Integrand size = 24, antiderivative size = 334 \[ \int \frac {x^4 \left (a+b x^3\right )^{4/3}}{c+d x^3} \, dx=-\frac {(6 b c-7 a d) x^2 \sqrt [3]{a+b x^3}}{18 d^2}+\frac {b x^5 \sqrt [3]{a+b x^3}}{6 d}-\frac {\left (9 b^2 c^2-12 a b c d+2 a^2 d^2\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} b^{2/3} d^3}+\frac {c^{2/3} (b c-a d)^{4/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} d^3}-\frac {c^{2/3} (b c-a d)^{4/3} \log \left (c+d x^3\right )}{6 d^3}-\frac {\left (9 b^2 c^2-12 a b c d+2 a^2 d^2\right ) \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{18 b^{2/3} d^3}+\frac {c^{2/3} (b c-a d)^{4/3} \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 d^3} \] Output:
-1/18*(-7*a*d+6*b*c)*x^2*(b*x^3+a)^(1/3)/d^2+1/6*b*x^5*(b*x^3+a)^(1/3)/d-1 /27*(2*a^2*d^2-12*a*b*c*d+9*b^2*c^2)*arctan(1/3*(1+2*b^(1/3)*x/(b*x^3+a)^( 1/3))*3^(1/2))*3^(1/2)/b^(2/3)/d^3+1/3*c^(2/3)*(-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)/d^3-1/6 *c^(2/3)*(-a*d+b*c)^(4/3)*ln(d*x^3+c)/d^3-1/18*(2*a^2*d^2-12*a*b*c*d+9*b^2 *c^2)*ln(b^(1/3)*x-(b*x^3+a)^(1/3))/b^(2/3)/d^3+1/2*c^(2/3)*(-a*d+b*c)^(4/ 3)*ln((-a*d+b*c)^(1/3)*x/c^(1/3)-(b*x^3+a)^(1/3))/d^3
Result contains complex when optimal does not.
Time = 6.81 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.57 \[ \int \frac {x^4 \left (a+b x^3\right )^{4/3}}{c+d x^3} \, dx=\frac {6 d x^2 \sqrt [3]{a+b x^3} \left (-6 b c+7 a d+3 b d x^3\right )-\frac {4 \sqrt {3} \left (9 b^2 c^2-12 a b c d+2 a^2 d^2\right ) \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )}{b^{2/3}}-18 \sqrt {-6-6 i \sqrt {3}} c^{2/3} (b c-a d)^{4/3} \arctan \left (\frac {3 \sqrt [3]{b c-a d} x}{\sqrt {3} \sqrt [3]{b c-a d} x-\left (3 i+\sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )-\frac {4 \left (9 b^2 c^2-12 a b c d+2 a^2 d^2\right ) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{b^{2/3}}+18 i \left (i+\sqrt {3}\right ) c^{2/3} (b c-a d)^{4/3} \log \left (2 \sqrt [3]{b c-a d} x+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )+\frac {2 \left (9 b^2 c^2-12 a b c d+2 a^2 d^2\right ) \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{b^{2/3}}+9 \left (1-i \sqrt {3}\right ) c^{2/3} (b c-a d)^{4/3} \log \left (2 (b c-a d)^{2/3} x^2+\left (-1-i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{b c-a d} x \sqrt [3]{a+b x^3}+i \left (i+\sqrt {3}\right ) c^{2/3} \left (a+b x^3\right )^{2/3}\right )}{108 d^3} \] Input:
Integrate[(x^4*(a + b*x^3)^(4/3))/(c + d*x^3),x]
Output:
(6*d*x^2*(a + b*x^3)^(1/3)*(-6*b*c + 7*a*d + 3*b*d*x^3) - (4*Sqrt[3]*(9*b^ 2*c^2 - 12*a*b*c*d + 2*a^2*d^2)*ArcTan[(Sqrt[3]*b^(1/3)*x)/(b^(1/3)*x + 2* (a + b*x^3)^(1/3))])/b^(2/3) - 18*Sqrt[-6 - (6*I)*Sqrt[3]]*c^(2/3)*(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))] - (4*(9*b^2*c^2 - 12*a*b*c*d + 2*a^2*d^2)*Log[-(b^(1/3)*x) + (a + b*x^3)^(1/3)])/b^(2/3) + (18*I)*(I + Sq rt[3])*c^(2/3)*(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)] + (2*(9*b^2*c^2 - 12*a*b*c*d + 2*a^2*d^2)*Lo g[b^(2/3)*x^2 + b^(1/3)*x*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)])/b^(2/3) + 9*(1 - I*Sqrt[3])*c^(2/3)*(b*c - a*d)^(4/3)*Log[2*(b*c - a*d)^(2/3)*x^2 + (-1 - I*Sqrt[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)])/(108*d^3)
Time = 1.05 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {977, 25, 1052, 27, 1054, 2009}
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.
