Integrand size = 26, antiderivative size = 434 \[ \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^3} \, dx=\frac {3 d (a+b x)^{7/3}}{(b c-a d) (d e-c f) \sqrt [3]{c+d x} (e+f x)^2}-\frac {(6 b d e+b c f-7 a d f) (a+b x)^{4/3} (c+d x)^{2/3}}{2 (b c-a d) (d e-c f)^2 (e+f x)^2}+\frac {2 (6 b d e+b c f-7 a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{3 (d e-c f)^3 (e+f x)}+\frac {2 (b c-a d) (6 b d e+b c f-7 a d f) \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}\right )}{3 \sqrt {3} (b e-a f)^{2/3} (d e-c f)^{10/3}}-\frac {(b c-a d) (6 b d e+b c f-7 a d f) \log (e+f x)}{9 (b e-a f)^{2/3} (d e-c f)^{10/3}}+\frac {(b c-a d) (6 b d e+b c f-7 a d f) \log \left (-\sqrt [3]{a+b x}+\frac {\sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt [3]{d e-c f}}\right )}{3 (b e-a f)^{2/3} (d e-c f)^{10/3}} \]
3*d*(b*x+a)^(7/3)/(-a*d+b*c)/(-c*f+d*e)/(d*x+c)^(1/3)/(f*x+e)^2-1/2*(-7*a* d*f+b*c*f+6*b*d*e)*(b*x+a)^(4/3)*(d*x+c)^(2/3)/(-a*d+b*c)/(-c*f+d*e)^2/(f* x+e)^2+2/3*(-7*a*d*f+b*c*f+6*b*d*e)*(b*x+a)^(1/3)*(d*x+c)^(2/3)/(-c*f+d*e) ^3/(f*x+e)-1/9*(-a*d+b*c)*(-7*a*d*f+b*c*f+6*b*d*e)*ln(f*x+e)/(-a*f+b*e)^(2 /3)/(-c*f+d*e)^(10/3)+1/3*(-a*d+b*c)*(-7*a*d*f+b*c*f+6*b*d*e)*ln(-(b*x+a)^ (1/3)+(-a*f+b*e)^(1/3)*(d*x+c)^(1/3)/(-c*f+d*e)^(1/3))/(-a*f+b*e)^(2/3)/(- c*f+d*e)^(10/3)+2/9*(-a*d+b*c)*(-7*a*d*f+b*c*f+6*b*d*e)*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) )/(-a*f+b*e)^(2/3)/(-c*f+d*e)^(10/3)*3^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.28 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.49 \[ \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^3} \, dx=\frac {-6 d (a+b x)^{7/3}+\frac {(6 b d e+b c f-7 a d f) \sqrt [3]{a+b x} \left (3 (b e-a f) (d e-c f) (a+b x) (c+d x)-4 (b c-a d) (e+f x) \left ((b e-a f) (c+d x)-(b c-a d) (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {4}{3},\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )\right )\right )}{3 (b e-a f) (d e-c f)^2}}{2 (b c-a d) (-d e+c f) \sqrt [3]{c+d x} (e+f x)^2} \]
(-6*d*(a + b*x)^(7/3) + ((6*b*d*e + b*c*f - 7*a*d*f)*(a + b*x)^(1/3)*(3*(b *e - a*f)*(d*e - c*f)*(a + b*x)*(c + d*x) - 4*(b*c - a*d)*(e + f*x)*((b*e - a*f)*(c + d*x) - (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))])))/(3*(b*e - a*f)*(d*e - c* f)^2))/(2*(b*c - a*d)*(-(d*e) + c*f)*(c + d*x)^(1/3)*(e + f*x)^2)
Time = 0.36 (sec) , antiderivative size = 404, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {107, 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.
\(\displaystyle \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^3} \, dx\) |
\(\Big \downarrow \) 107 |
\(\displaystyle \frac {3 d (a+b x)^{7/3}}{\sqrt [3]{c+d x} (e+f x)^2 (b c-a d) (d e-c f)}-\frac {(-7 a d f+b c f+6 b d e) \int \frac {(a+b x)^{4/3}}{\sqrt [3]{c+d x} (e+f x)^3}dx}{(b c-a d) (d e-c f)}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {3 d (a+b x)^{7/3}}{\sqrt [3]{c+d x} (e+f x)^2 (b c-a d) (d e-c f)}-\frac {(-7 a d f+b c f+6 b d e) \left (\frac {(a+b x)^{4/3} (c+d x)^{2/3}}{2 (e+f x)^2 (d e-c f)}-\frac {2 (b c-a d) \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^2}dx}{3 (d e-c f)}\right )}{(b c-a d) (d e-c f)}\) |
\(\Big \downarrow \) 105 |
\(\displaystyle \frac {3 d (a+b x)^{7/3}}{\sqrt [3]{c+d x} (e+f x)^2 (b c-a d) (d e-c f)}-\frac {(-7 a d f+b c f+6 b d e) \left (\frac {(a+b x)^{4/3} (c+d x)^{2/3}}{2 (e+f x)^2 (d e-c f)}-\frac {2 (b c-a d) \left (\frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x) (d e-c f)}-\frac {(b c-a d) \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)}dx}{3 (d e-c f)}\right )}{3 (d e-c f)}\right )}{(b c-a d) (d e-c f)}\) |
\(\Big \downarrow \) 102 |
\(\displaystyle \frac {3 d (a+b x)^{7/3}}{\sqrt [3]{c+d x} (e+f x)^2 (b c-a d) (d e-c f)}-\frac {(-7 a d f+b c f+6 b d e) \left (\frac {(a+b x)^{4/3} (c+d x)^{2/3}}{2 (e+f x)^2 (d e-c f)}-\frac {2 (b c-a d) \left (\frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x) (d e-c f)}-\frac {(b c-a d) \left (-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt {3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac {1}{\sqrt {3}}\right )}{(b e-a f)^{2/3} \sqrt [3]{d e-c f}}+\frac {\log (e+f x)}{2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}-\frac {3 \log \left (\frac {\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}\right )}{3 (d e-c f)}\right )}{3 (d e-c f)}\right )}{(b c-a d) (d e-c f)}\) |
(3*d*(a + b*x)^(7/3))/((b*c - a*d)*(d*e - c*f)*(c + d*x)^(1/3)*(e + f*x)^2 ) - ((6*b*d*e + b*c*f - 7*a*d*f)*(((a + b*x)^(4/3)*(c + d*x)^(2/3))/(2*(d* e - c*f)*(e + f*x)^2) - (2*(b*c - a*d)*(((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))))/(3*(d*e - c*f))))/((b*c - a*d)*(d*e - c*f))
3.31.42.3.1 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[((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[(a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x ] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
\[\int \frac {\left (b x +a \right )^{\frac {4}{3}}}{\left (d x +c \right )^{\frac {4}{3}} \left (f x +e \right )^{3}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 3335 vs. \(2 (380) = 760\).
Time = 0.91 (sec) , antiderivative size = 6825, normalized size of antiderivative = 15.73 \[ \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^3} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^3} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {4}{3}}}{{\left (d x + c\right )}^{\frac {4}{3}} {\left (f x + e\right )}^{3}} \,d x } \]
\[ \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^3} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {4}{3}}}{{\left (d x + c\right )}^{\frac {4}{3}} {\left (f x + e\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^3} \, dx=\int \frac {{\left (a+b\,x\right )}^{4/3}}{{\left (e+f\,x\right )}^3\,{\left (c+d\,x\right )}^{4/3}} \,d x \]