3.2334 \(\int \frac {(a+b \sqrt [3]{x})^{10}}{x^6} \, dx\)

Optimal. Leaf size=122 \[ -\frac {b^4 \left (a+b \sqrt [3]{x}\right )^{11}}{5005 a^5 x^{11/3}}+\frac {b^3 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^4 x^4}-\frac {6 b^2 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^3 x^{13/3}}+\frac {2 b \left (a+b \sqrt [3]{x}\right )^{11}}{35 a^2 x^{14/3}}-\frac {\left (a+b \sqrt [3]{x}\right )^{11}}{5 a x^5} \]

[Out]

-1/5*(a+b*x^(1/3))^11/a/x^5+2/35*b*(a+b*x^(1/3))^11/a^2/x^(14/3)-6/455*b^2*(a+b*x^(1/3))^11/a^3/x^(13/3)+1/455
*b^3*(a+b*x^(1/3))^11/a^4/x^4-1/5005*b^4*(a+b*x^(1/3))^11/a^5/x^(11/3)

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Rubi [A]  time = 0.05, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {266, 45, 37} \[ -\frac {b^4 \left (a+b \sqrt [3]{x}\right )^{11}}{5005 a^5 x^{11/3}}+\frac {b^3 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^4 x^4}-\frac {6 b^2 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^3 x^{13/3}}+\frac {2 b \left (a+b \sqrt [3]{x}\right )^{11}}{35 a^2 x^{14/3}}-\frac {\left (a+b \sqrt [3]{x}\right )^{11}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^10/x^6,x]

[Out]

-(a + b*x^(1/3))^11/(5*a*x^5) + (2*b*(a + b*x^(1/3))^11)/(35*a^2*x^(14/3)) - (6*b^2*(a + b*x^(1/3))^11)/(455*a
^3*x^(13/3)) + (b^3*(a + b*x^(1/3))^11)/(455*a^4*x^4) - (b^4*(a + b*x^(1/3))^11)/(5005*a^5*x^(11/3))

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]

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 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^6} \, dx &=3 \operatorname {Subst}\left (\int \frac {(a+b x)^{10}}{x^{16}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {\left (a+b \sqrt [3]{x}\right )^{11}}{5 a x^5}-\frac {(4 b) \operatorname {Subst}\left (\int \frac {(a+b x)^{10}}{x^{15}} \, dx,x,\sqrt [3]{x}\right )}{5 a}\\ &=-\frac {\left (a+b \sqrt [3]{x}\right )^{11}}{5 a x^5}+\frac {2 b \left (a+b \sqrt [3]{x}\right )^{11}}{35 a^2 x^{14/3}}+\frac {\left (6 b^2\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{10}}{x^{14}} \, dx,x,\sqrt [3]{x}\right )}{35 a^2}\\ &=-\frac {\left (a+b \sqrt [3]{x}\right )^{11}}{5 a x^5}+\frac {2 b \left (a+b \sqrt [3]{x}\right )^{11}}{35 a^2 x^{14/3}}-\frac {6 b^2 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^3 x^{13/3}}-\frac {\left (12 b^3\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^{10}}{x^{13}} \, dx,x,\sqrt [3]{x}\right )}{455 a^3}\\ &=-\frac {\left (a+b \sqrt [3]{x}\right )^{11}}{5 a x^5}+\frac {2 b \left (a+b \sqrt [3]{x}\right )^{11}}{35 a^2 x^{14/3}}-\frac {6 b^2 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^3 x^{13/3}}+\frac {b^3 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^4 x^4}+\frac {b^4 \operatorname {Subst}\left (\int \frac {(a+b x)^{10}}{x^{12}} \, dx,x,\sqrt [3]{x}\right )}{455 a^4}\\ &=-\frac {\left (a+b \sqrt [3]{x}\right )^{11}}{5 a x^5}+\frac {2 b \left (a+b \sqrt [3]{x}\right )^{11}}{35 a^2 x^{14/3}}-\frac {6 b^2 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^3 x^{13/3}}+\frac {b^3 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^4 x^4}-\frac {b^4 \left (a+b \sqrt [3]{x}\right )^{11}}{5005 a^5 x^{11/3}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 67, normalized size = 0.55 \[ -\frac {\left (a+b \sqrt [3]{x}\right )^{11} \left (1001 a^4-286 a^3 b \sqrt [3]{x}+66 a^2 b^2 x^{2/3}-11 a b^3 x+b^4 x^{4/3}\right )}{5005 a^5 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^10/x^6,x]