| \(\displaystyle \int \frac {x^4 \left (a+b x^3\right )^{4/3}}{c+d x^3} \, dx\) |
| \(\Big \downarrow \) 977 |
| \(\displaystyle \frac {\int -\frac {x^4 \left (b (6 b c-7 a d) x^3+a (5 b c-6 a d)\right )}{\left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx}{6 d}+\frac {b x^5 \sqrt [3]{a+b x^3}}{6 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b x^5 \sqrt [3]{a+b x^3}}{6 d}-\frac {\int \frac {x^4 \left (b (6 b c-7 a d) x^3+a (5 b c-6 a d)\right )}{\left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx}{6 d}\) |
| \(\Big \downarrow \) 1052 |
| \(\displaystyle \frac {b x^5 \sqrt [3]{a+b x^3}}{6 d}-\frac {\frac {x^2 \sqrt [3]{a+b x^3} (6 b c-7 a d)}{3 d}-\frac {\int \frac {2 b x \left (\left (9 b^2 c^2-12 a b d c+2 a^2 d^2\right ) x^3+a c (6 b c-7 a d)\right )}{\left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx}{3 b d}}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b x^5 \sqrt [3]{a+b x^3}}{6 d}-\frac {\frac {x^2 \sqrt [3]{a+b x^3} (6 b c-7 a d)}{3 d}-\frac {2 \int \frac {x \left (\left (9 b^2 c^2-12 a b d c+2 a^2 d^2\right ) x^3+a c (6 b c-7 a d)\right )}{\left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx}{3 d}}{6 d}\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle \frac {b x^5 \sqrt [3]{a+b x^3}}{6 d}-\frac {\frac {x^2 \sqrt [3]{a+b x^3} (6 b c-7 a d)}{3 d}-\frac {2 \int \left (\frac {\left (9 b^2 c^2-12 a b d c+2 a^2 d^2\right ) x}{d \left (b x^3+a\right )^{2/3}}-\frac {9 \left (b^2 c^3-2 a b d c^2+a^2 d^2 c\right ) x}{d \left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}\right )dx}{3 d}}{6 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b x^5 \sqrt [3]{a+b x^3}}{6 d}-\frac {\frac {x^2 \sqrt [3]{a+b x^3} (6 b c-7 a d)}{3 d}-\frac {2 \left (-\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right ) \left (2 a^2 d^2-12 a b c d+9 b^2 c^2\right )}{\sqrt {3} b^{2/3} d}-\frac {\left (2 a^2 d^2-12 a b c d+9 b^2 c^2\right ) \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2 b^{2/3} d}+\frac {3 \sqrt {3} c^{2/3} (b c-a d)^{4/3} \arctan \left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{d}-\frac {3 c^{2/3} (b c-a d)^{4/3} \log \left (c+d x^3\right )}{2 d}+\frac {9 c^{2/3} (b c-a d)^{4/3} \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 d}\right )}{3 d}}{6 d}\) |
Input:
Int[(x^4*(a + b*x^3)^(4/3))/(c + d*x^3),x]
Output:
(b*x^5*(a + b*x^3)^(1/3))/(6*d) - (((6*b*c - 7*a*d)*x^2*(a + b*x^3)^(1/3)) /(3*d) - (2*(-(((9*b^2*c^2 - 12*a*b*c*d + 2*a^2*d^2)*ArcTan[(1 + (2*b^(1/3 )*x)/(a + b*x^3)^(1/3))/Sqrt[3]])/(Sqrt[3]*b^(2/3)*d)) + (3*Sqrt[3]*c^(2/3 )*(b*c - a*d)^(4/3)*ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^ 3)^(1/3)))/Sqrt[3]])/d - (3*c^(2/3)*(b*c - a*d)^(4/3)*Log[c + d*x^3])/(2*d ) - ((9*b^2*c^2 - 12*a*b*c*d + 2*a^2*d^2)*Log[b^(1/3)*x - (a + b*x^3)^(1/3 )])/(2*b^(2/3)*d) + (9*c^(2/3)*(b*c - a*d)^(4/3)*Log[((b*c - a*d)^(1/3)*x) /c^(1/3) - (a + b*x^3)^(1/3)])/(2*d)))/(3*d))/(6*d)
