3.2378 \(\int \frac {1}{(a+b \sqrt [3]{x})^3 x^2} \, dx\)

Optimal. Leaf size=103 \[ \frac {30 b^3 \log \left (a+b \sqrt [3]{x}\right )}{a^6}-\frac {10 b^3 \log (x)}{a^6}-\frac {12 b^3}{a^5 \left (a+b \sqrt [3]{x}\right )}-\frac {18 b^2}{a^5 \sqrt [3]{x}}-\frac {3 b^3}{2 a^4 \left (a+b \sqrt [3]{x}\right )^2}+\frac {9 b}{2 a^4 x^{2/3}}-\frac {1}{a^3 x} \]

[Out]

-3/2*b^3/a^4/(a+b*x^(1/3))^2-12*b^3/a^5/(a+b*x^(1/3))-1/a^3/x+9/2*b/a^4/x^(2/3)-18*b^2/a^5/x^(1/3)+30*b^3*ln(a
+b*x^(1/3))/a^6-10*b^3*ln(x)/a^6

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Rubi [A]  time = 0.07, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 44} \[ -\frac {12 b^3}{a^5 \left (a+b \sqrt [3]{x}\right )}-\frac {3 b^3}{2 a^4 \left (a+b \sqrt [3]{x}\right )^2}-\frac {18 b^2}{a^5 \sqrt [3]{x}}+\frac {30 b^3 \log \left (a+b \sqrt [3]{x}\right )}{a^6}-\frac {10 b^3 \log (x)}{a^6}+\frac {9 b}{2 a^4 x^{2/3}}-\frac {1}{a^3 x} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^(1/3))^3*x^2),x]

[Out]

(-3*b^3)/(2*a^4*(a + b*x^(1/3))^2) - (12*b^3)/(a^5*(a + b*x^(1/3))) - 1/(a^3*x) + (9*b)/(2*a^4*x^(2/3)) - (18*
b^2)/(a^5*x^(1/3)) + (30*b^3*Log[a + b*x^(1/3)])/a^6 - (10*b^3*Log[x])/a^6

Rule 44

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] && L
tQ[m + n + 2, 0])

Rule 266

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 {1}{\left (a+b \sqrt [3]{x}\right )^3 x^2} \, dx &=3 \operatorname {Subst}\left (\int \frac {1}{x^4 (a+b x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (\frac {1}{a^3 x^4}-\frac {3 b}{a^4 x^3}+\frac {6 b^2}{a^5 x^2}-\frac {10 b^3}{a^6 x}+\frac {b^4}{a^4 (a+b x)^3}+\frac {4 b^4}{a^5 (a+b x)^2}+\frac {10 b^4}{a^6 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {3 b^3}{2 a^4 \left (a+b \sqrt [3]{x}\right )^2}-\frac {12 b^3}{a^5 \left (a+b \sqrt [3]{x}\right )}-\frac {1}{a^3 x}+\frac {9 b}{2 a^4 x^{2/3}}-\frac {18 b^2}{a^5 \sqrt [3]{x}}+\frac {30 b^3 \log \left (a+b \sqrt [3]{x}\right )}{a^6}-\frac {10 b^3 \log (x)}{a^6}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 93, normalized size = 0.90 \[ -\frac {\frac {a \left (2 a^4-5 a^3 b \sqrt [3]{x}+20 a^2 b^2 x^{2/3}+90 a b^3 x+60 b^4 x^{4/3}\right )}{x \left (a+b \sqrt [3]{x}\right )^2}-60 b^3 \log \left (a+b \sqrt [3]{x}\right )+20 b^3 \log (x)}{2 a^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((a*(2*a^4 - 5*a^3*b*x^(1/3) + 20*a^2*b^2*x^(2/3) + 90*a*b^3*x + 60*b^4*x^(4/3)))/((a + b*x^(1/3))^2*x) -
60*b^3*Log[a + b*x^(1/3)] + 20*b^3*Log[x])/a^6

