3.3043 \(\int \frac{(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^4} \, dx\)

Optimal. Leaf size=645 \[ \frac{\sqrt [3]{a+b x} (c+d x)^{2/3} \left (140 a^2 d^2 f^2-7 a b d f (19 c f+21 d e)+b^2 \left (2 c^2 f^2+129 c d e f+9 d^2 e^2\right )\right )}{27 (e+f x) (b e-a f) (d e-c f)^4}-\frac{2 (b c-a d) \log (e+f x) \left (35 a^2 d^2 f^2-7 a b d f (c f+9 d e)+b^2 \left (-c^2 f^2+9 c d e f+27 d^2 e^2\right )\right )}{81 (b e-a f)^{5/3} (d e-c f)^{13/3}}+\frac{2 (b c-a d) \left (35 a^2 d^2 f^2-7 a b d f (c f+9 d e)+b^2 \left (-c^2 f^2+9 c d e f+27 d^2 e^2\right )\right ) \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{27 (b e-a f)^{5/3} (d e-c f)^{13/3}}+\frac{4 (b c-a d) \left (35 a^2 d^2 f^2-7 a b d f (c f+9 d e)+b^2 \left (-c^2 f^2+9 c d e f+27 d^2 e^2\right )\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3} (b e-a f)^{5/3} (d e-c f)^{13/3}}+\frac{3 \sqrt [3]{a+b x} (b c-a d)}{d \sqrt [3]{c+d x} (e+f x)^3 (d e-c f)}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (-35 a d f+32 b c f+3 b d e)}{9 (e+f x)^2 (d e-c f)^3}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (-10 a d f+9 b c f+b d e)}{3 d (e+f x)^3 (d e-c f)^2} \]

[Out]

(3*(b*c - a*d)*(a + b*x)^(1/3))/(d*(d*e - c*f)*(c + d*x)^(1/3)*(e + f*x)^3) + ((b*d*e + 9*b*c*f - 10*a*d*f)*(a
+ b*x)^(1/3)*(c + d*x)^(2/3))/(3*d*(d*e - c*f)^2*(e + f*x)^3) + ((3*b*d*e + 32*b*c*f - 35*a*d*f)*(a + b*x)^(1
/3)*(c + d*x)^(2/3))/(9*(d*e - c*f)^3*(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))/(27*(b*e - a*f)*(d*e - c*f)^4*(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))*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))])/(27*Sqrt[3]*(b*e
- a*f)^(5/3)*(d*e - c*f)^(13/3)) - (2*(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))*Log[e + f*x])/(81*(b*e - a*f)^(5/3)*(d*e - c*f)^(13/3)) + (2*(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))*Log[-(a + b*x)^(1/3) + ((b*e - a*f)^(1
/3)*(c + d*x)^(1/3))/(d*e - c*f)^(1/3)])/(27*(b*e - a*f)^(5/3)*(d*e - c*f)^(13/3))

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Rubi [A]  time = 1.48976, antiderivative size = 645, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {98, 151, 12, 91} \[ \frac{\sqrt [3]{a+b x} (c+d x)^{2/3} \left (140 a^2 d^2 f^2-7 a b d f (19 c f+21 d e)+b^2 \left (2 c^2 f^2+129 c d e f+9 d^2 e^2\right )\right )}{27 (e+f x) (b e-a f) (d e-c f)^4}-\frac{2 (b c-a d) \log (e+f x) \left (35 a^2 d^2 f^2-7 a b d f (c f+9 d e)+b^2 \left (-c^2 f^2+9 c d e f+27 d^2 e^2\right )\right )}{81 (b e-a f)^{5/3} (d e-c f)^{13/3}}+\frac{2 (b c-a d) \left (35 a^2 d^2 f^2-7 a b d f (c f+9 d e)+b^2 \left (-c^2 f^2+9 c d e f+27 d^2 e^2\right )\right ) \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{27 (b e-a f)^{5/3} (d e-c f)^{13/3}}+\frac{4 (b c-a d) \left (35 a^2 d^2 f^2-7 a b d f (c f+9 d e)+b^2 \left (-c^2 f^2+9 c d e f+27 d^2 e^2\right )\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3} (b e-a f)^{5/3} (d e-c f)^{13/3}}+\frac{3 \sqrt [3]{a+b x} (b c-a d)}{d \sqrt [3]{c+d x} (e+f x)^3 (d e-c f)}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (-35 a d f+32 b c f+3 b d e)}{9 (e+f x)^2 (d e-c f)^3}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (-10 a d f+9 b c f+b d e)}{3 d (e+f x)^3 (d e-c f)^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(4/3)/((c + d*x)^(4/3)*(e + f*x)^4),x]

[Out]

(3*(b*c - a*d)*(a + b*x)^(1/3))/(d*(d*e - c*f)*(c + d*x)^(1/3)*(e + f*x)^3) + ((b*d*e + 9*b*c*f - 10*a*d*f)*(a
+ b*x)^(1/3)*(c + d*x)^(2/3))/(3*d*(d*e - c*f)^2*(e + f*x)^3) + ((3*b*d*e + 32*b*c*f - 35*a*d*f)*(a + b*x)^(1
/3)*(c + d*x)^(2/3))/(9*(d*e - c*f)^3*(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))/(27*(b*e - a*f)*(d*e - c*f)^4*(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))*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))])/(27*Sqrt[3]*(b*e
- a*f)^(5/3)*(d*e - c*f)^(13/3)) - (2*(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))*Log[e + f*x])/(81*(b*e - a*f)^(5/3)*(d*e - c*f)^(13/3)) + (2*(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))*Log[-(a + b*x)^(1/3) + ((b*e - a*f)^(1
/3)*(c + d*x)^(1/3))/(d*e - c*f)^(1/3)])/(27*(b*e - a*f)^(5/3)*(d*e - c*f)^(13/3))

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> 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] + Dist[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])

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> 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] + Dist[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] && LtQ[m, -1] && IntegerQ[m]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 91

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.)*(x_))), x_Symbol] :> 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]

Rubi steps

\begin{align*} \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{3 \int \frac{\frac{1}{3} \left (b^2 c e-2 a b d e-9 a b c f+10 a^2 d f\right )-\frac{1}{3} 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)}\\ &=\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{\int \frac{-\frac{2}{9} d (b e-a f) \left (5 b^2 c e+35 a^2 d f-8 a b (d e+4 c f)\right )+\frac{2}{3} b d (b e-a f) (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}{d (b e-a f) (d e-c f)^2}\\ &=\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{\int \frac{\frac{2}{27} d (b e-a f)^2 \left (140 a^2 d^2 f+b^2 c (33 d e+2 c f)-7 a b d (6 d e+19 c f)\right )-\frac{2}{9} b d^2 (b e-a f)^2 (3 b d e+32 b c f-35 a d f) x}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)^2} \, dx}{2 d (b e-a f)^2 (d e-c f)^3}\\ &=\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{\int -\frac{8 d (b c-a d) (b e-a f)^2 \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 )}{81 (a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)} \, dx}{2 d (b e-a f)^3 (d e-c f)^4}\\ &=\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{\left (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 )\right ) \int \frac{1}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)} \, dx}{81 (b e-a f) (d e-c f)^4}\\ &=\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 ) \tan ^{-1}\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}}\\ \end{align*}