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[d*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n) ^(q - 1)/(b*e*(m + n*(p + q) + 1))), x] + Simp[1/(b*(m + n*(p + q) + 1)) Int[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*((c*b - a*d)*(m + 1) + c*b*n*(p + q)) + (d*(c*b - a*d)*(m + 1) + d*n*(q - 1)*(b*c - a*d) + c*b*d* n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[f*g^(n - 1)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q + 1) + 1))), x] - Simp[g^n/(b*d*(m + n*(p + q + 1) + 1)) Int[(g*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m - n + 1) + (a*f*d*(m + n*q + 1) + b*( f*c*(m + n*p + 1) - e*d*(m + n*(p + q + 1) + 1)))*x^n, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && GtQ[m, n - 1]
Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n _)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0]
Time = 2.25 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.37
| method | result | size |
| pseudoelliptic | \(\frac {\left (-\frac {b^{\frac {2}{3}} a^{2} d^{2}}{2}+b^{\frac {5}{3}} a c d -\frac {b^{\frac {8}{3}} c^{2}}{2}\right ) \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )+\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} \left (a^{2} d^{2}-6 a b c d +\frac {9}{2} b^{2} c^{2}\right ) \ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{9}-\sqrt {3}\, \left (-2 b^{\frac {5}{3}} a c d +b^{\frac {8}{3}} c^{2}+b^{\frac {2}{3}} a^{2} d^{2}\right ) \arctan \left (\frac {\sqrt {3}\, \left (-\frac {2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}}+x \right )}{3 x}\right )+\left (-2 b^{\frac {5}{3}} a c d +b^{\frac {8}{3}} c^{2}+b^{\frac {2}{3}} a^{2} d^{2}\right ) \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )-\frac {2 \left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} \left (-\sqrt {3}\, \left (a^{2} d^{2}-6 a b c d +\frac {9}{2} b^{2} c^{2}\right ) \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}+x \right )}{3 x}\right )+\left (a^{2} d^{2}-6 a b c d +\frac {9}{2} b^{2} c^{2}\right ) \ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )-\frac {21 \left (\frac {3 \left (d \,x^{3}-2 c \right ) b^{\frac {5}{3}}}{7}+a d \,b^{\frac {2}{3}}\right ) d \left (b \,x^{3}+a \right )^{\frac {1}{3}} x^{2}}{4}\right )}{9}}{3 \left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} b^{\frac {2}{3}} d^{3}}\) | \(459\) |
Input:
int(x^4*(b*x^3+a)^(4/3)/(d*x^3+c),x,method=_RETURNVERBOSE)
Output:
1/3/((a*d-b*c)/c)^(2/3)*((-1/2*b^(2/3)*a^2*d^2+b^(5/3)*a*c*d-1/2*b^(8/3)*c ^2)*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)+1/9*((a*d-b*c)/c)^(2/3)*(a^2*d^2-6*a*b*c*d+9/2*b^2*c^2)* ln((b^(2/3)*x^2+b^(1/3)*(b*x^3+a)^(1/3)*x+(b*x^3+a)^(2/3))/x^2)-3^(1/2)*(- 2*b^(5/3)*a*c*d+b^(8/3)*c^2+b^(2/3)*a^2*d^2)*arctan(1/3*3^(1/2)*(-2/((a*d- b*c)/c)^(1/3)*(b*x^3+a)^(1/3)+x)/x)+(-2*b^(5/3)*a*c*d+b^(8/3)*c^2+b^(2/3)* a^2*d^2)*ln((((a*d-b*c)/c)^(1/3)*x+(b*x^3+a)^(1/3))/x)-2/9*((a*d-b*c)/c)^( 2/3)*(-3^(1/2)*(a^2*d^2-6*a*b*c*d+9/2*b^2*c^2)*arctan(1/3*3^(1/2)*(2*(b*x^ 3+a)^(1/3)/b^(1/3)+x)/x)+(a^2*d^2-6*a*b*c*d+9/2*b^2*c^2)*ln((-b^(1/3)*x+(b *x^3+a)^(1/3))/x)-21/4*(3/7*(d*x^3-2*c)*b^(5/3)+a*d*b^(2/3))*d*(b*x^3+a)^( 1/3)*x^2))/b^(2/3)/d^3
