Integrand size = 26, antiderivative size = 409 \[ \int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{e+f x} \, dx=\frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{f}+\frac {(3 b d e-2 b c f-a d f) \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{\sqrt {3} b^{2/3} \sqrt [3]{d} f^2}-\frac {\sqrt {3} \sqrt [3]{b e-a f} (d e-c f)^{2/3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}\right )}{f^2}+\frac {(3 b d e-2 b c f-a d f) \log (a+b x)}{6 b^{2/3} \sqrt [3]{d} f^2}+\frac {\sqrt [3]{b e-a f} (d e-c f)^{2/3} \log (e+f x)}{2 f^2}-\frac {3 \sqrt [3]{b e-a f} (d e-c f)^{2/3} \log \left (-\sqrt [3]{a+b x}+\frac {\sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt [3]{d e-c f}}\right )}{2 f^2}+\frac {(3 b d e-2 b c f-a d f) \log \left (-1+\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{2 b^{2/3} \sqrt [3]{d} f^2} \]
(b*x+a)^(1/3)*(d*x+c)^(2/3)/f+1/6*(-a*d*f-2*b*c*f+3*b*d*e)*ln(b*x+a)/b^(2/ 3)/d^(1/3)/f^2+1/2*(-a*f+b*e)^(1/3)*(-c*f+d*e)^(2/3)*ln(f*x+e)/f^2-3/2*(-a *f+b*e)^(1/3)*(-c*f+d*e)^(2/3)*ln(-(b*x+a)^(1/3)+(-a*f+b*e)^(1/3)*(d*x+c)^ (1/3)/(-c*f+d*e)^(1/3))/f^2+1/2*(-a*d*f-2*b*c*f+3*b*d*e)*ln(-1+b^(1/3)*(d* x+c)^(1/3)/d^(1/3)/(b*x+a)^(1/3))/b^(2/3)/d^(1/3)/f^2+1/3*(-a*d*f-2*b*c*f+ 3*b*d*e)*arctan(1/3*3^(1/2)+2/3*b^(1/3)*(d*x+c)^(1/3)/d^(1/3)/(b*x+a)^(1/3 )*3^(1/2))/b^(2/3)/d^(1/3)/f^2*3^(1/2)-(-a*f+b*e)^(1/3)*(-c*f+d*e)^(2/3)*a rctan(1/3*3^(1/2)+2/3*(-a*f+b*e)^(1/3)*(d*x+c)^(1/3)/(-c*f+d*e)^(1/3)/(b*x +a)^(1/3)*3^(1/2))*3^(1/2)/f^2
Time = 1.84 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.26 \[ \int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{e+f x} \, dx=\frac {6 f \sqrt [3]{a+b x} (c+d x)^{2/3}-\frac {2 \sqrt {3} (3 b d e-2 b c f-a d f) \arctan \left (\frac {1+\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}}{\sqrt {3}}\right )}{b^{2/3} \sqrt [3]{d}}+6 \sqrt {3} \sqrt [3]{b e-a f} (-d e+c f)^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{-d e+c f} \sqrt [3]{a+b x}}{\sqrt [3]{b e-a f} \sqrt [3]{c+d x}}}{\sqrt {3}}\right )+\frac {2 (3 b d e-2 b c f-a d f) \log \left (\sqrt [3]{b}-\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}\right )}{b^{2/3} \sqrt [3]{d}}+\frac {(-3 b d e+2 b c f+a d f) \log \left (b^{2/3}+\frac {d^{2/3} (a+b x)^{2/3}}{(c+d x)^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}\right )}{b^{2/3} \sqrt [3]{d}}-6 \sqrt [3]{b e-a f} (-d e+c f)^{2/3} \log \left (\sqrt [3]{b e-a f}+\frac {\sqrt [3]{-d e+c f} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}\right )+3 \sqrt [3]{b e-a f} (-d e+c f)^{2/3} \log \left ((b e-a f)^{2/3}+\frac {(-d e+c f)^{2/3} (a+b x)^{2/3}}{(c+d x)^{2/3}}-\frac {\sqrt [3]{b e-a f} \sqrt [3]{-d e+c f} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}\right )}{6 f^2} \]
(6*f*(a + b*x)^(1/3)*(c + d*x)^(2/3) - (2*Sqrt[3]*(3*b*d*e - 2*b*c*f - a*d *f)*ArcTan[(1 + (2*d^(1/3)*(a + b*x)^(1/3))/(b^(1/3)*(c + d*x)^(1/3)))/Sqr t[3]])/(b^(2/3)*d^(1/3)) + 6*Sqrt[3]*(b*e - a*f)^(1/3)*(-(d*e) + c*f)^(2/3 )*ArcTan[(1 - (2*(-(d*e) + c*f)^(1/3)*(a + b*x)^(1/3))/((b*e - a*f)^(1/3)* (c + d*x)^(1/3)))/Sqrt[3]] + (2*(3*b*d*e - 2*b*c*f - a*d*f)*Log[b^(1/3) - (d^(1/3)*(a + b*x)^(1/3))/(c + d*x)^(1/3)])/(b^(2/3)*d^(1/3)) + ((-3*b*d*e + 2*b*c*f + a*d*f)*Log[b^(2/3) + (d^(2/3)*(a + b*x)^(2/3))/(c + d*x)^(2/3 ) + (b^(1/3)*d^(1/3)*(a + b*x)^(1/3))/(c + d*x)^(1/3)])/(b^(2/3)*d^(1/3)) - 6*(b*e - a*f)^(1/3)*(-(d*e) + c*f)^(2/3)*Log[(b*e - a*f)^(1/3) + ((-(d*e ) + c*f)^(1/3)*(a + b*x)^(1/3))/(c + d*x)^(1/3)] + 3*(b*e - a*f)^(1/3)*(-( d*e) + c*f)^(2/3)*Log[(b*e - a*f)^(2/3) + ((-(d*e) + c*f)^(2/3)*(a + b*x)^ (2/3))/(c + d*x)^(2/3) - ((b*e - a*f)^(1/3)*(-(d*e) + c*f)^(1/3)*(a + b*x) ^(1/3))/(c + d*x)^(1/3)])/(6*f^2)
