Integrand size = 26, antiderivative size = 645 \[ \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^4} \, dx=\frac {3 (b c-a d) \sqrt [3]{a+b x}}{d (d e-c f) \sqrt [3]{c+d x} (e+f x)^3}+\frac {(b d e+9 b c f-10 a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{3 d (d e-c f)^2 (e+f x)^3}+\frac {(3 b d e+32 b c f-35 a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{9 (d e-c f)^3 (e+f x)^2}+\frac {\left (140 a^2 d^2 f^2-7 a b d f (21 d e+19 c f)+b^2 \left (9 d^2 e^2+129 c d e f+2 c^2 f^2\right )\right ) \sqrt [3]{a+b x} (c+d x)^{2/3}}{27 (b e-a f) (d e-c f)^4 (e+f x)}+\frac {4 (b c-a d) \left (35 a^2 d^2 f^2-7 a b d f (9 d e+c f)+b^2 \left (27 d^2 e^2+9 c d e f-c^2 f^2\right )\right ) \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}\right )}{27 \sqrt {3} (b e-a f)^{5/3} (d e-c f)^{13/3}}-\frac {2 (b c-a d) \left (35 a^2 d^2 f^2-7 a b d f (9 d e+c f)+b^2 \left (27 d^2 e^2+9 c d e f-c^2 f^2\right )\right ) \log (e+f x)}{81 (b e-a f)^{5/3} (d e-c f)^{13/3}}+\frac {2 (b c-a d) \left (35 a^2 d^2 f^2-7 a b d f (9 d e+c f)+b^2 \left (27 d^2 e^2+9 c d e f-c^2 f^2\right )\right ) \log \left (-\sqrt [3]{a+b x}+\frac {\sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt [3]{d e-c f}}\right )}{27 (b e-a f)^{5/3} (d e-c f)^{13/3}} \]
3*(-a*d+b*c)*(b*x+a)^(1/3)/d/(-c*f+d*e)/(d*x+c)^(1/3)/(f*x+e)^3+1/3*(-10*a *d*f+9*b*c*f+b*d*e)*(b*x+a)^(1/3)*(d*x+c)^(2/3)/d/(-c*f+d*e)^2/(f*x+e)^3+1 /9*(-35*a*d*f+32*b*c*f+3*b*d*e)*(b*x+a)^(1/3)*(d*x+c)^(2/3)/(-c*f+d*e)^3/( f*x+e)^2+1/27*(140*a^2*d^2*f^2-7*a*b*d*f*(19*c*f+21*d*e)+b^2*(2*c^2*f^2+12 9*c*d*e*f+9*d^2*e^2))*(b*x+a)^(1/3)*(d*x+c)^(2/3)/(-a*f+b*e)/(-c*f+d*e)^4/ (f*x+e)-2/81*(-a*d+b*c)*(35*a^2*d^2*f^2-7*a*b*d*f*(c*f+9*d*e)+b^2*(-c^2*f^ 2+9*c*d*e*f+27*d^2*e^2))*ln(f*x+e)/(-a*f+b*e)^(5/3)/(-c*f+d*e)^(13/3)+2/27 *(-a*d+b*c)*(35*a^2*d^2*f^2-7*a*b*d*f*(c*f+9*d*e)+b^2*(-c^2*f^2+9*c*d*e*f+ 27*d^2*e^2))*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)^(5/3)/(-c*f+d*e)^(13/3)+4/81*(-a*d+b*c)*(35*a^2*d^2*f^2-7 *a*b*d*f*(c*f+9*d*e)+b^2*(-c^2*f^2+9*c*d*e*f+27*d^2*e^2))*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)^(5/3)/(-c*f+d*e)^(13/3)*3^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.60 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.50 \[ \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^4} \, dx=\frac {\sqrt [3]{a+b x} \left (-81 d (b e-a f)^2 (d e-c f)^3 (a+b x)^2+9 f (-b e+a f) (d e-c f)^2 (9 b d e+b c f-10 a d f) (a+b x)^2 (c+d x)+\left (35 a^2 d^2 f^2-7 a b d f (9 d e+c f)+b^2 \left (27 d^2 e^2+9 c d e f-c^2 f^2\right )\right ) (e+f x) \left (3 (b e-a f) (d e-c f) (a+b x) (c+d x)-4 (b c-a d) (e+f x) \left ((b e-a f) (c+d x)-(b c-a d) (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {4}{3},\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )\right )\right )\right )}{27 (b c-a d) (b e-a f)^2 (d e-c f)^3 (-d e+c f) \sqrt [3]{c+d x} (e+f x)^3} \]
