\(\int \frac {1}{(a+b \sqrt [3]{x})^2 x^4} \, dx\) [2372]

Optimal result

Integrand size = 15, antiderivative size = 162 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^4} \, dx=-\frac {3 b^9}{a^{10} \left (a+b \sqrt [3]{x}\right )}-\frac {1}{3 a^2 x^3}+\frac {3 b}{4 a^3 x^{8/3}}-\frac {9 b^2}{7 a^4 x^{7/3}}+\frac {2 b^3}{a^5 x^2}-\frac {3 b^4}{a^6 x^{5/3}}+\frac {9 b^5}{2 a^7 x^{4/3}}-\frac {7 b^6}{a^8 x}+\frac {12 b^7}{a^9 x^{2/3}}-\frac {27 b^8}{a^{10} \sqrt [3]{x}}+\frac {30 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{11}}-\frac {10 b^9 \log (x)}{a^{11}} \]

[Out]

-3*b^9/a^10/(a+b*x^(1/3))-1/3/a^2/x^3+3/4*b/a^3/x^(8/3)-9/7*b^2/a^4/x^(7/3)+2*b^3/a^5/x^2-3*b^4/a^6/x^(5/3)+9/
2*b^5/a^7/x^(4/3)-7*b^6/a^8/x+12*b^7/a^9/x^(2/3)-27*b^8/a^10/x^(1/3)+30*b^9*ln(a+b*x^(1/3))/a^11-10*b^9*ln(x)/
a^11

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 162, 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} \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^4} \, dx=\frac {30 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{11}}-\frac {10 b^9 \log (x)}{a^{11}}-\frac {3 b^9}{a^{10} \left (a+b \sqrt [3]{x}\right )}-\frac {27 b^8}{a^{10} \sqrt [3]{x}}+\frac {12 b^7}{a^9 x^{2/3}}-\frac {7 b^6}{a^8 x}+\frac {9 b^5}{2 a^7 x^{4/3}}-\frac {3 b^4}{a^6 x^{5/3}}+\frac {2 b^3}{a^5 x^2}-\frac {9 b^2}{7 a^4 x^{7/3}}+\frac {3 b}{4 a^3 x^{8/3}}-\frac {1}{3 a^2 x^3} \]

[In]

Int[1/((a + b*x^(1/3))^2*x^4),x]

[Out]

(-3*b^9)/(a^10*(a + b*x^(1/3))) - 1/(3*a^2*x^3) + (3*b)/(4*a^3*x^(8/3)) - (9*b^2)/(7*a^4*x^(7/3)) + (2*b^3)/(a
^5*x^2) - (3*b^4)/(a^6*x^(5/3)) + (9*b^5)/(2*a^7*x^(4/3)) - (7*b^6)/(a^8*x) + (12*b^7)/(a^9*x^(2/3)) - (27*b^8
)/(a^10*x^(1/3)) + (30*b^9*Log[a + b*x^(1/3)])/a^11 - (10*b^9*Log[x])/a^11

