3.3014 \(\int \frac{\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^4} \, dx\)

Optimal. Leaf size=591 \[ \frac{\sqrt [3]{a+b x} (c+d x)^{2/3} \left (28 a^2 d^2 f^2-a b d f (5 c f+51 d e)+b^2 \left (-5 c^2 f^2+15 c d e f+18 d^2 e^2\right )\right )}{54 (e+f x) (b e-a f)^2 (d e-c f)^3}-\frac{(b c-a d) \log (e+f x) \left (14 a^2 d^2 f^2-4 a b d f (9 d e-2 c f)+b^2 \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )\right )}{162 (b e-a f)^{8/3} (d e-c f)^{10/3}}+\frac{(b c-a d) \left (14 a^2 d^2 f^2-4 a b d f (9 d e-2 c f)+b^2 \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )\right ) \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{54 (b e-a f)^{8/3} (d e-c f)^{10/3}}+\frac{(b c-a d) \left (14 a^2 d^2 f^2-4 a b d f (9 d e-2 c f)+b^2 \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3} (b e-a f)^{8/3} (d e-c f)^{10/3}}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (-7 a d f+b c f+6 b d e)}{18 (e+f x)^2 (b e-a f) (d e-c f)^2}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{3 (e+f x)^3 (d e-c f)} \]

[Out]

((a + b*x)^(1/3)*(c + d*x)^(2/3))/(3*(d*e - c*f)*(e + f*x)^3) + ((6*b*d*e + b*c*f - 7*a*d*f)*(a + b*x)^(1/3)*(
c + d*x)^(2/3))/(18*(b*e - a*f)*(d*e - c*f)^2*(e + f*x)^2) + ((28*a^2*d^2*f^2 - a*b*d*f*(51*d*e + 5*c*f) + b^2
*(18*d^2*e^2 + 15*c*d*e*f - 5*c^2*f^2))*(a + b*x)^(1/3)*(c + d*x)^(2/3))/(54*(b*e - a*f)^2*(d*e - c*f)^3*(e +
f*x)) + ((b*c - a*d)*(14*a^2*d^2*f^2 - 4*a*b*d*f*(9*d*e - 2*c*f) + b^2*(27*d^2*e^2 - 18*c*d*e*f + 5*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*Sqr
t[3]*(b*e - a*f)^(8/3)*(d*e - c*f)^(10/3)) - ((b*c - a*d)*(14*a^2*d^2*f^2 - 4*a*b*d*f*(9*d*e - 2*c*f) + b^2*(2
7*d^2*e^2 - 18*c*d*e*f + 5*c^2*f^2))*Log[e + f*x])/(162*(b*e - a*f)^(8/3)*(d*e - c*f)^(10/3)) + ((b*c - a*d)*(
14*a^2*d^2*f^2 - 4*a*b*d*f*(9*d*e - 2*c*f) + b^2*(27*d^2*e^2 - 18*c*d*e*f + 5*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)])/(54*(b*e - a*f)^(8/3)*(d*e - c*f)^(10/3))

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Rubi [A]  time = 0.991362, antiderivative size = 591, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {99, 151, 12, 91} \[ \frac{\sqrt [3]{a+b x} (c+d x)^{2/3} \left (28 a^2 d^2 f^2-a b d f (5 c f+51 d e)+b^2 \left (-5 c^2 f^2+15 c d e f+18 d^2 e^2\right )\right )}{54 (e+f x) (b e-a f)^2 (d e-c f)^3}-\frac{(b c-a d) \log (e+f x) \left (14 a^2 d^2 f^2-4 a b d f (9 d e-2 c f)+b^2 \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )\right )}{162 (b e-a f)^{8/3} (d e-c f)^{10/3}}+\frac{(b c-a d) \left (14 a^2 d^2 f^2-4 a b d f (9 d e-2 c f)+b^2 \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )\right ) \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{54 (b e-a f)^{8/3} (d e-c f)^{10/3}}+\frac{(b c-a d) \left (14 a^2 d^2 f^2-4 a b d f (9 d e-2 c f)+b^2 \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3} (b e-a f)^{8/3} (d e-c f)^{10/3}}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (-7 a d f+b c f+6 b d e)}{18 (e+f x)^2 (b e-a f) (d e-c f)^2}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{3 (e+f x)^3 (d e-c f)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(1/3)/((c + d*x)^(1/3)*(e + f*x)^4),x]