((a + b*x)^(1/3)*(-81*d*(b*e - a*f)^2*(d*e - c*f)^3*(a + b*x)^2 + 9*f*(-(b *e) + a*f)*(d*e - c*f)^2*(9*b*d*e + b*c*f - 10*a*d*f)*(a + b*x)^2*(c + d*x ) + (35*a^2*d^2*f^2 - 7*a*b*d*f*(9*d*e + c*f) + b^2*(27*d^2*e^2 + 9*c*d*e* f - c^2*f^2))*(e + f*x)*(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)*Hype rgeometric2F1[1/3, 1, 4/3, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x)) ]))))/(27*(b*c - a*d)*(b*e - a*f)^2*(d*e - c*f)^3*(-(d*e) + c*f)*(c + d*x) ^(1/3)*(e + f*x)^3)
Time = 0.73 (sec) , antiderivative size = 588, normalized size of antiderivative = 0.91, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {109, 27, 168, 27, 168, 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.
| \(\displaystyle \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^4} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {3 \sqrt [3]{a+b x} (b c-a d)}{d \sqrt [3]{c+d x} (e+f x)^3 (d e-c f)}-\frac {3 \int \frac {10 d f a^2-2 b d e a-9 b c f a+b^2 c e-b (b d e+8 b c f-9 a d f) x}{3 (a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)^4}dx}{d (d e-c f)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \sqrt [3]{a+b x} (b c-a d)}{d \sqrt [3]{c+d x} (e+f x)^3 (d e-c f)}-\frac {\int \frac {10 d f a^2-2 b d e a-9 b c f a+b^2 c e-b (b d e+8 b c f-9 a d f) x}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)^4}dx}{d (d e-c f)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {3 \sqrt [3]{a+b x} (b c-a d)}{d \sqrt [3]{c+d x} (e+f x)^3 (d e-c f)}-\frac {-\frac {\int -\frac {2 d (b e-a f) \left (35 d f a^2-8 b (d e+4 c f) a+5 b^2 c e-3 b (b d e+9 b c f-10 a d f) x\right )}{3 (a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)^3}dx}{3 (b e-a f) (d e-c f)}-\frac {\sqrt [3]{a+b x} (c+d x)^{2/3} (-10 a d f+9 b c f+b d e)}{3 (e+f x)^3 (d e-c f)}}{d (d e-c f)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \sqrt [3]{a+b x} (b c-a d)}{d \sqrt [3]{c+d x} (e+f x)^3 (d e-c f)}-\frac {\frac {2 d \int \frac {35 d f a^2-8 b (d e+4 c f) a+5 b^2 c e-3 b (b d e+9 b c f-10 a d f) x}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)^3}dx}{9 (d e-c f)}-\frac {\sqrt [3]{a+b x} (c+d x)^{2/3} (-10 a d f+9 b c f+b d e)}{3 (e+f x)^3 (d e-c f)}}{d (d e-c f)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {3 \sqrt [3]{a+b x} (b c-a d)}{d \sqrt [3]{c+d x} (e+f x)^3 (d e-c f)}-\frac {\frac {2 d \left (-\frac {\int -\frac {(b e-a f) \left (c (33 d e+2 c f) b^2-7 a d (6 d e+19 c f) b-3 d (3 b d e+32 b c f-35 a d f) x