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^{10} (a+b x)^2} \, dx,x,\sqrt [3]{x}\right ) \\ & = 3 \text {Subst}\left (\int \left (\frac {1}{a^2 x^{10}}-\frac {2 b}{a^3 x^9}+\frac {3 b^2}{a^4 x^8}-\frac {4 b^3}{a^5 x^7}+\frac {5 b^4}{a^6 x^6}-\frac {6 b^5}{a^7 x^5}+\frac {7 b^6}{a^8 x^4}-\frac {8 b^7}{a^9 x^3}+\frac {9 b^8}{a^{10} x^2}-\frac {10 b^9}{a^{11} x}+\frac {b^{10}}{a^{10} (a+b x)^2}+\frac {10 b^{10}}{a^{11} (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {3 b^9}{a^{10} \left (a+b \sqrt [3]{x}\right )}-\frac {1}{3 a^2 x^3}+\frac {3 b}{4 a^3 x^{8/3}}-\frac {9 b^2}{7 a^4 x^{7/3}}+\frac {2 b^3}{a^5 x^2}-\frac {3 b^4}{a^6 x^{5/3}}+\frac {9 b^5}{2 a^7 x^{4/3}}-\frac {7 b^6}{a^8 x}+\frac {12 b^7}{a^9 x^{2/3}}-\frac {27 b^8}{a^{10} \sqrt [3]{x}}+\frac {30 b^9 \log \left (a+b \sqrt [3]{x}\right )}{a^{11}}-\frac {10 b^9 \log (x)}{a^{11}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^4} \, dx=-\frac {\frac {a \left (28 a^9-35 a^8 b \sqrt [3]{x}+45 a^7 b^2 x^{2/3}-60 a^6 b^3 x+84 a^5 b^4 x^{4/3}-126 a^4 b^5 x^{5/3}+210 a^3 b^6 x^2-420 a^2 b^7 x^{7/3}+1260 a b^8 x^{8/3}+2520 b^9 x^3\right )}{\left (a+b \sqrt [3]{x}\right ) x^3}-2520 b^9 \log \left (a+b \sqrt [3]{x}\right )+840 b^9 \log (x)}{84 a^{11}} \]

[In]

Integrate[1/((a + b*x^(1/3))^2*x^4),x]

[Out]

-1/84*((a*(28*a^9 - 35*a^8*b*x^(1/3) + 45*a^7*b^2*x^(2/3) - 60*a^6*b^3*x + 84*a^5*b^4*x^(4/3) - 126*a^4*b^5*x^
(5/3) + 210*a^3*b^6*x^2 - 420*a^2*b^7*x^(7/3) + 1260*a*b^8*x^(8/3) + 2520*b^9*x^3))/((a + b*x^(1/3))*x^3) - 25
20*b^9*Log[a + b*x^(1/3)] + 840*b^9*Log[x])/a^11

Maple [A] (verified)

Time = 3.66 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.86

method	result	size
derivativedivides	\(-\frac {3 b^{9}}{a^{10} \left (a +b \,x^{\frac {1}{3}}\right )}-\frac {1}{3 a^{2} x^{3}}+\frac {3 b}{4 a^{3} x^{\frac {8}{3}}}-\frac {9 b^{2}}{7 a^{4} x^{\frac {7}{3}}}+\frac {2 b^{3}}{a^{5} x^{2}}-\frac {3 b^{4}}{a^{6} x^{\frac {5}{3}}}+\frac {9 b^{5}}{2 a^{7} x^{\frac {4}{3}}}-\frac {7 b^{6}}{a^{8} x}+\frac {12 b^{7}}{a^{9} x^{\frac {2}{3}}}-\frac {27 b^{8}}{a^{10} x^{\frac {1}{3}}}+\frac {30 b^{9} \ln \left (a +b \,x^{\frac {1}{3}}\right )}{a^{11}}-\frac {10 b^{9} \ln \left (x \right )}{a^{11}}\)	\(139\)
default	\(-\frac {3 b^{9}}{a^{10} \left (a +b \,x^{\frac {1}{3}}\right )}-\frac {1}{3 a^{2} x^{3}}+\frac {3 b}{4 a^{3} x^{\frac {8}{3}}}-\frac {9 b^{2}}{7 a^{4} x^{\frac {7}{3}}}+\frac {2 b^{3}}{a^{5} x^{2}}-\frac {3 b^{4}}{a^{6} x^{\frac {5}{3}}}+\frac {9 b^{5}}{2 a^{7} x^{\frac {4}{3}}}-\frac {7 b^{6}}{a^{8} x}+\frac {12 b^{7}}{a^{9} x^{\frac {2}{3}}}-\frac {27 b^{8}}{a^{10} x^{\frac {1}{3}}}+\frac {30 b^{9} \ln \left (a +b \,x^{\frac {1}{3}}\right )}{a^{11}}-\frac {10 b^{9} \ln \left (x \right )}{a^{11}}\)	\(139\)

[In]

int(1/(a+b*x^(1/3))^2/x^4,x,method=_RETURNVERBOSE)

