3.2335 \(\int \frac{(a+b \sqrt [3]{x})^{10}}{x^7} \, dx\)

Optimal. Leaf size=144 \[ -\frac{135 a^8 b^2}{16 x^{16/3}}-\frac{24 a^7 b^3}{x^5}-\frac{45 a^6 b^4}{x^{14/3}}-\frac{756 a^5 b^5}{13 x^{13/3}}-\frac{105 a^4 b^6}{2 x^4}-\frac{360 a^3 b^7}{11 x^{11/3}}-\frac{27 a^2 b^8}{2 x^{10/3}}-\frac{30 a^9 b}{17 x^{17/3}}-\frac{a^{10}}{6 x^6}-\frac{10 a b^9}{3 x^3}-\frac{3 b^{10}}{8 x^{8/3}} \]

[Out]

-a^10/(6*x^6) - (30*a^9*b)/(17*x^(17/3)) - (135*a^8*b^2)/(16*x^(16/3)) - (24*a^7*b^3)/x^5 - (45*a^6*b^4)/x^(14
/3) - (756*a^5*b^5)/(13*x^(13/3)) - (105*a^4*b^6)/(2*x^4) - (360*a^3*b^7)/(11*x^(11/3)) - (27*a^2*b^8)/(2*x^(1
0/3)) - (10*a*b^9)/(3*x^3) - (3*b^10)/(8*x^(8/3))

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Rubi [A]  time = 0.073725, antiderivative size = 144, 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} \[ -\frac{135 a^8 b^2}{16 x^{16/3}}-\frac{24 a^7 b^3}{x^5}-\frac{45 a^6 b^4}{x^{14/3}}-\frac{756 a^5 b^5}{13 x^{13/3}}-\frac{105 a^4 b^6}{2 x^4}-\frac{360 a^3 b^7}{11 x^{11/3}}-\frac{27 a^2 b^8}{2 x^{10/3}}-\frac{30 a^9 b}{17 x^{17/3}}-\frac{a^{10}}{6 x^6}-\frac{10 a b^9}{3 x^3}-\frac{3 b^{10}}{8 x^{8/3}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^10/x^7,x]

[Out]

-a^10/(6*x^6) - (30*a^9*b)/(17*x^(17/3)) - (135*a^8*b^2)/(16*x^(16/3)) - (24*a^7*b^3)/x^5 - (45*a^6*b^4)/x^(14
/3) - (756*a^5*b^5)/(13*x^(13/3)) - (105*a^4*b^6)/(2*x^4) - (360*a^3*b^7)/(11*x^(11/3)) - (27*a^2*b^8)/(2*x^(1
0/3)) - (10*a*b^9)/(3*x^3) - (3*b^10)/(8*x^(8/3))

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 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^7} \, dx &=3 \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^{19}} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{a^{10}}{x^{19}}+\frac{10 a^9 b}{x^{18}}+\frac{45 a^8 b^2}{x^{17}}+\frac{120 a^7 b^3}{x^{16}}+\frac{210 a^6 b^4}{x^{15}}+\frac{252 a^5 b^5}{x^{14}}+\frac{210 a^4 b^6}{x^{13}}+\frac{120 a^3 b^7}{x^{12}}+\frac{45 a^2 b^8}{x^{11}}+\frac{10 a b^9}{x^{10}}+\frac{b^{10}}{x^9}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{a^{10}}{6 x^6}-\frac{30 a^9 b}{17 x^{17/3}}-\frac{135 a^8 b^2}{16 x^{16/3}}-\frac{24 a^7 b^3}{x^5}-\frac{45 a^6 b^4}{x^{14/3}}-\frac{756 a^5 b^5}{13 x^{13/3}}-\frac{105 a^4 b^6}{2 x^4}-\frac{360 a^3 b^7}{11 x^{11/3}}-\frac{27 a^2 b^8}{2 x^{10/3}}-\frac{10 a b^9}{3 x^3}-\frac{3 b^{10}}{8 x^{8/3}}\\ \end{align*}