b+140 a^2 d^2 f\right )}{3 (a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)^2}dx}{2 (b e-a f) (d e-c f)}-\frac {\sqrt [3]{a+b x} (c+d x)^{2/3} (-35 a d f+32 b c f+3 b d e)}{2 (e+f x)^2 (d e-c f)}\right )}{9 (d e-c f)}-\frac {\sqrt [3]{a+b x} (c+d x)^{2/3} (-10 a d f+9 b c f+b d e)}{3 (e+f x)^3 (d e-c f)}}{d (d e-c f)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \sqrt [3]{a+b x} (b c-a d)}{d \sqrt [3]{c+d x} (e+f x)^3 (d e-c f)}-\frac {\frac {2 d \left (\frac {\int \frac {c (33 d e+2 c f) b^2-7 a d (6 d e+19 c f) b-3 d (3 b d e+32 b c f-35 a d f) x b+140 a^2 d^2 f}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)^2}dx}{6 (d e-c f)}-\frac {\sqrt [3]{a+b x} (c+d x)^{2/3} (-35 a d f+32 b c f+3 b d e)}{2 (e+f x)^2 (d e-c f)}\right )}{9 (d e-c f)}-\frac {\sqrt [3]{a+b x} (c+d x)^{2/3} (-10 a d f+9 b c f+b d e)}{3 (e+f x)^3 (d e-c f)}}{d (d e-c f)}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {3 \sqrt [3]{a+b x} (b c-a d)}{d \sqrt [3]{c+d x} (e+f x)^3 (d e-c f)}-\frac {\frac {2 d \left (\frac {-\frac {\int -\frac {4 (b c-a d) \left (\left (27 d^2 e^2+9 c d f e-c^2 f^2\right ) b^2-7 a d f (9 d e+c f) b+35 a^2 d^2 f^2\right )}{3 (a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)}dx}{(b e-a f) (d e-c f)}-\frac {\sqrt [3]{a+b x} (c+d x)^{2/3} \left (140 a^2 d^2 f^2-7 a b d f (19 c f+21 d e)+b^2 \left (2 c^2 f^2+129 c d e f+9 d^2 e^2\right )\right )}{(e+f x) (b e-a f) (d e-c f)}}{6 (d e-c f)}-\frac {\sqrt [3]{a+b x} (c+d x)^{2/3} (-35 a d f+32 b c f+3 b d e)}{2 (e+f x)^2 (d e-c f)}\right )}{9 (d e-c f)}-\frac {\sqrt [3]{a+b x} (c+d x)^{2/3} (-10 a d f+9 b c f+b d e)}{3 (e+f x)^3 (d e-c f)}}{d (d e-c f)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \sqrt [3]{a+b x} (b c-a d)}{d \sqrt [3]{c+d x} (e+f x)^3 (d e-c f)}-\frac {\frac {2 d \left (\frac {\frac {4 (b c-a d) \left (35 a^2 d^2 f^2-7 a b d f (c f+9 d e)+b^2 \left (-c^2 f^2+9 c d e f+27 d^2 e^2\right )\right ) \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)}dx}{3 (b e-a f) (d e-c f)}-\frac {\sqrt [3]{a+b x} (c+d x)^{2/3} \left (140 a^2 d^2 f^2-7 a b d f (19 c f+21 d e)+b^2 \left (2 c^2 f^2+129 c d e f+9 d^2 e^2\right )\right )}{(e+f x) (b e-a f) (d e-c f)}}{6 (d e-c f)}-\frac {\sqrt [3]{a+b x} (c+d x)^{2/3} (-35 a d f+32 b c f+3 b d e)}{2 (e+f x)^2 (d e-c f)}\right )}{9 (d e-c f)}-\frac {\sqrt [3]{a+b x} (c+d x)^{2/3} (-10 a d f+9 b c f+b d e)}{3 (e+f x)^3 (d e-c f)}}{d (d e-c f)}\) |
| \(\Big \downarrow \) 102 |