Mathematica [C]  time = 0.80636, size = 320, normalized size = 0.5 \[ \frac{\sqrt [3]{a+b x} \left ((e+f x) \left (35 a^2 d^2 f^2-7 a b d f (c f+9 d e)+b^2 \left (-c^2 f^2+9 c d e f+27 d^2 e^2\right )\right ) \left (3 (a+b x) (c+d x) (b e-a f) (d e-c f)-4 (e+f x) (b c-a d) \left ((c+d x) (b e-a f)-(e+f x) (b c-a d) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )\right )\right )-81 d (a+b x)^2 (b e-a f)^2 (d e-c f)^3+9 f (a+b x)^2 (c+d x) (a f-b e) (d e-c f)^2 (-10 a d f+b c f+9 b d e)\right )}{27 \sqrt [3]{c+d x} (e+f x)^3 (b c-a d) (b e-a f)^2 (d e-c f)^3 (c f-d e)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(4/3)/((c + d*x)^(4/3)*(e + f*x)^4),x]

[Out]

((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)*Hypergeometric2F1[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)

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Maple [F]  time = 0.058, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( fx+e \right ) ^{4}} \left ( bx+a \right ) ^{{\frac{4}{3}}} \left ( dx+c \right ) ^{-{\frac{4}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(4/3)/(d*x+c)^(4/3)/(f*x+e)^4,x)

[Out]

int((b*x+a)^(4/3)/(d*x+c)^(4/3)/(f*x+e)^4,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(4/3)/(d*x+c)^(4/3)/(f*x+e)^4,x, algorithm="maxima")

[Out]

integrate((b*x + a)^(4/3)/((d*x + c)^(4/3)*(f*x + e)^4), x)

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Fricas [B]  time = 62.6846, size = 26298, normalized size = 40.77 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(4/3)/(d*x+c)^(4/3)/(f*x+e)^4,x, algorithm="fricas")

[Out]

