3.2331 \(\int \frac {(a+b \sqrt [3]{x})^{10}}{x^3} \, dx\)

Optimal. Leaf size=131 \[ -\frac {a^{10}}{2 x^2}-\frac {6 a^9 b}{x^{5/3}}-\frac {135 a^8 b^2}{4 x^{4/3}}-\frac {120 a^7 b^3}{x}-\frac {315 a^6 b^4}{x^{2/3}}-\frac {756 a^5 b^5}{\sqrt [3]{x}}+210 a^4 b^6 \log (x)+360 a^3 b^7 \sqrt [3]{x}+\frac {135}{2} a^2 b^8 x^{2/3}+10 a b^9 x+\frac {3}{4} b^{10} x^{4/3} \]

[Out]

-1/2*a^10/x^2-6*a^9*b/x^(5/3)-135/4*a^8*b^2/x^(4/3)-120*a^7*b^3/x-315*a^6*b^4/x^(2/3)-756*a^5*b^5/x^(1/3)+360*
a^3*b^7*x^(1/3)+135/2*a^2*b^8*x^(2/3)+10*a*b^9*x+3/4*b^10*x^(4/3)+210*a^4*b^6*ln(x)

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Rubi [A]  time = 0.07, antiderivative size = 131, 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} \[ -\frac {135 a^8 b^2}{4 x^{4/3}}-\frac {315 a^6 b^4}{x^{2/3}}+\frac {135}{2} a^2 b^8 x^{2/3}-\frac {120 a^7 b^3}{x}-\frac {756 a^5 b^5}{\sqrt [3]{x}}+360 a^3 b^7 \sqrt [3]{x}+210 a^4 b^6 \log (x)-\frac {6 a^9 b}{x^{5/3}}-\frac {a^{10}}{2 x^2}+10 a b^9 x+\frac {3}{4} b^{10} x^{4/3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^10/x^3,x]

[Out]

-a^10/(2*x^2) - (6*a^9*b)/x^(5/3) - (135*a^8*b^2)/(4*x^(4/3)) - (120*a^7*b^3)/x - (315*a^6*b^4)/x^(2/3) - (756
*a^5*b^5)/x^(1/3) + 360*a^3*b^7*x^(1/3) + (135*a^2*b^8*x^(2/3))/2 + 10*a*b^9*x + (3*b^10*x^(4/3))/4 + 210*a^4*
b^6*Log[x]

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 {\left (a+b \sqrt [3]{x}\right )^{10}}{x^3} \, dx &=3 \operatorname {Subst}\left (\int \frac {(a+b x)^{10}}{x^7} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (120 a^3 b^7+\frac {a^{10}}{x^7}+\frac {10 a^9 b}{x^6}+\frac {45 a^8 b^2}{x^5}+\frac {120 a^7 b^3}{x^4}+\frac {210 a^6 b^4}{x^3}+\frac {252 a^5 b^5}{x^2}+\frac {210 a^4 b^6}{x}+45 a^2 b^8 x+10 a b^9 x^2+b^{10} x^3\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {a^{10}}{2 x^2}-\frac {6 a^9 b}{x^{5/3}}-\frac {135 a^8 b^2}{4 x^{4/3}}-\frac {120 a^7 b^3}{x}-\frac {315 a^6 b^4}{x^{2/3}}-\frac {756 a^5 b^5}{\sqrt [3]{x}}+360 a^3 b^7 \sqrt [3]{x}+\frac {135}{2} a^2 b^8 x^{2/3}+10 a b^9 x+\frac {3}{4} b^{10} x^{4/3}+210 a^4 b^6 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 131, normalized size = 1.00 \[ -\frac {a^{10}}{2 x^2}-\frac {6 a^9 b}{x^{5/3}}-\frac {135 a^8 b^2}{4 x^{4/3}}-\frac {120 a^7 b^3}{x}-\frac {315 a^6 b^4}{x^{2/3}}-\frac {756 a^5 b^5}{\sqrt [3]{x}}+210 a^4 b^6 \log (x)+360 a^3 b^7 \sqrt [3]{x}+\frac {135}{2} a^2 b^8 x^{2/3}+10 a b^9 x+\frac {3}{4} b^{10} x^{4/3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^10/x^3,x]

