Integrand size = 21, antiderivative size = 331 \[ \int \frac {\left (a+b x^3\right )^{8/3}}{c+d x^3} \, dx=-\frac {b (6 b c-11 a d) x \left (a+b x^3\right )^{2/3}}{18 d^2}+\frac {b x \left (a+b x^3\right )^{5/3}}{6 d}+\frac {b^{2/3} \left (9 b^2 c^2-24 a b c d+20 a^2 d^2\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} d^3}-\frac {(b c-a d)^{8/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} c^{2/3} d^3}-\frac {(b c-a d)^{8/3} \log \left (c+d x^3\right )}{6 c^{2/3} d^3}+\frac {(b c-a d)^{8/3} \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{2/3} d^3}-\frac {b^{2/3} \left (9 b^2 c^2-24 a b c d+20 a^2 d^2\right ) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{18 d^3} \] Output:
-1/18*b*(-11*a*d+6*b*c)*x*(b*x^3+a)^(2/3)/d^2+1/6*b*x*(b*x^3+a)^(5/3)/d+1/ 27*b^(2/3)*(20*a^2*d^2-24*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)/d^3-1/3*(-a*d+b*c)^(8/3)*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^(2/3)/d^3-1/6 *(-a*d+b*c)^(8/3)*ln(d*x^3+c)/c^(2/3)/d^3+1/2*(-a*d+b*c)^(8/3)*ln((-a*d+b* c)^(1/3)*x/c^(1/3)-(b*x^3+a)^(1/3))/c^(2/3)/d^3-1/18*b^(2/3)*(20*a^2*d^2-2 4*a*b*c*d+9*b^2*c^2)*ln(-b^(1/3)*x+(b*x^3+a)^(1/3))/d^3
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 10.86 (sec) , antiderivative size = 655, normalized size of antiderivative = 1.98 \[ \int \frac {\left (a+b x^3\right )^{8/3}}{c+d x^3} \, dx=\frac {3 b \sqrt [3]{b c-a d} \left (9 b^2 c^2-24 a b c d+20 a^2 d^2\right ) x^4 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 \sqrt [3]{c} \left (-18 a b^2 c^{5/3} \sqrt [3]{b c-a d} x+42 a^2 b c^{2/3} d \sqrt [3]{b c-a d} x-18 b^3 c^{5/3} \sqrt [3]{b c-a d} x^4+51 a b^2 c^{2/3} d \sqrt [3]{b c-a d} x^4+9 b^3 c^{2/3} d \sqrt [3]{b c-a d} x^7+2 \sqrt {3} a \left (3 b^2 c^2-7 a b c d+9 a^2 d^2\right ) \sqrt [3]{a+b x^3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{b+a x^3}}}{\sqrt {3}}\right )-2 a \left (3 b^2 c^2-7 a b c d+9 a^2 d^2\right ) \sqrt [3]{a+b x^3} \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )+3 a b^2 c^2 \sqrt [3]{a+b x^3} \log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )-7 a^2 b c d \sqrt [3]{a+b x^3} \log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )+9 a^3 d^2 \sqrt [3]{a+b x^3} \log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )\right )}{108 c d^2 \sqrt [3]{b c-a d} \sqrt [3]{a+b x^3}} \] Input:
Integrate[(a + b*x^3)^(8/3)/(c + d*x^3),x]
Output:
(3*b*(b*c - a*d)^(1/3)*(9*b^2*c^2 - 24*a*b*c*d + 20*a^2*d^2)*x^4*(1 + (b*x ^3)/a)^(1/3)*AppellF1[4/3, 1/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + 2*c^ (1/3)*(-18*a*b^2*c^(5/3)*(b*c - a*d)^(1/3)*x + 42*a^2*b*c^(2/3)*d*(b*c - a *d)^(1/3)*x - 18*b^3*c^(5/3)*(b*c - a*d)^(1/3)*x^4 + 51*a*b^2*c^(2/3)*d*(b *c - a*d)^(1/3)*x^4 + 9*b^3*c^(2/3)*d*(b*c - a*d)^(1/3)*x^7 + 2*Sqrt[3]*a* (3*b^2*c^2 - 7*a*b*c*d + 9*a^2*d^2)*(a + b*x^3)^(1/3)*ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(b + a*x^3)^(1/3)))/Sqrt[3]] - 2*a*(3*b^2*c^2 - 7 *a*b*c*d + 9*a^2*d^2)*(a + b*x^3)^(1/3)*Log[c^(1/3) - ((b*c - a*d)^(1/3)*x )/(b + a*x^3)^(1/3)] + 3*a*b^2*c^2*(a + b*x^3)^(1/3)*Log[c^(2/3) + ((b*c - a*d)^(2/3)*x^2)/(b + a*x^3)^(2/3) + (c^(1/3)*(b*c - a*d)^(1/3)*x)/(b + a* x^3)^(1/3)] - 7*a^2*b*c*d*(a + b*x^3)^(1/3)*Log[c^(2/3) + ((b*c - a*d)^(2/ 3)*x^2)/(b + a*x^3)^(2/3) + (c^(1/3)*(b*c - a*d)^(1/3)*x)/(b + a*x^3)^(1/3 )] + 9*a^3*d^2*(a + b*x^3)^(1/3)*Log[c^(2/3) + ((b*c - a*d)^(2/3)*x^2)/(b + a*x^3)^(2/3) + (c^(1/3)*(b*c - a*d)^(1/3)*x)/(b + a*x^3)^(1/3)]))/(108*c *d^2*(b*c - a*d)^(1/3)*(a + b*x^3)^(1/3))
Time = 0.83 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {933, 25, 1025, 27, 1026, 769, 901}
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.
