Integrand size = 21, antiderivative size = 280 \[ \int \frac {1}{\left (a+b x^3\right )^{10/3} \left (c+d x^3\right )} \, dx=\frac {b x}{7 a (b c-a d) \left (a+b x^3\right )^{7/3}}+\frac {b (6 b c-13 a d) x}{28 a^2 (b c-a d)^2 \left (a+b x^3\right )^{4/3}}+\frac {b \left (18 b^2 c^2-57 a b c d+67 a^2 d^2\right ) x}{28 a^3 (b c-a d)^3 \sqrt [3]{a+b x^3}}-\frac {d^3 \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} c^{2/3} (b c-a d)^{10/3}}-\frac {d^3 \log \left (c+d x^3\right )}{6 c^{2/3} (b c-a d)^{10/3}}+\frac {d^3 \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{2/3} (b c-a d)^{10/3}} \]
1/7*b*x/a/(-a*d+b*c)/(b*x^3+a)^(7/3)+1/28*b*(-13*a*d+6*b*c)*x/a^2/(-a*d+b* c)^2/(b*x^3+a)^(4/3)+1/28*b*(67*a^2*d^2-57*a*b*c*d+18*b^2*c^2)*x/a^3/(-a*d +b*c)^3/(b*x^3+a)^(1/3)-1/6*d^3*ln(d*x^3+c)/c^(2/3)/(-a*d+b*c)^(10/3)+1/2* d^3*ln((-a*d+b*c)^(1/3)*x/c^(1/3)-(b*x^3+a)^(1/3))/c^(2/3)/(-a*d+b*c)^(10/ 3)-1/3*d^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))/c^(2/3)/(-a*d+b*c)^(10/3)*3^(1/2)
Result contains complex when optimal does not.
Time = 6.43 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.50 \[ \int \frac {1}{\left (a+b x^3\right )^{10/3} \left (c+d x^3\right )} \, dx=\frac {1}{84} \left (-\frac {3 b x \left (84 a^4 d^2+18 b^4 c^2 x^6+3 a b^3 c x^3 \left (14 c-19 d x^3\right )+21 a^3 b d \left (-4 c+7 d x^3\right )+a^2 b^2 \left (28 c^2-133 c d x^3+67 d^2 x^6\right )\right )}{a^3 (-b c+a d)^3 \left (a+b x^3\right )^{7/3}}+\frac {14 \sqrt {-6+6 i \sqrt {3}} d^3 \arctan \left (\frac {3 \sqrt [3]{b c-a d} x}{\sqrt {3} \sqrt [3]{b c-a d} x-\left (3 i+\sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )}{c^{2/3} (b c-a d)^{10/3}}-\frac {14 i \left (-i+\sqrt {3}\right ) d^3 \log \left (2 \sqrt [3]{b c-a d} x+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )}{c^{2/3} (b c-a d)^{10/3}}+\frac {7 \left (1+i \sqrt {3}\right ) d^3 \log \left (2 (b c-a d)^{2/3} x^2+\left (-1-i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{b c-a d} x \sqrt [3]{a+b x^3}+i \left (i+\sqrt {3}\right ) c^{2/3} \left (a+b x^3\right )^{2/3}\right )}{c^{2/3} (b c-a d)^{10/3}}\right ) \]
((-3*b*x*(84*a^4*d^2 + 18*b^4*c^2*x^6 + 3*a*b^3*c*x^3*(14*c - 19*d*x^3) + 21*a^3*b*d*(-4*c + 7*d*x^3) + a^2*b^2*(28*c^2 - 133*c*d*x^3 + 67*d^2*x^6)) )/(a^3*(-(b*c) + a*d)^3*(a + b*x^3)^(7/3)) + (14*Sqrt[-6 + (6*I)*Sqrt[3]]* d^3*ArcTan[(3*(b*c - a*d)^(1/3)*x)/(Sqrt[3]*(b*c - a*d)^(1/3)*x - (3*I + S qrt[3])*c^(1/3)*(a + b*x^3)^(1/3))])/(c^(2/3)*(b*c - a*d)^(10/3)) - ((14*I )*(-I + Sqrt[3])*d^3*Log[2*(b*c - a*d)^(1/3)*x + (1 + I*Sqrt[3])*c^(1/3)*( a + b*x^3)^(1/3)])/(c^(2/3)*(b*c - a*d)^(10/3)) + (7*(1 + I*Sqrt[3])*d^3*L og[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)])/(c^(2/3)*( b*c - a*d)^(10/3)))/84
Time = 0.45 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {931, 25, 1024, 25, 1024, 27, 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.
