Integrand size = 26, antiderivative size = 477 \[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)^3} \, dx=-\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x}}{2 (b e-a f) (d e-c f) (e+f x)^2}-\frac {f (9 b d e-4 b c f-5 a d f) (a+b x)^{2/3} \sqrt [3]{c+d x}}{6 (b e-a f)^2 (d e-c f)^2 (e+f x)}-\frac {\left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (9 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right ) \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}\right )}{3 \sqrt {3} (b e-a f)^{7/3} (d e-c f)^{8/3}}+\frac {\left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (9 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right ) \log (e+f x)}{18 (b e-a f)^{7/3} (d e-c f)^{8/3}}-\frac {\left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (9 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right ) \log \left (\frac {\sqrt [3]{d e-c f} \sqrt [3]{a+b x}}{\sqrt [3]{b e-a f}}-\sqrt [3]{c+d x}\right )}{6 (b e-a f)^{7/3} (d e-c f)^{8/3}} \]
-1/2*f*(b*x+a)^(2/3)*(d*x+c)^(1/3)/(-a*f+b*e)/(-c*f+d*e)/(f*x+e)^2-1/6*f*( -5*a*d*f-4*b*c*f+9*b*d*e)*(b*x+a)^(2/3)*(d*x+c)^(1/3)/(-a*f+b*e)^2/(-c*f+d *e)^2/(f*x+e)+1/18*(5*a^2*d^2*f^2-2*a*b*d*f*(-c*f+6*d*e)+b^2*(2*c^2*f^2-6* c*d*e*f+9*d^2*e^2))*ln(f*x+e)/(-a*f+b*e)^(7/3)/(-c*f+d*e)^(8/3)-1/6*(5*a^2 *d^2*f^2-2*a*b*d*f*(-c*f+6*d*e)+b^2*(2*c^2*f^2-6*c*d*e*f+9*d^2*e^2))*ln((- c*f+d*e)^(1/3)*(b*x+a)^(1/3)/(-a*f+b*e)^(1/3)-(d*x+c)^(1/3))/(-a*f+b*e)^(7 /3)/(-c*f+d*e)^(8/3)-1/9*(5*a^2*d^2*f^2-2*a*b*d*f*(-c*f+6*d*e)+b^2*(2*c^2* f^2-6*c*d*e*f+9*d^2*e^2))*arctan(1/3*3^(1/2)+2/3*(-c*f+d*e)^(1/3)*(b*x+a)^ (1/3)/(-a*f+b*e)^(1/3)/(d*x+c)^(1/3)*3^(1/2))/(-a*f+b*e)^(7/3)/(-c*f+d*e)^ (8/3)*3^(1/2)
Time = 3.78 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)^3} \, dx=\frac {1}{18} \left (\frac {3 f (a+b x)^{2/3} \sqrt [3]{c+d x} (-3 b d e (4 e+3 f x)+b c f (7 e+4 f x)+a f (8 d e-3 c f+5 d f x))}{(b e-a f)^2 (d e-c f)^2 (e+f x)^2}-\frac {2 \sqrt {3} \left (5 a^2 d^2 f^2+2 a b d f (-6 d e+c f)+b^2 \left (9 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{-b e+a f} \sqrt [3]{c+d x}}{\sqrt [3]{d e-c f} \sqrt [3]{a+b x}}}{\sqrt {3}}\right )}{(-b e+a f)^{7/3} (d e-c f)^{8/3}}+\frac {2 \left (5 a^2 d^2 f^2+2 a b d f (-6 d e+c f)+b^2 \left (9 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right ) \log \left (\sqrt [3]{d e-c f}+\frac {\sqrt [3]{-b e+a f} \sqrt [3]{c+d x}}{\sqrt [3]{a+b x}}\right )}{(-b e+a f)^{7/3} (d e-c f)^{8/3}}-\frac {\left (5 a^2 d^2 f^2+2 a b d f (-6 d e+c f)+b^2 \left (9 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right ) \log \left ((d e-c f)^{2/3}-\frac {\sqrt [3]{-b e+a f} \sqrt [3]{d e-c f} \sqrt [3]{c+d x}}{\sqrt [3]{a+b x}}+\frac {(-b e+a f)^{2/3} (c+d x)^{2/3}}{(a+b x)^{2/3}}\right )}{(-b e+a f)^{7/3} (d e-c f)^{8/3}}\right ) \]