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fricas [B]  time = 0.67, size = 191, normalized size = 1.85 \[ -\frac {20 \, a^{3} b^{6} x^{2} + 31 \, a^{6} b^{3} x + 2 \, a^{9} - 60 \, {\left (b^{9} x^{3} + 2 \, a^{3} b^{6} x^{2} + a^{6} b^{3} x\right )} \log \left (b x^{\frac {1}{3}} + a\right ) + 60 \, {\left (b^{9} x^{3} + 2 \, a^{3} b^{6} x^{2} + a^{6} b^{3} x\right )} \log \left (x^{\frac {1}{3}}\right ) + 3 \, {\left (20 \, a b^{8} x^{2} + 35 \, a^{4} b^{5} x + 12 \, a^{7} b^{2}\right )} x^{\frac {2}{3}} - 3 \, {\left (10 \, a^{2} b^{7} x^{2} + 16 \, a^{5} b^{4} x + 3 \, a^{8} b\right )} x^{\frac {1}{3}}}{2 \, {\left (a^{6} b^{6} x^{3} + 2 \, a^{9} b^{3} x^{2} + a^{12} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(20*a^3*b^6*x^2 + 31*a^6*b^3*x + 2*a^9 - 60*(b^9*x^3 + 2*a^3*b^6*x^2 + a^6*b^3*x)*log(b*x^(1/3) + a) + 60
*(b^9*x^3 + 2*a^3*b^6*x^2 + a^6*b^3*x)*log(x^(1/3)) + 3*(20*a*b^8*x^2 + 35*a^4*b^5*x + 12*a^7*b^2)*x^(2/3) - 3
*(10*a^2*b^7*x^2 + 16*a^5*b^4*x + 3*a^8*b)*x^(1/3))/(a^6*b^6*x^3 + 2*a^9*b^3*x^2 + a^12*x)

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giac [A]  time = 0.17, size = 90, normalized size = 0.87 \[ \frac {30 \, b^{3} \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{a^{6}} - \frac {10 \, b^{3} \log \left ({\left | x \right |}\right )}{a^{6}} - \frac {60 \, a b^{4} x^{\frac {4}{3}} + 90 \, a^{2} b^{3} x + 20 \, a^{3} b^{2} x^{\frac {2}{3}} - 5 \, a^{4} b x^{\frac {1}{3}} + 2 \, a^{5}}{2 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2} a^{6} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

30*b^3*log(abs(b*x^(1/3) + a))/a^6 - 10*b^3*log(abs(x))/a^6 - 1/2*(60*a*b^4*x^(4/3) + 90*a^2*b^3*x + 20*a^3*b^
2*x^(2/3) - 5*a^4*b*x^(1/3) + 2*a^5)/((b*x^(1/3) + a)^2*a^6*x)

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maple [A]  time = 0.01, size = 90, normalized size = 0.87 \[ -\frac {3 b^{3}}{2 \left (b \,x^{\frac {1}{3}}+a \right )^{2} a^{4}}-\frac {12 b^{3}}{\left (b \,x^{\frac {1}{3}}+a \right ) a^{5}}-\frac {10 b^{3} \ln \relax (x )}{a^{6}}+\frac {30 b^{3} \ln \left (b \,x^{\frac {1}{3}}+a \right )}{a^{6}}-\frac {18 b^{2}}{a^{5} x^{\frac {1}{3}}}+\frac {9 b}{2 a^{4} x^{\frac {2}{3}}}-\frac {1}{a^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*b^3/a^4/(b*x^(1/3)+a)^2-12*b^3/a^5/(b*x^(1/3)+a)-1/a^3/x+9/2*b/a^4/x^(2/3)-18*b^2/a^5/x^(1/3)+30*b^3*ln(b
*x^(1/3)+a)/a^6-10*b^3*ln(x)/a^6

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maxima [A]  time = 0.64, size = 97, normalized size = 0.94 \[ -\frac {60 \, b^{4} x^{\frac {4}{3}} + 90 \, a b^{3} x + 20 \, a^{2} b^{2} x^{\frac {2}{3}} - 5 \, a^{3} b x^{\frac {1}{3}} + 2 \, a^{4}}{2 \, {\left (a^{5} b^{2} x^{\frac {5}{3}} + 2 \, a^{6} b x^{\frac {4}{3}} + a^{7} x\right )}} + \frac {30 \, b^{3} \log \left (b x^{\frac {1}{3}} + a\right )}{a^{6}} - \frac {10 \, b^{3} \log \relax (x)}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(60*b^4*x^(4/3) + 90*a*b^3*x + 20*a^2*b^2*x^(2/3) - 5*a^3*b*x^(1/3) + 2*a^4)/(a^5*b^2*x^(5/3) + 2*a^6*b*x
^(4/3) + a^7*x) + 30*b^3*log(b*x^(1/3) + a)/a^6 - 10*b^3*log(x)/a^6

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mupad [B]  time = 1.16, size = 89, normalized size = 0.86 \[ \frac {60\,b^3\,\mathrm {atanh}\left (\frac {2\,b\,x^{1/3}}{a}+1\right )}{a^6}-\frac {\frac {1}{a}-\frac {5\,b\,x^{1/3}}{2\,a^2}+\frac {45\,b^3\,x}{a^4}+\frac {10\,b^2\,x^{2/3}}{a^3}+\frac {30\,b^4\,x^{4/3}}{a^5}}{a^2\,x+b^2\,x^{5/3}+2\,a\,b\,x^{4/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b*x^(1/3))^3),x)