| \(\displaystyle \int \frac {1}{\left (a+b x^3\right )^{10/3} \left (c+d x^3\right )} \, dx\) |
| \(\Big \downarrow \) 931 |
| \(\displaystyle \frac {b x}{7 a \left (a+b x^3\right )^{7/3} (b c-a d)}-\frac {\int -\frac {6 b d x^3+6 b c-7 a d}{\left (b x^3+a\right )^{7/3} \left (d x^3+c\right )}dx}{7 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {6 b d x^3+6 b c-7 a d}{\left (b x^3+a\right )^{7/3} \left (d x^3+c\right )}dx}{7 a (b c-a d)}+\frac {b x}{7 a \left (a+b x^3\right )^{7/3} (b c-a d)}\) |
| \(\Big \downarrow \) 1024 |
| \(\displaystyle \frac {\frac {b x (6 b c-13 a d)}{4 a \left (a+b x^3\right )^{4/3} (b c-a d)}-\frac {\int -\frac {3 b d (6 b c-13 a d) x^3+18 b^2 c^2+28 a^2 d^2-39 a b c d}{\left (b x^3+a\right )^{4/3} \left (d x^3+c\right )}dx}{4 a (b c-a d)}}{7 a (b c-a d)}+\frac {b x}{7 a \left (a+b x^3\right )^{7/3} (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {3 b d (6 b c-13 a d) x^3+18 b^2 c^2+28 a^2 d^2-39 a b c d}{\left (b x^3+a\right )^{4/3} \left (d x^3+c\right )}dx}{4 a (b c-a d)}+\frac {b x (6 b c-13 a d)}{4 a \left (a+b x^3\right )^{4/3} (b c-a d)}}{7 a (b c-a d)}+\frac {b x}{7 a \left (a+b x^3\right )^{7/3} (b c-a d)}\) |
| \(\Big \downarrow \) 1024 |
| \(\displaystyle \frac {\frac {\frac {b x \left (67 a^2 d^2-57 a b c d+18 b^2 c^2\right )}{a \sqrt [3]{a+b x^3} (b c-a d)}-\frac {\int \frac {28 a^3 d^3}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{a (b c-a d)}}{4 a (b c-a d)}+\frac {b x (6 b c-13 a d)}{4 a \left (a+b x^3\right )^{4/3} (b c-a d)}}{7 a (b c-a d)}+\frac {b x}{7 a \left (a+b x^3\right )^{7/3} (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {b x \left (67 a^2 d^2-57 a b c d+18 b^2 c^2\right )}{a \sqrt [3]{a+b x^3} (b c-a d)}-\frac {28 a^2 d^3 \int \frac {1}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{b c-a d}}{4 a (b c-a d)}+\frac {b x (6 b c-13 a d)}{4 a \left (a+b x^3\right )^{4/3} (b c-a d)}}{7 a (b c-a d)}+\frac {b x}{7 a \left (a+b x^3\right )^{7/3} (b c-a d)}\) |
| \(\Big \downarrow \) 901 |
| \(\displaystyle \frac {\frac {\frac {b x \left (67 a^2 d^2-57 a b c d+18 b^2 c^2\right )}{a \sqrt [3]{a+b x^3} (b c-a d)}-\frac {28 a^2 d^3 \left (\frac {\arctan \left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} c^{2/3} \sqrt [3]{b c-a d}}+\frac {\log \left (c+d x^3\right )}{6 c^{2/3} \sqrt [3]{b c-a d}}-\frac {\log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{2/3} \sqrt [3]{b c-a d}}\right )}{b c-a d}}{4 a (b c-a d)}+\frac {b x (6 b c-13 a d)}{4 a \left (a+b x^3\right )^{4/3} (b c-a d)}}{7 a (b c-a d)}+\frac {b x}{7 a \left (a+b x^3\right )^{7/3} (b c-a d)}\) |
(b*x)/(7*a*(b*c - a*d)*(a + b*x^3)^(7/3)) + ((b*(6*b*c - 13*a*d)*x)/(4*a*( b*c - a*d)*(a + b*x^3)^(4/3)) + ((b*(18*b^2*c^2 - 57*a*b*c*d + 67*a^2*d^2) *x)/(a*(b*c - a*d)*(a + b*x^3)^(1/3)) - (28*a^2*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)) ))/(b*c - a*d))/(4*a*(b*c - a*d)))/(7*a*(b*c - a*d))
