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

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

[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]

________________________________________________________________________________________

Rubi [A]  time = 0.0722625, antiderivative size = 131, normalized size of antiderivative = 1., 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 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]]

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])

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*}

Mathematica [A]  time = 0.0506021, size = 131, normalized size = 1. \[ -\frac{135 a^8 b^2}{4 x^{4/3}}-\frac{315 a^6 b^4}{x^{2/3}}+\frac{135}{2} a^2 b^8 x^{2/3}-\frac{120 a^7 b^3}{x}-\frac{756 a^5 b^5}{\sqrt [3]{x}}+360 a^3 b^7 \sqrt [3]{x}+210 a^4 b^6 \log (x)-\frac{6 a^9 b}{x^{5/3}}-\frac{a^{10}}{2 x^2}+10 a b^9 x+\frac{3}{4} b^{10} x^{4/3} \]

Antiderivative was successfully verified.

[In]

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) + 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]

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 110, normalized size = 0.8 \begin{align*} -{\frac{{a}^{10}}{2\,{x}^{2}}}-6\,{\frac{{a}^{9}b}{{x}^{5/3}}}-{\frac{135\,{a}^{8}{b}^{2}}{4}{x}^{-{\frac{4}{3}}}}-120\,{\frac{{a}^{7}{b}^{3}}{x}}-315\,{\frac{{a}^{6}{b}^{4}}{{x}^{2/3}}}-756\,{\frac{{a}^{5}{b}^{5}}{\sqrt [3]{x}}}+360\,{a}^{3}{b}^{7}\sqrt [3]{x}+{\frac{135\,{a}^{2}{b}^{8}}{2}{x}^{{\frac{2}{3}}}}+10\,a{b}^{9}x+{\frac{3\,{b}^{10}}{4}{x}^{{\frac{4}{3}}}}+210\,{a}^{4}{b}^{6}\ln \left ( x \right ) \end{align*}

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)

________________________________________________________________________________________

Maxima [A]  time = 0.98386, size = 149, normalized size = 1.14 \begin{align*} \frac{3}{4} \, b^{10} x^{\frac{4}{3}} + 10 \, a b^{9} x + 210 \, a^{4} b^{6} \log \left (x\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}} \end{align*}

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

________________________________________________________________________________________

Fricas [A]  time = 1.54349, size = 278, normalized size = 2.12 \begin{align*} \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}} \end{align*}

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

________________________________________________________________________________________

Sympy [A]  time = 11.7769, size = 136, normalized size = 1.04 \begin{align*} - \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} \end{align*}

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

________________________________________________________________________________________

Giac [A]  time = 1.20726, size = 150, normalized size = 1.15 \begin{align*} \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}} \end{align*}

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