[-1/81*(6*sqrt(1/3)*(27*(b^4*c^2*d^3 - a*b^3*c*d^4)*e^7 - 18*(b^4*c^3*d^2 + 4*a*b^3*c^2*d^3 - 5*a^2*b^2*c*d^4)
*e^6*f - 2*(5*b^4*c^4*d - 42*a*b^3*c^3*d^2 - 12*a^2*b^2*c^2*d^3 + 49*a^3*b*c*d^4)*e^5*f^2 + (b^4*c^5 + 16*a*b^
3*c^4*d - 108*a^2*b^2*c^3*d^2 + 56*a^3*b*c^2*d^3 + 35*a^4*c*d^4)*e^4*f^3 - (a*b^3*c^5 + 6*a^2*b^2*c^4*d - 42*a
^3*b*c^3*d^2 + 35*a^4*c^2*d^3)*e^3*f^4 + (27*(b^4*c*d^4 - a*b^3*d^5)*e^4*f^3 - 18*(b^4*c^2*d^3 + 4*a*b^3*c*d^4
- 5*a^2*b^2*d^5)*e^3*f^4 - 2*(5*b^4*c^3*d^2 - 42*a*b^3*c^2*d^3 - 12*a^2*b^2*c*d^4 + 49*a^3*b*d^5)*e^2*f^5 + (
b^4*c^4*d + 16*a*b^3*c^3*d^2 - 108*a^2*b^2*c^2*d^3 + 56*a^3*b*c*d^4 + 35*a^4*d^5)*e*f^6 - (a*b^3*c^4*d + 6*a^2
*b^2*c^3*d^2 - 42*a^3*b*c^2*d^3 + 35*a^4*c*d^4)*f^7)*x^4 + (81*(b^4*c*d^4 - a*b^3*d^5)*e^5*f^2 - 27*(b^4*c^2*d
^3 + 9*a*b^3*c*d^4 - 10*a^2*b^2*d^5)*e^4*f^3 - 6*(8*b^4*c^3*d^2 - 30*a*b^3*c^2*d^3 - 27*a^2*b^2*c*d^4 + 49*a^3
*b*d^5)*e^3*f^4 - (7*b^4*c^4*d - 132*a*b^3*c^3*d^2 + 300*a^2*b^2*c^2*d^3 - 70*a^3*b*c*d^4 - 105*a^4*d^5)*e^2*f
^5 + (b^4*c^5 + 13*a*b^3*c^4*d - 126*a^2*b^2*c^3*d^2 + 182*a^3*b*c^2*d^3 - 70*a^4*c*d^4)*e*f^6 - (a*b^3*c^5 +
6*a^2*b^2*c^4*d - 42*a^3*b*c^3*d^2 + 35*a^4*c^2*d^3)*f^7)*x^3 + 3*(27*(b^4*c*d^4 - a*b^3*d^5)*e^6*f + 9*(b^4*c
^2*d^3 - 11*a*b^3*c*d^4 + 10*a^2*b^2*d^5)*e^5*f^2 - 2*(14*b^4*c^3*d^2 - 6*a*b^3*c^2*d^3 - 57*a^2*b^2*c*d^4 + 4
9*a^3*b*d^5)*e^4*f^3 - (9*b^4*c^4*d - 100*a*b^3*c^3*d^2 + 84*a^2*b^2*c^2*d^3 + 42*a^3*b*c*d^4 - 35*a^4*d^5)*e^
3*f^4 + (b^4*c^5 + 15*a*b^3*c^4*d - 114*a^2*b^2*c^3*d^2 + 98*a^3*b*c^2*d^3)*e^2*f^5 - (a*b^3*c^5 + 6*a^2*b^2*c
^4*d - 42*a^3*b*c^3*d^2 + 35*a^4*c^2*d^3)*e*f^6)*x^2 + (27*(b^4*c*d^4 - a*b^3*d^5)*e^7 + 9*(7*b^4*c^2*d^3 - 17
*a*b^3*c*d^4 + 10*a^2*b^2*d^5)*e^6*f - 2*(32*b^4*c^3*d^2 + 66*a*b^3*c^2*d^3 - 147*a^2*b^2*c*d^4 + 49*a^3*b*d^5
)*e^5*f^2 - (29*b^4*c^4*d - 268*a*b^3*c^3*d^2 + 36*a^2*b^2*c^2*d^3 + 238*a^3*b*c*d^4 - 35*a^4*d^5)*e^4*f^3 + (
3*b^4*c^5 + 47*a*b^3*c^4*d - 330*a^2*b^2*c^3*d^2 + 210*a^3*b*c^2*d^3 + 70*a^4*c*d^4)*e^3*f^4 - 3*(a*b^3*c^5 +
6*a^2*b^2*c^4*d - 42*a^3*b*c^3*d^2 + 35*a^4*c^2*d^3)*e^2*f^5)*x)*sqrt((-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b
*d)*e^2*f - (2*a*b*c + a^2*d)*e*f^2)^(1/3)/(d*e - c*f))*log(-(3*a^2*c*f^2 + (b^2*c + 2*a*b*d)*e^2 - 2*(2*a*b*c
+ a^2*d)*e*f + 3*(-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b*d)*e^2*f - (2*a*b*c + a^2*d)*e*f^2)^(1/3)*(b*e - a*
f)*(b*x + a)^(1/3)*(d*x + c)^(2/3) + (3*b^2*d*e^2 - 2*(b^2*c + 2*a*b*d)*e*f + (2*a*b*c + a^2*d)*f^2)*x + 3*sqr
t(1/3)*(2*(b*d*e^2 + a*c*f^2 - (b*c + a*d)*e*f)*(b*x + a)^(2/3)*(d*x + c)^(1/3) - (-b^2*d*e^3 + a^2*c*f^3 + (b
^2*c + 2*a*b*d)*e^2*f - (2*a*b*c + a^2*d)*e*f^2)^(2/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3) + (-b^2*d*e^3 + a^2*c*f
^3 + (b^2*c + 2*a*b*d)*e^2*f - (2*a*b*c + a^2*d)*e*f^2)^(1/3)*(b*c*e - a*c*f + (b*d*e - a*d*f)*x))*sqrt((-b^2*
d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b*d)*e^2*f - (2*a*b*c + a^2*d)*e*f^2)^(1/3)/(d*e - c*f)))/(f*x + e)) + 2*(-b^
2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b*d)*e^2*f - (2*a*b*c + a^2*d)*e*f^2)^(2/3)*(27*(b^3*c^2*d^2 - a*b^2*c*d^3)
*e^5 + 9*(b^3*c^3*d - 8*a*b^2*c^2*d^2 + 7*a^2*b*c*d^3)*e^4*f - (b^3*c^4 + 6*a*b^2*c^3*d - 42*a^2*b*c^2*d^2 + 3
5*a^3*c*d^3)*e^3*f^2 + (27*(b^3*c*d^3 - a*b^2*d^4)*e^2*f^3 + 9*(b^3*c^2*d^2 - 8*a*b^2*c*d^3 + 7*a^2*b*d^4)*e*f
^4 - (b^3*c^3*d + 6*a*b^2*c^2*d^2 - 42*a^2*b*c*d^3 + 35*a^3*d^4)*f^5)*x^4 + (81*(b^3*c*d^3 - a*b^2*d^4)*e^3*f^