3.1.92.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[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[(-b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Simp[1/(a*n*(p + 1)*(b*c - a*d)) Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -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[(-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*( p + 1)) Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b *c - a*d)*(p + 1) + d*(b*e - a*f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; Fr eeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]
Time = 4.42 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.15
| method | result | size |
| pseudoelliptic | \(\frac {\ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right ) a^{3} d^{3} \left (b \,x^{3}+a \right )^{\frac {7}{3}}-9 x b c \left (a^{4} d^{2}-\left (-\frac {7 d \,x^{3}}{4}+c \right ) b d \,a^{3}+\frac {b^{2} \left (\frac {67}{28} d^{2} x^{6}-\frac {19}{4} c d \,x^{3}+c^{2}\right ) a^{2}}{3}+\frac {x^{3} b^{3} \left (-\frac {19 d \,x^{3}}{14}+c \right ) c a}{2}+\frac {3 b^{4} c^{2} x^{6}}{14}\right ) \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}+\left (\arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x -2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x}\right ) \sqrt {3}-\frac {\ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}\right ) \left (b \,x^{3}+a \right )^{\frac {7}{3}} d^{3} a^{3}}{3 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {7}{3}} \left (a d -b c \right )^{3} c \,a^{3}}\) | \(321\) |
1/3/((a*d-b*c)/c)^(1/3)/(b*x^3+a)^(7/3)*(ln((((a*d-b*c)/c)^(1/3)*x+(b*x^3+ a)^(1/3))/x)*a^3*d^3*(b*x^3+a)^(7/3)-9*x*b*c*(a^4*d^2-(-7/4*d*x^3+c)*b*d*a ^3+1/3*b^2*(67/28*d^2*x^6-19/4*c*d*x^3+c^2)*a^2+1/2*x^3*b^3*(-19/14*d*x^3+ c)*c*a+3/14*b^4*c^2*x^6)*((a*d-b*c)/c)^(1/3)+(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)*3^(1/2)-1/2*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))*(b*x^3+a)^(7/3)*d^3*a^3)/(a*d-b*c)^3/c/a^3
Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^{10/3} \left (c+d x^3\right )} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\left (a+b x^3\right )^{10/3} \left (c+d x^3\right )} \, dx=\int \frac {1}{\left (a + b x^{3}\right )^{\frac {10}{3}} \left (c + d x^{3}\right )}\, dx \]
\[ \int \frac {1}{\left (a+b x^3\right )^{10/3} \left (c+d x^3\right )} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {10}{3}} {\left (d x^{3} + c\right )}} \,d x } \]
\[ \int \frac {1}{\left (a+b x^3\right )^{10/3} \left (c+d x^3\right )} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {10}{3}} {\left (d x^{3} + c\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^{10/3} \left (c+d x^3\right )} \, dx=\int \frac {1}{{\left (b\,x^3+a\right )}^{10/3}\,\left (d\,x^3+c\right )} \,d x \]