Integrand size = 15, antiderivative size = 92 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^{14}} \, dx=-\frac {\left (a+b x^3\right )^{4/3}}{13 a x^{13}}+\frac {9 b \left (a+b x^3\right )^{4/3}}{130 a^2 x^{10}}-\frac {27 b^2 \left (a+b x^3\right )^{4/3}}{455 a^3 x^7}+\frac {81 b^3 \left (a+b x^3\right )^{4/3}}{1820 a^4 x^4} \]
-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
Time = 0.12 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.58 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^{14}} \, dx=\frac {\left (a+b x^3\right )^{4/3} \left (-140 a^3+126 a^2 b x^3-108 a b^2 x^6+81 b^3 x^9\right )}{1820 a^4 x^{13}} \]
Time = 0.22 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.13, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {803, 803, 803, 796}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sqrt [3]{a+b x^3}}{x^{14}} \, dx\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {9 b \int \frac {\sqrt [3]{b x^3+a}}{x^{11}}dx}{13 a}-\frac {\left (a+b x^3\right )^{4/3}}{13 a x^{13}}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {9 b \left (-\frac {3 b \int \frac {\sqrt [3]{b x^3+a}}{x^8}dx}{5 a}-\frac {\left (a+b x^3\right )^{4/3}}{10 a x^{10}}\right )}{13 a}-\frac {\left (a+b x^3\right )^{4/3}}{13 a x^{13}}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {9 b \left (-\frac {3 b \left (-\frac {3 b \int \frac {\sqrt [3]{b x^3+a}}{x^5}dx}{7 a}-\frac {\left (a+b x^3\right )^{4/3}}{7 a x^7}\right )}{5 a}-\frac {\left (a+b x^3\right )^{4/3}}{10 a x^{10}}\right )}{13 a}-\frac {\left (a+b x^3\right )^{4/3}}{13 a x^{13}}\) |
\(\Big \downarrow \) 796 |
\(\displaystyle -\frac {9 b \left (-\frac {3 b \left (\frac {3 b \left (a+b x^3\right )^{4/3}}{28 a^2 x^4}-\frac {\left (a+b x^3\right )^{4/3}}{7 a x^7}\right )}{5 a}-\frac {\left (a+b x^3\right )^{4/3}}{10 a x^{10}}\right )}{13 a}-\frac {\left (a+b x^3\right )^{4/3}}{13 a x^{13}}\) |
-1/13*(a + b*x^3)^(4/3)/(a*x^13) - (9*b*(-1/10*(a + b*x^3)^(4/3)/(a*x^10) - (3*b*(-1/7*(a + b*x^3)^(4/3)/(a*x^7) + (3*b*(a + b*x^3)^(4/3))/(28*a^2*x ^4)))/(5*a)))/(13*a)
3.6.22.3.1 Defintions of rubi rules used
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]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*(( a + b*x^n)^(p + 1)/(a*(m + 1))), x] - Simp[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] && I LtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]
Time = 3.95 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.54
method | result | size |
gosper | \(-\frac {\left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (-81 b^{3} x^{9}+108 a \,b^{2} x^{6}-126 a^{2} b \,x^{3}+140 a^{3}\right )}{1820 x^{13} a^{4}}\) | \(50\) |
pseudoelliptic | \(-\frac {\left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (-81 b^{3} x^{9}+108 a \,b^{2} x^{6}-126 a^{2} b \,x^{3}+140 a^{3}\right )}{1820 x^{13} a^{4}}\) | \(50\) |
trager | \(-\frac {\left (-81 b^{4} x^{12}+27 a \,b^{3} x^{9}-18 a^{2} b^{2} x^{6}+14 a^{3} b \,x^{3}+140 a^{4}\right ) \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{1820 x^{13} a^{4}}\) | \(61\) |
risch | \(-\frac {\left (-81 b^{4} x^{12}+27 a \,b^{3} x^{9}-18 a^{2} b^{2} x^{6}+14 a^{3} b \,x^{3}+140 a^{4}\right ) \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{1820 x^{13} a^{4}}\) | \(61\) |
Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^{14}} \, dx=\frac {{\left (81 \, b^{4} x^{12} - 27 \, a b^{3} x^{9} + 18 \, a^{2} b^{2} x^{6} - 14 \, a^{3} b x^{3} - 140 \, a^{4}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{1820 \, a^{4} x^{13}} \]
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)
Leaf count of result is larger than twice the leaf count of optimal. 847 vs. \(2 (85) = 170\).
Time = 1.18 (sec) , antiderivative size = 847, normalized size of antiderivative = 9.21 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^{14}} \, dx=- \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 )} \]
-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**15*gamma(-1/3) + 243*a**5*b**11*x**18*g amma(-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*gamma(-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*gamma(-1/3) + 243*a**6*b**10*x**15*gamma(-1/3) + 243*a**5*b**11*x**18*gamma(-1/3) + 8 1*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*ga mma(-1/3)) + 432*a*b**(46/3)*x**18*(a/(b*x**3) + 1)**(1/3)*gamma(-13/3)/(8 1*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...
Time = 0.20 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^{14}} \, dx=\frac {\frac {455 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} b^{3}}{x^{4}} - \frac {780 \, {\left (b x^{3} + a\right )}^{\frac {7}{3}} b^{2}}{x^{7}} + \frac {546 \, {\left (b x^{3} + a\right )}^{\frac {10}{3}} b}{x^{10}} - \frac {140 \, {\left (b x^{3} + a\right )}^{\frac {13}{3}}}{x^{13}}}{1820 \, a^{4}} \]
1/1820*(455*(b*x^3 + a)^(4/3)*b^3/x^4 - 780*(b*x^3 + a)^(7/3)*b^2/x^7 + 54 6*(b*x^3 + a)^(10/3)*b/x^10 - 140*(b*x^3 + a)^(13/3)/x^13)/a^4
\[ \int \frac {\sqrt [3]{a+b x^3}}{x^{14}} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x^{14}} \,d x } \]
Time = 6.35 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^{14}} \, dx=\frac {81\,b^4\,{\left (b\,x^3+a\right )}^{1/3}}{1820\,a^4\,x}-\frac {b\,{\left (b\,x^3+a\right )}^{1/3}}{130\,a\,x^{10}}-\frac {{\left (b\,x^3+a\right )}^{1/3}}{13\,x^{13}}-\frac {27\,b^3\,{\left (b\,x^3+a\right )}^{1/3}}{1820\,a^3\,x^4}+\frac {9\,b^2\,{\left (b\,x^3+a\right )}^{1/3}}{910\,a^2\,x^7} \]