[Out]

(60*b^3*atanh((2*b*x^(1/3))/a + 1))/a^6 - (1/a - (5*b*x^(1/3))/(2*a^2) + (45*b^3*x)/a^4 + (10*b^2*x^(2/3))/a^3
+ (30*b^4*x^(4/3))/a^5)/(a^2*x + b^2*x^(5/3) + 2*a*b*x^(4/3))

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sympy [A]  time = 4.87, size = 561, normalized size = 5.45 \[ \begin {cases} \frac {\tilde {\infty }}{x^{2}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{a^{3} x} & \text {for}\: b = 0 \\- \frac {1}{2 b^{3} x^{2}} & \text {for}\: a = 0 \\- \frac {2 a^{5} x^{\frac {2}{3}}}{2 a^{8} x^{\frac {5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac {7}{3}}} + \frac {5 a^{4} b x}{2 a^{8} x^{\frac {5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac {7}{3}}} - \frac {20 a^{3} b^{2} x^{\frac {4}{3}}}{2 a^{8} x^{\frac {5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac {7}{3}}} - \frac {20 a^{2} b^{3} x^{\frac {5}{3}} \log {\relax (x )}}{2 a^{8} x^{\frac {5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac {7}{3}}} + \frac {60 a^{2} b^{3} x^{\frac {5}{3}} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{2 a^{8} x^{\frac {5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac {7}{3}}} - \frac {90 a^{2} b^{3} x^{\frac {5}{3}}}{2 a^{8} x^{\frac {5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac {7}{3}}} - \frac {40 a b^{4} x^{2} \log {\relax (x )}}{2 a^{8} x^{\frac {5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac {7}{3}}} + \frac {120 a b^{4} x^{2} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{2 a^{8} x^{\frac {5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac {7}{3}}} - \frac {60 a b^{4} x^{2}}{2 a^{8} x^{\frac {5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac {7}{3}}} - \frac {20 b^{5} x^{\frac {7}{3}} \log {\relax (x )}}{2 a^{8} x^{\frac {5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac {7}{3}}} + \frac {60 b^{5} x^{\frac {7}{3}} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{2 a^{8} x^{\frac {5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac {7}{3}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/x**2, Eq(a, 0) & Eq(b, 0)), (-1/(a**3*x), Eq(b, 0)), (-1/(2*b**3*x**2), Eq(a, 0)), (-2*a**5*x**
(2/3)/(2*a**8*x**(5/3) + 4*a**7*b*x**2 + 2*a**6*b**2*x**(7/3)) + 5*a**4*b*x/(2*a**8*x**(5/3) + 4*a**7*b*x**2 +
2*a**6*b**2*x**(7/3)) - 20*a**3*b**2*x**(4/3)/(2*a**8*x**(5/3) + 4*a**7*b*x**2 + 2*a**6*b**2*x**(7/3)) - 20*a
**2*b**3*x**(5/3)*log(x)/(2*a**8*x**(5/3) + 4*a**7*b*x**2 + 2*a**6*b**2*x**(7/3)) + 60*a**2*b**3*x**(5/3)*log(
a/b + x**(1/3))/(2*a**8*x**(5/3) + 4*a**7*b*x**2 + 2*a**6*b**2*x**(7/3)) - 90*a**2*b**3*x**(5/3)/(2*a**8*x**(5
/3) + 4*a**7*b*x**2 + 2*a**6*b**2*x**(7/3)) - 40*a*b**4*x**2*log(x)/(2*a**8*x**(5/3) + 4*a**7*b*x**2 + 2*a**6*
b**2*x**(7/3)) + 120*a*b**4*x**2*log(a/b + x**(1/3))/(2*a**8*x**(5/3) + 4*a**7*b*x**2 + 2*a**6*b**2*x**(7/3))
- 60*a*b**4*x**2/(2*a**8*x**(5/3) + 4*a**7*b*x**2 + 2*a**6*b**2*x**(7/3)) - 20*b**5*x**(7/3)*log(x)/(2*a**8*x*
*(5/3) + 4*a**7*b*x**2 + 2*a**6*b**2*x**(7/3)) + 60*b**5*x**(7/3)*log(a/b + x**(1/3))/(2*a**8*x**(5/3) + 4*a**
7*b*x**2 + 2*a**6*b**2*x**(7/3)), True))

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