Time = 1.25 (sec) , antiderivative size = 547, normalized size of antiderivative = 1.64 \[ \int \frac {x^4 \left (a+b x^3\right )^{4/3}}{c+d x^3} \, dx=\frac {6 \, \sqrt {\frac {1}{3}} {\left (9 \, b^{3} c^{2} - 12 \, a b^{2} c d + 2 \, a^{2} b d^{2}\right )} \sqrt {-\left (-b^{2}\right )^{\frac {1}{3}}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (\left (-b^{2}\right )^{\frac {1}{3}} b x - 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b^{2}\right )^{\frac {2}{3}}\right )} \sqrt {-\left (-b^{2}\right )^{\frac {1}{3}}}}{b^{2} x}\right ) - 18 \, \sqrt {3} {\left (b^{3} c - a b^{2} d\right )} {\left (-b c^{3} + a c^{2} d\right )}^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} {\left (b c^{2} - a c d\right )} x + 2 \, \sqrt {3} {\left (-b c^{3} + a c^{2} d\right )}^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{3 \, {\left (b c^{2} - a c d\right )} x}\right ) - 2 \, {\left (9 \, b^{2} c^{2} - 12 \, a b c d + 2 \, a^{2} d^{2}\right )} \left (-b^{2}\right )^{\frac {2}{3}} \log \left (-\frac {\left (-b^{2}\right )^{\frac {2}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}} b}{x}\right ) + {\left (9 \, b^{2} c^{2} - 12 \, a b c d + 2 \, a^{2} d^{2}\right )} \left (-b^{2}\right )^{\frac {2}{3}} \log \left (-\frac {\left (-b^{2}\right )^{\frac {1}{3}} b x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b^{2}\right )^{\frac {2}{3}} x - {\left (b x^{3} + a\right )}^{\frac {2}{3}} b}{x^{2}}\right ) - 18 \, {\left (b^{3} c - a b^{2} d\right )} {\left (-b c^{3} + a c^{2} d\right )}^{\frac {1}{3}} \log \left (\frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} c + {\left (-b c^{3} + a c^{2} d\right )}^{\frac {1}{3}} x}{x}\right ) + 9 \, {\left (b^{3} c - a b^{2} d\right )} {\left (-b c^{3} + a c^{2} d\right )}^{\frac {1}{3}} \log \left (\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} c^{2} - {\left (-b c^{3} + a c^{2} d\right )}^{\frac {1}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} c x + {\left (-b c^{3} + a c^{2} d\right )}^{\frac {2}{3}} x^{2}}{x^{2}}\right ) + 3 \, {\left (3 \, b^{3} d^{2} x^{5} - {\left (6 \, b^{3} c d - 7 \, a b^{2} d^{2}\right )} x^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{54 \, b^{2} d^{3}} \] Input:
integrate(x^4*(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="fricas")
Output:
1/54*(6*sqrt(1/3)*(9*b^3*c^2 - 12*a*b^2*c*d + 2*a^2*b*d^2)*sqrt(-(-b^2)^(1 /3))*arctan(-sqrt(1/3)*((-b^2)^(1/3)*b*x - 2*(b*x^3 + a)^(1/3)*(-b^2)^(2/3 ))*sqrt(-(-b^2)^(1/3))/(b^2*x)) - 18*sqrt(3)*(b^3*c - a*b^2*d)*(-b*c^3 + a *c^2*d)^(1/3)*arctan(-1/3*(sqrt(3)*(b*c^2 - a*c*d)*x + 2*sqrt(3)*(-b*c^3 + a*c^2*d)^(2/3)*(b*x^3 + a)^(1/3))/((b*c^2 - a*c*d)*x)) - 2*(9*b^2*c^2 - 1 2*a*b*c*d + 2*a^2*d^2)*(-b^2)^(2/3)*log(-((-b^2)^(2/3)*x - (b*x^3 + a)^(1/ 3)*b)/x) + (9*b^2*c^2 - 12*a*b*c*d + 2*a^2*d^2)*(-b^2)^(2/3)*log(-((-b^2)^ (1/3)*b*x^2 - (b*x^3 + a)^(1/3)*(-b^2)^(2/3)*x - (b*x^3 + a)^(2/3)*b)/x^2) - 18*(b^3*c - a*b^2*d)*(-b*c^3 + a*c^2*d)^(1/3)*log(((b*x^3 + a)^(1/3)*c + (-b*c^3 + a*c^2*d)^(1/3)*x)/x) + 9*(b^3*c - a*b^2*d)*(-b*c^3 + a*c^2*d)^ (1/3)*log(((b*x^3 + a)^(2/3)*c^2 - (-b*c^3 + a*c^2*d)^(1/3)*(b*x^3 + a)^(1 /3)*c*x + (-b*c^3 + a*c^2*d)^(2/3)*x^2)/x^2) + 3*(3*b^3*d^2*x^5 - (6*b^3*c *d - 7*a*b^2*d^2)*x^2)*(b*x^3 + a)^(1/3))/(b^2*d^3)
