3.522 \(\int \frac {\sqrt [3]{a+b x^3}}{x^{14}} \, dx\)

Optimal. Leaf size=92 \[ \frac {81 b^3 \left (a+b x^3\right )^{4/3}}{1820 a^4 x^4}-\frac {27 b^2 \left (a+b x^3\right )^{4/3}}{455 a^3 x^7}+\frac {9 b \left (a+b x^3\right )^{4/3}}{130 a^2 x^{10}}-\frac {\left (a+b x^3\right )^{4/3}}{13 a x^{13}} \]

[Out]

-1/13*(b*x^3+a)^(4/3)/a/x^13+9/130*b*(b*x^3+a)^(4/3)/a^2/x^10-27/455*b^2*(b*x^3+a)^(4/3)/a^3/x^7+81/1820*b^3*(
b*x^3+a)^(4/3)/a^4/x^4

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Rubi [A]  time = 0.03, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac {81 b^3 \left (a+b x^3\right )^{4/3}}{1820 a^4 x^4}-\frac {27 b^2 \left (a+b x^3\right )^{4/3}}{455 a^3 x^7}+\frac {9 b \left (a+b x^3\right )^{4/3}}{130 a^2 x^{10}}-\frac {\left (a+b x^3\right )^{4/3}}{13 a x^{13}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^(1/3)/x^14,x]

[Out]

-(a + b*x^3)^(4/3)/(13*a*x^13) + (9*b*(a + b*x^3)^(4/3))/(130*a^2*x^10) - (27*b^2*(a + b*x^3)^(4/3))/(455*a^3*
x^7) + (81*b^3*(a + b*x^3)^(4/3))/(1820*a^4*x^4)

Rule 264

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 271

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 {\sqrt [3]{a+b x^3}}{x^{14}} \, dx &=-\frac {\left (a+b x^3\right )^{4/3}}{13 a x^{13}}-\frac {(9 b) \int \frac {\sqrt [3]{a+b x^3}}{x^{11}} \, dx}{13 a}\\ &=-\frac {\left (a+b x^3\right )^{4/3}}{13 a x^{13}}+\frac {9 b \left (a+b x^3\right )^{4/3}}{130 a^2 x^{10}}+\frac {\left (27 b^2\right ) \int \frac {\sqrt [3]{a+b x^3}}{x^8} \, dx}{65 a^2}\\ &=-\frac {\left (a+b x^3\right )^{4/3}}{13 a x^{13}}+\frac {9 b \left (a+b x^3\right )^{4/3}}{130 a^2 x^{10}}-\frac {27 b^2 \left (a+b x^3\right )^{4/3}}{455 a^3 x^7}-\frac {\left (81 b^3\right ) \int \frac {\sqrt [3]{a+b x^3}}{x^5} \, dx}{455 a^3}\\ &=-\frac {\left (a+b x^3\right )^{4/3}}{13 a x^{13}}+\frac {9 b \left (a+b x^3\right )^{4/3}}{130 a^2 x^{10}}-\frac {27 b^2 \left (a+b x^3\right )^{4/3}}{455 a^3 x^7}+\frac {81 b^3 \left (a+b x^3\right )^{4/3}}{1820 a^4 x^4}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 53, normalized size = 0.58 \[ \frac {\left (a+b x^3\right )^{4/3} \left (-140 a^3+126 a^2 b x^3-108 a b^2 x^6+81 b^3 x^9\right )}{1820 a^4 x^{13}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^(1/3)/x^14,x]

[Out]

((a + b*x^3)^(4/3)*(-140*a^3 + 126*a^2*b*x^3 - 108*a*b^2*x^6 + 81*b^3*x^9))/(1820*a^4*x^13)

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fricas [A]  time = 0.70, size = 60, normalized size = 0.65 \[ \frac {{\left (81 \, b^{4} x^{12} - 27 \, a b^{3} x^{9} + 18 \, a^{2} b^{2} x^{6} - 14 \, a^{3} b x^{3} - 140 \, a^{4}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{1820 \, a^{4} x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(1/3)/x^14,x, algorithm="fricas")

[Out]

1/1820*(81*b^4*x^12 - 27*a*b^3*x^9 + 18*a^2*b^2*x^6 - 14*a^3*b*x^3 - 140*a^4)*(b*x^3 + a)^(1/3)/(a^4*x^13)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x^{14}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(1/3)/x^14,x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(1/3)/x^14, x)

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maple [A]  time = 0.01, size = 50, normalized size = 0.54 \[ -\frac {\left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (-81 b^{3} x^{9}+108 a \,b^{2} x^{6}-126 a^{2} b \,x^{3}+140 a^{3}\right )}{1820 a^{4} x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^(1/3)/x^14,x)