2 + 27*(2*b^3*c^2*d^2 - 9*a*b^2*c*d^3 + 7*a^2*b*d^4)*e^2*f^3 + 3*(2*b^3*c^3*d - 30*a*b^2*c^2*d^2 + 63*a^2*b*c*
d^3 - 35*a^3*d^4)*e*f^4 - (b^3*c^4 + 6*a*b^2*c^3*d - 42*a^2*b*c^2*d^2 + 35*a^3*c*d^3)*f^5)*x^3 + 3*(27*(b^3*c*
d^3 - a*b^2*d^4)*e^4*f + 9*(4*b^3*c^2*d^2 - 11*a*b^2*c*d^3 + 7*a^2*b*d^4)*e^3*f^2 + (8*b^3*c^3*d - 78*a*b^2*c^
2*d^2 + 105*a^2*b*c*d^3 - 35*a^3*d^4)*e^2*f^3 - (b^3*c^4 + 6*a*b^2*c^3*d - 42*a^2*b*c^2*d^2 + 35*a^3*c*d^3)*e*
f^4)*x^2 + (27*(b^3*c*d^3 - a*b^2*d^4)*e^5 + 9*(10*b^3*c^2*d^2 - 17*a*b^2*c*d^3 + 7*a^2*b*d^4)*e^4*f + (26*b^3
*c^3*d - 222*a*b^2*c^2*d^2 + 231*a^2*b*c*d^3 - 35*a^3*d^4)*e^3*f^2 - 3*(b^3*c^4 + 6*a*b^2*c^3*d - 42*a^2*b*c^2
*d^2 + 35*a^3*c*d^3)*e^2*f^3)*x)*log(((b*d*e^2 + a*c*f^2 - (b*c + a*d)*e*f)*(b*x + a)^(2/3)*(d*x + c)^(1/3) +
(-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b*d)*e^2*f - (2*a*b*c + a^2*d)*e*f^2)^(2/3)*(b*x + a)^(1/3)*(d*x + c)^(
2/3) - (-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b*d)*e^2*f - (2*a*b*c + a^2*d)*e*f^2)^(1/3)*(b*c*e - a*c*f + (b*
d*e - a*d*f)*x))/(d*x + c)) - 4*(-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b*d)*e^2*f - (2*a*b*c + a^2*d)*e*f^2)^(
2/3)*(27*(b^3*c^2*d^2 - a*b^2*c*d^3)*e^5 + 9*(b^3*c^3*d - 8*a*b^2*c^2*d^2 + 7*a^2*b*c*d^3)*e^4*f - (b^3*c^4 +
6*a*b^2*c^3*d - 42*a^2*b*c^2*d^2 + 35*a^3*c*d^3)*e^3*f^2 + (27*(b^3*c*d^3 - a*b^2*d^4)*e^2*f^3 + 9*(b^3*c^2*d^
2 - 8*a*b^2*c*d^3 + 7*a^2*b*d^4)*e*f^4 - (b^3*c^3*d + 6*a*b^2*c^2*d^2 - 42*a^2*b*c*d^3 + 35*a^3*d^4)*f^5)*x^4
+ (81*(b^3*c*d^3 - a*b^2*d^4)*e^3*f^2 + 27*(2*b^3*c^2*d^2 - 9*a*b^2*c*d^3 + 7*a^2*b*d^4)*e^2*f^3 + 3*(2*b^3*c^
3*d - 30*a*b^2*c^2*d^2 + 63*a^2*b*c*d^3 - 35*a^3*d^4)*e*f^4 - (b^3*c^4 + 6*a*b^2*c^3*d - 42*a^2*b*c^2*d^2 + 35
*a^3*c*d^3)*f^5)*x^3 + 3*(27*(b^3*c*d^3 - a*b^2*d^4)*e^4*f + 9*(4*b^3*c^2*d^2 - 11*a*b^2*c*d^3 + 7*a^2*b*d^4)*
e^3*f^2 + (8*b^3*c^3*d - 78*a*b^2*c^2*d^2 + 105*a^2*b*c*d^3 - 35*a^3*d^4)*e^2*f^3 - (b^3*c^4 + 6*a*b^2*c^3*d -
42*a^2*b*c^2*d^2 + 35*a^3*c*d^3)*e*f^4)*x^2 + (27*(b^3*c*d^3 - a*b^2*d^4)*e^5 + 9*(10*b^3*c^2*d^2 - 17*a*b^2*
c*d^3 + 7*a^2*b*d^4)*e^4*f + (26*b^3*c^3*d - 222*a*b^2*c^2*d^2 + 231*a^2*b*c*d^3 - 35*a^3*d^4)*e^3*f^2 - 3*(b^
3*c^4 + 6*a*b^2*c^3*d - 42*a^2*b*c^2*d^2 + 35*a^3*c*d^3)*e^2*f^3)*x)*log(((b*d*e^2 + a*c*f^2 - (b*c + a*d)*e*f
)*(b*x + a)^(1/3)*(d*x + c)^(2/3) - (-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b*d)*e^2*f - (2*a*b*c + a^2*d)*e*f^
2)^(2/3)*(d*x + c))/(d*x + c)) + 3*(9*a^4*c^4*f^7 - 27*(4*b^4*c*d^3 - 3*a*b^3*d^4)*e^7 + 9*(8*b^4*c^2*d^2 + 37
*a*b^3*c*d^3 - 27*a^2*b^2*d^4)*e^6*f + (40*b^4*c^3*d - 344*a*b^3*c^2*d^2 - 353*a^2*b^2*c*d^3 + 243*a^3*b*d^4)*
e^5*f^2 - (4*b^4*c^4 + 75*a*b^3*c^3*d - 606*a^2*b^2*c^2*d^2 - 139*a^3*b*c*d^3 + 81*a^4*d^4)*e^4*f^3 + (5*a*b^3
*c^4 - 21*a^2*b^2*c^3*d - 468*a^3*b*c^2*d^2 - 11*a^4*c*d^3)*e^3*f^4 + (11*a^2*b^2*c^4 + 107*a^3*b*c^3*d + 134*
a^4*c^2*d^2)*e^2*f^5 - 3*(7*a^3*b*c^4 + 17*a^4*c^3*d)*e*f^6 - (9*b^4*d^4*e^5*f^2 + 15*(8*b^4*c*d^3 - 11*a*b^3*
d^4)*e^4*f^3 - (127*b^4*c^2*d^2 + 226*a*b^3*c*d^3 - 443*a^2*b^2*d^4)*e^3*f^4 - (2*b^4*c^3*d - 387*a*b^3*c^2*d^
2 + 48*a^2*b^2*c*d^3 + 427*a^3*b*d^4)*e^2*f^5 + (4*a*b^3*c^3*d - 393*a^2*b^2*c^2*d^2 + 294*a^3*b*c*d^3 + 140*a
^4*d^4)*e*f^6 - (2*a^2*b^2*c^3*d - 133*a^3*b*c^2*d^2 + 140*a^4*c*d^3)*f^7)*x^3 - (27*b^4*d^4*e^6*f + 3*(109*b^
