + \frac {7 \, b^{6} \log \left ({\left | x \right |}\right )}{a^{8}} + \frac {420 \, a b^{6} x^{2} + 210 \, a^{2} b^{5} x^{\frac {5}{3}} - 70 \, a^{3} b^{4} x^{\frac {4}{3}} + 35 \, a^{4} b^{3} x - 21 \, a^{5} b^{2} x^{\frac {2}{3}} + 14 \, a^{6} b x^{\frac {1}{3}} - 10 \, a^{7}}{20 \, {\left (b x^{\frac {1}{3}} + a\right )} a^{8} x^{2}} \]

[In]

integrate(1/(a+b*x^(1/3))^2/x^3,x, algorithm="giac")

[Out]

-21*b^6*log(abs(b*x^(1/3) + a))/a^8 + 7*b^6*log(abs(x))/a^8 + 1/20*(420*a*b^6*x^2 + 210*a^2*b^5*x^(5/3) - 70*a
^3*b^4*x^(4/3) + 35*a^4*b^3*x - 21*a^5*b^2*x^(2/3) + 14*a^6*b*x^(1/3) - 10*a^7)/((b*x^(1/3) + a)*a^8*x^2)

Mupad [B] (verification not implemented)

Time = 5.89 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^3} \, dx=\frac {\frac {7\,b\,x^{1/3}}{10\,a^2}-\frac {1}{2\,a}+\frac {7\,b^3\,x}{4\,a^4}-\frac {21\,b^2\,x^{2/3}}{20\,a^3}+\frac {21\,b^6\,x^2}{a^7}-\frac {7\,b^4\,x^{4/3}}{2\,a^5}+\frac {21\,b^5\,x^{5/3}}{2\,a^6}}{a\,x^2+b\,x^{7/3}}-\frac {42\,b^6\,\mathrm {atanh}\left (\frac {2\,b\,x^{1/3}}{a}+1\right )}{a^8} \]

[In]

int(1/(x^3*(a + b*x^(1/3))^2),x)

[Out]

((7*b*x^(1/3))/(10*a^2) - 1/(2*a) + (7*b^3*x)/(4*a^4) - (21*b^2*x^(2/3))/(20*a^3) + (21*b^6*x^2)/a^7 - (7*b^4*
x^(4/3))/(2*a^5) + (21*b^5*x^(5/3))/(2*a^6))/(a*x^2 + b*x^(7/3)) - (42*b^6*atanh((2*b*x^(1/3))/a + 1))/a^8