3.2374 \(\int \frac {x^2}{(a+b \sqrt [3]{x})^3} \, dx\)

Optimal. Leaf size=134 \[ -\frac {3 a^8}{2 b^9 \left (a+b \sqrt [3]{x}\right )^2}+\frac {24 a^7}{b^9 \left (a+b \sqrt [3]{x}\right )}+\frac {84 a^6 \log \left (a+b \sqrt [3]{x}\right )}{b^9}-\frac {63 a^5 \sqrt [3]{x}}{b^8}+\frac {45 a^4 x^{2/3}}{2 b^7}-\frac {10 a^3 x}{b^6}+\frac {9 a^2 x^{4/3}}{2 b^5}-\frac {9 a x^{5/3}}{5 b^4}+\frac {x^2}{2 b^3} \]

[Out]

-3/2*a^8/b^9/(a+b*x^(1/3))^2+24*a^7/b^9/(a+b*x^(1/3))-63*a^5*x^(1/3)/b^8+45/2*a^4*x^(2/3)/b^7-10*a^3*x/b^6+9/2
*a^2*x^(4/3)/b^5-9/5*a*x^(5/3)/b^4+1/2*x^2/b^3+84*a^6*ln(a+b*x^(1/3))/b^9

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Rubi [A]  time = 0.10, antiderivative size = 134, 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, 43} \[ \frac {45 a^4 x^{2/3}}{2 b^7}+\frac {9 a^2 x^{4/3}}{2 b^5}-\frac {3 a^8}{2 b^9 \left (a+b \sqrt [3]{x}\right )^2}+\frac {24 a^7}{b^9 \left (a+b \sqrt [3]{x}\right )}-\frac {63 a^5 \sqrt [3]{x}}{b^8}-\frac {10 a^3 x}{b^6}+\frac {84 a^6 \log \left (a+b \sqrt [3]{x}\right )}{b^9}-\frac {9 a x^{5/3}}{5 b^4}+\frac {x^2}{2 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*x^(1/3))^3,x]

[Out]

(-3*a^8)/(2*b^9*(a + b*x^(1/3))^2) + (24*a^7)/(b^9*(a + b*x^(1/3))) - (63*a^5*x^(1/3))/b^8 + (45*a^4*x^(2/3))/
(2*b^7) - (10*a^3*x)/b^6 + (9*a^2*x^(4/3))/(2*b^5) - (9*a*x^(5/3))/(5*b^4) + x^2/(2*b^3) + (84*a^6*Log[a + b*x
^(1/3)])/b^9

Rule 43

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 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 {x^2}{\left (a+b \sqrt [3]{x}\right )^3} \, dx &=3 \operatorname {Subst}\left (\int \frac {x^8}{(a+b x)^3} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (-\frac {21 a^5}{b^8}+\frac {15 a^4 x}{b^7}-\frac {10 a^3 x^2}{b^6}+\frac {6 a^2 x^3}{b^5}-\frac {3 a x^4}{b^4}+\frac {x^5}{b^3}+\frac {a^8}{b^8 (a+b x)^3}-\frac {8 a^7}{b^8 (a+b x)^2}+\frac {28 a^6}{b^8 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {3 a^8}{2 b^9 \left (a+b \sqrt [3]{x}\right )^2}+\frac {24 a^7}{b^9 \left (a+b \sqrt [3]{x}\right )}-\frac {63 a^5 \sqrt [3]{x}}{b^8}+\frac {45 a^4 x^{2/3}}{2 b^7}-\frac {10 a^3 x}{b^6}+\frac {9 a^2 x^{4/3}}{2 b^5}-\frac {9 a x^{5/3}}{5 b^4}+\frac {x^2}{2 b^3}+\frac {84 a^6 \log \left (a+b \sqrt [3]{x}\right )}{b^9}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 120, normalized size = 0.90 \[ \frac {-\frac {15 a^8}{\left (a+b \sqrt [3]{x}\right )^2}+\frac {240 a^7}{a+b \sqrt [3]{x}}+840 a^6 \log \left (a+b \sqrt [3]{x}\right )-630 a^5 b \sqrt [3]{x}+225 a^4 b^2 x^{2/3}-100 a^3 b^3 x+45 a^2 b^4 x^{4/3}-18 a b^5 x^{5/3}+5 b^6 x^2}{10 b^9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*x^(1/3))^3,x]

[Out]

((-15*a^8)/(a + b*x^(1/3))^2 + (240*a^7)/(a + b*x^(1/3)) - 630*a^5*b*x^(1/3) + 225*a^4*b^2*x^(2/3) - 100*a^3*b
^3*x + 45*a^2*b^4*x^(4/3) - 18*a*b^5*x^(5/3) + 5*b^6*x^2 + 840*a^6*Log[a + b*x^(1/3)])/(10*b^9)