4*c*d^3 - 154*a*b^3*d^4)*e^5*f^2 - (317*b^4*c^2*d^2 + 641*a*b^3*c*d^3 - 1228*a^2*b^2*d^4)*e^4*f^3 - (35*b^4*c^
3*d - 992*a*b^3*c^2*d^2 + 49*a^2*b^2*c*d^3 + 1178*a^3*b*d^4)*e^3*f^4 - (2*b^4*c^4 - 107*a*b^3*c^3*d + 1068*a^2
*b^2*c^2*d^2 - 713*a^3*b*c*d^3 - 385*a^4*d^4)*e^2*f^5 + (4*a*b^3*c^4 - 109*a^2*b^2*c^3*d + 428*a^3*b*c^2*d^2 -
350*a^4*c*d^3)*e*f^6 - (2*a^2*b^2*c^4 - 37*a^3*b*c^3*d + 35*a^4*c^2*d^2)*f^7)*x^2 - (27*b^4*d^4*e^7 + 9*(31*b
^4*c*d^3 - 46*a*b^3*d^4)*e^6*f - 2*(104*b^4*c^2*d^2 + 302*a*b^3*c*d^3 - 541*a^2*b^2*d^4)*e^5*f^2 - (109*b^4*c^
3*d - 733*a*b^3*c^2*d^2 - 136*a^2*b^2*c*d^3 + 1030*a^3*b*d^4)*e^4*f^3 + (11*b^4*c^4 + 322*a*b^3*c^3*d - 957*a^
2*b^2*c^2*d^2 + 424*a^3*b*c*d^3 + 335*a^4*d^4)*e^3*f^4 - (37*a*b^3*c^4 + 302*a^2*b^2*c^3*d - 547*a^3*b*c^2*d^2
+ 235*a^4*c*d^3)*e^2*f^5 + (41*a^2*b^2*c^4 + 74*a^3*b*c^3*d - 115*a^4*c^2*d^2)*e*f^6 - 15*(a^3*b*c^4 - a^4*c^
3*d)*f^7)*x)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(b^3*c*d^5*e^11 + a^3*c^6*e^3*f^8 - (5*b^3*c^2*d^4 + 3*a*b^2*c*d
^5)*e^10*f + (10*b^3*c^3*d^3 + 15*a*b^2*c^2*d^4 + 3*a^2*b*c*d^5)*e^9*f^2 - (10*b^3*c^4*d^2 + 30*a*b^2*c^3*d^3
+ 15*a^2*b*c^2*d^4 + a^3*c*d^5)*e^8*f^3 + 5*(b^3*c^5*d + 6*a*b^2*c^4*d^2 + 6*a^2*b*c^3*d^3 + a^3*c^2*d^4)*e^7*
f^4 - (b^3*c^6 + 15*a*b^2*c^5*d + 30*a^2*b*c^4*d^2 + 10*a^3*c^3*d^3)*e^6*f^5 + (3*a*b^2*c^6 + 15*a^2*b*c^5*d +
10*a^3*c^4*d^2)*e^5*f^6 - (3*a^2*b*c^6 + 5*a^3*c^5*d)*e^4*f^7 + (b^3*d^6*e^8*f^3 + a^3*c^5*d*f^11 - (5*b^3*c*
d^5 + 3*a*b^2*d^6)*e^7*f^4 + (10*b^3*c^2*d^4 + 15*a*b^2*c*d^5 + 3*a^2*b*d^6)*e^6*f^5 - (10*b^3*c^3*d^3 + 30*a*
b^2*c^2*d^4 + 15*a^2*b*c*d^5 + a^3*d^6)*e^5*f^6 + 5*(b^3*c^4*d^2 + 6*a*b^2*c^3*d^3 + 6*a^2*b*c^2*d^4 + a^3*c*d
^5)*e^4*f^7 - (b^3*c^5*d + 15*a*b^2*c^4*d^2 + 30*a^2*b*c^3*d^3 + 10*a^3*c^2*d^4)*e^3*f^8 + (3*a*b^2*c^5*d + 15
*a^2*b*c^4*d^2 + 10*a^3*c^3*d^3)*e^2*f^9 - (3*a^2*b*c^5*d + 5*a^3*c^4*d^2)*e*f^10)*x^4 + (3*b^3*d^6*e^9*f^2 +
a^3*c^6*f^11 - (14*b^3*c*d^5 + 9*a*b^2*d^6)*e^8*f^3 + (25*b^3*c^2*d^4 + 42*a*b^2*c*d^5 + 9*a^2*b*d^6)*e^7*f^4
- (20*b^3*c^3*d^3 + 75*a*b^2*c^2*d^4 + 42*a^2*b*c*d^5 + 3*a^3*d^6)*e^6*f^5 + (5*b^3*c^4*d^2 + 60*a*b^2*c^3*d^3
+ 75*a^2*b*c^2*d^4 + 14*a^3*c*d^5)*e^5*f^6 + (2*b^3*c^5*d - 15*a*b^2*c^4*d^2 - 60*a^2*b*c^3*d^3 - 25*a^3*c^2*
d^4)*e^4*f^7 - (b^3*c^6 + 6*a*b^2*c^5*d - 15*a^2*b*c^4*d^2 - 20*a^3*c^3*d^3)*e^3*f^8 + (3*a*b^2*c^6 + 6*a^2*b*
c^5*d - 5*a^3*c^4*d^2)*e^2*f^9 - (3*a^2*b*c^6 + 2*a^3*c^5*d)*e*f^10)*x^3 + 3*(b^3*d^6*e^10*f + a^3*c^6*e*f^10
- (4*b^3*c*d^5 + 3*a*b^2*d^6)*e^9*f^2 + (5*b^3*c^2*d^4 + 12*a*b^2*c*d^5 + 3*a^2*b*d^6)*e^8*f^3 - (15*a*b^2*c^2
*d^4 + 12*a^2*b*c*d^5 + a^3*d^6)*e^7*f^4 - (5*b^3*c^4*d^2 - 15*a^2*b*c^2*d^4 - 4*a^3*c*d^5)*e^6*f^5 + (4*b^3*c
^5*d + 15*a*b^2*c^4*d^2 - 5*a^3*c^2*d^4)*e^5*f^6 - (b^3*c^6 + 12*a*b^2*c^5*d + 15*a^2*b*c^4*d^2)*e^4*f^7 + (3*
a*b^2*c^6 + 12*a^2*b*c^5*d + 5*a^3*c^4*d^2)*e^3*f^8 - (3*a^2*b*c^6 + 4*a^3*c^5*d)*e^2*f^9)*x^2 + (b^3*d^6*e^11
+ 3*a^3*c^6*e^2*f^9 - (2*b^3*c*d^5 + 3*a*b^2*d^6)*e^10*f - (5*b^3*c^2*d^4 - 6*a*b^2*c*d^5 - 3*a^2*b*d^6)*e^9*
f^2 + (20*b^3*c^3*d^3 + 15*a*b^2*c^2*d^4 - 6*a^2*b*c*d^5 - a^3*d^6)*e^8*f^3 - (25*b^3*c^4*d^2 + 60*a*b^2*c^3*d
^3 + 15*a^2*b*c^2*d^4 - 2*a^3*c*d^5)*e^7*f^4 + (14*b^3*c^5*d + 75*a*b^2*c^4*d^2 + 60*a^2*b*c^3*d^3 + 5*a^3*c^2
*d^4)*e^6*f^5 - (3*b^3*c^6 + 42*a*b^2*c^5*d + 75*a^2*b*c^4*d^2 + 20*a^3*c^3*d^3)*e^5*f^6 + (9*a*b^2*c^6 + 42*a
^2*b*c^5*d + 25*a^3*c^4*d^2)*e^4*f^7 - (9*a^2*b*c^6 + 14*a^3*c^5*d)*e^3*f^8)*x), -1/81*(12*sqrt(1/3)*(27*(b^4*
