3.6.42 \(\int \frac {(a+b x^3)^{2/3}}{x^{12}} \, dx\) [542]

Optimal. Leaf size=68 \[ -\frac {\left (a+b x^3\right )^{5/3}}{11 a x^{11}}+\frac {3 b \left (a+b x^3\right )^{5/3}}{44 a^2 x^8}-\frac {9 b^2 \left (a+b x^3\right )^{5/3}}{220 a^3 x^5} \]

[Out]

-1/11*(b*x^3+a)^(5/3)/a/x^11+3/44*b*(b*x^3+a)^(5/3)/a^2/x^8-9/220*b^2*(b*x^3+a)^(5/3)/a^3/x^5

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Rubi [A]
time = 0.01, antiderivative size = 68, 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 = {277, 270} \begin {gather*} -\frac {9 b^2 \left (a+b x^3\right )^{5/3}}{220 a^3 x^5}+\frac {3 b \left (a+b x^3\right )^{5/3}}{44 a^2 x^8}-\frac {\left (a+b x^3\right )^{5/3}}{11 a x^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^(2/3)/x^12,x]

[Out]

-1/11*(a + b*x^3)^(5/3)/(a*x^11) + (3*b*(a + b*x^3)^(5/3))/(44*a^2*x^8) - (9*b^2*(a + b*x^3)^(5/3))/(220*a^3*x
^5)

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]

Rule 277

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

Rubi steps

\begin {align*} \int \frac {\left (a+b x^3\right )^{2/3}}{x^{12}} \, dx &=-\frac {\left (a+b x^3\right )^{5/3}}{11 a x^{11}}-\frac {(6 b) \int \frac {\left (a+b x^3\right )^{2/3}}{x^9} \, dx}{11 a}\\ &=-\frac {\left (a+b x^3\right )^{5/3}}{11 a x^{11}}+\frac {3 b \left (a+b x^3\right )^{5/3}}{44 a^2 x^8}+\frac {\left (9 b^2\right ) \int \frac {\left (a+b x^3\right )^{2/3}}{x^6} \, dx}{44 a^2}\\ &=-\frac {\left (a+b x^3\right )^{5/3}}{11 a x^{11}}+\frac {3 b \left (a+b x^3\right )^{5/3}}{44 a^2 x^8}-\frac {9 b^2 \left (a+b x^3\right )^{5/3}}{220 a^3 x^5}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 53, normalized size = 0.78 \begin {gather*} \frac {\left (a+b x^3\right )^{2/3} \left (-20 a^3-5 a^2 b x^3+6 a b^2 x^6-9 b^3 x^9\right )}{220 a^3 x^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^(2/3)/x^12,x]

[Out]

((a + b*x^3)^(2/3)*(-20*a^3 - 5*a^2*b*x^3 + 6*a*b^2*x^6 - 9*b^3*x^9))/(220*a^3*x^11)

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Maple [A]
time = 0.15, size = 39, normalized size = 0.57

method	result	size
gosper	\(-\frac {\left (b \,x^{3}+a \right )^{\frac {5}{3}} \left (9 b^{2} x^{6}-15 a b \,x^{3}+20 a^{2}\right )}{220 x^{11} a^{3}}\)	\(39\)
trager	\(-\frac {\left (9 b^{3} x^{9}-6 a \,b^{2} x^{6}+5 a^{2} b \,x^{3}+20 a^{3}\right ) \left (b \,x^{3}+a \right )^{\frac {2}{3}}}{220 x^{11} a^{3}}\)	\(50\)
risch	\(-\frac {\left (9 b^{3} x^{9}-6 a \,b^{2} x^{6}+5 a^{2} b \,x^{3}+20 a^{3}\right ) \left (b \,x^{3}+a \right )^{\frac {2}{3}}}{220 x^{11} a^{3}}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^(2/3)/x^12,x,method=_RETURNVERBOSE)

[Out]

-1/220*(b*x^3+a)^(5/3)*(9*b^2*x^6-15*a*b*x^3+20*a^2)/x^11/a^3

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Maxima [A]
time = 0.29, size = 52, normalized size = 0.76 \begin {gather*} -\frac {\frac {44 \, {\left (b x^{3} + a\right )}^{\frac {5}{3}} b^{2}}{x^{5}} - \frac {55 \, {\left (b x^{3} + a\right )}^{\frac {8}{3}} b}{x^{8}} + \frac {20 \, {\left (b x^{3} + a\right )}^{\frac {11}{3}}}{x^{11}}}{220 \, a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(2/3)/x^12,x, algorithm="maxima")

[Out]

-1/220*(44*(b*x^3 + a)^(5/3)*b^2/x^5 - 55*(b*x^3 + a)^(8/3)*b/x^8 + 20*(b*x^3 + a)^(11/3)/x^11)/a^3