[Out]

-1/5005*((a + b*x^(1/3))^11*(1001*a^4 - 286*a^3*b*x^(1/3) + 66*a^2*b^2*x^(2/3) - 11*a*b^3*x + b^4*x^(4/3)))/(a
^5*x^5)

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fricas [A]  time = 0.76, size = 114, normalized size = 0.93 \[ -\frac {25025 \, a b^{9} x^{3} + 350350 \, a^{4} b^{6} x^{2} + 150150 \, a^{7} b^{3} x + 1001 \, a^{10} + 297 \, {\left (325 \, a^{2} b^{8} x^{2} + 1274 \, a^{5} b^{5} x + 175 \, a^{8} b^{2}\right )} x^{\frac {2}{3}} + 39 \, {\left (77 \, b^{10} x^{3} + 5775 \, a^{3} b^{7} x^{2} + 7350 \, a^{6} b^{4} x + 275 \, a^{9} b\right )} x^{\frac {1}{3}}}{5005 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^10/x^6,x, algorithm="fricas")

[Out]

-1/5005*(25025*a*b^9*x^3 + 350350*a^4*b^6*x^2 + 150150*a^7*b^3*x + 1001*a^10 + 297*(325*a^2*b^8*x^2 + 1274*a^5
*b^5*x + 175*a^8*b^2)*x^(2/3) + 39*(77*b^10*x^3 + 5775*a^3*b^7*x^2 + 7350*a^6*b^4*x + 275*a^9*b)*x^(1/3))/x^5

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giac [A]  time = 0.19, size = 112, normalized size = 0.92 \[ -\frac {3003 \, b^{10} x^{\frac {10}{3}} + 25025 \, a b^{9} x^{3} + 96525 \, a^{2} b^{8} x^{\frac {8}{3}} + 225225 \, a^{3} b^{7} x^{\frac {7}{3}} + 350350 \, a^{4} b^{6} x^{2} + 378378 \, a^{5} b^{5} x^{\frac {5}{3}} + 286650 \, a^{6} b^{4} x^{\frac {4}{3}} + 150150 \, a^{7} b^{3} x + 51975 \, a^{8} b^{2} x^{\frac {2}{3}} + 10725 \, a^{9} b x^{\frac {1}{3}} + 1001 \, a^{10}}{5005 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^10/x^6,x, algorithm="giac")

[Out]

-1/5005*(3003*b^10*x^(10/3) + 25025*a*b^9*x^3 + 96525*a^2*b^8*x^(8/3) + 225225*a^3*b^7*x^(7/3) + 350350*a^4*b^
6*x^2 + 378378*a^5*b^5*x^(5/3) + 286650*a^6*b^4*x^(4/3) + 150150*a^7*b^3*x + 51975*a^8*b^2*x^(2/3) + 10725*a^9
*b*x^(1/3) + 1001*a^10)/x^5

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maple [A]  time = 0.01, size = 113, normalized size = 0.93 \[ -\frac {3 b^{10}}{5 x^{\frac {5}{3}}}-\frac {5 a \,b^{9}}{x^{2}}-\frac {135 a^{2} b^{8}}{7 x^{\frac {7}{3}}}-\frac {45 a^{3} b^{7}}{x^{\frac {8}{3}}}-\frac {70 a^{4} b^{6}}{x^{3}}-\frac {378 a^{5} b^{5}}{5 x^{\frac {10}{3}}}-\frac {630 a^{6} b^{4}}{11 x^{\frac {11}{3}}}-\frac {30 a^{7} b^{3}}{x^{4}}-\frac {135 a^{8} b^{2}}{13 x^{\frac {13}{3}}}-\frac {15 a^{9} b}{7 x^{\frac {14}{3}}}-\frac {a^{10}}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/3))^10/x^6,x)

[Out]

-3/5*b^10/x^(5/3)-15/7*a^9*b/x^(14/3)-45*a^3*b^7/x^(8/3)-1/5*a^10/x^5-135/13*a^8*b^2/x^(13/3)-70*a^4*b^6/x^3-1
35/7*a^2*b^8/x^(7/3)-5*a*b^9/x^2-30*a^7*b^3/x^4-378/5*a^5*b^5/x^(10/3)-630/11*a^6*b^4/x^(11/3)