c^2*d^3 - a*b^3*c*d^4)*e^7 - 18*(b^4*c^3*d^2 + 4*a*b^3*c^2*d^3 - 5*a^2*b^2*c*d^4)*e^6*f - 2*(5*b^4*c^4*d - 42*
a*b^3*c^3*d^2 - 12*a^2*b^2*c^2*d^3 + 49*a^3*b*c*d^4)*e^5*f^2 + (b^4*c^5 + 16*a*b^3*c^4*d - 108*a^2*b^2*c^3*d^2
+ 56*a^3*b*c^2*d^3 + 35*a^4*c*d^4)*e^4*f^3 - (a*b^3*c^5 + 6*a^2*b^2*c^4*d - 42*a^3*b*c^3*d^2 + 35*a^4*c^2*d^3
)*e^3*f^4 + (27*(b^4*c*d^4 - a*b^3*d^5)*e^4*f^3 - 18*(b^4*c^2*d^3 + 4*a*b^3*c*d^4 - 5*a^2*b^2*d^5)*e^3*f^4 - 2
*(5*b^4*c^3*d^2 - 42*a*b^3*c^2*d^3 - 12*a^2*b^2*c*d^4 + 49*a^3*b*d^5)*e^2*f^5 + (b^4*c^4*d + 16*a*b^3*c^3*d^2
- 108*a^2*b^2*c^2*d^3 + 56*a^3*b*c*d^4 + 35*a^4*d^5)*e*f^6 - (a*b^3*c^4*d + 6*a^2*b^2*c^3*d^2 - 42*a^3*b*c^2*d
^3 + 35*a^4*c*d^4)*f^7)*x^4 + (81*(b^4*c*d^4 - a*b^3*d^5)*e^5*f^2 - 27*(b^4*c^2*d^3 + 9*a*b^3*c*d^4 - 10*a^2*b
^2*d^5)*e^4*f^3 - 6*(8*b^4*c^3*d^2 - 30*a*b^3*c^2*d^3 - 27*a^2*b^2*c*d^4 + 49*a^3*b*d^5)*e^3*f^4 - (7*b^4*c^4*
d - 132*a*b^3*c^3*d^2 + 300*a^2*b^2*c^2*d^3 - 70*a^3*b*c*d^4 - 105*a^4*d^5)*e^2*f^5 + (b^4*c^5 + 13*a*b^3*c^4*
d - 126*a^2*b^2*c^3*d^2 + 182*a^3*b*c^2*d^3 - 70*a^4*c*d^4)*e*f^6 - (a*b^3*c^5 + 6*a^2*b^2*c^4*d - 42*a^3*b*c^
3*d^2 + 35*a^4*c^2*d^3)*f^7)*x^3 + 3*(27*(b^4*c*d^4 - a*b^3*d^5)*e^6*f + 9*(b^4*c^2*d^3 - 11*a*b^3*c*d^4 + 10*
a^2*b^2*d^5)*e^5*f^2 - 2*(14*b^4*c^3*d^2 - 6*a*b^3*c^2*d^3 - 57*a^2*b^2*c*d^4 + 49*a^3*b*d^5)*e^4*f^3 - (9*b^4
*c^4*d - 100*a*b^3*c^3*d^2 + 84*a^2*b^2*c^2*d^3 + 42*a^3*b*c*d^4 - 35*a^4*d^5)*e^3*f^4 + (b^4*c^5 + 15*a*b^3*c
^4*d - 114*a^2*b^2*c^3*d^2 + 98*a^3*b*c^2*d^3)*e^2*f^5 - (a*b^3*c^5 + 6*a^2*b^2*c^4*d - 42*a^3*b*c^3*d^2 + 35*
a^4*c^2*d^3)*e*f^6)*x^2 + (27*(b^4*c*d^4 - a*b^3*d^5)*e^7 + 9*(7*b^4*c^2*d^3 - 17*a*b^3*c*d^4 + 10*a^2*b^2*d^5
)*e^6*f - 2*(32*b^4*c^3*d^2 + 66*a*b^3*c^2*d^3 - 147*a^2*b^2*c*d^4 + 49*a^3*b*d^5)*e^5*f^2 - (29*b^4*c^4*d - 2
68*a*b^3*c^3*d^2 + 36*a^2*b^2*c^2*d^3 + 238*a^3*b*c*d^4 - 35*a^4*d^5)*e^4*f^3 + (3*b^4*c^5 + 47*a*b^3*c^4*d -
330*a^2*b^2*c^3*d^2 + 210*a^3*b*c^2*d^3 + 70*a^4*c*d^4)*e^3*f^4 - 3*(a*b^3*c^5 + 6*a^2*b^2*c^4*d - 42*a^3*b*c^
3*d^2 + 35*a^4*c^2*d^3)*e^2*f^5)*x)*sqrt(-(-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b*d)*e^2*f - (2*a*b*c + a^2*d
)*e*f^2)^(1/3)/(d*e - c*f))*arctan(sqrt(1/3)*(2*(-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b*d)*e^2*f - (2*a*b*c +
a^2*d)*e*f^2)^(2/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3) - (-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b*d)*e^2*f - (2*
a*b*c + a^2*d)*e*f^2)^(1/3)*(b*c*e - a*c*f + (b*d*e - a*d*f)*x))*sqrt(-(-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*
b*d)*e^2*f - (2*a*b*c + a^2*d)*e*f^2)^(1/3)/(d*e - c*f))/(b^2*c*e^2 - 2*a*b*c*e*f + a^2*c*f^2 + (b^2*d*e^2 - 2
*a*b*d*e*f + a^2*d*f^2)*x)) + 2*(-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b*d)*e^2*f - (2*a*b*c + a^2*d)*e*f^2)^(
2/3)*(27*(b^3*c^2*d^2 - a*b^2*c*d^3)*e^5 + 9*(b^3*c^3*d - 8*a*b^2*c^2*d^2 + 7*a^2*b*c*d^3)*e^4*f - (b^3*c^4 +
6*a*b^2*c^3*d - 42*a^2*b*c^2*d^2 + 35*a^3*c*d^3)*e^3*f^2 + (27*(b^3*c*d^3 - a*b^2*d^4)*e^2*f^3 + 9*(b^3*c^2*d^
2 - 8*a*b^2*c*d^3 + 7*a^2*b*d^4)*e*f^4 - (b^3*c^3*d + 6*a*b^2*c^2*d^2 - 42*a^2*b*c*d^3 + 35*a^3*d^4)*f^5)*x^4
+ (81*(b^3*c*d^3 - a*b^2*d^4)*e^3*f^2 + 27*(2*b^3*c^2*d^2 - 9*a*b^2*c*d^3 + 7*a^2*b*d^4)*e^2*f^3 + 3*(2*b^3*c^
3*d - 30*a*b^2*c^2*d^2 + 63*a^2*b*c*d^3 - 35*a^3*d^4)*e*f^4 - (b^3*c^4 + 6*a*b^2*c^3*d - 42*a^2*b*c^2*d^2 + 35
