Integrand size = 24, antiderivative size = 392 \[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^{14} \left (c+d x^3\right )} \, dx=-\frac {a \sqrt [3]{a+b x^3}}{13 c x^{13}}-\frac {(14 b c-13 a d) \sqrt [3]{a+b x^3}}{130 c^2 x^{10}}-\frac {\left (4 b^2 c^2-143 a b c d+130 a^2 d^2\right ) \sqrt [3]{a+b x^3}}{910 a c^3 x^7}+\frac {\left (12 b^3 c^3+26 a b^2 c^2 d-520 a^2 b c d^2+455 a^3 d^3\right ) \sqrt [3]{a+b x^3}}{1820 a^2 c^4 x^4}-\frac {\left (36 b^4 c^4+78 a b^3 c^3 d+260 a^2 b^2 c^2 d^2-2275 a^3 b c d^3+1820 a^4 d^4\right ) \sqrt [3]{a+b x^3}}{1820 a^3 c^5 x}+\frac {d^3 (b c-a d)^{4/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} c^{16/3}}-\frac {d^3 (b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 c^{16/3}}+\frac {d^3 (b c-a d)^{4/3} \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{16/3}} \] Output:
-1/13*a*(b*x^3+a)^(1/3)/c/x^13-1/130*(-13*a*d+14*b*c)*(b*x^3+a)^(1/3)/c^2/ x^10-1/910*(130*a^2*d^2-143*a*b*c*d+4*b^2*c^2)*(b*x^3+a)^(1/3)/a/c^3/x^7+1 /1820*(455*a^3*d^3-520*a^2*b*c*d^2+26*a*b^2*c^2*d+12*b^3*c^3)*(b*x^3+a)^(1 /3)/a^2/c^4/x^4-1/1820*(1820*a^4*d^4-2275*a^3*b*c*d^3+260*a^2*b^2*c^2*d^2+ 78*a*b^3*c^3*d+36*b^4*c^4)*(b*x^3+a)^(1/3)/a^3/c^5/x+1/3*d^3*(-a*d+b*c)^(4 /3)*arctan(1/3*(1+2*(-a*d+b*c)^(1/3)*x/c^(1/3)/(b*x^3+a)^(1/3))*3^(1/2))*3 ^(1/2)/c^(16/3)-1/6*d^3*(-a*d+b*c)^(4/3)*ln(d*x^3+c)/c^(16/3)+1/2*d^3*(-a* d+b*c)^(4/3)*ln((-a*d+b*c)^(1/3)*x/c^(1/3)-(b*x^3+a)^(1/3))/c^(16/3)
Result contains complex when optimal does not.
Time = 3.23 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.22 \[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^{14} \left (c+d x^3\right )} \, dx=\frac {-\frac {3 \sqrt [3]{c} \sqrt [3]{a+b x^3} \left (36 b^4 c^4 x^{12}+6 a b^3 c^3 x^9 \left (-2 c+13 d x^3\right )+2 a^2 b^2 c^2 x^6 \left (4 c^2-13 c d x^3+130 d^2 x^6\right )+a^3 b c x^3 \left (196 c^3-286 c^2 d x^3+520 c d^2 x^6-2275 d^3 x^9\right )+a^4 \left (140 c^4-182 c^3 d x^3+260 c^2 d^2 x^6-455 c d^3 x^9+1820 d^4 x^{12}\right )\right )}{a^3 x^{13}}-910 \sqrt {-6-6 i \sqrt {3}} d^3 (b c-a d)^{4/3} \arctan \left (\frac {3 \sqrt [3]{b c-a d} x}{\sqrt {3} \sqrt [3]{b c-a d} x-\left (3 i+\sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )+910 i \left (i+\sqrt {3}\right ) d^3 (b c-a d)^{4/3} \log \left (2 \sqrt [3]{b c-a d} x+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )+455 \left (1-i \sqrt {3}\right ) d^3 (b c-a d)^{4/3} \log \left (2 (b c-a d)^{2/3} x^2+\left (-1-i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{b c-a d} x \sqrt [3]{a+b x^3}+i \left (i+\sqrt {3}\right ) c^{2/3} \left (a+b x^3\right )^{2/3}\right )}{5460 c^{16/3}} \] Input:
Integrate[(a + b*x^3)^(4/3)/(x^14*(c + d*x^3)),x]
Output:
((-3*c^(1/3)*(a + b*x^3)^(1/3)*(36*b^4*c^4*x^12 + 6*a*b^3*c^3*x^9*(-2*c + 13*d*x^3) + 2*a^2*b^2*c^2*x^6*(4*c^2 - 13*c*d*x^3 + 130*d^2*x^6) + a^3*b*c *x^3*(196*c^3 - 286*c^2*d*x^3 + 520*c*d^2*x^6 - 2275*d^3*x^9) + a^4*(140*c ^4 - 182*c^3*d*x^3 + 260*c^2*d^2*x^6 - 455*c*d^3*x^9 + 1820*d^4*x^12)))/(a ^3*x^13) - 910*Sqrt[-6 - (6*I)*Sqrt[3]]*d^3*(b*c - a*d)^(4/3)*ArcTan[(3*(b *c - a*d)^(1/3)*x)/(Sqrt[3]*(b*c - a*d)^(1/3)*x - (3*I + Sqrt[3])*c^(1/3)* (a + b*x^3)^(1/3))] + (910*I)*(I + Sqrt[3])*d^3*(b*c - a*d)^(4/3)*Log[2*(b *c - a*d)^(1/3)*x + (1 + I*Sqrt[3])*c^(1/3)*(a + b*x^3)^(1/3)] + 455*(1 - I*Sqrt[3])*d^3*(b*c - a*d)^(4/3)*Log[2*(b*c - a*d)^(2/3)*x^2 + (-1 - I*Sqr t[3])*c^(1/3)*(b*c - a*d)^(1/3)*x*(a + b*x^3)^(1/3) + I*(I + Sqrt[3])*c^(2 /3)*(a + b*x^3)^(2/3)])/(5460*c^(16/3))