[Out]

((a + b*x)^(1/3)*(c + d*x)^(2/3))/(3*(d*e - c*f)*(e + f*x)^3) + ((6*b*d*e + b*c*f - 7*a*d*f)*(a + b*x)^(1/3)*(
c + d*x)^(2/3))/(18*(b*e - a*f)*(d*e - c*f)^2*(e + f*x)^2) + ((28*a^2*d^2*f^2 - a*b*d*f*(51*d*e + 5*c*f) + b^2
*(18*d^2*e^2 + 15*c*d*e*f - 5*c^2*f^2))*(a + b*x)^(1/3)*(c + d*x)^(2/3))/(54*(b*e - a*f)^2*(d*e - c*f)^3*(e +
f*x)) + ((b*c - a*d)*(14*a^2*d^2*f^2 - 4*a*b*d*f*(9*d*e - 2*c*f) + b^2*(27*d^2*e^2 - 18*c*d*e*f + 5*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*Sqr
t[3]*(b*e - a*f)^(8/3)*(d*e - c*f)^(10/3)) - ((b*c - a*d)*(14*a^2*d^2*f^2 - 4*a*b*d*f*(9*d*e - 2*c*f) + b^2*(2
7*d^2*e^2 - 18*c*d*e*f + 5*c^2*f^2))*Log[e + f*x])/(162*(b*e - a*f)^(8/3)*(d*e - c*f)^(10/3)) + ((b*c - a*d)*(
14*a^2*d^2*f^2 - 4*a*b*d*f*(9*d*e - 2*c*f) + b^2*(27*d^2*e^2 - 18*c*d*e*f + 5*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)])/(54*(b*e - a*f)^(8/3)*(d*e - c*f)^(10/3))

Rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (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{\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^4} \, dx &=\frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{3 (d e-c f) (e+f x)^3}-\frac{\int \frac{\frac{1}{3} (b c-7 a d)-2 b d x}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)^3} \, dx}{3 (d e-c f)}\\ &=\frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{3 (d e-c f) (e+f x)^3}+\frac{(6 b d e+b c f-7 a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{18 (b e-a f) (d e-c f)^2 (e+f x)^2}+\frac{\int \frac{\frac{1}{9} \left (-28 a^2 d^2 f-b^2 c (12 d e-5 c f)+5 a b d (6 d e+c f)\right )+\frac{1}{3} b d (6 b d e+b c f-7 a d f) x}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)^2} \, dx}{6 (b e-a f) (d e-c f)^2}\\ &=\frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{3 (d e-c f) (e+f x)^3}+\frac{(6 b d e+b c f-7 a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{18 (b e-a f) (d e-c f)^2 (e+f x)^2}+\frac{\left (28 a^2 d^2 f^2-a b d f (51 d e+5 c f)+b^2 \left (18 d^2 e^2+15 c d e f-5 c^2 f^2\right )\right ) \sqrt [3]{a+b x} (c+d x)^{2/3}}{54 (b e-a f)^2 (d e-c f)^3 (e+f x)}-\frac{\int \frac{2 (b c-a d) \left (14 a^2 d^2 f^2-4 a b d f (9 d e-2 c f)+b^2 \left (27 d^2 e^2-18 c d e f+5 c^2 f^2\right )\right )}{27 (a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)} \, dx}{6 (b e-a f)^2 (d e-c f)^3}\\ &=\frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{3 (d e-c f) (e+f x)^3}+\frac{(6 b d e+b c f-7 a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{18 (b e-a f) (d e-c f)^2 (e+f x)^2}+\frac{\left (28 a^2 d^2 f^2-a b d f (51 d e+5 c f)+b^2 \left (18 d^2 e^2+15 c d e f-5 c^2 f^2\right )\right ) \sqrt [3]{a+b x} (c+d x)^{2/3}}{54 (b e-a f)^2 (d e-c f)^3 (e+f x)}-\frac{\left ((b c-a d) \left (14 a^2 d^2 f^2-4 a b d f (9 d e-2 c f)+b^2 \left (27 d^2 e^2-18 c d e f+5 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)^2 (d e-c f)^3}\\ &=\frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{3 (d e-c f) (e+f x)^3}+\frac{(6 b d e+b c f-7 a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{18 (b e-a f) (d e-c f)^2 (e+f x)^2}+\frac{\left (28 a^2 d^2 f^2-a b d f (51 d e+5 c f)+b^2 \left (18 d^2 e^2+15 c d e f-5 c^2 f^2\right )\right ) \sqrt [3]{a+b x} (c+d x)^{2/3}}{54 (b e-a f)^2 (d e-c f)^3 (e+f x)}+\frac{(b c-a d) \left (14 a^2 d^2 f^2-4 a b d f (9 d e-2 c f)+b^2 \left (27 d^2 e^2-18 c d e f+5 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)^{8/3} (d e-c f)^{10/3}}-\frac{(b c-a d) \left (14 a^2 d^2 f^2-4 a b d f (9 d e-2 c f)+b^2 \left (27 d^2 e^2-18 c d e f+5 c^2 f^2\right )\right ) \log (e+f x)}{162 (b e-a f)^{8/3} (d e-c f)^{10/3}}+\frac{(b c-a d) \left (14 a^2 d^2 f^2-4 a b d f (9 d e-2 c f)+b^2 \left (27 d^2 e^2-18 c d e f+5 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 )}{54 (b e-a f)^{8/3} (d e-c f)^{10/3}}\\ \end{align*}

