3.24.19 \(\int \frac {(a+b \sqrt [3]{x})^5}{x^3} \, dx\) [2319]

Optimal. Leaf size=21 \[ -\frac {\left (a+b \sqrt [3]{x}\right )^6}{2 a x^2} \]

[Out]

-1/2*(a+b*x^(1/3))^6/a/x^2

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Rubi [A]
time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {270} \begin {gather*} -\frac {\left (a+b \sqrt [3]{x}\right )^6}{2 a x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^(1/3))^5/x^3,x]

[Out]

-1/2*(a + b*x^(1/3))^6/(a*x^2)

Rule 270

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

Rubi steps

\begin {align*} \int \frac {\left (a+b \sqrt [3]{x}\right )^5}{x^3} \, dx &=-\frac {\left (a+b \sqrt [3]{x}\right )^6}{2 a x^2}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(67\) vs. \(2(21)=42\).
time = 0.02, size = 67, normalized size = 3.19 \begin {gather*} \frac {-a^5-6 a^4 b \sqrt [3]{x}-15 a^3 b^2 x^{2/3}-20 a^2 b^3 x-15 a b^4 x^{4/3}-6 b^5 x^{5/3}}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^(1/3))^5/x^3,x]

[Out]

(-a^5 - 6*a^4*b*x^(1/3) - 15*a^3*b^2*x^(2/3) - 20*a^2*b^3*x - 15*a*b^4*x^(4/3) - 6*b^5*x^(5/3))/(2*x^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(57\) vs. \(2(17)=34\).
time = 0.20, size = 58, normalized size = 2.76

method	result	size
derivativedivides	\(-\frac {15 a \,b^{4}}{2 x^{\frac {2}{3}}}-\frac {10 a^{2} b^{3}}{x}-\frac {3 a^{4} b}{x^{\frac {5}{3}}}-\frac {15 a^{3} b^{2}}{2 x^{\frac {4}{3}}}-\frac {a^{5}}{2 x^{2}}-\frac {3 b^{5}}{x^{\frac {1}{3}}}\)	\(58\)
default	\(-\frac {15 a \,b^{4}}{2 x^{\frac {2}{3}}}-\frac {10 a^{2} b^{3}}{x}-\frac {3 a^{4} b}{x^{\frac {5}{3}}}-\frac {15 a^{3} b^{2}}{2 x^{\frac {4}{3}}}-\frac {a^{5}}{2 x^{2}}-\frac {3 b^{5}}{x^{\frac {1}{3}}}\)	\(58\)
trager	\(\frac {\left (x -1\right ) \left (a^{3} x +20 b^{3} x +a^{3}\right ) a^{2}}{2 x^{2}}-\frac {3 \left (5 b^{3} x +2 a^{3}\right ) a b}{2 x^{\frac {5}{3}}}-\frac {3 \left (2 b^{3} x +5 a^{3}\right ) b^{2}}{2 x^{\frac {4}{3}}}\)	\(67\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-15/2*a*b^4/x^(2/3)-10*a^2*b^3/x-3*a^4*b/x^(5/3)-15/2*a^3*b^2/x^(4/3)-1/2*a^5/x^2-3*b^5/x^(1/3)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (17) = 34\).
time = 0.29, size = 55, normalized size = 2.62 \begin {gather*} -\frac {6 \, b^{5} x^{\frac {5}{3}} + 15 \, a b^{4} x^{\frac {4}{3}} + 20 \, a^{2} b^{3} x + 15 \, a^{3} b^{2} x^{\frac {2}{3}} + 6 \, a^{4} b x^{\frac {1}{3}} + a^{5}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(6*b^5*x^(5/3) + 15*a*b^4*x^(4/3) + 20*a^2*b^3*x + 15*a^3*b^2*x^(2/3) + 6*a^4*b*x^(1/3) + a^5)/x^2

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (17) = 34\).
time = 0.36, size = 57, normalized size = 2.71 \begin {gather*} -\frac {20 \, a^{2} b^{3} x + a^{5} + 3 \, {\left (2 \, b^{5} x + 5 \, a^{3} b^{2}\right )} x^{\frac {2}{3}} + 3 \, {\left (5 \, a b^{4} x + 2 \, a^{4} b\right )} x^{\frac {1}{3}}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(20*a^2*b^3*x + a^5 + 3*(2*b^5*x + 5*a^3*b^2)*x^(2/3) + 3*(5*a*b^4*x + 2*a^4*b)*x^(1/3))/x^2

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (17) = 34\).
time = 0.29, size = 70, normalized size = 3.33 \begin {gather*} - \frac {a^{5}}{2 x^{2}} - \frac {3 a^{4} b}{x^{\frac {5}{3}}} - \frac {15 a^{3} b^{2}}{2 x^{\frac {4}{3}}} - \frac {10 a^{2} b^{3}}{x} - \frac {15 a b^{4}}{2 x^{\frac {2}{3}}} - \frac {3 b^{5}}{\sqrt [3]{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**5/(2*x**2) - 3*a**4*b/x**(5/3) - 15*a**3*b**2/(2*x**(4/3)) - 10*a**2*b**3/x - 15*a*b**4/(2*x**(2/3)) - 3*b
**5/x**(1/3)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (17) = 34\).
time = 1.62, size = 55, normalized size = 2.62 \begin {gather*} -\frac {6 \, b^{5} x^{\frac {5}{3}} + 15 \, a b^{4} x^{\frac {4}{3}} + 20 \, a^{2} b^{3} x + 15 \, a^{3} b^{2} x^{\frac {2}{3}} + 6 \, a^{4} b x^{\frac {1}{3}} + a^{5}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(6*b^5*x^(5/3) + 15*a*b^4*x^(4/3) + 20*a^2*b^3*x + 15*a^3*b^2*x^(2/3) + 6*a^4*b*x^(1/3) + a^5)/x^2

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Mupad [B]
time = 1.12, size = 55, normalized size = 2.62 \begin {gather*} -\frac {a^5+6\,b^5\,x^{5/3}+20\,a^2\,b^3\,x+6\,a^4\,b\,x^{1/3}+15\,a\,b^4\,x^{4/3}+15\,a^3\,b^2\,x^{2/3}}{2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/3))^5/x^3,x)

[Out]

-(a^5 + 6*b^5*x^(5/3) + 20*a^2*b^3*x + 6*a^4*b*x^(1/3) + 15*a*b^4*x^(4/3) + 15*a^3*b^2*x^(2/3))/(2*x^2)

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