| \(\displaystyle \int \frac {\left (a+b x^3\right )^{8/3}}{c+d x^3} \, dx\) |
| \(\Big \downarrow \) 933 |
| \(\displaystyle \frac {\int -\frac {\left (b x^3+a\right )^{2/3} \left (b (6 b c-11 a d) x^3+a (b c-6 a d)\right )}{d x^3+c}dx}{6 d}+\frac {b x \left (a+b x^3\right )^{5/3}}{6 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b x \left (a+b x^3\right )^{5/3}}{6 d}-\frac {\int \frac {\left (b x^3+a\right )^{2/3} \left (b (6 b c-11 a d) x^3+a (b c-6 a d)\right )}{d x^3+c}dx}{6 d}\) |
| \(\Big \downarrow \) 1025 |
| \(\displaystyle \frac {b x \left (a+b x^3\right )^{5/3}}{6 d}-\frac {\frac {\int -\frac {2 \left (b \left (9 b^2 c^2-24 a b d c+20 a^2 d^2\right ) x^3+a \left (3 b^2 c^2-7 a b d c+9 a^2 d^2\right )\right )}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{3 d}+\frac {b x \left (a+b x^3\right )^{2/3} (6 b c-11 a d)}{3 d}}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b x \left (a+b x^3\right )^{5/3}}{6 d}-\frac {\frac {b x \left (a+b x^3\right )^{2/3} (6 b c-11 a d)}{3 d}-\frac {2 \int \frac {b \left (9 b^2 c^2-24 a b d c+20 a^2 d^2\right ) x^3+a \left (3 b^2 c^2-7 a b d c+9 a^2 d^2\right )}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{3 d}}{6 d}\) |
| \(\Big \downarrow \) 1026 |
| \(\displaystyle \frac {b x \left (a+b x^3\right )^{5/3}}{6 d}-\frac {\frac {b x \left (a+b x^3\right )^{2/3} (6 b c-11 a d)}{3 d}-\frac {2 \left (\frac {b \left (20 a^2 d^2-24 a b c d+9 b^2 c^2\right ) \int \frac {1}{\sqrt [3]{b x^3+a}}dx}{d}-\frac {9 (b c-a d)^3 \int \frac {1}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{d}\right )}{3 d}}{6 d}\) |
| \(\Big \downarrow \) 769 |
| \(\displaystyle \frac {b x \left (a+b x^3\right )^{5/3}}{6 d}-\frac {\frac {b x \left (a+b x^3\right )^{2/3} (6 b c-11 a d)}{3 d}-\frac {2 \left (\frac {b \left (20 a^2 d^2-24 a b c d+9 b^2 c^2\right ) \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}\right )}{d}-\frac {9 (b c-a d)^3 \int \frac {1}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{d}\right )}{3 d}}{6 d}\) |
| \(\Big \downarrow \) 901 |
| \(\displaystyle \frac {b x \left (a+b x^3\right )^{5/3}}{6 d}-\frac {\frac {b x \left (a+b x^3\right )^{2/3} (6 b c-11 a d)}{3 d}-\frac {2 \left (\frac {b \left (20 a^2 d^2-24 a b c d+9 b^2 c^2\right ) \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}\right )}{d}-\frac {9 (b c-a d)^3 \left (\frac {\arctan \left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} c^{2/3} \sqrt [3]{b c-a d}}+\frac {\log \left (c+d x^3\right )}{6 c^{2/3} \sqrt [3]{b c-a d}}-\frac {\log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{2/3} \sqrt [3]{b c-a d}}\right )}{d}\right )}{3 d}}{6 d}\) |
Input:
Int[(a + b*x^3)^(8/3)/(c + d*x^3),x]
Output:
(b*x*(a + b*x^3)^(5/3))/(6*d) - ((b*(6*b*c - 11*a*d)*x*(a + b*x^3)^(2/3))/ (3*d) - (2*((-9*(b*c - a*d)^3*(ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3 )*(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*c^(2/3)*(b*c - a*d)^(1/3)) + Log[c + d*x^3]/(6*c^(2/3)*(b*c - a*d)^(1/3)) - Log[((b*c - a*d)^(1/3)*x)/c^(1/3 ) - (a + b*x^3)^(1/3)]/(2*c^(2/3)*(b*c - a*d)^(1/3))))/d + (b*(9*b^2*c^2 - 24*a*b*c*d + 20*a^2*d^2)*(ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sq rt[3]]/(Sqrt[3]*b^(1/3)) - Log[-(b^(1/3)*x) + (a + b*x^3)^(1/3)]/(2*b^(1/3 ))))/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[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]* (x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^ 3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]
Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> Wit h[{q = Rt[(b*c - a*d)/c, 3]}, Simp[ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/S qrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), x] + Simp[1/(b*(n*(p + q) + 1)) Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Sim p[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d , 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[ a, b, c, d, n, p, q, x]
Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + ( f_.)*(x_)^(n_)), x_Symbol] :> Simp[f*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/( b*(n*(p + q + 1) + 1))), x] + Simp[1/(b*(n*(p + q + 1) + 1)) Int[(a + b*x ^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e - a*f) + f*n*q*(b*c - a*d) + b*d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[ {a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1, 0]
Int[(((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)* (x_)^(n_)), x_Symbol] :> Simp[f/d Int[(a + b*x^n)^p, x], x] + Simp[(d*e - c*f)/d Int[(a + b*x^n)^p/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, p, n}, x]
Time = 3.01 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.02
| method | result | size |
| pseudoelliptic | \(-\frac {2 c \left (-\frac {7 \left (\frac {3}{14} b d \,x^{3}+a d -\frac {3}{7} b c \right ) d b x \left (b \,x^{3}+a \right )^{\frac {2}{3}}}{3}+\left (\frac {20 b^{\frac {2}{3}} a^{2} d^{2}}{9}+c \,b^{\frac {5}{3}} \left (b c -\frac {8 a d}{3}\right )\right ) \left (\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right )+\ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}\right )\right ) \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}-\left (a d -b c \right )^{3} \left (2 \arctan \left (\frac {\sqrt {3}\, \left (-\frac {2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}}+x \right )}{3 x}\right ) \sqrt {3}+2 \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{6 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} d^{3} c}\) | \(338\) |
Input:
int((b*x^3+a)^(8/3)/(d*x^3+c),x,method=_RETURNVERBOSE)
Output:
-1/6/((a*d-b*c)/c)^(1/3)*(2*c*(-7/3*(3/14*b*d*x^3+a*d-3/7*b*c)*d*b*x*(b*x^ 3+a)^(2/3)+(20/9*b^(2/3)*a^2*d^2+c*b^(5/3)*(b*c-8/3*a*d))*(3^(1/2)*arctan( 1/3*3^(1/2)*(b^(1/3)*x+2*(b*x^3+a)^(1/3))/b^(1/3)/x)+ln((-b^(1/3)*x+(b*x^3 +a)^(1/3))/x)-1/2*ln((b^(2/3)*x^2+b^(1/3)*(b*x^3+a)^(1/3)*x+(b*x^3+a)^(2/3 ))/x^2)))*((a*d-b*c)/c)^(1/3)-(a*d-b*c)^3*(2*arctan(1/3*3^(1/2)*(-2/((a*d- b*c)/c)^(1/3)*(b*x^3+a)^(1/3)+x)/x)*3^(1/2)+2*ln((((a*d-b*c)/c)^(1/3)*x+(b *x^3+a)^(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)))/d^3/c
Leaf count of result is larger than twice the leaf count of optimal. 643 vs. \(2 (273) = 546\).