Mathematica [C]  time = 0.636484, size = 265, normalized size = 0.45 \[ \frac{\sqrt [3]{a+b x} \left (\frac{2 (e+f x)^2 \left (14 a^2 d^2 f^2+4 a b d f (2 c f-9 d e)+b^2 \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )\right ) \left ((c+d x) (b e-a f)-(e+f x) (b c-a d) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )\right )}{(b e-a f) (d e-c f)}+3 f (a+b x) (c+d x) (e+f x) (7 a d f+5 b c f-12 b d e)-18 f (a+b x) (c+d x) (a f-b e) (c f-d e)\right )}{54 \sqrt [3]{c+d x} (e+f x)^3 (b e-a f)^2 (d e-c f)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(1/3)/((c + d*x)^(1/3)*(e + f*x)^4),x]

[Out]

((a + b*x)^(1/3)*(-18*f*(-(b*e) + a*f)*(-(d*e) + c*f)*(a + b*x)*(c + d*x) + 3*f*(-12*b*d*e + 5*b*c*f + 7*a*d*f
)*(a + b*x)*(c + d*x)*(e + f*x) + (2*(14*a^2*d^2*f^2 + 4*a*b*d*f*(-9*d*e + 2*c*f) + b^2*(27*d^2*e^2 - 18*c*d*e
*f + 5*c^2*f^2))*(e + f*x)^2*((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))]))/((b*e - a*f)*(d*e - c*f))))/(54*(b*e - a*f)^2*(d*e - c*f)^2*(c
+ d*x)^(1/3)*(e + f*x)^3)

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Maple [F]  time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( fx+e \right ) ^{4}}\sqrt [3]{bx+a}{\frac{1}{\sqrt [3]{dx+c}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(1/3)/(d*x+c)^(1/3)/(f*x+e)^4,x)

[Out]

int((b*x+a)^(1/3)/(d*x+c)^(1/3)/(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{1}{3}}}{{\left (d x + c\right )}^{\frac{1}{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)^(1/3)/(d*x+c)^(1/3)/(f*x+e)^4,x, algorithm="maxima")

