Integrand size = 26, antiderivative size = 301 \[ \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^2} \, dx=-\frac {3 (a+b x)^{4/3}}{(d e-c f) \sqrt [3]{c+d x} (e+f x)}+\frac {4 (b e-a f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{(d e-c f)^2 (e+f x)}+\frac {4 (b c-a d) \sqrt [3]{b e-a f} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}\right )}{\sqrt {3} (d e-c f)^{7/3}}-\frac {2 (b c-a d) \sqrt [3]{b e-a f} \log (e+f x)}{3 (d e-c f)^{7/3}}+\frac {2 (b c-a d) \sqrt [3]{b e-a f} \log \left (-\sqrt [3]{a+b x}+\frac {\sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt [3]{d e-c f}}\right )}{(d e-c f)^{7/3}} \]
-3*(b*x+a)^(4/3)/(-c*f+d*e)/(d*x+c)^(1/3)/(f*x+e)+4*(-a*f+b*e)*(b*x+a)^(1/ 3)*(d*x+c)^(2/3)/(-c*f+d*e)^2/(f*x+e)-2/3*(-a*d+b*c)*(-a*f+b*e)^(1/3)*ln(f *x+e)/(-c*f+d*e)^(7/3)+2*(-a*d+b*c)*(-a*f+b*e)^(1/3)*ln(-(b*x+a)^(1/3)+(-a *f+b*e)^(1/3)*(d*x+c)^(1/3)/(-c*f+d*e)^(1/3))/(-c*f+d*e)^(7/3)+4/3*(-a*d+b *c)*(-a*f+b*e)^(1/3)*arctan(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))/(-c*f+d*e)^(7/3)*3^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.05 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.41 \[ \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^2} \, dx=\frac {\sqrt [3]{a+b x} \left (b (4 c e+d e x+3 c f x)-a (3 d e+c f+4 d f x)-4 (b c-a d) (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {4}{3},\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )\right )}{(d e-c f)^2 \sqrt [3]{c+d x} (e+f x)} \]
((a + b*x)^(1/3)*(b*(4*c*e + d*e*x + 3*c*f*x) - a*(3*d*e + c*f + 4*d*f*x) - 4*(b*c - a*d)*(e + f*x)*Hypergeometric2F1[1/3, 1, 4/3, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))]))/((d*e - c*f)^2*(c + d*x)^(1/3)*(e + f*x) )
Time = 0.30 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {105, 105, 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.
| \(\displaystyle \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^2} \, dx\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {4 (b e-a f) \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^2}dx}{d e-c f}-\frac {3 (a+b x)^{4/3}}{\sqrt [3]{c+d x} (e+f x) (d e-c f)}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {4 (b e-a f) \left (\frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x) (d e-c f)}-\frac {(b c-a d) \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)}dx}{3 (d e-c f)}\right )}{d e-c f}-\frac {3 (a+b x)^{4/3}}{\sqrt [3]{c+d x} (e+f x) (d e-c f)}\) |
| \(\Big \downarrow \) 102 |
| \(\displaystyle \frac {4 (b e-a f) \left (\frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x) (d e-c f)}-\frac {(b c-a d) \left (-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt {3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac {1}{\sqrt {3}}\right )}{(b e-a f)^{2/3} \sqrt [3]{d e-c f}}+\frac {\log (e+f x)}{2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}-\frac {3 \log \left (\frac {\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}\right )}{3 (d e-c f)}\right )}{d e-c f}-\frac {3 (a+b x)^{4/3}}{\sqrt [3]{c+d x} (e+f x) (d e-c f)}\) |
(-3*(a + b*x)^(4/3))/((d*e - c*f)*(c + d*x)^(1/3)*(e + f*x)) + (4*(b*e - a *f)*(((a + b*x)^(1/3)*(c + d*x)^(2/3))/((d*e - c*f)*(e + f*x)) - ((b*c - a *d)*(-((Sqrt[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))))/(3*(d*e - c*f))))/(d*e - c*f)
3.31.41.3.1 Defintions of rubi rules used
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 + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
\[\int \frac {\left (b x +a \right )^{\frac {4}{3}}}{\left (d x +c \right )^{\frac {4}{3}} \left (f x +e \right )^{2}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 651 vs. \(2 (259) = 518\).
