3.2356 \(\int \frac {x^3}{a+b \sqrt [3]{x}} \, dx\)

Optimal. Leaf size=166 \[ -\frac {3 a^{11} \log \left (a+b \sqrt [3]{x}\right )}{b^{12}}+\frac {3 a^{10} \sqrt [3]{x}}{b^{11}}-\frac {3 a^9 x^{2/3}}{2 b^{10}}+\frac {a^8 x}{b^9}-\frac {3 a^7 x^{4/3}}{4 b^8}+\frac {3 a^6 x^{5/3}}{5 b^7}-\frac {a^5 x^2}{2 b^6}+\frac {3 a^4 x^{7/3}}{7 b^5}-\frac {3 a^3 x^{8/3}}{8 b^4}+\frac {a^2 x^3}{3 b^3}-\frac {3 a x^{10/3}}{10 b^2}+\frac {3 x^{11/3}}{11 b} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 166, 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} \[ -\frac {3 a^9 x^{2/3}}{2 b^{10}}-\frac {3 a^7 x^{4/3}}{4 b^8}+\frac {3 a^6 x^{5/3}}{5 b^7}-\frac {a^5 x^2}{2 b^6}+\frac {3 a^4 x^{7/3}}{7 b^5}-\frac {3 a^3 x^{8/3}}{8 b^4}+\frac {a^2 x^3}{3 b^3}+\frac {3 a^{10} \sqrt [3]{x}}{b^{11}}+\frac {a^8 x}{b^9}-\frac {3 a^{11} \log \left (a+b \sqrt [3]{x}\right )}{b^{12}}-\frac {3 a x^{10/3}}{10 b^2}+\frac {3 x^{11/3}}{11 b} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*x^(1/3)),x]

[Out]

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

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

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Mathematica [A]  time = 0.14, size = 166, normalized size = 1.00 \[ -\frac {3 a^{11} \log \left (a+b \sqrt [3]{x}\right )}{b^{12}}+\frac {3 a^{10} \sqrt [3]{x}}{b^{11}}-\frac {3 a^9 x^{2/3}}{2 b^{10}}+\frac {a^8 x}{b^9}-\frac {3 a^7 x^{4/3}}{4 b^8}+\frac {3 a^6 x^{5/3}}{5 b^7}-\frac {a^5 x^2}{2 b^6}+\frac {3 a^4 x^{7/3}}{7 b^5}-\frac {3 a^3 x^{8/3}}{8 b^4}+\frac {a^2 x^3}{3 b^3}-\frac {3 a x^{10/3}}{10 b^2}+\frac {3 x^{11/3}}{11 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*x^(1/3)),x]

[Out]

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

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fricas [A]  time = 0.82, size = 133, normalized size = 0.80 \[ \frac {3080 \, a^{2} b^{9} x^{3} - 4620 \, a^{5} b^{6} x^{2} + 9240 \, a^{8} b^{3} x - 27720 \, a^{11} \log \left (b x^{\frac {1}{3}} + a\right ) + 63 \, {\left (40 \, b^{11} x^{3} - 55 \, a^{3} b^{8} x^{2} + 88 \, a^{6} b^{5} x - 220 \, a^{9} b^{2}\right )} x^{\frac {2}{3}} - 198 \, {\left (14 \, a b^{10} x^{3} - 20 \, a^{4} b^{7} x^{2} + 35 \, a^{7} b^{4} x - 140 \, a^{10} b\right )} x^{\frac {1}{3}}}{9240 \, b^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9240*(3080*a^2*b^9*x^3 - 4620*a^5*b^6*x^2 + 9240*a^8*b^3*x - 27720*a^11*log(b*x^(1/3) + a) + 63*(40*b^11*x^3
- 55*a^3*b^8*x^2 + 88*a^6*b^5*x - 220*a^9*b^2)*x^(2/3) - 198*(14*a*b^10*x^3 - 20*a^4*b^7*x^2 + 35*a^7*b^4*x -
140*a^10*b)*x^(1/3))/b^12

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giac [A]  time = 0.19, size = 133, normalized size = 0.80 \[ -\frac {3 \, a^{11} \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{b^{12}} + \frac {2520 \, b^{10} x^{\frac {11}{3}} - 2772 \, a b^{9} x^{\frac {10}{3}} + 3080 \, a^{2} b^{8} x^{3} - 3465 \, a^{3} b^{7} x^{\frac {8}{3}} + 3960 \, a^{4} b^{6} x^{\frac {7}{3}} - 4620 \, a^{5} b^{5} x^{2} + 5544 \, a^{6} b^{4} x^{\frac {5}{3}} - 6930 \, a^{7} b^{3} x^{\frac {4}{3}} + 9240 \, a^{8} b^{2} x - 13860 \, a^{9} b x^{\frac {2}{3}} + 27720 \, a^{10} x^{\frac {1}{3}}}{9240 \, b^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*a^11*log(abs(b*x^(1/3) + a))/b^12 + 1/9240*(2520*b^10*x^(11/3) - 2772*a*b^9*x^(10/3) + 3080*a^2*b^8*x^3 - 3
465*a^3*b^7*x^(8/3) + 3960*a^4*b^6*x^(7/3) - 4620*a^5*b^5*x^2 + 5544*a^6*b^4*x^(5/3) - 6930*a^7*b^3*x^(4/3) +
9240*a^8*b^2*x - 13860*a^9*b*x^(2/3) + 27720*a^10*x^(1/3))/b^11