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maxima [A]  time = 0.91, size = 112, normalized size = 0.92 \[ -\frac {3003 \, b^{10} x^{\frac {10}{3}} + 25025 \, a b^{9} x^{3} + 96525 \, a^{2} b^{8} x^{\frac {8}{3}} + 225225 \, a^{3} b^{7} x^{\frac {7}{3}} + 350350 \, a^{4} b^{6} x^{2} + 378378 \, a^{5} b^{5} x^{\frac {5}{3}} + 286650 \, a^{6} b^{4} x^{\frac {4}{3}} + 150150 \, a^{7} b^{3} x + 51975 \, a^{8} b^{2} x^{\frac {2}{3}} + 10725 \, a^{9} b x^{\frac {1}{3}} + 1001 \, a^{10}}{5005 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^10/x^6,x, algorithm="maxima")

[Out]

-1/5005*(3003*b^10*x^(10/3) + 25025*a*b^9*x^3 + 96525*a^2*b^8*x^(8/3) + 225225*a^3*b^7*x^(7/3) + 350350*a^4*b^
6*x^2 + 378378*a^5*b^5*x^(5/3) + 286650*a^6*b^4*x^(4/3) + 150150*a^7*b^3*x + 51975*a^8*b^2*x^(2/3) + 10725*a^9
*b*x^(1/3) + 1001*a^10)/x^5

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mupad [B]  time = 1.17, size = 112, normalized size = 0.92 \[ -\frac {\frac {a^{10}}{5}+\frac {3\,b^{10}\,x^{10/3}}{5}+30\,a^7\,b^3\,x+5\,a\,b^9\,x^3+\frac {15\,a^9\,b\,x^{1/3}}{7}+70\,a^4\,b^6\,x^2+\frac {135\,a^8\,b^2\,x^{2/3}}{13}+\frac {630\,a^6\,b^4\,x^{4/3}}{11}+\frac {378\,a^5\,b^5\,x^{5/3}}{5}+45\,a^3\,b^7\,x^{7/3}+\frac {135\,a^2\,b^8\,x^{8/3}}{7}}{x^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/3))^10/x^6,x)

[Out]

-(a^10/5 + (3*b^10*x^(10/3))/5 + 30*a^7*b^3*x + 5*a*b^9*x^3 + (15*a^9*b*x^(1/3))/7 + 70*a^4*b^6*x^2 + (135*a^8
*b^2*x^(2/3))/13 + (630*a^6*b^4*x^(4/3))/11 + (378*a^5*b^5*x^(5/3))/5 + 45*a^3*b^7*x^(7/3) + (135*a^2*b^8*x^(8
/3))/7)/x^5

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sympy [A]  time = 6.12, size = 143, normalized size = 1.17 \[ - \frac {a^{10}}{5 x^{5}} - \frac {15 a^{9} b}{7 x^{\frac {14}{3}}} - \frac {135 a^{8} b^{2}}{13 x^{\frac {13}{3}}} - \frac {30 a^{7} b^{3}}{x^{4}} - \frac {630 a^{6} b^{4}}{11 x^{\frac {11}{3}}} - \frac {378 a^{5} b^{5}}{5 x^{\frac {10}{3}}} - \frac {70 a^{4} b^{6}}{x^{3}} - \frac {45 a^{3} b^{7}}{x^{\frac {8}{3}}} - \frac {135 a^{2} b^{8}}{7 x^{\frac {7}{3}}} - \frac {5 a b^{9}}{x^{2}} - \frac {3 b^{10}}{5 x^{\frac {5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))**10/x**6,x)

[Out]

-a**10/(5*x**5) - 15*a**9*b/(7*x**(14/3)) - 135*a**8*b**2/(13*x**(13/3)) - 30*a**7*b**3/x**4 - 630*a**6*b**4/(
11*x**(11/3)) - 378*a**5*b**5/(5*x**(10/3)) - 70*a**4*b**6/x**3 - 45*a**3*b**7/x**(8/3) - 135*a**2*b**8/(7*x**
(7/3)) - 5*a*b**9/x**2 - 3*b**10/(5*x**(5/3))

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