Time = 5.59 (sec) , antiderivative size = 643, normalized size of antiderivative = 1.94 \[ \int \frac {\left (a+b x^3\right )^{8/3}}{c+d x^3} \, dx =\text {Too large to display} \] Input:
integrate((b*x^3+a)^(8/3)/(d*x^3+c),x, algorithm="fricas")
Output:
-1/54*(18*sqrt(3)*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)/c^2)^(1/3)*arctan(-1/3*(sqrt(3)*(b*c - a*d)*x + 2*sqrt(3)*(b*x^3 + a)^(1/3)*c*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)/c^2)^(1/3))/((b*c - a*d)*x)) + 2*sqrt(3)*(9*b^2*c^2 - 24*a*b*c*d + 20*a^2*d^2)*(-b^2)^(1/3)*arctan(-1/ 3*(sqrt(3)*b*x - 2*sqrt(3)*(b*x^3 + a)^(1/3)*(-b^2)^(1/3))/(b*x)) - 18*(b^ 2*c^2 - 2*a*b*c*d + a^2*d^2)*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)/c^2)^(1/3)*l og((c*x*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)/c^2)^(2/3) - (b*x^3 + a)^(1/3)*(b *c - a*d))/x) - 2*(9*b^2*c^2 - 24*a*b*c*d + 20*a^2*d^2)*(-b^2)^(1/3)*log(- ((-b^2)^(2/3)*x - (b*x^3 + a)^(1/3)*b)/x) + (9*b^2*c^2 - 24*a*b*c*d + 20*a ^2*d^2)*(-b^2)^(1/3)*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) + 9*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*((b ^2*c^2 - 2*a*b*c*d + a^2*d^2)/c^2)^(1/3)*log(-((b*c - a*d)*x^2*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)/c^2)^(1/3) + (b*x^3 + a)^(1/3)*c*x*((b^2*c^2 - 2*a*b *c*d + a^2*d^2)/c^2)^(2/3) + (b*x^3 + a)^(2/3)*(b*c - a*d))/x^2) - 3*(3*b^ 2*d^2*x^4 - 2*(3*b^2*c*d - 7*a*b*d^2)*x)*(b*x^3 + a)^(2/3))/d^3
\[ \int \frac {\left (a+b x^3\right )^{8/3}}{c+d x^3} \, dx=\int \frac {\left (a + b x^{3}\right )^{\frac {8}{3}}}{c + d x^{3}}\, dx \] Input:
integrate((b*x**3+a)**(8/3)/(d*x**3+c),x)
Output:
Integral((a + b*x**3)**(8/3)/(c + d*x**3), x)
\[ \int \frac {\left (a+b x^3\right )^{8/3}}{c+d x^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {8}{3}}}{d x^{3} + c} \,d x } \] Input:
integrate((b*x^3+a)^(8/3)/(d*x^3+c),x, algorithm="maxima")
Output:
integrate((b*x^3 + a)^(8/3)/(d*x^3 + c), x)
\[ \int \frac {\left (a+b x^3\right )^{8/3}}{c+d x^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {8}{3}}}{d x^{3} + c} \,d x } \] Input:
integrate((b*x^3+a)^(8/3)/(d*x^3+c),x, algorithm="giac")
Output:
integrate((b*x^3 + a)^(8/3)/(d*x^3 + c), x)
Timed out. \[ \int \frac {\left (a+b x^3\right )^{8/3}}{c+d x^3} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{8/3}}{d\,x^3+c} \,d x \] Input:
int((a + b*x^3)^(8/3)/(c + d*x^3),x)
Output:
int((a + b*x^3)^(8/3)/(c + d*x^3), x)
\[ \int \frac {\left (a+b x^3\right )^{8/3}}{c+d x^3} \, dx=\frac {14 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a b d x -6 \left (b \,x^{3}+a \right )^{\frac {2}{3}} b^{2} c x +3 \left (b \,x^{3}+a \right )^{\frac {2}{3}} b^{2} d \,x^{4}+18 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a^{3} d^{2}-14 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a^{2} b c d +6 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a \,b^{2} c^{2}+40 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} x^{3}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a^{2} b \,d^{2}-48 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} x^{3}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a \,b^{2} c d +18 \left (\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} x^{3}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) b^{3} c^{2}}{18 d^{2}} \] Input:
int((b*x^3+a)^(8/3)/(d*x^3+c),x)
Output:
(14*(a + b*x**3)**(2/3)*a*b*d*x - 6*(a + b*x**3)**(2/3)*b**2*c*x + 3*(a + b*x**3)**(2/3)*b**2*d*x**4 + 18*int((a + b*x**3)**(2/3)/(a*c + a*d*x**3 + b*c*x**3 + b*d*x**6),x)*a**3*d**2 - 14*int((a + b*x**3)**(2/3)/(a*c + a*d* x**3 + b*c*x**3 + b*d*x**6),x)*a**2*b*c*d + 6*int((a + b*x**3)**(2/3)/(a*c + a*d*x**3 + b*c*x**3 + b*d*x**6),x)*a*b**2*c**2 + 40*int(((a + b*x**3)** (2/3)*x**3)/(a*c + a*d*x**3 + b*c*x**3 + b*d*x**6),x)*a**2*b*d**2 - 48*int (((a + b*x**3)**(2/3)*x**3)/(a*c + a*d*x**3 + b*c*x**3 + b*d*x**6),x)*a*b* *2*c*d + 18*int(((a + b*x**3)**(2/3)*x**3)/(a*c + a*d*x**3 + b*c*x**3 + b* d*x**6),x)*b**3*c**2)/(18*d**2)