3.2333 \(\int \frac{(a+b \sqrt [3]{x})^{10}}{x^5} \, dx\)

Optimal. Leaf size=46 \[ \frac{b \left (a+b \sqrt [3]{x}\right )^{11}}{44 a^2 x^{11/3}}-\frac{\left (a+b \sqrt [3]{x}\right )^{11}}{4 a x^4} \]

[Out]

-(a + b*x^(1/3))^11/(4*a*x^4) + (b*(a + b*x^(1/3))^11)/(44*a^2*x^(11/3))

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Rubi [A]  time = 0.0136786, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {266, 45, 37} \[ \frac{b \left (a+b \sqrt [3]{x}\right )^{11}}{44 a^2 x^{11/3}}-\frac{\left (a+b \sqrt [3]{x}\right )^{11}}{4 a x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^10/x^5,x]

[Out]

-(a + b*x^(1/3))^11/(4*a*x^4) + (b*(a + b*x^(1/3))^11)/(44*a^2*x^(11/3))

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 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (a+b \sqrt [3]{x}\right )^{10}}{x^5} \, dx &=3 \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^{13}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{\left (a+b \sqrt [3]{x}\right )^{11}}{4 a x^4}-\frac{b \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^{12}} \, dx,x,\sqrt [3]{x}\right )}{4 a}\\ &=-\frac{\left (a+b \sqrt [3]{x}\right )^{11}}{4 a x^4}+\frac{b \left (a+b \sqrt [3]{x}\right )^{11}}{44 a^2 x^{11/3}}\\ \end{align*}

Mathematica [A]  time = 0.0080046, size = 32, normalized size = 0.7 \[ \frac{\left (b \sqrt [3]{x}-11 a\right ) \left (a+b \sqrt [3]{x}\right )^{11}}{44 a^2 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^10/x^5,x]

[Out]

((-11*a + b*x^(1/3))*(a + b*x^(1/3))^11)/(44*a^2*x^4)

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Maple [B]  time = 0.007, size = 113, normalized size = 2.5 \begin{align*} -10\,{\frac{a{b}^{9}}{x}}-72\,{\frac{{a}^{3}{b}^{7}}{{x}^{5/3}}}-{\frac{{a}^{10}}{4\,{x}^{4}}}-{\frac{30\,{a}^{9}b}{11}{x}^{-{\frac{11}{3}}}}-{\frac{135\,{a}^{2}{b}^{8}}{4}{x}^{-{\frac{4}{3}}}}-{\frac{27\,{a}^{8}{b}^{2}}{2}{x}^{-{\frac{10}{3}}}}-{\frac{3\,{b}^{10}}{2}{x}^{-{\frac{2}{3}}}}-{\frac{315\,{a}^{6}{b}^{4}}{4}{x}^{-{\frac{8}{3}}}}-105\,{\frac{{a}^{4}{b}^{6}}{{x}^{2}}}-108\,{\frac{{a}^{5}{b}^{5}}{{x}^{7/3}}}-40\,{\frac{{a}^{7}{b}^{3}}{{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/3))^10/x^5,x)

[Out]

-10*a*b^9/x-72*a^3*b^7/x^(5/3)-1/4*a^10/x^4-30/11*a^9*b/x^(11/3)-135/4*a^2*b^8/x^(4/3)-27/2*a^8*b^2/x^(10/3)-3
/2*b^10/x^(2/3)-315/4*a^6*b^4/x^(8/3)-105*a^4*b^6/x^2-108*a^5*b^5/x^(7/3)-40*a^7*b^3/x^3

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Maxima [B]  time = 0.97133, size = 151, normalized size = 3.28 \begin{align*} -\frac{66 \, b^{10} x^{\frac{10}{3}} + 440 \, a b^{9} x^{3} + 1485 \, a^{2} b^{8} x^{\frac{8}{3}} + 3168 \, a^{3} b^{7} x^{\frac{7}{3}} + 4620 \, a^{4} b^{6} x^{2} + 4752 \, a^{5} b^{5} x^{\frac{5}{3}} + 3465 \, a^{6} b^{4} x^{\frac{4}{3}} + 1760 \, a^{7} b^{3} x + 594 \, a^{8} b^{2} x^{\frac{2}{3}} + 120 \, a^{9} b x^{\frac{1}{3}} + 11 \, a^{10}}{44 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^10/x^5,x, algorithm="maxima")

[Out]