[Out]

integrate((b*x + a)^(1/3)/((d*x + c)^(1/3)*(f*x + e)^4), x)

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Fricas [B]  time = 80.3265, size = 18777, normalized size = 31.77 \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)^(1/3)/(d*x+c)^(1/3)/(f*x+e)^4,x, algorithm="fricas")

[Out]

[1/162*(3*sqrt(1/3)*(27*(b^4*c*d^3 - a*b^3*d^4)*e^7 - 9*(5*b^4*c^2*d^2 + 2*a*b^3*c*d^3 - 7*a^2*b^2*d^4)*e^6*f
+ (23*b^4*c^3*d + 66*a*b^3*c^2*d^2 - 39*a^2*b^2*c*d^3 - 50*a^3*b*d^4)*e^5*f^2 - (5*b^4*c^4 + 26*a*b^3*c^3*d +
27*a^2*b^2*c^2*d^2 - 44*a^3*b*c*d^3 - 14*a^4*d^4)*e^4*f^3 + (5*a*b^3*c^4 + 3*a^2*b^2*c^3*d + 6*a^3*b*c^2*d^2 -
14*a^4*c*d^3)*e^3*f^4 + (27*(b^4*c*d^3 - a*b^3*d^4)*e^4*f^3 - 9*(5*b^4*c^2*d^2 + 2*a*b^3*c*d^3 - 7*a^2*b^2*d^
4)*e^3*f^4 + (23*b^4*c^3*d + 66*a*b^3*c^2*d^2 - 39*a^2*b^2*c*d^3 - 50*a^3*b*d^4)*e^2*f^5 - (5*b^4*c^4 + 26*a*b
^3*c^3*d + 27*a^2*b^2*c^2*d^2 - 44*a^3*b*c*d^3 - 14*a^4*d^4)*e*f^6 + (5*a*b^3*c^4 + 3*a^2*b^2*c^3*d + 6*a^3*b*
c^2*d^2 - 14*a^4*c*d^3)*f^7)*x^3 + 3*(27*(b^4*c*d^3 - a*b^3*d^4)*e^5*f^2 - 9*(5*b^4*c^2*d^2 + 2*a*b^3*c*d^3 -
7*a^2*b^2*d^4)*e^4*f^3 + (23*b^4*c^3*d + 66*a*b^3*c^2*d^2 - 39*a^2*b^2*c*d^3 - 50*a^3*b*d^4)*e^3*f^4 - (5*b^4*
c^4 + 26*a*b^3*c^3*d + 27*a^2*b^2*c^2*d^2 - 44*a^3*b*c*d^3 - 14*a^4*d^4)*e^2*f^5 + (5*a*b^3*c^4 + 3*a^2*b^2*c^
3*d + 6*a^3*b*c^2*d^2 - 14*a^4*c*d^3)*e*f^6)*x^2 + 3*(27*(b^4*c*d^3 - a*b^3*d^4)*e^6*f - 9*(5*b^4*c^2*d^2 + 2*
a*b^3*c*d^3 - 7*a^2*b^2*d^4)*e^5*f^2 + (23*b^4*c^3*d + 66*a*b^3*c^2*d^2 - 39*a^2*b^2*c*d^3 - 50*a^3*b*d^4)*e^4
*f^3 - (5*b^4*c^4 + 26*a*b^3*c^3*d + 27*a^2*b^2*c^2*d^2 - 44*a^3*b*c*d^3 - 14*a^4*d^4)*e^3*f^4 + (5*a*b^3*c^4
+ 3*a^2*b^2*c^3*d + 6*a^3*b*c^2*d^2 - 14*a^4*c*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*sqrt(
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)) - (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*d^2 - a*b^2*d^3)*e^5 - 18*(