[Out]

-3*b^9/a^10/(a+b*x^(1/3))-1/3/a^2/x^3+3/4*b/a^3/x^(8/3)-9/7*b^2/a^4/x^(7/3)+2*b^3/a^5/x^2-3*b^4/a^6/x^(5/3)+9/
2*b^5/a^7/x^(4/3)-7*b^6/a^8/x+12*b^7/a^9/x^(2/3)-27*b^8/a^10/x^(1/3)+30*b^9*ln(a+b*x^(1/3))/a^11-10*b^9*ln(x)/
a^11

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.21 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^4} \, dx=-\frac {840 \, a^{3} b^{9} x^{3} + 420 \, a^{6} b^{6} x^{2} - 140 \, a^{9} b^{3} x + 28 \, a^{12} - 2520 \, {\left (b^{12} x^{4} + a^{3} b^{9} x^{3}\right )} \log \left (b x^{\frac {1}{3}} + a\right ) + 2520 \, {\left (b^{12} x^{4} + a^{3} b^{9} x^{3}\right )} \log \left (x^{\frac {1}{3}}\right ) + 18 \, {\left (140 \, a b^{11} x^{3} + 105 \, a^{4} b^{8} x^{2} - 15 \, a^{7} b^{5} x + 6 \, a^{10} b^{2}\right )} x^{\frac {2}{3}} - 63 \, {\left (20 \, a^{2} b^{10} x^{3} + 12 \, a^{5} b^{7} x^{2} - 3 \, a^{8} b^{4} x + a^{11} b\right )} x^{\frac {1}{3}}}{84 \, {\left (a^{11} b^{3} x^{4} + a^{14} x^{3}\right )}} \]

[In]

integrate(1/(a+b*x^(1/3))^2/x^4,x, algorithm="fricas")

[Out]

-1/84*(840*a^3*b^9*x^3 + 420*a^6*b^6*x^2 - 140*a^9*b^3*x + 28*a^12 - 2520*(b^12*x^4 + a^3*b^9*x^3)*log(b*x^(1/
3) + a) + 2520*(b^12*x^4 + a^3*b^9*x^3)*log(x^(1/3)) + 18*(140*a*b^11*x^3 + 105*a^4*b^8*x^2 - 15*a^7*b^5*x + 6
*a^10*b^2)*x^(2/3) - 63*(20*a^2*b^10*x^3 + 12*a^5*b^7*x^2 - 3*a^8*b^4*x + a^11*b)*x^(1/3))/(a^11*b^3*x^4 + a^1
4*x^3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 505 vs. \(2 (163) = 326\).

Time = 5.46 (sec) , antiderivative size = 505, normalized size of antiderivative = 3.12 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^4} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {11}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{3 a^{2} x^{3}} & \text {for}\: b = 0 \\- \frac {3}{11 b^{2} x^{\frac {11}{3}}} & \text {for}\: a = 0 \\- \frac {28 a^{10} x^{\frac {2}{3}}}{84 a^{12} x^{\frac {11}{3}} + 84 a^{11} b x^{4}} + \frac {35 a^{9} b x}{84 a^{12} x^{\frac {11}{3}} + 84 a^{11} b x^{4}} - \frac {45 a^{8} b^{2} x^{\frac {4}{3}}}{84 a^{12} x^{\frac {11}{3}} + 84 a^{11} b x^{4}} + \frac {60 a^{7} b^{3} x^{\frac {5}{3}}}{84 a^{12} x^{\frac {11}{3}} + 84 a^{11} b x^{4}} - \frac {84 a^{6} b^{4} x^{2}}{84 a^{12} x^{\frac {11}{3}} + 84 a^{11} b x^{4}} + \frac {126 a^{5} b^{5} x^{\frac {7}{3}}}{84 a^{12} x^{\frac {11}{3}} + 84 a^{11} b x^{4}} - \frac {210 a^{4} b^{6} x^{\frac {8}{3}}}{84 a^{12} x^{\frac {11}{3}} + 84 a^{11} b x^{4}} + \frac {420 a^{3} b^{7} x^{3}}{84 a^{12} x^{\frac {11}{3}} + 84 a^{11} b x^{4}} - \frac {1260 a^{2} b^{8} x^{\frac {10}{3}}}{84 a^{12} x^{\frac {11}{3}} + 84 a^{11} b x^{4}} - \frac {840 a b^{9} x^{\frac {11}{3}} \log {\left (x \right )}}{84 a^{12} x^{\frac {11}{3}} + 84 a^{11} b x^{4}} + \frac {2520 a b^{9} x^{\frac {11}{3}} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{84 a^{12} x^{\frac {11}{3}} + 84 a^{11} b x^{4}} - \frac {2520 a b^{9} x^{\frac {11}{3}}}{84 a^{12} x^{\frac {11}{3}} + 84 a^{11} b x^{4}} - \frac {840 b^{10} x^{4} \log {\left (x \right )}}{84 a^{12} x^{\frac {11}{3}} + 84 a^{11} b x^{4}} + \frac {2520 b^{10} x^{4} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{84 a^{12} x^{\frac {11}{3}} + 84 a^{11} b x^{4}} & \text {otherwise} \end {cases} \]