((3*f*(a + b*x)^(2/3)*(c + d*x)^(1/3)*(-3*b*d*e*(4*e + 3*f*x) + b*c*f*(7*e + 4*f*x) + a*f*(8*d*e - 3*c*f + 5*d*f*x)))/((b*e - a*f)^2*(d*e - c*f)^2*( e + f*x)^2) - (2*Sqrt[3]*(5*a^2*d^2*f^2 + 2*a*b*d*f*(-6*d*e + c*f) + b^2*( 9*d^2*e^2 - 6*c*d*e*f + 2*c^2*f^2))*ArcTan[(1 - (2*(-(b*e) + a*f)^(1/3)*(c + d*x)^(1/3))/((d*e - c*f)^(1/3)*(a + b*x)^(1/3)))/Sqrt[3]])/((-(b*e) + a *f)^(7/3)*(d*e - c*f)^(8/3)) + (2*(5*a^2*d^2*f^2 + 2*a*b*d*f*(-6*d*e + c*f ) + b^2*(9*d^2*e^2 - 6*c*d*e*f + 2*c^2*f^2))*Log[(d*e - c*f)^(1/3) + ((-(b *e) + a*f)^(1/3)*(c + d*x)^(1/3))/(a + b*x)^(1/3)])/((-(b*e) + a*f)^(7/3)* (d*e - c*f)^(8/3)) - ((5*a^2*d^2*f^2 + 2*a*b*d*f*(-6*d*e + c*f) + b^2*(9*d ^2*e^2 - 6*c*d*e*f + 2*c^2*f^2))*Log[(d*e - c*f)^(2/3) - ((-(b*e) + a*f)^( 1/3)*(d*e - c*f)^(1/3)*(c + d*x)^(1/3))/(a + b*x)^(1/3) + ((-(b*e) + a*f)^ (2/3)*(c + d*x)^(2/3))/(a + b*x)^(2/3)])/((-(b*e) + a*f)^(7/3)*(d*e - c*f) ^(8/3)))/18
Time = 0.46 (sec) , antiderivative size = 415, normalized size of antiderivative = 0.87, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {114, 27, 168, 27, 102}
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.
| \(\displaystyle \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)^3} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {\int -\frac {6 b d e-4 b c f-5 a d f-3 b d f x}{3 \sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)^2}dx}{2 (b e-a f) (d e-c f)}-\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x}}{2 (e+f x)^2 (b e-a f) (d e-c f)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {6 b d e-4 b c f-5 a d f-3 b d f x}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)^2}dx}{6 (b e-a f) (d e-c f)}-\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x}}{2 (e+f x)^2 (b e-a f) (d e-c f)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {-\frac {\int -\frac {2 \left (\left (9 d^2 e^2-6 c d f e+2 c^2 f^2\right ) b^2-2 a d f (6 d e-c f) b+5 a^2 d^2 f^2\right )}{3 \sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)}dx}{(b e-a f) (d e-c f)}-\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x} (-5 a d f-4 b c f+9 b d e)}{(e+f x) (b e-a f) (d e-c f)}}{6 (b e-a f) (d e-c f)}-\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x}}{2 (e+f x)^2 (b e-a f) (d e-c f)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (2 c^2 f^2-6 c d e f+9 d^2 e^2\right )\right ) \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)}dx}{3 (b e-a f) (d e-c f)}-\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x} (-5 a d f-4 b c f+9 b d e)}{(e+f x) (b e-a f) (d e-c f)}}{6 (b e-a f) (d e-c f)}-\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x}}{2 (e+f x)^2 (b e-a f) (d e-c f)}\) |
| \(\Big \downarrow \) 102 |