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fricas [A]  time = 0.65, size = 192, normalized size = 1.43 \[ \frac {5 \, b^{12} x^{4} - 90 \, a^{3} b^{9} x^{3} - 195 \, a^{6} b^{6} x^{2} + 170 \, a^{9} b^{3} x + 225 \, a^{12} + 840 \, {\left (a^{6} b^{6} x^{2} + 2 \, a^{9} b^{3} x + a^{12}\right )} \log \left (b x^{\frac {1}{3}} + a\right ) - 3 \, {\left (6 \, a b^{11} x^{3} - 63 \, a^{4} b^{8} x^{2} - 224 \, a^{7} b^{5} x - 140 \, a^{10} b^{2}\right )} x^{\frac {2}{3}} + 15 \, {\left (3 \, a^{2} b^{10} x^{3} - 36 \, a^{5} b^{7} x^{2} - 98 \, a^{8} b^{4} x - 56 \, a^{11} b\right )} x^{\frac {1}{3}}}{10 \, {\left (b^{15} x^{2} + 2 \, a^{3} b^{12} x + a^{6} b^{9}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(5*b^12*x^4 - 90*a^3*b^9*x^3 - 195*a^6*b^6*x^2 + 170*a^9*b^3*x + 225*a^12 + 840*(a^6*b^6*x^2 + 2*a^9*b^3*
x + a^12)*log(b*x^(1/3) + a) - 3*(6*a*b^11*x^3 - 63*a^4*b^8*x^2 - 224*a^7*b^5*x - 140*a^10*b^2)*x^(2/3) + 15*(
3*a^2*b^10*x^3 - 36*a^5*b^7*x^2 - 98*a^8*b^4*x - 56*a^11*b)*x^(1/3))/(b^15*x^2 + 2*a^3*b^12*x + a^6*b^9)

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giac [A]  time = 0.21, size = 112, normalized size = 0.84 \[ \frac {84 \, a^{6} \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{b^{9}} + \frac {3 \, {\left (16 \, a^{7} b x^{\frac {1}{3}} + 15 \, a^{8}\right )}}{2 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2} b^{9}} + \frac {5 \, b^{15} x^{2} - 18 \, a b^{14} x^{\frac {5}{3}} + 45 \, a^{2} b^{13} x^{\frac {4}{3}} - 100 \, a^{3} b^{12} x + 225 \, a^{4} b^{11} x^{\frac {2}{3}} - 630 \, a^{5} b^{10} x^{\frac {1}{3}}}{10 \, b^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

84*a^6*log(abs(b*x^(1/3) + a))/b^9 + 3/2*(16*a^7*b*x^(1/3) + 15*a^8)/((b*x^(1/3) + a)^2*b^9) + 1/10*(5*b^15*x^
2 - 18*a*b^14*x^(5/3) + 45*a^2*b^13*x^(4/3) - 100*a^3*b^12*x + 225*a^4*b^11*x^(2/3) - 630*a^5*b^10*x^(1/3))/b^
18

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maple [A]  time = 0.01, size = 111, normalized size = 0.83 \[ -\frac {3 a^{8}}{2 \left (b \,x^{\frac {1}{3}}+a \right )^{2} b^{9}}+\frac {x^{2}}{2 b^{3}}-\frac {9 a \,x^{\frac {5}{3}}}{5 b^{4}}+\frac {9 a^{2} x^{\frac {4}{3}}}{2 b^{5}}+\frac {24 a^{7}}{\left (b \,x^{\frac {1}{3}}+a \right ) b^{9}}+\frac {84 a^{6} \ln \left (b \,x^{\frac {1}{3}}+a \right )}{b^{9}}-\frac {10 a^{3} x}{b^{6}}+\frac {45 a^{4} x^{\frac {2}{3}}}{2 b^{7}}-\frac {63 a^{5} x^{\frac {1}{3}}}{b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*a^8/b^9/(b*x^(1/3)+a)^2+24*a^7/b^9/(b*x^(1/3)+a)-63*a^5*x^(1/3)/b^8+45/2*a^4*x^(2/3)/b^7-10*a^3/b^6*x+9/2
*a^2*x^(4/3)/b^5-9/5*a*x^(5/3)/b^4+1/2*x^2/b^3+84*a^6*ln(b*x^(1/3)+a)/b^9

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maxima [A]  time = 0.51, size = 146, normalized size = 1.09 \[ \frac {84 \, a^{6} \log \left (b x^{\frac {1}{3}} + a\right )}{b^{9}} + \frac {{\left (b x^{\frac {1}{3}} + a\right )}^{6}}{2 \, b^{9}} - \frac {24 \, {\left (b x^{\frac {1}{3}} + a\right )}^{5} a}{5 \, b^{9}} + \frac {21 \, {\left (b x^{\frac {1}{3}} + a\right )}^{4} a^{2}}{b^{9}} - \frac {56 \, {\left (b x^{\frac {1}{3}} + a\right )}^{3} a^{3}}{b^{9}} + \frac {105 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2} a^{4}}{b^{9}} - \frac {168 \, {\left (b x^{\frac {1}{3}} + a\right )} a^{5}}{b^{9}} + \frac {24 \, a^{7}}{{\left (b x^{\frac {1}{3}} + a\right )} b^{9}} - \frac {3 \, a^{8}}{2 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2} b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