[In]

integrate(1/(a+b*x**(1/3))**2/x**4,x)

[Out]

Piecewise((zoo/x**(11/3), Eq(a, 0) & Eq(b, 0)), (-1/(3*a**2*x**3), Eq(b, 0)), (-3/(11*b**2*x**(11/3)), Eq(a, 0
)), (-28*a**10*x**(2/3)/(84*a**12*x**(11/3) + 84*a**11*b*x**4) + 35*a**9*b*x/(84*a**12*x**(11/3) + 84*a**11*b*
x**4) - 45*a**8*b**2*x**(4/3)/(84*a**12*x**(11/3) + 84*a**11*b*x**4) + 60*a**7*b**3*x**(5/3)/(84*a**12*x**(11/
3) + 84*a**11*b*x**4) - 84*a**6*b**4*x**2/(84*a**12*x**(11/3) + 84*a**11*b*x**4) + 126*a**5*b**5*x**(7/3)/(84*
a**12*x**(11/3) + 84*a**11*b*x**4) - 210*a**4*b**6*x**(8/3)/(84*a**12*x**(11/3) + 84*a**11*b*x**4) + 420*a**3*
b**7*x**3/(84*a**12*x**(11/3) + 84*a**11*b*x**4) - 1260*a**2*b**8*x**(10/3)/(84*a**12*x**(11/3) + 84*a**11*b*x
**4) - 840*a*b**9*x**(11/3)*log(x)/(84*a**12*x**(11/3) + 84*a**11*b*x**4) + 2520*a*b**9*x**(11/3)*log(a/b + x*
*(1/3))/(84*a**12*x**(11/3) + 84*a**11*b*x**4) - 2520*a*b**9*x**(11/3)/(84*a**12*x**(11/3) + 84*a**11*b*x**4)
- 840*b**10*x**4*log(x)/(84*a**12*x**(11/3) + 84*a**11*b*x**4) + 2520*b**10*x**4*log(a/b + x**(1/3))/(84*a**12
*x**(11/3) + 84*a**11*b*x**4), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^4} \, dx=-\frac {2520 \, b^{9} x^{3} + 1260 \, a b^{8} x^{\frac {8}{3}} - 420 \, a^{2} b^{7} x^{\frac {7}{3}} + 210 \, a^{3} b^{6} x^{2} - 126 \, a^{4} b^{5} x^{\frac {5}{3}} + 84 \, a^{5} b^{4} x^{\frac {4}{3}} - 60 \, a^{6} b^{3} x + 45 \, a^{7} b^{2} x^{\frac {2}{3}} - 35 \, a^{8} b x^{\frac {1}{3}} + 28 \, a^{9}}{84 \, {\left (a^{10} b x^{\frac {10}{3}} + a^{11} x^{3}\right )}} + \frac {30 \, b^{9} \log \left (b x^{\frac {1}{3}} + a\right )}{a^{11}} - \frac {10 \, b^{9} \log \left (x\right )}{a^{11}} \]