| \(\displaystyle \frac {\frac {2 \left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (2 c^2 f^2-6 c d e f+9 d^2 e^2\right )\right ) \left (-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}{\sqrt {3} \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{b e-a f} (d e-c f)^{2/3}}+\frac {\log (e+f x)}{2 \sqrt [3]{b e-a f} (d e-c f)^{2/3}}-\frac {3 \log \left (\frac {\sqrt [3]{a+b x} \sqrt [3]{d e-c f}}{\sqrt [3]{b e-a f}}-\sqrt [3]{c+d x}\right )}{2 \sqrt [3]{b e-a f} (d e-c f)^{2/3}}\right )}{3 (b e-a f) (d e-c f)}-\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x} (-5 a d f-4 b c f+9 b d e)}{(e+f x) (b e-a f) (d e-c f)}}{6 (b e-a f) (d e-c f)}-\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x}}{2 (e+f x)^2 (b e-a f) (d e-c f)}\) |
-1/2*(f*(a + b*x)^(2/3)*(c + d*x)^(1/3))/((b*e - a*f)*(d*e - c*f)*(e + f*x )^2) + (-((f*(9*b*d*e - 4*b*c*f - 5*a*d*f)*(a + b*x)^(2/3)*(c + d*x)^(1/3) )/((b*e - a*f)*(d*e - c*f)*(e + f*x))) + (2*(5*a^2*d^2*f^2 - 2*a*b*d*f*(6* d*e - c*f) + b^2*(9*d^2*e^2 - 6*c*d*e*f + 2*c^2*f^2))*(-((Sqrt[3]*ArcTan[1 /Sqrt[3] + (2*(d*e - c*f)^(1/3)*(a + b*x)^(1/3))/(Sqrt[3]*(b*e - a*f)^(1/3 )*(c + d*x)^(1/3))])/((b*e - a*f)^(1/3)*(d*e - c*f)^(2/3))) + Log[e + f*x] /(2*(b*e - a*f)^(1/3)*(d*e - c*f)^(2/3)) - (3*Log[((d*e - c*f)^(1/3)*(a + b*x)^(1/3))/(b*e - a*f)^(1/3) - (c + d*x)^(1/3)])/(2*(b*e - a*f)^(1/3)*(d* e - c*f)^(2/3))))/(3*(b*e - a*f)*(d*e - c*f)))/(6*(b*e - a*f)*(d*e - c*f))
3.31.21.3.1 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_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.) *(x_))), x_] :> With[{q = Rt[(d*e - c*f)/(b*e - a*f), 3]}, Simp[(-Sqrt[3])* q*(ArcTan[1/Sqrt[3] + 2*q*((a + b*x)^(1/3)/(Sqrt[3]*(c + d*x)^(1/3)))]/(d*e - c*f)), x] + (Simp[q*(Log[e + f*x]/(2*(d*e - c*f))), x] - Simp[3*q*(Log[q *(a + b*x)^(1/3) - (c + d*x)^(1/3)]/(2*(d*e - c*f))), x])] /; FreeQ[{a, b, c, d, e, f}, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
\[\int \frac {1}{\left (b x +a \right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {2}{3}} \left (f x +e \right )^{3}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 2645 vs. \(2 (427) = 854\).
Time = 3.74 (sec) , antiderivative size = 5445, normalized size of antiderivative = 11.42 \[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)^3} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)^3} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} {\left (f x + e\right )}^{3}} \,d x } \]
\[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)^3} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} {\left (f x + e\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)^3} \, dx=\int \frac {1}{{\left (e+f\,x\right )}^3\,{\left (a+b\,x\right )}^{1/3}\,{\left (c+d\,x\right )}^{2/3}} \,d x \]