Time = 0.44 (sec) , antiderivative size = 395, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {112, 27, 175, 71, 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.
| \(\displaystyle \int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{e+f x} \, dx\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{f}-\frac {\int \frac {b c e+2 a d e-3 a c f+(3 b d e-2 b c f-a d f) x}{3 (a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)}dx}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{f}-\frac {\int \frac {b c e+2 a d e-3 a c f+(3 b d e-2 b c f-a d f) x}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)}dx}{3 f}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{f}-\frac {\frac {(-a d f-2 b c f+3 b d e) \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x}}dx}{f}-\frac {3 (b e-a f) (d e-c f) \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)}dx}{f}}{3 f}\) |
| \(\Big \downarrow \) 71 |
| \(\displaystyle \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{f}-\frac {\frac {(-a d f-2 b c f+3 b d e) \left (-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac {1}{\sqrt {3}}\right )}{b^{2/3} \sqrt [3]{d}}-\frac {3 \log \left (\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 b^{2/3} \sqrt [3]{d}}-\frac {\log (a+b x)}{2 b^{2/3} \sqrt [3]{d}}\right )}{f}-\frac {3 (b e-a f) (d e-c f) \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)}dx}{f}}{3 f}\) |
| \(\Big \downarrow \) 102 |
| \(\displaystyle \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{f}-\frac {\frac {(-a d f-2 b c f+3 b d e) \left (-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac {1}{\sqrt {3}}\right )}{b^{2/3} \sqrt [3]{d}}-\frac {3 \log \left (\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 b^{2/3} \sqrt [3]{d}}-\frac {\log (a+b x)}{2 b^{2/3} \sqrt [3]{d}}\right )}{f}-\frac {3 (b e-a f) (d e-c f) \left (-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt {3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac {1}{\sqrt {3}}\right )}{(b e-a f)^{2/3} \sqrt [3]{d e-c f}}+\frac {\log (e+f x)}{2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}-\frac {3 \log \left (\frac {\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}\right )}{f}}{3 f}\) |
((a + b*x)^(1/3)*(c + d*x)^(2/3))/f - ((-3*(b*e - a*f)*(d*e - c*f)*(-((Sqr t[3]*ArcTan[1/Sqrt[3] + (2*(b*e - a*f)^(1/3)*(c + d*x)^(1/3))/(Sqrt[3]*(d* e - c*f)^(1/3)*(a + b*x)^(1/3))])/((b*e - a*f)^(2/3)*(d*e - c*f)^(1/3))) + Log[e + f*x]/(2*(b*e - a*f)^(2/3)*(d*e - c*f)^(1/3)) - (3*Log[-(a + b*x)^ (1/3) + ((b*e - a*f)^(1/3)*(c + d*x)^(1/3))/(d*e - c*f)^(1/3)])/(2*(b*e - a*f)^(2/3)*(d*e - c*f)^(1/3))))/f + ((3*b*d*e - 2*b*c*f - a*d*f)*(-((Sqrt[ 3]*ArcTan[1/Sqrt[3] + (2*b^(1/3)*(c + d*x)^(1/3))/(Sqrt[3]*d^(1/3)*(a + b* x)^(1/3))])/(b^(2/3)*d^(1/3))) - Log[a + b*x]/(2*b^(2/3)*d^(1/3)) - (3*Log [-1 + (b^(1/3)*(c + d*x)^(1/3))/(d^(1/3)*(a + b*x)^(1/3))])/(2*b^(2/3)*d^( 1/3))))/f)/(3*f)