[In]

integrate(1/(a+b*x^(1/3))^2/x^4,x, algorithm="maxima")

[Out]

-1/84*(2520*b^9*x^3 + 1260*a*b^8*x^(8/3) - 420*a^2*b^7*x^(7/3) + 210*a^3*b^6*x^2 - 126*a^4*b^5*x^(5/3) + 84*a^
5*b^4*x^(4/3) - 60*a^6*b^3*x + 45*a^7*b^2*x^(2/3) - 35*a^8*b*x^(1/3) + 28*a^9)/(a^10*b*x^(10/3) + a^11*x^3) +
30*b^9*log(b*x^(1/3) + a)/a^11 - 10*b^9*log(x)/a^11

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^4} \, dx=\frac {30 \, b^{9} \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{a^{11}} - \frac {10 \, b^{9} \log \left ({\left | x \right |}\right )}{a^{11}} - \frac {2520 \, a b^{9} x^{3} + 1260 \, a^{2} b^{8} x^{\frac {8}{3}} - 420 \, a^{3} b^{7} x^{\frac {7}{3}} + 210 \, a^{4} b^{6} x^{2} - 126 \, a^{5} b^{5} x^{\frac {5}{3}} + 84 \, a^{6} b^{4} x^{\frac {4}{3}} - 60 \, a^{7} b^{3} x + 45 \, a^{8} b^{2} x^{\frac {2}{3}} - 35 \, a^{9} b x^{\frac {1}{3}} + 28 \, a^{10}}{84 \, {\left (b x^{\frac {1}{3}} + a\right )} a^{11} x^{3}} \]

[In]

integrate(1/(a+b*x^(1/3))^2/x^4,x, algorithm="giac")

[Out]

30*b^9*log(abs(b*x^(1/3) + a))/a^11 - 10*b^9*log(abs(x))/a^11 - 1/84*(2520*a*b^9*x^3 + 1260*a^2*b^8*x^(8/3) -
420*a^3*b^7*x^(7/3) + 210*a^4*b^6*x^2 - 126*a^5*b^5*x^(5/3) + 84*a^6*b^4*x^(4/3) - 60*a^7*b^3*x + 45*a^8*b^2*x
^(2/3) - 35*a^9*b*x^(1/3) + 28*a^10)/((b*x^(1/3) + a)*a^11*x^3)

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^4} \, dx=\frac {60\,b^9\,\mathrm {atanh}\left (\frac {2\,b\,x^{1/3}}{a}+1\right )}{a^{11}}-\frac {\frac {1}{3\,a}-\frac {5\,b\,x^{1/3}}{12\,a^2}-\frac {5\,b^3\,x}{7\,a^4}+\frac {15\,b^2\,x^{2/3}}{28\,a^3}+\frac {5\,b^6\,x^2}{2\,a^7}+\frac {b^4\,x^{4/3}}{a^5}-\frac {3\,b^5\,x^{5/3}}{2\,a^6}+\frac {30\,b^9\,x^3}{a^{10}}-\frac {5\,b^7\,x^{7/3}}{a^8}+\frac {15\,b^8\,x^{8/3}}{a^9}}{a\,x^3+b\,x^{10/3}} \]

[In]

int(1/(x^4*(a + b*x^(1/3))^2),x)

[Out]

(60*b^9*atanh((2*b*x^(1/3))/a + 1))/a^11 - (1/(3*a) - (5*b*x^(1/3))/(12*a^2) - (5*b^3*x)/(7*a^4) + (15*b^2*x^(
2/3))/(28*a^3) + (5*b^6*x^2)/(2*a^7) + (b^4*x^(4/3))/a^5 - (3*b^5*x^(5/3))/(2*a^6) + (30*b^9*x^3)/a^10 - (5*b^
7*x^(7/3))/a^8 + (15*b^8*x^(8/3))/a^9)/(a*x^3 + b*x^(10/3))