84*a^6*log(b*x^(1/3) + a)/b^9 + 1/2*(b*x^(1/3) + a)^6/b^9 - 24/5*(b*x^(1/3) + a)^5*a/b^9 + 21*(b*x^(1/3) + a)^
4*a^2/b^9 - 56*(b*x^(1/3) + a)^3*a^3/b^9 + 105*(b*x^(1/3) + a)^2*a^4/b^9 - 168*(b*x^(1/3) + a)*a^5/b^9 + 24*a^
7/((b*x^(1/3) + a)*b^9) - 3/2*a^8/((b*x^(1/3) + a)^2*b^9)

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mupad [B]  time = 0.05, size = 120, normalized size = 0.90 \[ \frac {\frac {45\,a^8}{2\,b}+24\,a^7\,x^{1/3}}{a^2\,b^8+b^{10}\,x^{2/3}+2\,a\,b^9\,x^{1/3}}+\frac {x^2}{2\,b^3}-\frac {10\,a^3\,x}{b^6}-\frac {9\,a\,x^{5/3}}{5\,b^4}+\frac {84\,a^6\,\ln \left (a+b\,x^{1/3}\right )}{b^9}+\frac {9\,a^2\,x^{4/3}}{2\,b^5}+\frac {45\,a^4\,x^{2/3}}{2\,b^7}-\frac {63\,a^5\,x^{1/3}}{b^8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + b*x^(1/3))^3,x)

[Out]

((45*a^8)/(2*b) + 24*a^7*x^(1/3))/(a^2*b^8 + b^10*x^(2/3) + 2*a*b^9*x^(1/3)) + x^2/(2*b^3) - (10*a^3*x)/b^6 -
(9*a*x^(5/3))/(5*b^4) + (84*a^6*log(a + b*x^(1/3)))/b^9 + (9*a^2*x^(4/3))/(2*b^5) + (45*a^4*x^(2/3))/(2*b^7) -
(63*a^5*x^(1/3))/b^8

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sympy [A]  time = 2.25, size = 493, normalized size = 3.68 \[ \begin {cases} \frac {840 a^{8} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac {2}{3}}} + \frac {1260 a^{8}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac {2}{3}}} + \frac {1680 a^{7} b \sqrt [3]{x} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac {2}{3}}} + \frac {1680 a^{7} b \sqrt [3]{x}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac {2}{3}}} + \frac {840 a^{6} b^{2} x^{\frac {2}{3}} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac {2}{3}}} - \frac {280 a^{5} b^{3} x}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac {2}{3}}} + \frac {70 a^{4} b^{4} x^{\frac {4}{3}}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac {2}{3}}} - \frac {28 a^{3} b^{5} x^{\frac {5}{3}}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac {2}{3}}} + \frac {14 a^{2} b^{6} x^{2}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac {2}{3}}} - \frac {8 a b^{7} x^{\frac {7}{3}}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac {2}{3}}} + \frac {5 b^{8} x^{\frac {8}{3}}}{10 a^{2} b^{9} + 20 a b^{10} \sqrt [3]{x} + 10 b^{11} x^{\frac {2}{3}}} & \text {for}\: b \neq 0 \\\frac {x^{3}}{3 a^{3}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((840*a**8*log(a/b + x**(1/3))/(10*a**2*b**9 + 20*a*b**10*x**(1/3) + 10*b**11*x**(2/3)) + 1260*a**8/(
10*a**2*b**9 + 20*a*b**10*x**(1/3) + 10*b**11*x**(2/3)) + 1680*a**7*b*x**(1/3)*log(a/b + x**(1/3))/(10*a**2*b*
*9 + 20*a*b**10*x**(1/3) + 10*b**11*x**(2/3)) + 1680*a**7*b*x**(1/3)/(10*a**2*b**9 + 20*a*b**10*x**(1/3) + 10*
b**11*x**(2/3)) + 840*a**6*b**2*x**(2/3)*log(a/b + x**(1/3))/(10*a**2*b**9 + 20*a*b**10*x**(1/3) + 10*b**11*x*
*(2/3)) - 280*a**5*b**3*x/(10*a**2*b**9 + 20*a*b**10*x**(1/3) + 10*b**11*x**(2/3)) + 70*a**4*b**4*x**(4/3)/(10
*a**2*b**9 + 20*a*b**10*x**(1/3) + 10*b**11*x**(2/3)) - 28*a**3*b**5*x**(5/3)/(10*a**2*b**9 + 20*a*b**10*x**(1
/3) + 10*b**11*x**(2/3)) + 14*a**2*b**6*x**2/(10*a**2*b**9 + 20*a*b**10*x**(1/3) + 10*b**11*x**(2/3)) - 8*a*b*
*7*x**(7/3)/(10*a**2*b**9 + 20*a*b**10*x**(1/3) + 10*b**11*x**(2/3)) + 5*b**8*x**(8/3)/(10*a**2*b**9 + 20*a*b*
*10*x**(1/3) + 10*b**11*x**(2/3)), Ne(b, 0)), (x**3/(3*a**3), True))

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