b^3*c^2*d + a*b^2*c*d^2 - 2*a^2*b*d^3)*e^4*f + (5*b^3*c^3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 14*a^3*d^3)*e^3*f^
2 + (27*(b^3*c*d^2 - a*b^2*d^3)*e^2*f^3 - 18*(b^3*c^2*d + a*b^2*c*d^2 - 2*a^2*b*d^3)*e*f^4 + (5*b^3*c^3 + 3*a*
b^2*c^2*d + 6*a^2*b*c*d^2 - 14*a^3*d^3)*f^5)*x^3 + 3*(27*(b^3*c*d^2 - a*b^2*d^3)*e^3*f^2 - 18*(b^3*c^2*d + a*b
^2*c*d^2 - 2*a^2*b*d^3)*e^2*f^3 + (5*b^3*c^3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 14*a^3*d^3)*e*f^4)*x^2 + 3*(27*
(b^3*c*d^2 - a*b^2*d^3)*e^4*f - 18*(b^3*c^2*d + a*b^2*c*d^2 - 2*a^2*b*d^3)*e^3*f^2 + (5*b^3*c^3 + 3*a*b^2*c^2*
d + 6*a^2*b*c*d^2 - 14*a^3*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)) + 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*d^2 - a*b^2*d^3)*e^5 - 18*(b^3*c^2*d + a*b^2*c*d^2 - 2*a^2*b*d^3)*e^4*f + (5*b^3*c^3 +
3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 14*a^3*d^3)*e^3*f^2 + (27*(b^3*c*d^2 - a*b^2*d^3)*e^2*f^3 - 18*(b^3*c^2*d + a
*b^2*c*d^2 - 2*a^2*b*d^3)*e*f^4 + (5*b^3*c^3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 14*a^3*d^3)*f^5)*x^3 + 3*(27*(b
^3*c*d^2 - a*b^2*d^3)*e^3*f^2 - 18*(b^3*c^2*d + a*b^2*c*d^2 - 2*a^2*b*d^3)*e^2*f^3 + (5*b^3*c^3 + 3*a*b^2*c^2*
d + 6*a^2*b*c*d^2 - 14*a^3*d^3)*e*f^4)*x^2 + 3*(27*(b^3*c*d^2 - a*b^2*d^3)*e^4*f - 18*(b^3*c^2*d + a*b^2*c*d^2
- 2*a^2*b*d^3)*e^3*f^2 + (5*b^3*c^3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 14*a^3*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*(54*b^4*d^3*e^7 - 18*a^4*c^3*f^7 - 18*(5*b^4*c*d^2
+ 13*a*b^3*d^3)*e^6*f + (46*b^4*c^2*d + 409*a*b^3*c*d^2 + 373*a^2*b^2*d^3)*e^5*f^2 - 2*(5*b^4*c^3 + 114*a*b^3
*c^2*d + 336*a^2*b^2*c*d^2 + 130*a^3*b*d^3)*e^4*f^3 + (53*a*b^3*c^3 + 393*a^2*b^2*c^2*d + 477*a^3*b*c*d^2 + 67
*a^4*d^3)*e^3*f^4 - 2*(47*a^2*b^2*c^3 + 143*a^3*b*c^2*d + 62*a^4*c*d^2)*e^2*f^5 + 3*(23*a^3*b*c^3 + 25*a^4*c^2
