3.2330 \(\int \frac{(a+b \sqrt [3]{x})^{10}}{x^2} \, dx\)

Optimal. Leaf size=125 \[ 378 a^5 b^5 x^{2/3}+90 a^3 b^7 x^{4/3}+27 a^2 b^8 x^{5/3}-\frac{135 a^8 b^2}{\sqrt [3]{x}}+630 a^6 b^4 \sqrt [3]{x}+210 a^4 b^6 x+120 a^7 b^3 \log (x)-\frac{15 a^9 b}{x^{2/3}}-\frac{a^{10}}{x}+5 a b^9 x^2+\frac{3}{7} b^{10} x^{7/3} \]

[Out]

-(a^10/x) - (15*a^9*b)/x^(2/3) - (135*a^8*b^2)/x^(1/3) + 630*a^6*b^4*x^(1/3) + 378*a^5*b^5*x^(2/3) + 210*a^4*b
^6*x + 90*a^3*b^7*x^(4/3) + 27*a^2*b^8*x^(5/3) + 5*a*b^9*x^2 + (3*b^10*x^(7/3))/7 + 120*a^7*b^3*Log[x]

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Rubi [A]  time = 0.0706347, antiderivative size = 125, 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} \[ 378 a^5 b^5 x^{2/3}+90 a^3 b^7 x^{4/3}+27 a^2 b^8 x^{5/3}-\frac{135 a^8 b^2}{\sqrt [3]{x}}+630 a^6 b^4 \sqrt [3]{x}+210 a^4 b^6 x+120 a^7 b^3 \log (x)-\frac{15 a^9 b}{x^{2/3}}-\frac{a^{10}}{x}+5 a b^9 x^2+\frac{3}{7} b^{10} x^{7/3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^10/x^2,x]

[Out]

-(a^10/x) - (15*a^9*b)/x^(2/3) - (135*a^8*b^2)/x^(1/3) + 630*a^6*b^4*x^(1/3) + 378*a^5*b^5*x^(2/3) + 210*a^4*b
^6*x + 90*a^3*b^7*x^(4/3) + 27*a^2*b^8*x^(5/3) + 5*a*b^9*x^2 + (3*b^10*x^(7/3))/7 + 120*a^7*b^3*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^2} \, dx &=3 \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^4} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (210 a^6 b^4+\frac{a^{10}}{x^4}+\frac{10 a^9 b}{x^3}+\frac{45 a^8 b^2}{x^2}+\frac{120 a^7 b^3}{x}+252 a^5 b^5 x+210 a^4 b^6 x^2+120 a^3 b^7 x^3+45 a^2 b^8 x^4+10 a b^9 x^5+b^{10} x^6\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{a^{10}}{x}-\frac{15 a^9 b}{x^{2/3}}-\frac{135 a^8 b^2}{\sqrt [3]{x}}+630 a^6 b^4 \sqrt [3]{x}+378 a^5 b^5 x^{2/3}+210 a^4 b^6 x+90 a^3 b^7 x^{4/3}+27 a^2 b^8 x^{5/3}+5 a b^9 x^2+\frac{3}{7} b^{10} x^{7/3}+120 a^7 b^3 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0612866, size = 125, normalized size = 1. \[ 378 a^5 b^5 x^{2/3}+90 a^3 b^7 x^{4/3}+27 a^2 b^8 x^{5/3}-\frac{135 a^8 b^2}{\sqrt [3]{x}}+630 a^6 b^4 \sqrt [3]{x}+210 a^4 b^6 x+120 a^7 b^3 \log (x)-\frac{15 a^9 b}{x^{2/3}}-\frac{a^{10}}{x}+5 a b^9 x^2+\frac{3}{7} b^{10} x^{7/3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^10/x^2,x]

[Out]

-(a^10/x) - (15*a^9*b)/x^(2/3) - (135*a^8*b^2)/x^(1/3) + 630*a^6*b^4*x^(1/3) + 378*a^5*b^5*x^(2/3) + 210*a^4*b
^6*x + 90*a^3*b^7*x^(4/3) + 27*a^2*b^8*x^(5/3) + 5*a*b^9*x^2 + (3*b^10*x^(7/3))/7 + 120*a^7*b^3*Log[x]

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Maple [A]  time = 0.008, size = 110, normalized size = 0.9 \begin{align*} -{\frac{{a}^{10}}{x}}-15\,{\frac{{a}^{9}b}{{x}^{2/3}}}-135\,{\frac{{a}^{8}{b}^{2}}{\sqrt [3]{x}}}+630\,{a}^{6}{b}^{4}\sqrt [3]{x}+378\,{a}^{5}{b}^{5}{x}^{2/3}+210\,{a}^{4}{b}^{6}x+90\,{a}^{3}{b}^{7}{x}^{4/3}+27\,{a}^{2}{b}^{8}{x}^{5/3}+5\,a{b}^{9}{x}^{2}+{\frac{3\,{b}^{10}}{7}{x}^{{\frac{7}{3}}}}+120\,{a}^{7}{b}^{3}\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^2,x)

[Out]

-a^10/x-15*a^9*b/x^(2/3)-135*a^8*b^2/x^(1/3)+630*a^6*b^4*x^(1/3)+378*a^5*b^5*x^(2/3)+210*a^4*b^6*x+90*a^3*b^7*
x^(4/3)+27*a^2*b^8*x^(5/3)+5*a*b^9*x^2+3/7*b^10*x^(7/3)+120*a^7*b^3*ln(x)

