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

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

[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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Rubi [A]  time = 0.07, antiderivative size = 125, 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} \[ 378 a^5 b^5 x^{2/3}+90 a^3 b^7 x^{4/3}+27 a^2 b^8 x^{5/3}-\frac {135 a^8 b^2}{\sqrt [3]{x}}+630 a^6 b^4 \sqrt [3]{x}+210 a^4 b^6 x+120 a^7 b^3 \log (x)-\frac {15 a^9 b}{x^{2/3}}-\frac {a^{10}}{x}+5 a b^9 x^2+\frac {3}{7} b^{10} x^{7/3} \]

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 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^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*}

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Mathematica [A]  time = 0.08, size = 125, normalized size = 1.00 \[ -\frac {a^{10}}{x}-\frac {15 a^9 b}{x^{2/3}}-\frac {135 a^8 b^2}{\sqrt [3]{x}}+120 a^7 b^3 \log (x)+630 a^6 b^4 \sqrt [3]{x}+378 a^5 b^5 x^{2/3}+210 a^4 b^6 x+90 a^3 b^7 x^{4/3}+27 a^2 b^8 x^{5/3}+5 a b^9 x^2+\frac {3}{7} b^{10} x^{7/3} \]

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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fricas [A]  time = 0.82, size = 116, normalized size = 0.93 \[ \frac {35 \, a b^{9} x^{3} + 1470 \, a^{4} b^{6} x^{2} + 2520 \, a^{7} b^{3} x \log \left (x^{\frac {1}{3}}\right ) - 7 \, a^{10} + 189 \, {\left (a^{2} b^{8} x^{2} + 14 \, a^{5} b^{5} x - 5 \, a^{8} b^{2}\right )} x^{\frac {2}{3}} + 3 \, {\left (b^{10} x^{3} + 210 \, a^{3} b^{7} x^{2} + 1470 \, a^{6} b^{4} x - 35 \, a^{9} b\right )} x^{\frac {1}{3}}}{7 \, x} \]

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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giac [A]  time = 0.16, size = 111, normalized size = 0.89 \[ \frac {3}{7} \, b^{10} x^{\frac {7}{3}} + 5 \, a b^{9} x^{2} + 27 \, a^{2} b^{8} x^{\frac {5}{3}} + 90 \, a^{3} b^{7} x^{\frac {4}{3}} + 210 \, a^{4} b^{6} x + 120 \, a^{7} b^{3} \log \left ({\left | x \right |}\right ) + 378 \, a^{5} b^{5} x^{\frac {2}{3}} + 630 \, a^{6} b^{4} x^{\frac {1}{3}} - \frac {135 \, a^{8} b^{2} x^{\frac {2}{3}} + 15 \, a^{9} b x^{\frac {1}{3}} + a^{10}}{x} \]

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

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maple [A]  time = 0.01, size = 110, normalized size = 0.88 \[ \frac {3 b^{10} x^{\frac {7}{3}}}{7}+5 a \,b^{9} x^{2}+27 a^{2} b^{8} x^{\frac {5}{3}}+90 a^{3} b^{7} x^{\frac {4}{3}}+120 a^{7} b^{3} \ln \relax (x )+210 a^{4} b^{6} x +378 a^{5} b^{5} x^{\frac {2}{3}}+630 a^{6} b^{4} x^{\frac {1}{3}}-\frac {135 a^{8} b^{2}}{x^{\frac {1}{3}}}-\frac {15 a^{9} b}{x^{\frac {2}{3}}}-\frac {a^{10}}{x} \]

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 = 0.93, size = 110, normalized size = 0.88 \[ \frac {3}{7} \, b^{10} x^{\frac {7}{3}} + 5 \, a b^{9} x^{2} + 27 \, a^{2} b^{8} x^{\frac {5}{3}} + 90 \, a^{3} b^{7} x^{\frac {4}{3}} + 210 \, a^{4} b^{6} x + 120 \, a^{7} b^{3} \log \relax (x) + 378 \, a^{5} b^{5} x^{\frac {2}{3}} + 630 \, a^{6} b^{4} x^{\frac {1}{3}} - \frac {135 \, a^{8} b^{2} x^{\frac {2}{3}} + 15 \, a^{9} b x^{\frac {1}{3}} + a^{10}}{x} \]

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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mupad [B]  time = 0.05, size = 112, normalized size = 0.90 \[ \frac {3\,b^{10}\,x^{7/3}}{7}-\frac {a^{10}+15\,a^9\,b\,x^{1/3}+135\,a^8\,b^2\,x^{2/3}}{x}+360\,a^7\,b^3\,\ln \left (x^{1/3}\right )+210\,a^4\,b^6\,x+5\,a\,b^9\,x^2+630\,a^6\,b^4\,x^{1/3}+378\,a^5\,b^5\,x^{2/3}+90\,a^3\,b^7\,x^{4/3}+27\,a^2\,b^8\,x^{5/3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/3))^10/x^2,x)

[Out]

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

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sympy [A]  time = 15.71, size = 131, normalized size = 1.05 \[ - \frac {a^{10}}{x} - \frac {15 a^{9} b}{x^{\frac {2}{3}}} - \frac {135 a^{8} b^{2}}{\sqrt [3]{x}} + 360 a^{7} b^{3} \log {\left (\sqrt [3]{x} \right )} + 630 a^{6} b^{4} \sqrt [3]{x} + 378 a^{5} b^{5} x^{\frac {2}{3}} + 210 a^{4} b^{6} x + 90 a^{3} b^{7} x^{\frac {4}{3}} + 27 a^{2} b^{8} x^{\frac {5}{3}} + 5 a b^{9} x^{2} + \frac {3 b^{10} x^{\frac {7}{3}}}{7} \]

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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