*a^3*c*d^3)*f^5)*x^3 + 3*(27*(b^3*c*d^3 - a*b^2*d^4)*e^4*f + 9*(4*b^3*c^2*d^2 - 11*a*b^2*c*d^3 + 7*a^2*b*d^4)*
e^3*f^2 + (8*b^3*c^3*d - 78*a*b^2*c^2*d^2 + 105*a^2*b*c*d^3 - 35*a^3*d^4)*e^2*f^3 - (b^3*c^4 + 6*a*b^2*c^3*d -
42*a^2*b*c^2*d^2 + 35*a^3*c*d^3)*e*f^4)*x^2 + (27*(b^3*c*d^3 - a*b^2*d^4)*e^5 + 9*(10*b^3*c^2*d^2 - 17*a*b^2*
c*d^3 + 7*a^2*b*d^4)*e^4*f + (26*b^3*c^3*d - 222*a*b^2*c^2*d^2 + 231*a^2*b*c*d^3 - 35*a^3*d^4)*e^3*f^2 - 3*(b^
3*c^4 + 6*a*b^2*c^3*d - 42*a^2*b*c^2*d^2 + 35*a^3*c*d^3)*e^2*f^3)*x)*log(((b*d*e^2 + a*c*f^2 - (b*c + a*d)*e*f
)*(b*x + a)^(2/3)*(d*x + c)^(1/3) + (-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b*d)*e^2*f - (2*a*b*c + a^2*d)*e*f^
2)^(2/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3) - (-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b*d)*e^2*f - (2*a*b*c + a^2*
d)*e*f^2)^(1/3)*(b*c*e - a*c*f + (b*d*e - a*d*f)*x))/(d*x + c)) - 4*(-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a*b*d
)*e^2*f - (2*a*b*c + a^2*d)*e*f^2)^(2/3)*(27*(b^3*c^2*d^2 - a*b^2*c*d^3)*e^5 + 9*(b^3*c^3*d - 8*a*b^2*c^2*d^2
+ 7*a^2*b*c*d^3)*e^4*f - (b^3*c^4 + 6*a*b^2*c^3*d - 42*a^2*b*c^2*d^2 + 35*a^3*c*d^3)*e^3*f^2 + (27*(b^3*c*d^3
- a*b^2*d^4)*e^2*f^3 + 9*(b^3*c^2*d^2 - 8*a*b^2*c*d^3 + 7*a^2*b*d^4)*e*f^4 - (b^3*c^3*d + 6*a*b^2*c^2*d^2 - 42
*a^2*b*c*d^3 + 35*a^3*d^4)*f^5)*x^4 + (81*(b^3*c*d^3 - a*b^2*d^4)*e^3*f^2 + 27*(2*b^3*c^2*d^2 - 9*a*b^2*c*d^3
+ 7*a^2*b*d^4)*e^2*f^3 + 3*(2*b^3*c^3*d - 30*a*b^2*c^2*d^2 + 63*a^2*b*c*d^3 - 35*a^3*d^4)*e*f^4 - (b^3*c^4 + 6
*a*b^2*c^3*d - 42*a^2*b*c^2*d^2 + 35*a^3*c*d^3)*f^5)*x^3 + 3*(27*(b^3*c*d^3 - a*b^2*d^4)*e^4*f + 9*(4*b^3*c^2*
d^2 - 11*a*b^2*c*d^3 + 7*a^2*b*d^4)*e^3*f^2 + (8*b^3*c^3*d - 78*a*b^2*c^2*d^2 + 105*a^2*b*c*d^3 - 35*a^3*d^4)*
e^2*f^3 - (b^3*c^4 + 6*a*b^2*c^3*d - 42*a^2*b*c^2*d^2 + 35*a^3*c*d^3)*e*f^4)*x^2 + (27*(b^3*c*d^3 - a*b^2*d^4)
*e^5 + 9*(10*b^3*c^2*d^2 - 17*a*b^2*c*d^3 + 7*a^2*b*d^4)*e^4*f + (26*b^3*c^3*d - 222*a*b^2*c^2*d^2 + 231*a^2*b
*c*d^3 - 35*a^3*d^4)*e^3*f^2 - 3*(b^3*c^4 + 6*a*b^2*c^3*d - 42*a^2*b*c^2*d^2 + 35*a^3*c*d^3)*e^2*f^3)*x)*log((
(b*d*e^2 + a*c*f^2 - (b*c + a*d)*e*f)*(b*x + a)^(1/3)*(d*x + c)^(2/3) - (-b^2*d*e^3 + a^2*c*f^3 + (b^2*c + 2*a
*b*d)*e^2*f - (2*a*b*c + a^2*d)*e*f^2)^(2/3)*(d*x + c))/(d*x + c)) + 3*(9*a^4*c^4*f^7 - 27*(4*b^4*c*d^3 - 3*a*
b^3*d^4)*e^7 + 9*(8*b^4*c^2*d^2 + 37*a*b^3*c*d^3 - 27*a^2*b^2*d^4)*e^6*f + (40*b^4*c^3*d - 344*a*b^3*c^2*d^2 -
353*a^2*b^2*c*d^3 + 243*a^3*b*d^4)*e^5*f^2 - (4*b^4*c^4 + 75*a*b^3*c^3*d - 606*a^2*b^2*c^2*d^2 - 139*a^3*b*c*
d^3 + 81*a^4*d^4)*e^4*f^3 + (5*a*b^3*c^4 - 21*a^2*b^2*c^3*d - 468*a^3*b*c^2*d^2 - 11*a^4*c*d^3)*e^3*f^4 + (11*
a^2*b^2*c^4 + 107*a^3*b*c^3*d + 134*a^4*c^2*d^2)*e^2*f^5 - 3*(7*a^3*b*c^4 + 17*a^4*c^3*d)*e*f^6 - (9*b^4*d^4*e
^5*f^2 + 15*(8*b^4*c*d^3 - 11*a*b^3*d^4)*e^4*f^3 - (127*b^4*c^2*d^2 + 226*a*b^3*c*d^3 - 443*a^2*b^2*d^4)*e^3*f
^4 - (2*b^4*c^3*d - 387*a*b^3*c^2*d^2 + 48*a^2*b^2*c*d^3 + 427*a^3*b*d^4)*e^2*f^5 + (4*a*b^3*c^3*d - 393*a^2*b
^2*c^2*d^2 + 294*a^3*b*c*d^3 + 140*a^4*d^4)*e*f^6 - (2*a^2*b^2*c^3*d - 133*a^3*b*c^2*d^2 + 140*a^4*c*d^3)*f^7)
*x^3 - (27*b^4*d^4*e^6*f + 3*(109*b^4*c*d^3 - 154*a*b^3*d^4)*e^5*f^2 - (317*b^4*c^2*d^2 + 641*a*b^3*c*d^3 - 12
28*a^2*b^2*d^4)*e^4*f^3 - (35*b^4*c^3*d - 992*a*b^3*c^2*d^2 + 49*a^2*b^2*c*d^3 + 1178*a^3*b*d^4)*e^3*f^4 - (2*
b^4*c^4 - 107*a*b^3*c^3*d + 1068*a^2*b^2*c^2*d^2 - 713*a^3*b*c*d^3 - 385*a^4*d^4)*e^2*f^5 + (4*a*b^3*c^4 - 109