Mathematica [A]  time = 0.0629793, size = 144, normalized size = 1. \[ -\frac{135 a^8 b^2}{16 x^{16/3}}-\frac{24 a^7 b^3}{x^5}-\frac{45 a^6 b^4}{x^{14/3}}-\frac{756 a^5 b^5}{13 x^{13/3}}-\frac{105 a^4 b^6}{2 x^4}-\frac{360 a^3 b^7}{11 x^{11/3}}-\frac{27 a^2 b^8}{2 x^{10/3}}-\frac{30 a^9 b}{17 x^{17/3}}-\frac{a^{10}}{6 x^6}-\frac{10 a b^9}{3 x^3}-\frac{3 b^{10}}{8 x^{8/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^10/x^7,x]

[Out]

-a^10/(6*x^6) - (30*a^9*b)/(17*x^(17/3)) - (135*a^8*b^2)/(16*x^(16/3)) - (24*a^7*b^3)/x^5 - (45*a^6*b^4)/x^(14
/3) - (756*a^5*b^5)/(13*x^(13/3)) - (105*a^4*b^6)/(2*x^4) - (360*a^3*b^7)/(11*x^(11/3)) - (27*a^2*b^8)/(2*x^(1
0/3)) - (10*a*b^9)/(3*x^3) - (3*b^10)/(8*x^(8/3))

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Maple [A]  time = 0.007, size = 113, normalized size = 0.8 \begin{align*} -{\frac{{a}^{10}}{6\,{x}^{6}}}-{\frac{30\,{a}^{9}b}{17}{x}^{-{\frac{17}{3}}}}-{\frac{135\,{a}^{8}{b}^{2}}{16}{x}^{-{\frac{16}{3}}}}-24\,{\frac{{a}^{7}{b}^{3}}{{x}^{5}}}-45\,{\frac{{a}^{6}{b}^{4}}{{x}^{14/3}}}-{\frac{756\,{a}^{5}{b}^{5}}{13}{x}^{-{\frac{13}{3}}}}-{\frac{105\,{a}^{4}{b}^{6}}{2\,{x}^{4}}}-{\frac{360\,{a}^{3}{b}^{7}}{11}{x}^{-{\frac{11}{3}}}}-{\frac{27\,{a}^{2}{b}^{8}}{2}{x}^{-{\frac{10}{3}}}}-{\frac{10\,a{b}^{9}}{3\,{x}^{3}}}-{\frac{3\,{b}^{10}}{8}{x}^{-{\frac{8}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/3))^10/x^7,x)

[Out]

-1/6*a^10/x^6-30/17*a^9*b/x^(17/3)-135/16*a^8*b^2/x^(16/3)-24*a^7*b^3/x^5-45*a^6*b^4/x^(14/3)-756/13*a^5*b^5/x
^(13/3)-105/2*a^4*b^6/x^4-360/11*a^3*b^7/x^(11/3)-27/2*a^2*b^8/x^(10/3)-10/3*a*b^9/x^3-3/8*b^10/x^(8/3)

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Maxima [A]  time = 0.975279, size = 151, normalized size = 1.05 \begin{align*} -\frac{43758 \, b^{10} x^{\frac{10}{3}} + 388960 \, a b^{9} x^{3} + 1575288 \, a^{2} b^{8} x^{\frac{8}{3}} + 3818880 \, a^{3} b^{7} x^{\frac{7}{3}} + 6126120 \, a^{4} b^{6} x^{2} + 6785856 \, a^{5} b^{5} x^{\frac{5}{3}} + 5250960 \, a^{6} b^{4} x^{\frac{4}{3}} + 2800512 \, a^{7} b^{3} x + 984555 \, a^{8} b^{2} x^{\frac{2}{3}} + 205920 \, a^{9} b x^{\frac{1}{3}} + 19448 \, a^{10}}{116688 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^10/x^7,x, algorithm="maxima")

[Out]

-1/116688*(43758*b^10*x^(10/3) + 388960*a*b^9*x^3 + 1575288*a^2*b^8*x^(8/3) + 3818880*a^3*b^7*x^(7/3) + 612612
0*a^4*b^6*x^2 + 6785856*a^5*b^5*x^(5/3) + 5250960*a^6*b^4*x^(4/3) + 2800512*a^7*b^3*x + 984555*a^8*b^2*x^(2/3)
+ 205920*a^9*b*x^(1/3) + 19448*a^10)/x^6