Time = 1.32 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {974, 1053, 25, 27, 1053, 27, 1053, 1053, 27, 992}
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.
| \(\displaystyle \int \frac {\left (a+b x^3\right )^{4/3}}{x^{14} \left (c+d x^3\right )} \, dx\) |
| \(\Big \downarrow \) 974 |
| \(\displaystyle \frac {\int \frac {b (13 b c-12 a d) x^3+a (14 b c-13 a d)}{x^{11} \left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx}{13 c}-\frac {a \sqrt [3]{a+b x^3}}{13 c x^{13}}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {-\frac {\int -\frac {a \left (-9 b d (14 b c-13 a d) x^3+4 b^2 c^2+130 a^2 d^2-143 a b c d\right )}{x^8 \left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx}{10 a c}-\frac {\sqrt [3]{a+b x^3} (14 b c-13 a d)}{10 c x^{10}}}{13 c}-\frac {a \sqrt [3]{a+b x^3}}{13 c x^{13}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {a \left (-9 b d (14 b c-13 a d) x^3+4 b^2 c^2+130 a^2 d^2-143 a b c d\right )}{x^8 \left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx}{10 a c}-\frac {\sqrt [3]{a+b x^3} (14 b c-13 a d)}{10 c x^{10}}}{13 c}-\frac {a \sqrt [3]{a+b x^3}}{13 c x^{13}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {-9 b d (14 b c-13 a d) x^3+4 b^2 c^2+130 a^2 d^2-143 a b c d}{x^8 \left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx}{10 c}-\frac {\sqrt [3]{a+b x^3} (14 b c-13 a d)}{10 c x^{10}}}{13 c}-\frac {a \sqrt [3]{a+b x^3}}{13 c x^{13}}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {2 \left (12 b^3 c^3+26 a b^2 d c^2-520 a^2 b d^2 c+455 a^3 d^3+3 b d \left (4 b^2 c^2-143 a b d c+130 a^2 d^2\right ) x^3\right )}{x^5 \left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx}{7 a c}-\frac {\sqrt [3]{a+b x^3} \left (\frac {4 b^2 c}{a}+\frac {130 a d^2}{c}-143 b d\right )}{7 x^7}}{10 c}-\frac {\sqrt [3]{a+b x^3} (14 b c-13 a d)}{10 c x^{10}}}{13 c}-\frac {a \sqrt [3]{a+b x^3}}{13 c x^{13}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {2 \int \frac {12 b^3 c^3+26 a b^2 d c^2-520 a^2 b d^2 c+455 a^3 d^3+3 b d \left (4 b^2 c^2-143 a b d c+130 a^2 d^2\right ) x^3}{x^5 \left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx}{7 a c}-\frac {\sqrt [3]{a+b x^3} \left (\frac {4 b^2 c}{a}+\frac {130 a d^2}{c}-143 b d\right )}{7 x^7}}{10 c}-\frac {\sqrt [3]{a+b x^3} (14 b c-13 a d)}{10 c x^{10}}}{13 c}-\frac {a \sqrt [3]{a+b x^3}}{13 c x^{13}}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {\frac {-\frac {2 \left (-\frac {\int \frac {36 b^4 c^4+78 a b^3 d c^3+260 a^2 b^2 d^2 c^2-2275 a^3 b d^3 c+1820 a^4 d^4+3 b d \left (12 b^3 c^3+26 a b^2 d c^2-520 a^2 b d^2 c+455 a^3 d^3\right ) x^3}{x^2 \left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx}{4 a c}-\frac {\sqrt [3]{a+b x^3} \left (455 a^3 d^3-520 a^2 b c d^2+26 a b^2 c^2 d+12 b^3 c^3\right )}{4 a c x^4}\right )}{7 a c}-\frac {\sqrt [3]{a+b x^3} \left (\frac {4 b^2 c}{a}+\frac {130 a d^2}{c}-143 b d\right )}{7 x^7}}{10 c}-\frac {\sqrt [3]{a+b x^3} (14 b c-13 a d)}{10 c x^{10}}}{13 c}-\frac {a \sqrt [3]{a+b x^3}}{13 c x^{13}}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {\frac {-\frac {2 \left (-\frac {-\frac {\int \frac {1820 a^3 d^3 (b c-a d)^2 x}{\left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx}{a c}-\frac {\sqrt [3]{a+b x^3} \left (1820 a^4 d^4-2275 a^3 b c d^3+260 a^2 b^2 c^2 d^2+78 a b^3 c^3 d+36 b^4 c^4\right )}{a c x}}{4 a c}-\frac {\sqrt [3]{a+b x^3} \left (455 a^3 d^3-520 a^2 b c d^2+26 a b^2 c^2 d+12 b^3 c^3\right )}{4 a c x^4}\right )}{7 a c}-\frac {\sqrt [3]{a+b x^3} \left (\frac {4 b^2 c}{a}+\frac {130 a d^2}{c}-143 b d\right )}{7 x^7}}{10 c}-\frac {\sqrt [3]{a+b x^3} (14 b c-13 a d)}{10 c x^{10}}}{13 c}-\frac {a \sqrt [3]{a+b x^3}}{13 c x^{13}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {2 \left (-\frac {-\frac {1820 a^2 d^3 (b c-a d)^2 \int \frac {x}{\left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx}{c}-\frac {\sqrt [3]{a+b x^3} \left (1820 a^4 d^4-2275 a^3 b c d^3+260 a^2 b^2 c^2 d^2+78 a b^3 c^3 d+36 b^4 c^4\right )}{a c x}}{4 a c}-\frac {\sqrt [3]{a+b x^3} \left (455 a^3 d^3-520 a^2 b c d^2+26 a b^2 c^2 d+12 b^3 c^3\right )}{4 a c x^4}\right )}{7 a c}-\frac {\sqrt [3]{a+b x^3} \left (\frac {4 b^2 c}{a}+\frac {130 a d^2}{c}-143 b d\right )}{7 x^7}}{10 c}-\frac {\sqrt [3]{a+b x^3} (14 b c-13 a d)}{10 c x^{10}}}{13 c}-\frac {a \sqrt [3]{a+b x^3}}{13 c x^{13}}\) |
| \(\Big \downarrow \) 992 |
| \(\displaystyle \frac {\frac {-\frac {2 \left (-\frac {\sqrt [3]{a+b x^3} \left (455 a^3 d^3-520 a^2 b c d^2+26 a b^2 c^2 d+12 b^3 c^3\right )}{4 a c x^4}-\frac {-\frac {1820 a^2 d^3 (b c-a d)^2 \left (-\frac {\arctan \left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{c} (b c-a d)^{2/3}}+\frac {\log \left (c+d x^3\right )}{6 \sqrt [3]{c} (b c-a d)^{2/3}}-\frac {\log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{c} (b c-a d)^{2/3}}\right )}{c}-\frac {\sqrt [3]{a+b x^3} \left (1820 a^4 d^4-2275 a^3 b c d^3+260 a^2 b^2 c^2 d^2+78 a b^3 c^3 d+36 b^4 c^4\right )}{a c x}}{4 a c}\right )}{7 a c}-\frac {\sqrt [3]{a+b x^3} \left (\frac {4 b^2 c}{a}+\frac {130 a d^2}{c}-143 b d\right )}{7 x^7}}{10 c}-\frac {\sqrt [3]{a+b x^3} (14 b c-13 a d)}{10 c x^{10}}}{13 c}-\frac {a \sqrt [3]{a+b x^3}}{13 c x^{13}}\) |
Input:
Int[(a + b*x^3)^(4/3)/(x^14*(c + d*x^3)),x]
Output:
-1/13*(a*(a + b*x^3)^(1/3))/(c*x^13) + (-1/10*((14*b*c - 13*a*d)*(a + b*x^ 3)^(1/3))/(c*x^10) + (-1/7*(((4*b^2*c)/a - 143*b*d + (130*a*d^2)/c)*(a + b *x^3)^(1/3))/x^7 - (2*(-1/4*((12*b^3*c^3 + 26*a*b^2*c^2*d - 520*a^2*b*c*d^ 2 + 455*a^3*d^3)*(a + b*x^3)^(1/3))/(a*c*x^4) - (-(((36*b^4*c^4 + 78*a*b^3 *c^3*d + 260*a^2*b^2*c^2*d^2 - 2275*a^3*b*c*d^3 + 1820*a^4*d^4)*(a + b*x^3 )^(1/3))/(a*c*x)) - (1820*a^2*d^3*(b*c - a*d)^2*(-(ArcTan[(1 + (2*(b*c - a *d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*c^(1/3)*(b*c - a*d)^(2/3))) + Log[c + d*x^3]/(6*c^(1/3)*(b*c - a*d)^(2/3)) - Log[((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)]/(2*c^(1/3)*(b*c - a*d)^(2/3))) )/c)/(4*a*c)))/(7*a*c))/(10*c))/(13*c)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_) )^(q_), x_Symbol] :> Simp[c*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^ (q - 1)/(a*e*(m + 1))), x] - Simp[1/(a*e^n*(m + 1)) Int[(e*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*(c*b - a*d)*(m + 1) + c*n*(b*c*(p + 1 ) + a*d*(q - 1)) + d*((c*b - a*d)*(m + 1) + c*b*n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q , 1] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[(x_)/(((a_) + (b_.)*(x_)^3)^(2/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[-ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3 ))/Sqrt[3]]/(Sqrt[3]*c*q^2), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c* q^2), x] + Simp[Log[c + d*x^3]/(6*c*q^2), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_ ))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b *x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^n*( m + 1)) Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2 ) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && LtQ[m, -1]
Time = 2.05 (sec) , antiderivative size = 336, normalized size of antiderivative = 0.86
| method | result | size |
| pseudoelliptic | \(-\frac {c \left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} \left (\left (b \,x^{3}+a \right )^{2} \left (\frac {9}{35} b^{2} x^{6}-\frac {3}{5} a b \,x^{3}+a^{2}\right ) c^{4}-\frac {13 a d \left (b \,x^{3}+a \right )^{2} \left (-\frac {3 b \,x^{3}}{7}+a \right ) x^{3} c^{3}}{10}+\frac {13 \left (b \,x^{3}+a \right )^{2} a^{2} c^{2} d^{2} x^{6}}{7}-\frac {13 a^{3} d^{3} x^{9} \left (5 b \,x^{3}+a \right ) c}{4}+13 a^{4} d^{4} x^{12}\right ) \left (b \,x^{3}+a \right )^{\frac {1}{3}}+\frac {13 a^{3} d^{3} x^{13} \left (a d -b c \right )^{2} \left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x -2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x}\right )+\ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )\right )}{6}}{13 \left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{13} c^{6} a^{3}}\) | \(336\) |
Input:
int((b*x^3+a)^(4/3)/x^14/(d*x^3+c),x,method=_RETURNVERBOSE)
Output:
-1/13/((a*d-b*c)/c)^(2/3)*(c*((a*d-b*c)/c)^(2/3)*((b*x^3+a)^2*(9/35*b^2*x^ 6-3/5*a*b*x^3+a^2)*c^4-13/10*a*d*(b*x^3+a)^2*(-3/7*b*x^3+a)*x^3*c^3+13/7*( b*x^3+a)^2*a^2*c^2*d^2*x^6-13/4*a^3*d^3*x^9*(5*b*x^3+a)*c+13*a^4*d^4*x^12) *(b*x^3+a)^(1/3)+13/6*a^3*d^3*x^13*(a*d-b*c)^2*(2*3^(1/2)*arctan(1/3*3^(1/ 2)*(((a*d-b*c)/c)^(1/3)*x-2*(b*x^3+a)^(1/3))/((a*d-b*c)/c)^(1/3)/x)+ln(((( a*d-b*c)/c)^(2/3)*x^2-((a*d-b*c)/c)^(1/3)*(b*x^3+a)^(1/3)*x+(b*x^3+a)^(2/3 ))/x^2)-2*ln((((a*d-b*c)/c)^(1/3)*x+(b*x^3+a)^(1/3))/x)))/x^13/c^6/a^3
Timed out. \[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^{14} \left (c+d x^3\right )} \, dx=\text {Timed out} \] Input:
integrate((b*x^3+a)^(4/3)/x^14/(d*x^3+c),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^{14} \left (c+d x^3\right )} \, dx=\text {Timed out} \] Input:
integrate((b*x**3+a)**(4/3)/x**14/(d*x**3+c),x)
Output:
Timed out
\[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^{14} \left (c+d x^3\right )} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}}}{{\left (d x^{3} + c\right )} x^{14}} \,d x } \] Input:
integrate((b*x^3+a)^(4/3)/x^14/(d*x^3+c),x, algorithm="maxima")
Output:
integrate((b*x^3 + a)^(4/3)/((d*x^3 + c)*x^14), x)
\[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^{14} \left (c+d x^3\right )} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}}}{{\left (d x^{3} + c\right )} x^{14}} \,d x } \] Input:
integrate((b*x^3+a)^(4/3)/x^14/(d*x^3+c),x, algorithm="giac")
Output:
integrate((b*x^3 + a)^(4/3)/((d*x^3 + c)*x^14), x)
Timed out. \[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^{14} \left (c+d x^3\right )} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{4/3}}{x^{14}\,\left (d\,x^3+c\right )} \,d x \] Input:
int((a + b*x^3)^(4/3)/(x^14*(c + d*x^3)),x)
Output:
int((a + b*x^3)^(4/3)/(x^14*(c + d*x^3)), x)
\[ \int \frac {\left (a+b x^3\right )^{4/3}}{x^{14} \left (c+d x^3\right )} \, dx=\frac {-140 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{4} c^{2}+182 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{4} c d \,x^{3}-260 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{4} d^{2} x^{6}-196 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{3} b \,c^{2} x^{3}+286 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{3} b c d \,x^{6}+390 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{3} b \,d^{2} x^{9}-8 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{2} b^{2} c^{2} x^{6}-429 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{2} b^{2} c d \,x^{9}-1170 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{2} b^{2} d^{2} x^{12}+12 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a \,b^{3} c^{2} x^{9}+1287 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a \,b^{3} c d \,x^{12}-36 \left (b \,x^{3}+a \right )^{\frac {1}{3}} b^{4} c^{2} x^{12}-1820 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{b d \,x^{11}+a d \,x^{8}+b c \,x^{8}+a c \,x^{5}}d x \right ) a^{5} d^{3} x^{13}+3640 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{b d \,x^{11}+a d \,x^{8}+b c \,x^{8}+a c \,x^{5}}d x \right ) a^{4} b c \,d^{2} x^{13}-1820 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{b d \,x^{11}+a d \,x^{8}+b c \,x^{8}+a c \,x^{5}}d x \right ) a^{3} b^{2} c^{2} d \,x^{13}}{1820 a^{3} c^{3} x^{13}} \] Input:
int((b*x^3+a)^(4/3)/x^14/(d*x^3+c),x)
Output:
( - 140*(a + b*x**3)**(1/3)*a**4*c**2 + 182*(a + b*x**3)**(1/3)*a**4*c*d*x **3 - 260*(a + b*x**3)**(1/3)*a**4*d**2*x**6 - 196*(a + b*x**3)**(1/3)*a** 3*b*c**2*x**3 + 286*(a + b*x**3)**(1/3)*a**3*b*c*d*x**6 + 390*(a + b*x**3) **(1/3)*a**3*b*d**2*x**9 - 8*(a + b*x**3)**(1/3)*a**2*b**2*c**2*x**6 - 429 *(a + b*x**3)**(1/3)*a**2*b**2*c*d*x**9 - 1170*(a + b*x**3)**(1/3)*a**2*b* *2*d**2*x**12 + 12*(a + b*x**3)**(1/3)*a*b**3*c**2*x**9 + 1287*(a + b*x**3 )**(1/3)*a*b**3*c*d*x**12 - 36*(a + b*x**3)**(1/3)*b**4*c**2*x**12 - 1820* int((a + b*x**3)**(1/3)/(a*c*x**5 + a*d*x**8 + b*c*x**8 + b*d*x**11),x)*a* *5*d**3*x**13 + 3640*int((a + b*x**3)**(1/3)/(a*c*x**5 + a*d*x**8 + b*c*x* *8 + b*d*x**11),x)*a**4*b*c*d**2*x**13 - 1820*int((a + b*x**3)**(1/3)/(a*c *x**5 + a*d*x**8 + b*c*x**8 + b*d*x**11),x)*a**3*b**2*c**2*d*x**13)/(1820* a**3*c**3*x**13)