3.24.23 \(\int \frac {(a+b \sqrt [3]{x})^5}{x^7} \, dx\) [2323]

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

[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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Rubi [A]
time = 0.02, antiderivative size = 75, 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 = {272, 45} \begin {gather*} -\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 {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^5/x^7,x]

[Out]

-1/6*a^5/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 45

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 272

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 )^5}{x^7} \, dx &=3 \text {Subst}\left (\int \frac {(a+b x)^5}{x^{19}} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \text {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*}

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Mathematica [A]
time = 0.03, size = 67, normalized size = 0.89 \begin {gather*} \frac {-6188 a^5-32760 a^4 b \sqrt [3]{x}-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} \end {gather*}

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 = 0.19, size = 58, normalized size = 0.77

method	result	size
derivativedivides	\(-\frac {a^{5}}{6 x^{6}}-\frac {15 a^{4} b}{17 x^{\frac {17}{3}}}-\frac {15 a^{3} b^{2}}{8 x^{\frac {16}{3}}}-\frac {2 a^{2} b^{3}}{x^{5}}-\frac {15 a \,b^{4}}{14 x^{\frac {14}{3}}}-\frac {3 b^{5}}{13 x^{\frac {13}{3}}}\)	\(58\)
default	\(-\frac {a^{5}}{6 x^{6}}-\frac {15 a^{4} b}{17 x^{\frac {17}{3}}}-\frac {15 a^{3} b^{2}}{8 x^{\frac {16}{3}}}-\frac {2 a^{2} b^{3}}{x^{5}}-\frac {15 a \,b^{4}}{14 x^{\frac {14}{3}}}-\frac {3 b^{5}}{13 x^{\frac {13}{3}}}\)	\(58\)
trager	\(\frac {\left (x -1\right ) \left (a^{3} x^{5}+12 b^{3} x^{5}+a^{3} x^{4}+12 b^{3} x^{4}+a^{3} x^{3}+12 b^{3} x^{3}+a^{3} x^{2}+12 b^{3} x^{2}+a^{3} x +12 b^{3} x +a^{3}\right ) a^{2}}{6 x^{6}}-\frac {15 \left (17 b^{3} x +14 a^{3}\right ) a b}{238 x^{\frac {17}{3}}}-\frac {3 \left (8 b^{3} x +65 a^{3}\right ) b^{2}}{104 x^{\frac {16}{3}}}\)	\(127\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.30, size = 57, normalized size = 0.76 \begin {gather*} -\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 {gather*}

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 = 0.35, size = 59, normalized size = 0.79 \begin {gather*} -\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 {gather*}

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 = 1.06, size = 75, normalized size = 1.00 \begin {gather*} - \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 {gather*}

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 = 1.98, size = 57, normalized size = 0.76 \begin {gather*} -\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 {gather*}

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

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Mupad [B]
time = 0.04, size = 57, normalized size = 0.76 \begin {gather*} -\frac {\frac {a^5}{6}+\frac {3\,b^5\,x^{5/3}}{13}+2\,a^2\,b^3\,x+\frac {15\,a^4\,b\,x^{1/3}}{17}+\frac {15\,a\,b^4\,x^{4/3}}{14}+\frac {15\,a^3\,b^2\,x^{2/3}}{8}}{x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/3))^5/x^7,x)

[Out]

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

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