[Out]

-1/2*a^10/x^2 - (6*a^9*b)/x^(5/3) - (135*a^8*b^2)/(4*x^(4/3)) - (120*a^7*b^3)/x - (315*a^6*b^4)/x^(2/3) - (756
*a^5*b^5)/x^(1/3) + 360*a^3*b^7*x^(1/3) + (135*a^2*b^8*x^(2/3))/2 + 10*a*b^9*x + (3*b^10*x^(4/3))/4 + 210*a^4*
b^6*Log[x]

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fricas [A]  time = 0.79, size = 117, normalized size = 0.89 \[ \frac {40 \, a b^{9} x^{3} + 2520 \, a^{4} b^{6} x^{2} \log \left (x^{\frac {1}{3}}\right ) - 480 \, a^{7} b^{3} x - 2 \, a^{10} + 27 \, {\left (10 \, a^{2} b^{8} x^{2} - 112 \, a^{5} b^{5} x - 5 \, a^{8} b^{2}\right )} x^{\frac {2}{3}} + 3 \, {\left (b^{10} x^{3} + 480 \, a^{3} b^{7} x^{2} - 420 \, a^{6} b^{4} x - 8 \, a^{9} b\right )} x^{\frac {1}{3}}}{4 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^10/x^3,x, algorithm="fricas")

[Out]

1/4*(40*a*b^9*x^3 + 2520*a^4*b^6*x^2*log(x^(1/3)) - 480*a^7*b^3*x - 2*a^10 + 27*(10*a^2*b^8*x^2 - 112*a^5*b^5*
x - 5*a^8*b^2)*x^(2/3) + 3*(b^10*x^3 + 480*a^3*b^7*x^2 - 420*a^6*b^4*x - 8*a^9*b)*x^(1/3))/x^2

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giac [A]  time = 0.20, size = 111, normalized size = 0.85 \[ \frac {3}{4} \, b^{10} x^{\frac {4}{3}} + 10 \, a b^{9} x + 210 \, a^{4} b^{6} \log \left ({\left | x \right |}\right ) + \frac {135}{2} \, a^{2} b^{8} x^{\frac {2}{3}} + 360 \, a^{3} b^{7} x^{\frac {1}{3}} - \frac {3024 \, a^{5} b^{5} x^{\frac {5}{3}} + 1260 \, a^{6} b^{4} x^{\frac {4}{3}} + 480 \, a^{7} b^{3} x + 135 \, a^{8} b^{2} x^{\frac {2}{3}} + 24 \, a^{9} b x^{\frac {1}{3}} + 2 \, a^{10}}{4 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^10/x^3,x, algorithm="giac")

[Out]

3/4*b^10*x^(4/3) + 10*a*b^9*x + 210*a^4*b^6*log(abs(x)) + 135/2*a^2*b^8*x^(2/3) + 360*a^3*b^7*x^(1/3) - 1/4*(3
024*a^5*b^5*x^(5/3) + 1260*a^6*b^4*x^(4/3) + 480*a^7*b^3*x + 135*a^8*b^2*x^(2/3) + 24*a^9*b*x^(1/3) + 2*a^10)/
x^2

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maple [A]  time = 0.01, size = 110, normalized size = 0.84 \[ \frac {3 b^{10} x^{\frac {4}{3}}}{4}+210 a^{4} b^{6} \ln \relax (x )+10 a \,b^{9} x +\frac {135 a^{2} b^{8} x^{\frac {2}{3}}}{2}+360 a^{3} b^{7} x^{\frac {1}{3}}-\frac {756 a^{5} b^{5}}{x^{\frac {1}{3}}}-\frac {315 a^{6} b^{4}}{x^{\frac {2}{3}}}-\frac {120 a^{7} b^{3}}{x}-\frac {135 a^{8} b^{2}}{4 x^{\frac {4}{3}}}-\frac {6 a^{9} b}{x^{\frac {5}{3}}}-\frac {a^{10}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/3))^10/x^3,x)

[Out]