\[ \int \frac {x^4 \left (a+b x^3\right )^{4/3}}{c+d x^3} \, dx=\int \frac {x^{4} \left (a + b x^{3}\right )^{\frac {4}{3}}}{c + d x^{3}}\, dx \] Input:
integrate(x**4*(b*x**3+a)**(4/3)/(d*x**3+c),x)
Output:
Integral(x**4*(a + b*x**3)**(4/3)/(c + d*x**3), x)
\[ \int \frac {x^4 \left (a+b x^3\right )^{4/3}}{c+d x^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}} x^{4}}{d x^{3} + c} \,d x } \] Input:
integrate(x^4*(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="maxima")
Output:
integrate((b*x^3 + a)^(4/3)*x^4/(d*x^3 + c), x)
\[ \int \frac {x^4 \left (a+b x^3\right )^{4/3}}{c+d x^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}} x^{4}}{d x^{3} + c} \,d x } \] Input:
integrate(x^4*(b*x^3+a)^(4/3)/(d*x^3+c),x, algorithm="giac")
Output:
integrate((b*x^3 + a)^(4/3)*x^4/(d*x^3 + c), x)
Timed out. \[ \int \frac {x^4 \left (a+b x^3\right )^{4/3}}{c+d x^3} \, dx=\int \frac {x^4\,{\left (b\,x^3+a\right )}^{4/3}}{d\,x^3+c} \,d x \] Input:
int((x^4*(a + b*x^3)^(4/3))/(c + d*x^3),x)
Output:
int((x^4*(a + b*x^3)^(4/3))/(c + d*x^3), x)
\[ \int \frac {x^4 \left (a+b x^3\right )^{4/3}}{c+d x^3} \, dx=\frac {7 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a d \,x^{2}-6 \left (b \,x^{3}+a \right )^{\frac {1}{3}} b c \,x^{2}+3 \left (b \,x^{3}+a \right )^{\frac {1}{3}} b d \,x^{5}+4 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}} x^{4}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a^{2} d^{2}-24 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}} x^{4}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a b c d +18 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}} x^{4}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) b^{2} c^{2}-14 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}} x}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a^{2} c d +12 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}} x}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a b \,c^{2}}{18 d^{2}} \] Input:
int(x^4*(b*x^3+a)^(4/3)/(d*x^3+c),x)
Output:
(7*(a + b*x**3)**(1/3)*a*d*x**2 - 6*(a + b*x**3)**(1/3)*b*c*x**2 + 3*(a + b*x**3)**(1/3)*b*d*x**5 + 4*int(((a + b*x**3)**(1/3)*x**4)/(a*c + a*d*x**3 + b*c*x**3 + b*d*x**6),x)*a**2*d**2 - 24*int(((a + b*x**3)**(1/3)*x**4)/( a*c + a*d*x**3 + b*c*x**3 + b*d*x**6),x)*a*b*c*d + 18*int(((a + b*x**3)**( 1/3)*x**4)/(a*c + a*d*x**3 + b*c*x**3 + b*d*x**6),x)*b**2*c**2 - 14*int((( a + b*x**3)**(1/3)*x)/(a*c + a*d*x**3 + b*c*x**3 + b*d*x**6),x)*a**2*c*d + 12*int(((a + b*x**3)**(1/3)*x)/(a*c + a*d*x**3 + b*c*x**3 + b*d*x**6),x)* a*b*c**2)/(18*d**2)