[Out]

-1/1820*(b*x^3+a)^(4/3)*(-81*b^3*x^9+108*a*b^2*x^6-126*a^2*b*x^3+140*a^3)/x^13/a^4

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maxima [A]  time = 1.37, size = 69, normalized size = 0.75 \[ \frac {\frac {455 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} b^{3}}{x^{4}} - \frac {780 \, {\left (b x^{3} + a\right )}^{\frac {7}{3}} b^{2}}{x^{7}} + \frac {546 \, {\left (b x^{3} + a\right )}^{\frac {10}{3}} b}{x^{10}} - \frac {140 \, {\left (b x^{3} + a\right )}^{\frac {13}{3}}}{x^{13}}}{1820 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^(1/3)/x^14,x, algorithm="maxima")

[Out]

1/1820*(455*(b*x^3 + a)^(4/3)*b^3/x^4 - 780*(b*x^3 + a)^(7/3)*b^2/x^7 + 546*(b*x^3 + a)^(10/3)*b/x^10 - 140*(b
*x^3 + a)^(13/3)/x^13)/a^4

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mupad [B]  time = 1.50, size = 93, normalized size = 1.01 \[ \frac {81\,b^4\,{\left (b\,x^3+a\right )}^{1/3}}{1820\,a^4\,x}-\frac {b\,{\left (b\,x^3+a\right )}^{1/3}}{130\,a\,x^{10}}-\frac {{\left (b\,x^3+a\right )}^{1/3}}{13\,x^{13}}-\frac {27\,b^3\,{\left (b\,x^3+a\right )}^{1/3}}{1820\,a^3\,x^4}+\frac {9\,b^2\,{\left (b\,x^3+a\right )}^{1/3}}{910\,a^2\,x^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^(1/3)/x^14,x)

[Out]

(81*b^4*(a + b*x^3)^(1/3))/(1820*a^4*x) - (b*(a + b*x^3)^(1/3))/(130*a*x^10) - (a + b*x^3)^(1/3)/(13*x^13) - (
27*b^3*(a + b*x^3)^(1/3))/(1820*a^3*x^4) + (9*b^2*(a + b*x^3)^(1/3))/(910*a^2*x^7)

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sympy [B]  time = 2.81, size = 847, normalized size = 9.21 \[ - \frac {280 a^{7} b^{\frac {28}{3}} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {13}{3}\right )}{81 a^{7} b^{9} x^{12} \Gamma \left (- \frac {1}{3}\right ) + 243 a^{6} b^{10} x^{15} \Gamma \left (- \frac {1}{3}\right ) + 243 a^{5} b^{11} x^{18} \Gamma \left (- \frac {1}{3}\right ) + 81 a^{4} b^{12} x^{21} \Gamma \left (- \frac {1}{3}\right )} - \frac {868 a^{6} b^{\frac {31}{3}} x^{3} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {13}{3}\right )}{81 a^{7} b^{9} x^{12} \Gamma \left (- \frac {1}{3}\right ) + 243 a^{6} b^{10} x^{15} \Gamma \left (- \frac {1}{3}\right ) + 243 a^{5} b^{11} x^{18} \Gamma \left (- \frac {1}{3}\right ) + 81 a^{4} b^{12} x^{21} \Gamma \left (- \frac {1}{3}\right )} - \frac {888 a^{5} b^{\frac {34}{3}} x^{6} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {13}{3}\right )}{81 a^{7} b^{9} x^{12} \Gamma \left (- \frac {1}{3}\right ) + 243 a^{6} b^{10} x^{15} \Gamma \left (- \frac {1}{3}\right ) + 243 a^{5} b^{11} x^{18} \Gamma \left (- \frac {1}{3}\right ) + 81 a^{4} b^{12} x^{21} \Gamma \left (- \frac {1}{3}\right )} - \frac {310 a^{4} b^{\frac {37}{3}} x^{9} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {13}{3}\right )}{81 a^{7} b^{9} x^{12} \Gamma \left (- \frac {1}{3}\right ) + 243 a^{6} b^{10} x^{15} \Gamma \left (- \frac {1}{3}\right ) + 243 a^{5} b^{11} x^{18} \Gamma \left (- \frac {1}{3}\right ) + 81 a^{4} b^{12} x^{21} \Gamma \left (- \frac {1}{3}\right )} + \frac {80 a^{3} b^{\frac {40}{3}} x^{12} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {13}{3}\right )}{81 a^{7} b^{9} x^{12} \Gamma \left (- \frac {1}{3}\right ) + 243 a^{6} b^{10} x^{15} \Gamma \left (- \frac {1}{3}\right ) + 243 a^{5} b^{11} x^{18} \Gamma \left (- \frac {1}{3}\right ) + 81 a^{4} b^{12} x^{21} \Gamma \left (- \frac {1}{3}\right )} + \frac {360 a^{2} b^{\frac {43}{3}} x^{15} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {13}{3}\right )}{81 a^{7} b^{9} x^{12} \Gamma \left (- \frac {1}{3}\right ) + 243 a^{6} b^{10} x^{15} \Gamma \left (- \frac {1}{3}\right ) + 243 a^{5} b^{11} x^{18} \Gamma \left (- \frac {1}{3}\right ) + 81 a^{4} b^{12} x^{21} \Gamma \left (- \frac {1}{3}\right )} + \frac {432 a b^{\frac {46}{3}} x^{18} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {13}{3}\right )}{81 a^{7} b^{9} x^{12} \Gamma \left (- \frac {1}{3}\right ) + 243 a^{6} b^{10} x^{15} \Gamma \left (- \frac {1}{3}\right ) + 243 a^{5} b^{11} x^{18} \Gamma \left (- \frac {1}{3}\right ) + 81 a^{4} b^{12} x^{21} \Gamma \left (- \frac {1}{3}\right )} + \frac {162 b^{\frac {49}{3}} x^{21} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {13}{3}\right )}{81 a^{7} b^{9} x^{12} \Gamma \left (- \frac {1}{3}\right ) + 243 a^{6} b^{10} x^{15} \Gamma \left (- \frac {1}{3}\right ) + 243 a^{5} b^{11} x^{18} \Gamma \left (- \frac {1}{3}\right ) + 81 a^{4} b^{12} x^{21} \Gamma \left (- \frac {1}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**(1/3)/x**14,x)