-1/2*a^10/x^2-6*a^9*b/x^(5/3)-135/4*a^8*b^2/x^(4/3)-120*a^7*b^3/x-315*a^6*b^4/x^(2/3)-756*a^5*b^5/x^(1/3)+360*
a^3*b^7*x^(1/3)+135/2*a^2*b^8*x^(2/3)+10*a*b^9*x+3/4*b^10*x^(4/3)+210*a^4*b^6*ln(x)

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maxima [A]  time = 0.91, size = 110, normalized size = 0.84 \[ \frac {3}{4} \, b^{10} x^{\frac {4}{3}} + 10 \, a b^{9} x + 210 \, a^{4} b^{6} \log \relax (x) + \frac {135}{2} \, a^{2} b^{8} x^{\frac {2}{3}} + 360 \, a^{3} b^{7} x^{\frac {1}{3}} - \frac {3024 \, a^{5} b^{5} x^{\frac {5}{3}} + 1260 \, a^{6} b^{4} x^{\frac {4}{3}} + 480 \, a^{7} b^{3} x + 135 \, a^{8} b^{2} x^{\frac {2}{3}} + 24 \, a^{9} b x^{\frac {1}{3}} + 2 \, a^{10}}{4 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^10/x^3,x, algorithm="maxima")

[Out]

3/4*b^10*x^(4/3) + 10*a*b^9*x + 210*a^4*b^6*log(x) + 135/2*a^2*b^8*x^(2/3) + 360*a^3*b^7*x^(1/3) - 1/4*(3024*a
^5*b^5*x^(5/3) + 1260*a^6*b^4*x^(4/3) + 480*a^7*b^3*x + 135*a^8*b^2*x^(2/3) + 24*a^9*b*x^(1/3) + 2*a^10)/x^2

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mupad [B]  time = 1.13, size = 112, normalized size = 0.85 \[ \frac {3\,b^{10}\,x^{4/3}}{4}-\frac {\frac {a^{10}}{2}+120\,a^7\,b^3\,x+6\,a^9\,b\,x^{1/3}+\frac {135\,a^8\,b^2\,x^{2/3}}{4}+315\,a^6\,b^4\,x^{4/3}+756\,a^5\,b^5\,x^{5/3}}{x^2}+630\,a^4\,b^6\,\ln \left (x^{1/3}\right )+360\,a^3\,b^7\,x^{1/3}+\frac {135\,a^2\,b^8\,x^{2/3}}{2}+10\,a\,b^9\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/3))^10/x^3,x)

[Out]

(3*b^10*x^(4/3))/4 - (a^10/2 + 120*a^7*b^3*x + 6*a^9*b*x^(1/3) + (135*a^8*b^2*x^(2/3))/4 + 315*a^6*b^4*x^(4/3)
+ 756*a^5*b^5*x^(5/3))/x^2 + 630*a^4*b^6*log(x^(1/3)) + 360*a^3*b^7*x^(1/3) + (135*a^2*b^8*x^(2/3))/2 + 10*a*
b^9*x

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sympy [A]  time = 15.85, size = 136, normalized size = 1.04 \[ - \frac {a^{10}}{2 x^{2}} - \frac {6 a^{9} b}{x^{\frac {5}{3}}} - \frac {135 a^{8} b^{2}}{4 x^{\frac {4}{3}}} - \frac {120 a^{7} b^{3}}{x} - \frac {315 a^{6} b^{4}}{x^{\frac {2}{3}}} - \frac {756 a^{5} b^{5}}{\sqrt [3]{x}} + 630 a^{4} b^{6} \log {\left (\sqrt [3]{x} \right )} + 360 a^{3} b^{7} \sqrt [3]{x} + \frac {135 a^{2} b^{8} x^{\frac {2}{3}}}{2} + 10 a b^{9} x + \frac {3 b^{10} x^{\frac {4}{3}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))**10/x**3,x)

[Out]

-a**10/(2*x**2) - 6*a**9*b/x**(5/3) - 135*a**8*b**2/(4*x**(4/3)) - 120*a**7*b**3/x - 315*a**6*b**4/x**(2/3) -
756*a**5*b**5/x**(1/3) + 630*a**4*b**6*log(x**(1/3)) + 360*a**3*b**7*x**(1/3) + 135*a**2*b**8*x**(2/3)/2 + 10*
a*b**9*x + 3*b**10*x**(4/3)/4

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