*d)*e*f^6 + (18*b^4*d^3*e^5*f^2 - 3*(b^4*c*d^2 + 29*a*b^3*d^3)*e^4*f^3 - 4*(5*b^4*c^2*d - 13*a*b^3*c*d^2 - 37*
a^2*b^2*d^3)*e^3*f^4 + (5*b^4*c^3 + 45*a*b^3*c^2*d - 123*a^2*b^2*c*d^2 - 107*a^3*b*d^3)*e^2*f^5 - 2*(5*a*b^3*c
^3 + 15*a^2*b^2*c^2*d - 51*a^3*b*c*d^2 - 14*a^4*d^3)*e*f^6 + (5*a^2*b^2*c^3 + 5*a^3*b*c^2*d - 28*a^4*c*d^2)*f^
7)*x^2 + (54*b^4*d^3*e^6*f - 3*(13*b^4*c*d^2 + 83*a*b^3*d^3)*e^5*f^2 - 7*(4*b^4*c^2*d - 35*a*b^3*c*d^2 - 59*a^
2*b^2*d^3)*e^4*f^3 + (13*b^4*c^3 + 33*a*b^3*c^2*d - 471*a^2*b^2*c*d^2 - 295*a^3*b*d^3)*e^3*f^4 - (29*a*b^3*c^3
- 39*a^2*b^2*c^2*d - 363*a^3*b*c*d^2 - 77*a^4*d^3)*e^2*f^5 + (19*a^2*b^2*c^3 - 65*a^3*b*c^2*d - 98*a^4*c*d^2)
*e*f^6 - 3*(a^3*b*c^3 - 7*a^4*c^2*d)*f^7)*x)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(b^4*d^4*e^11 + a^4*c^4*e^3*f^8
- 4*(b^4*c*d^3 + a*b^3*d^4)*e^10*f + 2*(3*b^4*c^2*d^2 + 8*a*b^3*c*d^3 + 3*a^2*b^2*d^4)*e^9*f^2 - 4*(b^4*c^3*d
+ 6*a*b^3*c^2*d^2 + 6*a^2*b^2*c*d^3 + a^3*b*d^4)*e^8*f^3 + (b^4*c^4 + 16*a*b^3*c^3*d + 36*a^2*b^2*c^2*d^2 + 16
*a^3*b*c*d^3 + a^4*d^4)*e^7*f^4 - 4*(a*b^3*c^4 + 6*a^2*b^2*c^3*d + 6*a^3*b*c^2*d^2 + a^4*c*d^3)*e^6*f^5 + 2*(3
*a^2*b^2*c^4 + 8*a^3*b*c^3*d + 3*a^4*c^2*d^2)*e^5*f^6 - 4*(a^3*b*c^4 + a^4*c^3*d)*e^4*f^7 + (b^4*d^4*e^8*f^3 +
a^4*c^4*f^11 - 4*(b^4*c*d^3 + a*b^3*d^4)*e^7*f^4 + 2*(3*b^4*c^2*d^2 + 8*a*b^3*c*d^3 + 3*a^2*b^2*d^4)*e^6*f^5
- 4*(b^4*c^3*d + 6*a*b^3*c^2*d^2 + 6*a^2*b^2*c*d^3 + a^3*b*d^4)*e^5*f^6 + (b^4*c^4 + 16*a*b^3*c^3*d + 36*a^2*b
^2*c^2*d^2 + 16*a^3*b*c*d^3 + a^4*d^4)*e^4*f^7 - 4*(a*b^3*c^4 + 6*a^2*b^2*c^3*d + 6*a^3*b*c^2*d^2 + a^4*c*d^3)
*e^3*f^8 + 2*(3*a^2*b^2*c^4 + 8*a^3*b*c^3*d + 3*a^4*c^2*d^2)*e^2*f^9 - 4*(a^3*b*c^4 + a^4*c^3*d)*e*f^10)*x^3 +
3*(b^4*d^4*e^9*f^2 + a^4*c^4*e*f^10 - 4*(b^4*c*d^3 + a*b^3*d^4)*e^8*f^3 + 2*(3*b^4*c^2*d^2 + 8*a*b^3*c*d^3 +
3*a^2*b^2*d^4)*e^7*f^4 - 4*(b^4*c^3*d + 6*a*b^3*c^2*d^2 + 6*a^2*b^2*c*d^3 + a^3*b*d^4)*e^6*f^5 + (b^4*c^4 + 16