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Fricas [A]  time = 1.47276, size = 312, normalized size = 2.17 \begin{align*} -\frac{388960 \, a b^{9} x^{3} + 6126120 \, a^{4} b^{6} x^{2} + 2800512 \, a^{7} b^{3} x + 19448 \, a^{10} + 15147 \,{\left (104 \, a^{2} b^{8} x^{2} + 448 \, a^{5} b^{5} x + 65 \, a^{8} b^{2}\right )} x^{\frac{2}{3}} + 234 \,{\left (187 \, b^{10} x^{3} + 16320 \, a^{3} b^{7} x^{2} + 22440 \, a^{6} b^{4} x + 880 \, a^{9} b\right )} x^{\frac{1}{3}}}{116688 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^10/x^7,x, algorithm="fricas")

[Out]

-1/116688*(388960*a*b^9*x^3 + 6126120*a^4*b^6*x^2 + 2800512*a^7*b^3*x + 19448*a^10 + 15147*(104*a^2*b^8*x^2 +
448*a^5*b^5*x + 65*a^8*b^2)*x^(2/3) + 234*(187*b^10*x^3 + 16320*a^3*b^7*x^2 + 22440*a^6*b^4*x + 880*a^9*b)*x^(
1/3))/x^6

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Sympy [A]  time = 13.8286, size = 146, normalized size = 1.01 \begin{align*} - \frac{a^{10}}{6 x^{6}} - \frac{30 a^{9} b}{17 x^{\frac{17}{3}}} - \frac{135 a^{8} b^{2}}{16 x^{\frac{16}{3}}} - \frac{24 a^{7} b^{3}}{x^{5}} - \frac{45 a^{6} b^{4}}{x^{\frac{14}{3}}} - \frac{756 a^{5} b^{5}}{13 x^{\frac{13}{3}}} - \frac{105 a^{4} b^{6}}{2 x^{4}} - \frac{360 a^{3} b^{7}}{11 x^{\frac{11}{3}}} - \frac{27 a^{2} b^{8}}{2 x^{\frac{10}{3}}} - \frac{10 a b^{9}}{3 x^{3}} - \frac{3 b^{10}}{8 x^{\frac{8}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))**10/x**7,x)

[Out]

-a**10/(6*x**6) - 30*a**9*b/(17*x**(17/3)) - 135*a**8*b**2/(16*x**(16/3)) - 24*a**7*b**3/x**5 - 45*a**6*b**4/x
**(14/3) - 756*a**5*b**5/(13*x**(13/3)) - 105*a**4*b**6/(2*x**4) - 360*a**3*b**7/(11*x**(11/3)) - 27*a**2*b**8
/(2*x**(10/3)) - 10*a*b**9/(3*x**3) - 3*b**10/(8*x**(8/3))

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Giac [A]  time = 1.18592, size = 151, normalized size = 1.05 \begin{align*} -\frac{43758 \, b^{10} x^{\frac{10}{3}} + 388960 \, a b^{9} x^{3} + 1575288 \, a^{2} b^{8} x^{\frac{8}{3}} + 3818880 \, a^{3} b^{7} x^{\frac{7}{3}} + 6126120 \, a^{4} b^{6} x^{2} + 6785856 \, a^{5} b^{5} x^{\frac{5}{3}} + 5250960 \, a^{6} b^{4} x^{\frac{4}{3}} + 2800512 \, a^{7} b^{3} x + 984555 \, a^{8} b^{2} x^{\frac{2}{3}} + 205920 \, a^{9} b x^{\frac{1}{3}} + 19448 \, a^{10}}{116688 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^10/x^7,x, algorithm="giac")

[Out]

-1/116688*(43758*b^10*x^(10/3) + 388960*a*b^9*x^3 + 1575288*a^2*b^8*x^(8/3) + 3818880*a^3*b^7*x^(7/3) + 612612
0*a^4*b^6*x^2 + 6785856*a^5*b^5*x^(5/3) + 5250960*a^6*b^4*x^(4/3) + 2800512*a^7*b^3*x + 984555*a^8*b^2*x^(2/3)
+ 205920*a^9*b*x^(1/3) + 19448*a^10)/x^6