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Maxima [A]  time = 0.996783, size = 149, normalized size = 1.19 \begin{align*} \frac{3}{7} \, b^{10} x^{\frac{7}{3}} + 5 \, a b^{9} x^{2} + 27 \, a^{2} b^{8} x^{\frac{5}{3}} + 90 \, a^{3} b^{7} x^{\frac{4}{3}} + 210 \, a^{4} b^{6} x + 120 \, a^{7} b^{3} \log \left (x\right ) + 378 \, a^{5} b^{5} x^{\frac{2}{3}} + 630 \, a^{6} b^{4} x^{\frac{1}{3}} - \frac{135 \, a^{8} b^{2} x^{\frac{2}{3}} + 15 \, a^{9} b x^{\frac{1}{3}} + a^{10}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^10/x^2,x, algorithm="maxima")

[Out]

3/7*b^10*x^(7/3) + 5*a*b^9*x^2 + 27*a^2*b^8*x^(5/3) + 90*a^3*b^7*x^(4/3) + 210*a^4*b^6*x + 120*a^7*b^3*log(x)
+ 378*a^5*b^5*x^(2/3) + 630*a^6*b^4*x^(1/3) - (135*a^8*b^2*x^(2/3) + 15*a^9*b*x^(1/3) + a^10)/x

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Fricas [A]  time = 1.53582, size = 275, normalized size = 2.2 \begin{align*} \frac{35 \, a b^{9} x^{3} + 1470 \, a^{4} b^{6} x^{2} + 2520 \, a^{7} b^{3} x \log \left (x^{\frac{1}{3}}\right ) - 7 \, a^{10} + 189 \,{\left (a^{2} b^{8} x^{2} + 14 \, a^{5} b^{5} x - 5 \, a^{8} b^{2}\right )} x^{\frac{2}{3}} + 3 \,{\left (b^{10} x^{3} + 210 \, a^{3} b^{7} x^{2} + 1470 \, a^{6} b^{4} x - 35 \, a^{9} b\right )} x^{\frac{1}{3}}}{7 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^10/x^2,x, algorithm="fricas")

[Out]

1/7*(35*a*b^9*x^3 + 1470*a^4*b^6*x^2 + 2520*a^7*b^3*x*log(x^(1/3)) - 7*a^10 + 189*(a^2*b^8*x^2 + 14*a^5*b^5*x
- 5*a^8*b^2)*x^(2/3) + 3*(b^10*x^3 + 210*a^3*b^7*x^2 + 1470*a^6*b^4*x - 35*a^9*b)*x^(1/3))/x

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Sympy [A]  time = 14.9041, size = 131, normalized size = 1.05 \begin{align*} - \frac{a^{10}}{x} - \frac{15 a^{9} b}{x^{\frac{2}{3}}} - \frac{135 a^{8} b^{2}}{\sqrt [3]{x}} + 360 a^{7} b^{3} \log{\left (\sqrt [3]{x} \right )} + 630 a^{6} b^{4} \sqrt [3]{x} + 378 a^{5} b^{5} x^{\frac{2}{3}} + 210 a^{4} b^{6} x + 90 a^{3} b^{7} x^{\frac{4}{3}} + 27 a^{2} b^{8} x^{\frac{5}{3}} + 5 a b^{9} x^{2} + \frac{3 b^{10} x^{\frac{7}{3}}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))**10/x**2,x)

[Out]

-a**10/x - 15*a**9*b/x**(2/3) - 135*a**8*b**2/x**(1/3) + 360*a**7*b**3*log(x**(1/3)) + 630*a**6*b**4*x**(1/3)
+ 378*a**5*b**5*x**(2/3) + 210*a**4*b**6*x + 90*a**3*b**7*x**(4/3) + 27*a**2*b**8*x**(5/3) + 5*a*b**9*x**2 + 3
*b**10*x**(7/3)/7

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Giac [A]  time = 1.20909, size = 150, normalized size = 1.2 \begin{align*} \frac{3}{7} \, b^{10} x^{\frac{7}{3}} + 5 \, a b^{9} x^{2} + 27 \, a^{2} b^{8} x^{\frac{5}{3}} + 90 \, a^{3} b^{7} x^{\frac{4}{3}} + 210 \, a^{4} b^{6} x + 120 \, a^{7} b^{3} \log \left ({\left | x \right |}\right ) + 378 \, a^{5} b^{5} x^{\frac{2}{3}} + 630 \, a^{6} b^{4} x^{\frac{1}{3}} - \frac{135 \, a^{8} b^{2} x^{\frac{2}{3}} + 15 \, a^{9} b x^{\frac{1}{3}} + a^{10}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^10/x^2,x, algorithm="giac")

[Out]

3/7*b^10*x^(7/3) + 5*a*b^9*x^2 + 27*a^2*b^8*x^(5/3) + 90*a^3*b^7*x^(4/3) + 210*a^4*b^6*x + 120*a^7*b^3*log(abs
(x)) + 378*a^5*b^5*x^(2/3) + 630*a^6*b^4*x^(1/3) - (135*a^8*b^2*x^(2/3) + 15*a^9*b*x^(1/3) + a^10)/x