Time = 0.28 (sec) , antiderivative size = 651, normalized size of antiderivative = 2.16 \[ \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^2} \, dx=\frac {4 \, \sqrt {3} {\left ({\left (b c d - a d^{2}\right )} f x^{2} + {\left (b c^{2} - a c d\right )} e + {\left ({\left (b c d - a d^{2}\right )} e + {\left (b c^{2} - a c d\right )} f\right )} x\right )} \left (-\frac {b e - a f}{d e - c f}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (d e - c f\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} \left (-\frac {b e - a f}{d e - c f}\right )^{\frac {2}{3}} + \sqrt {3} {\left (b c e - a c f + {\left (b d e - a d f\right )} x\right )}}{3 \, {\left (b c e - a c f + {\left (b d e - a d f\right )} x\right )}}\right ) + 2 \, {\left ({\left (b c d - a d^{2}\right )} f x^{2} + {\left (b c^{2} - a c d\right )} e + {\left ({\left (b c d - a d^{2}\right )} e + {\left (b c^{2} - a c d\right )} f\right )} x\right )} \left (-\frac {b e - a f}{d e - c f}\right )^{\frac {1}{3}} \log \left (\frac {{\left (d x + c\right )} \left (-\frac {b e - a f}{d e - c f}\right )^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} \left (-\frac {b e - a f}{d e - c f}\right )^{\frac {1}{3}} + {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{d x + c}\right ) - 4 \, {\left ({\left (b c d - a d^{2}\right )} f x^{2} + {\left (b c^{2} - a c d\right )} e + {\left ({\left (b c d - a d^{2}\right )} e + {\left (b c^{2} - a c d\right )} f\right )} x\right )} \left (-\frac {b e - a f}{d e - c f}\right )^{\frac {1}{3}} \log \left (\frac {{\left (d x + c\right )} \left (-\frac {b e - a f}{d e - c f}\right )^{\frac {1}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{d x + c}\right ) - 3 \, {\left (a c f - {\left (4 \, b c - 3 \, a d\right )} e - {\left (b d e + {\left (3 \, b c - 4 \, a d\right )} f\right )} x\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{3 \, {\left (c d^{2} e^{3} - 2 \, c^{2} d e^{2} f + c^{3} e f^{2} + {\left (d^{3} e^{2} f - 2 \, c d^{2} e f^{2} + c^{2} d f^{3}\right )} x^{2} + {\left (d^{3} e^{3} - c d^{2} e^{2} f - c^{2} d e f^{2} + c^{3} f^{3}\right )} x\right )}} \]
1/3*(4*sqrt(3)*((b*c*d - a*d^2)*f*x^2 + (b*c^2 - a*c*d)*e + ((b*c*d - a*d^ 2)*e + (b*c^2 - a*c*d)*f)*x)*(-(b*e - a*f)/(d*e - c*f))^(1/3)*arctan(1/3*( 2*sqrt(3)*(d*e - c*f)*(b*x + a)^(1/3)*(d*x + c)^(2/3)*(-(b*e - a*f)/(d*e - c*f))^(2/3) + sqrt(3)*(b*c*e - a*c*f + (b*d*e - a*d*f)*x))/(b*c*e - a*c*f + (b*d*e - a*d*f)*x)) + 2*((b*c*d - a*d^2)*f*x^2 + (b*c^2 - a*c*d)*e + (( b*c*d - a*d^2)*e + (b*c^2 - a*c*d)*f)*x)*(-(b*e - a*f)/(d*e - c*f))^(1/3)* log(((d*x + c)*(-(b*e - a*f)/(d*e - c*f))^(2/3) - (b*x + a)^(1/3)*(d*x + c )^(2/3)*(-(b*e - a*f)/(d*e - c*f))^(1/3) + (b*x + a)^(2/3)*(d*x + c)^(1/3) )/(d*x + c)) - 4*((b*c*d - a*d^2)*f*x^2 + (b*c^2 - a*c*d)*e + ((b*c*d - a* d^2)*e + (b*c^2 - a*c*d)*f)*x)*(-(b*e - a*f)/(d*e - c*f))^(1/3)*log(((d*x + c)*(-(b*e - a*f)/(d*e - c*f))^(1/3) + (b*x + a)^(1/3)*(d*x + c)^(2/3))/( d*x + c)) - 3*(a*c*f - (4*b*c - 3*a*d)*e - (b*d*e + (3*b*c - 4*a*d)*f)*x)* (b*x + a)^(1/3)*(d*x + c)^(2/3))/(c*d^2*e^3 - 2*c^2*d*e^2*f + c^3*e*f^2 + (d^3*e^2*f - 2*c*d^2*e*f^2 + c^2*d*f^3)*x^2 + (d^3*e^3 - c*d^2*e^2*f - c^2 *d*e*f^2 + c^3*f^3)*x)
\[ \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^2} \, dx=\int \frac {\left (a + b x\right )^{\frac {4}{3}}}{\left (c + d x\right )^{\frac {4}{3}} \left (e + f x\right )^{2}}\, dx \]
\[ \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^2} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {4}{3}}}{{\left (d x + c\right )}^{\frac {4}{3}} {\left (f x + e\right )}^{2}} \,d x } \]
\[ \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^2} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {4}{3}}}{{\left (d x + c\right )}^{\frac {4}{3}} {\left (f x + e\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^2} \, dx=\int \frac {{\left (a+b\,x\right )}^{4/3}}{{\left (e+f\,x\right )}^2\,{\left (c+d\,x\right )}^{4/3}} \,d x \]