3.2323 \(\int \frac{(a+b \sqrt [3]{x})^5}{x^7} \, dx\)

Optimal. Leaf size=75 \[ -\frac{15 a^3 b^2}{8 x^{16/3}}-\frac{2 a^2 b^3}{x^5}-\frac{15 a^4 b}{17 x^{17/3}}-\frac{a^5}{6 x^6}-\frac{15 a b^4}{14 x^{14/3}}-\frac{3 b^5}{13 x^{13/3}} \]

[Out]

-a^5/(6*x^6) - (15*a^4*b)/(17*x^(17/3)) - (15*a^3*b^2)/(8*x^(16/3)) - (2*a^2*b^3)/x^5 - (15*a*b^4)/(14*x^(14/3
)) - (3*b^5)/(13*x^(13/3))

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Rubi [A]  time = 0.0352752, antiderivative size = 75, 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} \[ -\frac{15 a^3 b^2}{8 x^{16/3}}-\frac{2 a^2 b^3}{x^5}-\frac{15 a^4 b}{17 x^{17/3}}-\frac{a^5}{6 x^6}-\frac{15 a b^4}{14 x^{14/3}}-\frac{3 b^5}{13 x^{13/3}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^5/x^7,x]

[Out]

-a^5/(6*x^6) - (15*a^4*b)/(17*x^(17/3)) - (15*a^3*b^2)/(8*x^(16/3)) - (2*a^2*b^3)/x^5 - (15*a*b^4)/(14*x^(14/3
)) - (3*b^5)/(13*x^(13/3))

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 )^5}{x^7} \, dx &=3 \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^{19}} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (\frac{a^5}{x^{19}}+\frac{5 a^4 b}{x^{18}}+\frac{10 a^3 b^2}{x^{17}}+\frac{10 a^2 b^3}{x^{16}}+\frac{5 a b^4}{x^{15}}+\frac{b^5}{x^{14}}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{a^5}{6 x^6}-\frac{15 a^4 b}{17 x^{17/3}}-\frac{15 a^3 b^2}{8 x^{16/3}}-\frac{2 a^2 b^3}{x^5}-\frac{15 a b^4}{14 x^{14/3}}-\frac{3 b^5}{13 x^{13/3}}\\ \end{align*}

Mathematica [A]  time = 0.0268487, size = 67, normalized size = 0.89 \[ -\frac{69615 a^3 b^2 x^{2/3}+74256 a^2 b^3 x+32760 a^4 b \sqrt [3]{x}+6188 a^5+39780 a b^4 x^{4/3}+8568 b^5 x^{5/3}}{37128 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^5/x^7,x]

[Out]

-(6188*a^5 + 32760*a^4*b*x^(1/3) + 69615*a^3*b^2*x^(2/3) + 74256*a^2*b^3*x + 39780*a*b^4*x^(4/3) + 8568*b^5*x^
(5/3))/(37128*x^6)

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Maple [A]  time = 0.007, size = 58, normalized size = 0.8 \begin{align*} -{\frac{{a}^{5}}{6\,{x}^{6}}}-{\frac{15\,{a}^{4}b}{17}{x}^{-{\frac{17}{3}}}}-{\frac{15\,{a}^{3}{b}^{2}}{8}{x}^{-{\frac{16}{3}}}}-2\,{\frac{{a}^{2}{b}^{3}}{{x}^{5}}}-{\frac{15\,a{b}^{4}}{14}{x}^{-{\frac{14}{3}}}}-{\frac{3\,{b}^{5}}{13}{x}^{-{\frac{13}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/3))^5/x^7,x)

[Out]

-1/6*a^5/x^6-15/17*a^4*b/x^(17/3)-15/8*a^3*b^2/x^(16/3)-2*a^2*b^3/x^5-15/14*a*b^4/x^(14/3)-3/13*b^5/x^(13/3)

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Maxima [A]  time = 0.959818, size = 77, normalized size = 1.03 \begin{align*} -\frac{8568 \, b^{5} x^{\frac{5}{3}} + 39780 \, a b^{4} x^{\frac{4}{3}} + 74256 \, a^{2} b^{3} x + 69615 \, a^{3} b^{2} x^{\frac{2}{3}} + 32760 \, a^{4} b x^{\frac{1}{3}} + 6188 \, a^{5}}{37128 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^5/x^7,x, algorithm="maxima")

[Out]

-1/37128*(8568*b^5*x^(5/3) + 39780*a*b^4*x^(4/3) + 74256*a^2*b^3*x + 69615*a^3*b^2*x^(2/3) + 32760*a^4*b*x^(1/
3) + 6188*a^5)/x^6

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Fricas [A]  time = 1.63932, size = 162, normalized size = 2.16 \begin{align*} -\frac{74256 \, a^{2} b^{3} x + 6188 \, a^{5} + 1071 \,{\left (8 \, b^{5} x + 65 \, a^{3} b^{2}\right )} x^{\frac{2}{3}} + 2340 \,{\left (17 \, a b^{4} x + 14 \, a^{4} b\right )} x^{\frac{1}{3}}}{37128 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^5/x^7,x, algorithm="fricas")

[Out]

-1/37128*(74256*a^2*b^3*x + 6188*a^5 + 1071*(8*b^5*x + 65*a^3*b^2)*x^(2/3) + 2340*(17*a*b^4*x + 14*a^4*b)*x^(1
/3))/x^6

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Sympy [A]  time = 14.9205, size = 75, normalized size = 1. \begin{align*} - \frac{a^{5}}{6 x^{6}} - \frac{15 a^{4} b}{17 x^{\frac{17}{3}}} - \frac{15 a^{3} b^{2}}{8 x^{\frac{16}{3}}} - \frac{2 a^{2} b^{3}}{x^{5}} - \frac{15 a b^{4}}{14 x^{\frac{14}{3}}} - \frac{3 b^{5}}{13 x^{\frac{13}{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/3))**5/x**7,x)

[Out]

-a**5/(6*x**6) - 15*a**4*b/(17*x**(17/3)) - 15*a**3*b**2/(8*x**(16/3)) - 2*a**2*b**3/x**5 - 15*a*b**4/(14*x**(
14/3)) - 3*b**5/(13*x**(13/3))

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Giac [A]  time = 1.18645, size = 77, normalized size = 1.03 \begin{align*} -\frac{8568 \, b^{5} x^{\frac{5}{3}} + 39780 \, a b^{4} x^{\frac{4}{3}} + 74256 \, a^{2} b^{3} x + 69615 \, a^{3} b^{2} x^{\frac{2}{3}} + 32760 \, a^{4} b x^{\frac{1}{3}} + 6188 \, a^{5}}{37128 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/3))^5/x^7,x, algorithm="giac")

[Out]

-1/37128*(8568*b^5*x^(5/3) + 39780*a*b^4*x^(4/3) + 74256*a^2*b^3*x + 69615*a^3*b^2*x^(2/3) + 32760*a^4*b*x^(1/
3) + 6188*a^5)/x^6