| \(\displaystyle \frac {3 \sqrt [3]{a+b x} (b c-a d)}{d \sqrt [3]{c+d x} (e+f x)^3 (d e-c f)}-\frac {\frac {2 d \left (\frac {\frac {4 (b c-a d) \left (35 a^2 d^2 f^2-7 a b d f (c f+9 d e)+b^2 \left (-c^2 f^2+9 c d e f+27 d^2 e^2\right )\right ) \left (-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt {3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac {1}{\sqrt {3}}\right )}{(b e-a f)^{2/3} \sqrt [3]{d e-c f}}+\frac {\log (e+f x)}{2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}-\frac {3 \log \left (\frac {\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}\right )}{3 (b e-a f) (d e-c f)}-\frac {\sqrt [3]{a+b x} (c+d x)^{2/3} \left (140 a^2 d^2 f^2-7 a b d f (19 c f+21 d e)+b^2 \left (2 c^2 f^2+129 c d e f+9 d^2 e^2\right )\right )}{(e+f x) (b e-a f) (d e-c f)}}{6 (d e-c f)}-\frac {\sqrt [3]{a+b x} (c+d x)^{2/3} (-35 a d f+32 b c f+3 b d e)}{2 (e+f x)^2 (d e-c f)}\right )}{9 (d e-c f)}-\frac {\sqrt [3]{a+b x} (c+d x)^{2/3} (-10 a d f+9 b c f+b d e)}{3 (e+f x)^3 (d e-c f)}}{d (d e-c f)}\) |
(3*(b*c - a*d)*(a + b*x)^(1/3))/(d*(d*e - c*f)*(c + d*x)^(1/3)*(e + f*x)^3 ) - (-1/3*((b*d*e + 9*b*c*f - 10*a*d*f)*(a + b*x)^(1/3)*(c + d*x)^(2/3))/( (d*e - c*f)*(e + f*x)^3) + (2*d*(-1/2*((3*b*d*e + 32*b*c*f - 35*a*d*f)*(a + b*x)^(1/3)*(c + d*x)^(2/3))/((d*e - c*f)*(e + f*x)^2) + (-(((140*a^2*d^2 *f^2 - 7*a*b*d*f*(21*d*e + 19*c*f) + b^2*(9*d^2*e^2 + 129*c*d*e*f + 2*c^2* f^2))*(a + b*x)^(1/3)*(c + d*x)^(2/3))/((b*e - a*f)*(d*e - c*f)*(e + f*x)) ) + (4*(b*c - a*d)*(35*a^2*d^2*f^2 - 7*a*b*d*f*(9*d*e + c*f) + b^2*(27*d^2 *e^2 + 9*c*d*e*f - c^2*f^2))*(-((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*(b* e - a*f)*(d*e - c*f)))/(6*(d*e - c*f))))/(9*(d*e - c*f)))/(d*(d*e - c*f))
3.31.43.3.1 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*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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]
\[\int \frac {\left (b x +a \right )^{\frac {4}{3}}}{\left (d x +c \right )^{\frac {4}{3}} \left (f x +e \right )^{4}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 6285 vs. \(2 (585) = 1170\).
Time = 5.62 (sec) , antiderivative size = 12725, normalized size of antiderivative = 19.73 \[ \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^4} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^4} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^4} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {4}{3}}}{{\left (d x + c\right )}^{\frac {4}{3}} {\left (f x + e\right )}^{4}} \,d x } \]
\[ \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^4} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {4}{3}}}{{\left (d x + c\right )}^{\frac {4}{3}} {\left (f x + e\right )}^{4}} \,d x } \]
Timed out. \[ \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^4} \, dx=\int \frac {{\left (a+b\,x\right )}^{4/3}}{{\left (e+f\,x\right )}^4\,{\left (c+d\,x\right )}^{4/3}} \,d x \]