-1/44*(66*b^10*x^(10/3) + 440*a*b^9*x^3 + 1485*a^2*b^8*x^(8/3) + 3168*a^3*b^7*x^(7/3) + 4620*a^4*b^6*x^2 + 475
2*a^5*b^5*x^(5/3) + 3465*a^6*b^4*x^(4/3) + 1760*a^7*b^3*x + 594*a^8*b^2*x^(2/3) + 120*a^9*b*x^(1/3) + 11*a^10)
/x^4

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Fricas [B]  time = 1.42985, size = 274, normalized size = 5.96 \begin{align*} -\frac{440 \, a b^{9} x^{3} + 4620 \, a^{4} b^{6} x^{2} + 1760 \, a^{7} b^{3} x + 11 \, a^{10} + 297 \,{\left (5 \, a^{2} b^{8} x^{2} + 16 \, a^{5} b^{5} x + 2 \, a^{8} b^{2}\right )} x^{\frac{2}{3}} + 3 \,{\left (22 \, b^{10} x^{3} + 1056 \, a^{3} b^{7} x^{2} + 1155 \, a^{6} b^{4} x + 40 \, a^{9} b\right )} x^{\frac{1}{3}}}{44 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^10/x^5,x, algorithm="fricas")

[Out]

-1/44*(440*a*b^9*x^3 + 4620*a^4*b^6*x^2 + 1760*a^7*b^3*x + 11*a^10 + 297*(5*a^2*b^8*x^2 + 16*a^5*b^5*x + 2*a^8
*b^2)*x^(2/3) + 3*(22*b^10*x^3 + 1056*a^3*b^7*x^2 + 1155*a^6*b^4*x + 40*a^9*b)*x^(1/3))/x^4

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Sympy [B]  time = 4.16753, size = 139, normalized size = 3.02 \begin{align*} - \frac{a^{10}}{4 x^{4}} - \frac{30 a^{9} b}{11 x^{\frac{11}{3}}} - \frac{27 a^{8} b^{2}}{2 x^{\frac{10}{3}}} - \frac{40 a^{7} b^{3}}{x^{3}} - \frac{315 a^{6} b^{4}}{4 x^{\frac{8}{3}}} - \frac{108 a^{5} b^{5}}{x^{\frac{7}{3}}} - \frac{105 a^{4} b^{6}}{x^{2}} - \frac{72 a^{3} b^{7}}{x^{\frac{5}{3}}} - \frac{135 a^{2} b^{8}}{4 x^{\frac{4}{3}}} - \frac{10 a b^{9}}{x} - \frac{3 b^{10}}{2 x^{\frac{2}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))**10/x**5,x)

[Out]

-a**10/(4*x**4) - 30*a**9*b/(11*x**(11/3)) - 27*a**8*b**2/(2*x**(10/3)) - 40*a**7*b**3/x**3 - 315*a**6*b**4/(4
*x**(8/3)) - 108*a**5*b**5/x**(7/3) - 105*a**4*b**6/x**2 - 72*a**3*b**7/x**(5/3) - 135*a**2*b**8/(4*x**(4/3))
- 10*a*b**9/x - 3*b**10/(2*x**(2/3))

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Giac [B]  time = 1.1558, size = 151, normalized size = 3.28 \begin{align*} -\frac{66 \, b^{10} x^{\frac{10}{3}} + 440 \, a b^{9} x^{3} + 1485 \, a^{2} b^{8} x^{\frac{8}{3}} + 3168 \, a^{3} b^{7} x^{\frac{7}{3}} + 4620 \, a^{4} b^{6} x^{2} + 4752 \, a^{5} b^{5} x^{\frac{5}{3}} + 3465 \, a^{6} b^{4} x^{\frac{4}{3}} + 1760 \, a^{7} b^{3} x + 594 \, a^{8} b^{2} x^{\frac{2}{3}} + 120 \, a^{9} b x^{\frac{1}{3}} + 11 \, a^{10}}{44 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^10/x^5,x, algorithm="giac")

[Out]

-1/44*(66*b^10*x^(10/3) + 440*a*b^9*x^3 + 1485*a^2*b^8*x^(8/3) + 3168*a^3*b^7*x^(7/3) + 4620*a^4*b^6*x^2 + 475
2*a^5*b^5*x^(5/3) + 3465*a^6*b^4*x^(4/3) + 1760*a^7*b^3*x + 594*a^8*b^2*x^(2/3) + 120*a^9*b*x^(1/3) + 11*a^10)
/x^4