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maple [A]  time = 0.00, size = 131, normalized size = 0.79 \[ \frac {3 x^{\frac {11}{3}}}{11 b}-\frac {3 a \,x^{\frac {10}{3}}}{10 b^{2}}+\frac {a^{2} x^{3}}{3 b^{3}}-\frac {3 a^{3} x^{\frac {8}{3}}}{8 b^{4}}+\frac {3 a^{4} x^{\frac {7}{3}}}{7 b^{5}}-\frac {a^{5} x^{2}}{2 b^{6}}+\frac {3 a^{6} x^{\frac {5}{3}}}{5 b^{7}}-\frac {3 a^{7} x^{\frac {4}{3}}}{4 b^{8}}-\frac {3 a^{11} \ln \left (b \,x^{\frac {1}{3}}+a \right )}{b^{12}}+\frac {a^{8} x}{b^{9}}-\frac {3 a^{9} x^{\frac {2}{3}}}{2 b^{10}}+\frac {3 a^{10} x^{\frac {1}{3}}}{b^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.92, size = 197, normalized size = 1.19 \[ -\frac {3 \, a^{11} \log \left (b x^{\frac {1}{3}} + a\right )}{b^{12}} + \frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{11}}{11 \, b^{12}} - \frac {33 \, {\left (b x^{\frac {1}{3}} + a\right )}^{10} a}{10 \, b^{12}} + \frac {55 \, {\left (b x^{\frac {1}{3}} + a\right )}^{9} a^{2}}{3 \, b^{12}} - \frac {495 \, {\left (b x^{\frac {1}{3}} + a\right )}^{8} a^{3}}{8 \, b^{12}} + \frac {990 \, {\left (b x^{\frac {1}{3}} + a\right )}^{7} a^{4}}{7 \, b^{12}} - \frac {231 \, {\left (b x^{\frac {1}{3}} + a\right )}^{6} a^{5}}{b^{12}} + \frac {1386 \, {\left (b x^{\frac {1}{3}} + a\right )}^{5} a^{6}}{5 \, b^{12}} - \frac {495 \, {\left (b x^{\frac {1}{3}} + a\right )}^{4} a^{7}}{2 \, b^{12}} + \frac {165 \, {\left (b x^{\frac {1}{3}} + a\right )}^{3} a^{8}}{b^{12}} - \frac {165 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2} a^{9}}{2 \, b^{12}} + \frac {33 \, {\left (b x^{\frac {1}{3}} + a\right )} a^{10}}{b^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*a^11*log(b*x^(1/3) + a)/b^12 + 3/11*(b*x^(1/3) + a)^11/b^12 - 33/10*(b*x^(1/3) + a)^10*a/b^12 + 55/3*(b*x^(
1/3) + a)^9*a^2/b^12 - 495/8*(b*x^(1/3) + a)^8*a^3/b^12 + 990/7*(b*x^(1/3) + a)^7*a^4/b^12 - 231*(b*x^(1/3) +
a)^6*a^5/b^12 + 1386/5*(b*x^(1/3) + a)^5*a^6/b^12 - 495/2*(b*x^(1/3) + a)^4*a^7/b^12 + 165*(b*x^(1/3) + a)^3*a
^8/b^12 - 165/2*(b*x^(1/3) + a)^2*a^9/b^12 + 33*(b*x^(1/3) + a)*a^10/b^12

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mupad [B]  time = 0.07, size = 130, normalized size = 0.78 \[ \frac {3\,x^{11/3}}{11\,b}-\frac {3\,a\,x^{10/3}}{10\,b^2}+\frac {a^8\,x}{b^9}-\frac {3\,a^{11}\,\ln \left (a+b\,x^{1/3}\right )}{b^{12}}+\frac {a^2\,x^3}{3\,b^3}-\frac {a^5\,x^2}{2\,b^6}-\frac {3\,a^3\,x^{8/3}}{8\,b^4}+\frac {3\,a^4\,x^{7/3}}{7\,b^5}+\frac {3\,a^6\,x^{5/3}}{5\,b^7}-\frac {3\,a^7\,x^{4/3}}{4\,b^8}-\frac {3\,a^9\,x^{2/3}}{2\,b^{10}}+\frac {3\,a^{10}\,x^{1/3}}{b^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + b*x^(1/3)),x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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