[Out]

-280*a**7*b**(28/3)*(a/(b*x**3) + 1)**(1/3)*gamma(-13/3)/(81*a**7*b**9*x**12*gamma(-1/3) + 243*a**6*b**10*x**1
5*gamma(-1/3) + 243*a**5*b**11*x**18*gamma(-1/3) + 81*a**4*b**12*x**21*gamma(-1/3)) - 868*a**6*b**(31/3)*x**3*
(a/(b*x**3) + 1)**(1/3)*gamma(-13/3)/(81*a**7*b**9*x**12*gamma(-1/3) + 243*a**6*b**10*x**15*gamma(-1/3) + 243*
a**5*b**11*x**18*gamma(-1/3) + 81*a**4*b**12*x**21*gamma(-1/3)) - 888*a**5*b**(34/3)*x**6*(a/(b*x**3) + 1)**(1
/3)*gamma(-13/3)/(81*a**7*b**9*x**12*gamma(-1/3) + 243*a**6*b**10*x**15*gamma(-1/3) + 243*a**5*b**11*x**18*gam
ma(-1/3) + 81*a**4*b**12*x**21*gamma(-1/3)) - 310*a**4*b**(37/3)*x**9*(a/(b*x**3) + 1)**(1/3)*gamma(-13/3)/(81
*a**7*b**9*x**12*gamma(-1/3) + 243*a**6*b**10*x**15*gamma(-1/3) + 243*a**5*b**11*x**18*gamma(-1/3) + 81*a**4*b
**12*x**21*gamma(-1/3)) + 80*a**3*b**(40/3)*x**12*(a/(b*x**3) + 1)**(1/3)*gamma(-13/3)/(81*a**7*b**9*x**12*gam
ma(-1/3) + 243*a**6*b**10*x**15*gamma(-1/3) + 243*a**5*b**11*x**18*gamma(-1/3) + 81*a**4*b**12*x**21*gamma(-1/
3)) + 360*a**2*b**(43/3)*x**15*(a/(b*x**3) + 1)**(1/3)*gamma(-13/3)/(81*a**7*b**9*x**12*gamma(-1/3) + 243*a**6
*b**10*x**15*gamma(-1/3) + 243*a**5*b**11*x**18*gamma(-1/3) + 81*a**4*b**12*x**21*gamma(-1/3)) + 432*a*b**(46/
3)*x**18*(a/(b*x**3) + 1)**(1/3)*gamma(-13/3)/(81*a**7*b**9*x**12*gamma(-1/3) + 243*a**6*b**10*x**15*gamma(-1/
3) + 243*a**5*b**11*x**18*gamma(-1/3) + 81*a**4*b**12*x**21*gamma(-1/3)) + 162*b**(49/3)*x**21*(a/(b*x**3) + 1
)**(1/3)*gamma(-13/3)/(81*a**7*b**9*x**12*gamma(-1/3) + 243*a**6*b**10*x**15*gamma(-1/3) + 243*a**5*b**11*x**1
8*gamma(-1/3) + 81*a**4*b**12*x**21*gamma(-1/3))

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