*a^2*b^2*c^3*d + 428*a^3*b*c^2*d^2 - 350*a^4*c*d^3)*e*f^6 - (2*a^2*b^2*c^4 - 37*a^3*b*c^3*d + 35*a^4*c^2*d^2)*
f^7)*x^2 - (27*b^4*d^4*e^7 + 9*(31*b^4*c*d^3 - 46*a*b^3*d^4)*e^6*f - 2*(104*b^4*c^2*d^2 + 302*a*b^3*c*d^3 - 54
1*a^2*b^2*d^4)*e^5*f^2 - (109*b^4*c^3*d - 733*a*b^3*c^2*d^2 - 136*a^2*b^2*c*d^3 + 1030*a^3*b*d^4)*e^4*f^3 + (1
1*b^4*c^4 + 322*a*b^3*c^3*d - 957*a^2*b^2*c^2*d^2 + 424*a^3*b*c*d^3 + 335*a^4*d^4)*e^3*f^4 - (37*a*b^3*c^4 + 3
02*a^2*b^2*c^3*d - 547*a^3*b*c^2*d^2 + 235*a^4*c*d^3)*e^2*f^5 + (41*a^2*b^2*c^4 + 74*a^3*b*c^3*d - 115*a^4*c^2
*d^2)*e*f^6 - 15*(a^3*b*c^4 - a^4*c^3*d)*f^7)*x)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(b^3*c*d^5*e^11 + a^3*c^6*e^
3*f^8 - (5*b^3*c^2*d^4 + 3*a*b^2*c*d^5)*e^10*f + (10*b^3*c^3*d^3 + 15*a*b^2*c^2*d^4 + 3*a^2*b*c*d^5)*e^9*f^2 -
(10*b^3*c^4*d^2 + 30*a*b^2*c^3*d^3 + 15*a^2*b*c^2*d^4 + a^3*c*d^5)*e^8*f^3 + 5*(b^3*c^5*d + 6*a*b^2*c^4*d^2 +
6*a^2*b*c^3*d^3 + a^3*c^2*d^4)*e^7*f^4 - (b^3*c^6 + 15*a*b^2*c^5*d + 30*a^2*b*c^4*d^2 + 10*a^3*c^3*d^3)*e^6*f
^5 + (3*a*b^2*c^6 + 15*a^2*b*c^5*d + 10*a^3*c^4*d^2)*e^5*f^6 - (3*a^2*b*c^6 + 5*a^3*c^5*d)*e^4*f^7 + (b^3*d^6*
e^8*f^3 + a^3*c^5*d*f^11 - (5*b^3*c*d^5 + 3*a*b^2*d^6)*e^7*f^4 + (10*b^3*c^2*d^4 + 15*a*b^2*c*d^5 + 3*a^2*b*d^
6)*e^6*f^5 - (10*b^3*c^3*d^3 + 30*a*b^2*c^2*d^4 + 15*a^2*b*c*d^5 + a^3*d^6)*e^5*f^6 + 5*(b^3*c^4*d^2 + 6*a*b^2
*c^3*d^3 + 6*a^2*b*c^2*d^4 + a^3*c*d^5)*e^4*f^7 - (b^3*c^5*d + 15*a*b^2*c^4*d^2 + 30*a^2*b*c^3*d^3 + 10*a^3*c^
2*d^4)*e^3*f^8 + (3*a*b^2*c^5*d + 15*a^2*b*c^4*d^2 + 10*a^3*c^3*d^3)*e^2*f^9 - (3*a^2*b*c^5*d + 5*a^3*c^4*d^2)
*e*f^10)*x^4 + (3*b^3*d^6*e^9*f^2 + a^3*c^6*f^11 - (14*b^3*c*d^5 + 9*a*b^2*d^6)*e^8*f^3 + (25*b^3*c^2*d^4 + 42
*a*b^2*c*d^5 + 9*a^2*b*d^6)*e^7*f^4 - (20*b^3*c^3*d^3 + 75*a*b^2*c^2*d^4 + 42*a^2*b*c*d^5 + 3*a^3*d^6)*e^6*f^5
+ (5*b^3*c^4*d^2 + 60*a*b^2*c^3*d^3 + 75*a^2*b*c^2*d^4 + 14*a^3*c*d^5)*e^5*f^6 + (2*b^3*c^5*d - 15*a*b^2*c^4*
d^2 - 60*a^2*b*c^3*d^3 - 25*a^3*c^2*d^4)*e^4*f^7 - (b^3*c^6 + 6*a*b^2*c^5*d - 15*a^2*b*c^4*d^2 - 20*a^3*c^3*d^
3)*e^3*f^8 + (3*a*b^2*c^6 + 6*a^2*b*c^5*d - 5*a^3*c^4*d^2)*e^2*f^9 - (3*a^2*b*c^6 + 2*a^3*c^5*d)*e*f^10)*x^3 +
3*(b^3*d^6*e^10*f + a^3*c^6*e*f^10 - (4*b^3*c*d^5 + 3*a*b^2*d^6)*e^9*f^2 + (5*b^3*c^2*d^4 + 12*a*b^2*c*d^5 +
3*a^2*b*d^6)*e^8*f^3 - (15*a*b^2*c^2*d^4 + 12*a^2*b*c*d^5 + a^3*d^6)*e^7*f^4 - (5*b^3*c^4*d^2 - 15*a^2*b*c^2*d
^4 - 4*a^3*c*d^5)*e^6*f^5 + (4*b^3*c^5*d + 15*a*b^2*c^4*d^2 - 5*a^3*c^2*d^4)*e^5*f^6 - (b^3*c^6 + 12*a*b^2*c^5
*d + 15*a^2*b*c^4*d^2)*e^4*f^7 + (3*a*b^2*c^6 + 12*a^2*b*c^5*d + 5*a^3*c^4*d^2)*e^3*f^8 - (3*a^2*b*c^6 + 4*a^3
*c^5*d)*e^2*f^9)*x^2 + (b^3*d^6*e^11 + 3*a^3*c^6*e^2*f^9 - (2*b^3*c*d^5 + 3*a*b^2*d^6)*e^10*f - (5*b^3*c^2*d^4
- 6*a*b^2*c*d^5 - 3*a^2*b*d^6)*e^9*f^2 + (20*b^3*c^3*d^3 + 15*a*b^2*c^2*d^4 - 6*a^2*b*c*d^5 - a^3*d^6)*e^8*f^
3 - (25*b^3*c^4*d^2 + 60*a*b^2*c^3*d^3 + 15*a^2*b*c^2*d^4 - 2*a^3*c*d^5)*e^7*f^4 + (14*b^3*c^5*d + 75*a*b^2*c^
4*d^2 + 60*a^2*b*c^3*d^3 + 5*a^3*c^2*d^4)*e^6*f^5 - (3*b^3*c^6 + 42*a*b^2*c^5*d + 75*a^2*b*c^4*d^2 + 20*a^3*c^
3*d^3)*e^5*f^6 + (9*a*b^2*c^6 + 42*a^2*b*c^5*d + 25*a^3*c^4*d^2)*e^4*f^7 - (9*a^2*b*c^6 + 14*a^3*c^5*d)*e^3*f^
8)*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(4/3)/(d*x+c)**(4/3)/(f*x+e)**4,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(4/3)/(d*x+c)^(4/3)/(f*x+e)^4,x, algorithm="giac")

[Out]

integrate((b*x + a)^(4/3)/((d*x + c)^(4/3)*(f*x + e)^4), x)