3.31.4.3.1 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)), x_Symbol] :> With[{q = Rt[d/b, 3]}, Simp[(-Sqrt[3])*(q/d)*ArcTan[2*q*((a + b*x)^(1/3)/( Sqrt[3]*(c + d*x)^(1/3))) + 1/Sqrt[3]], x] + (-Simp[3*(q/(2*d))*Log[q*((a + b*x)^(1/3)/(c + d*x)^(1/3)) - 1], x] - Simp[(q/(2*d))*Log[c + d*x], x])] / ; FreeQ[{a, b, c, d}, x] && PosQ[d/b]
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[(a + b*x)^m*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Simp[1/(f*(m + n + p + 1)) Int[(a + b*x)^(m - 1)*(c + d*x) ^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a *f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (IntegersQ[2*m, 2*n, 2*p ] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
\[\int \frac {\left (b x +a \right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {2}{3}}}{f x +e}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 984 vs. \(2 (334) = 668\).
Time = 2.25 (sec) , antiderivative size = 2014, normalized size of antiderivative = 4.92 \[ \int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{e+f x} \, dx=\text {Too large to display} \]
[1/6*(6*(b*x + a)^(1/3)*(d*x + c)^(2/3)*b^2*d*f - 6*sqrt(3)*(-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(1/3)*b^2*d *arctan(1/3*(2*sqrt(3)*(-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(2/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3) + sqrt(3)* (b*c*d*e^2 + a*c^2*f^2 - (b*c^2 + a*c*d)*e*f + (b*d^2*e^2 + a*c*d*f^2 - (b *c*d + a*d^2)*e*f)*x))/(b*c*d*e^2 + a*c^2*f^2 - (b*c^2 + a*c*d)*e*f + (b*d ^2*e^2 + a*c*d*f^2 - (b*c*d + a*d^2)*e*f)*x)) - 3*(-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(1/3)*b^2*d*log(-((-b *d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^ (1/3)*(d*e - c*f)*(b*x + a)^(1/3)*(d*x + c)^(2/3) - (d^2*e^2 - 2*c*d*e*f + c^2*f^2)*(b*x + a)^(2/3)*(d*x + c)^(1/3) - (-b*d^2*e^3 + a*c^2*f^3 + (2*b *c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2)^(2/3)*(d*x + c))/(d*x + c)) + 6*(-b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d) *e*f^2)^(1/3)*b^2*d*log(-((d*e - c*f)*(b*x + a)^(1/3)*(d*x + c)^(2/3) + (- b*d^2*e^3 + a*c^2*f^3 + (2*b*c*d + a*d^2)*e^2*f - (b*c^2 + 2*a*c*d)*e*f^2) ^(1/3)*(d*x + c))/(d*x + c)) - 3*sqrt(1/3)*(3*b^2*d^2*e - (2*b^2*c*d + a*b *d^2)*f)*sqrt((-b^2*d)^(1/3)/d)*log(3*b^2*d*x + b^2*c + 2*a*b*d + 3*(-b^2* d)^(1/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3)*b + 3*sqrt(1/3)*(2*(b*x + a)^(2/3 )*(d*x + c)^(1/3)*b*d - (-b^2*d)^(2/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3) + ( -b^2*d)^(1/3)*(b*d*x + b*c))*sqrt((-b^2*d)^(1/3)/d)) - (-b^2*d)^(2/3)*(...
\[ \int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{e+f x} \, dx=\int \frac {\sqrt [3]{a + b x} \left (c + d x\right )^{\frac {2}{3}}}{e + f x}\, dx \]
\[ \int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{e+f x} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{f x + e} \,d x } \]
\[ \int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{e+f x} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{f x + e} \,d x } \]
Timed out. \[ \int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{e+f x} \, dx=\int \frac {{\left (a+b\,x\right )}^{1/3}\,{\left (c+d\,x\right )}^{2/3}}{e+f\,x} \,d x \]