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Fricas [A]
time = 0.38, size = 49, normalized size = 0.72 \begin {gather*} -\frac {{\left (9 \, b^{3} x^{9} - 6 \, a b^{2} x^{6} + 5 \, a^{2} b x^{3} + 20 \, a^{3}\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{220 \, a^{3} x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(2/3)/x^12,x, algorithm="fricas")

[Out]

-1/220*(9*b^3*x^9 - 6*a*b^2*x^6 + 5*a^2*b*x^3 + 20*a^3)*(b*x^3 + a)^(2/3)/(a^3*x^11)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 520 vs. \(2 (61) = 122\).
time = 0.89, size = 520, normalized size = 7.65 \begin {gather*} \frac {40 a^{5} b^{\frac {14}{3}} \left (\frac {a}{b x^{3}} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {11}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac {2}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac {2}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac {2}{3}\right )} + \frac {90 a^{4} b^{\frac {17}{3}} x^{3} \left (\frac {a}{b x^{3}} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {11}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac {2}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac {2}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac {2}{3}\right )} + \frac {48 a^{3} b^{\frac {20}{3}} x^{6} \left (\frac {a}{b x^{3}} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {11}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac {2}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac {2}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac {2}{3}\right )} + \frac {4 a^{2} b^{\frac {23}{3}} x^{9} \left (\frac {a}{b x^{3}} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {11}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac {2}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac {2}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac {2}{3}\right )} + \frac {24 a b^{\frac {26}{3}} x^{12} \left (\frac {a}{b x^{3}} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {11}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac {2}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac {2}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac {2}{3}\right )} + \frac {18 b^{\frac {29}{3}} x^{15} \left (\frac {a}{b x^{3}} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {11}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac {2}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac {2}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac {2}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**(2/3)/x**12,x)

[Out]

40*a**5*b**(14/3)*(a/(b*x**3) + 1)**(2/3)*gamma(-11/3)/(27*a**5*b**4*x**9*gamma(-2/3) + 54*a**4*b**5*x**12*gam
ma(-2/3) + 27*a**3*b**6*x**15*gamma(-2/3)) + 90*a**4*b**(17/3)*x**3*(a/(b*x**3) + 1)**(2/3)*gamma(-11/3)/(27*a
**5*b**4*x**9*gamma(-2/3) + 54*a**4*b**5*x**12*gamma(-2/3) + 27*a**3*b**6*x**15*gamma(-2/3)) + 48*a**3*b**(20/
3)*x**6*(a/(b*x**3) + 1)**(2/3)*gamma(-11/3)/(27*a**5*b**4*x**9*gamma(-2/3) + 54*a**4*b**5*x**12*gamma(-2/3) +
27*a**3*b**6*x**15*gamma(-2/3)) + 4*a**2*b**(23/3)*x**9*(a/(b*x**3) + 1)**(2/3)*gamma(-11/3)/(27*a**5*b**4*x*
*9*gamma(-2/3) + 54*a**4*b**5*x**12*gamma(-2/3) + 27*a**3*b**6*x**15*gamma(-2/3)) + 24*a*b**(26/3)*x**12*(a/(b
*x**3) + 1)**(2/3)*gamma(-11/3)/(27*a**5*b**4*x**9*gamma(-2/3) + 54*a**4*b**5*x**12*gamma(-2/3) + 27*a**3*b**6
*x**15*gamma(-2/3)) + 18*b**(29/3)*x**15*(a/(b*x**3) + 1)**(2/3)*gamma(-11/3)/(27*a**5*b**4*x**9*gamma(-2/3) +
54*a**4*b**5*x**12*gamma(-2/3) + 27*a**3*b**6*x**15*gamma(-2/3))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(2/3)/x^12,x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(2/3)/x^12, x)

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Mupad [B]
time = 1.35, size = 73, normalized size = 1.07 \begin {gather*} \frac {3\,b^2\,{\left (b\,x^3+a\right )}^{2/3}}{110\,a^2\,x^5}-\frac {b\,{\left (b\,x^3+a\right )}^{2/3}}{44\,a\,x^8}-\frac {9\,b^3\,{\left (b\,x^3+a\right )}^{2/3}}{220\,a^3\,x^2}-\frac {{\left (b\,x^3+a\right )}^{2/3}}{11\,x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^(2/3)/x^12,x)

[Out]

(3*b^2*(a + b*x^3)^(2/3))/(110*a^2*x^5) - (b*(a + b*x^3)^(2/3))/(44*a*x^8) - (9*b^3*(a + b*x^3)^(2/3))/(220*a^
3*x^2) - (a + b*x^3)^(2/3)/(11*x^11)

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