*a*b^3*c^3*d + 36*a^2*b^2*c^2*d^2 + 16*a^3*b*c*d^3 + a^4*d^4)*e^5*f^6 - 4*(a*b^3*c^4 + 6*a^2*b^2*c^3*d + 6*a^3
*b*c^2*d^2 + a^4*c*d^3)*e^4*f^7 + 2*(3*a^2*b^2*c^4 + 8*a^3*b*c^3*d + 3*a^4*c^2*d^2)*e^3*f^8 - 4*(a^3*b*c^4 + a
^4*c^3*d)*e^2*f^9)*x^2 + 3*(b^4*d^4*e^10*f + a^4*c^4*e^2*f^9 - 4*(b^4*c*d^3 + a*b^3*d^4)*e^9*f^2 + 2*(3*b^4*c^
2*d^2 + 8*a*b^3*c*d^3 + 3*a^2*b^2*d^4)*e^8*f^3 - 4*(b^4*c^3*d + 6*a*b^3*c^2*d^2 + 6*a^2*b^2*c*d^3 + a^3*b*d^4)
*e^7*f^4 + (b^4*c^4 + 16*a*b^3*c^3*d + 36*a^2*b^2*c^2*d^2 + 16*a^3*b*c*d^3 + a^4*d^4)*e^6*f^5 - 4*(a*b^3*c^4 +
6*a^2*b^2*c^3*d + 6*a^3*b*c^2*d^2 + a^4*c*d^3)*e^5*f^6 + 2*(3*a^2*b^2*c^4 + 8*a^3*b*c^3*d + 3*a^4*c^2*d^2)*e^
4*f^7 - 4*(a^3*b*c^4 + a^4*c^3*d)*e^3*f^8)*x), -1/162*(6*sqrt(1/3)*(27*(b^4*c*d^3 - a*b^3*d^4)*e^7 - 9*(5*b^4*
c^2*d^2 + 2*a*b^3*c*d^3 - 7*a^2*b^2*d^4)*e^6*f + (23*b^4*c^3*d + 66*a*b^3*c^2*d^2 - 39*a^2*b^2*c*d^3 - 50*a^3*
b*d^4)*e^5*f^2 - (5*b^4*c^4 + 26*a*b^3*c^3*d + 27*a^2*b^2*c^2*d^2 - 44*a^3*b*c*d^3 - 14*a^4*d^4)*e^4*f^3 + (5*
a*b^3*c^4 + 3*a^2*b^2*c^3*d + 6*a^3*b*c^2*d^2 - 14*a^4*c*d^3)*e^3*f^4 + (27*(b^4*c*d^3 - a*b^3*d^4)*e^4*f^3 -
9*(5*b^4*c^2*d^2 + 2*a*b^3*c*d^3 - 7*a^2*b^2*d^4)*e^3*f^4 + (23*b^4*c^3*d + 66*a*b^3*c^2*d^2 - 39*a^2*b^2*c*d^
3 - 50*a^3*b*d^4)*e^2*f^5 - (5*b^4*c^4 + 26*a*b^3*c^3*d + 27*a^2*b^2*c^2*d^2 - 44*a^3*b*c*d^3 - 14*a^4*d^4)*e*
f^6 + (5*a*b^3*c^4 + 3*a^2*b^2*c^3*d + 6*a^3*b*c^2*d^2 - 14*a^4*c*d^3)*f^7)*x^3 + 3*(27*(b^4*c*d^3 - a*b^3*d^4
)*e^5*f^2 - 9*(5*b^4*c^2*d^2 + 2*a*b^3*c*d^3 - 7*a^2*b^2*d^4)*e^4*f^3 + (23*b^4*c^3*d + 66*a*b^3*c^2*d^2 - 39*
a^2*b^2*c*d^3 - 50*a^3*b*d^4)*e^3*f^4 - (5*b^4*c^4 + 26*a*b^3*c^3*d + 27*a^2*b^2*c^2*d^2 - 44*a^3*b*c*d^3 - 14
*a^4*d^4)*e^2*f^5 + (5*a*b^3*c^4 + 3*a^2*b^2*c^3*d + 6*a^3*b*c^2*d^2 - 14*a^4*c*d^3)*e*f^6)*x^2 + 3*(27*(b^4*c
*d^3 - a*b^3*d^4)*e^6*f - 9*(5*b^4*c^2*d^2 + 2*a*b^3*c*d^3 - 7*a^2*b^2*d^4)*e^5*f^2 + (23*b^4*c^3*d + 66*a*b^3
*c^2*d^2 - 39*a^2*b^2*c*d^3 - 50*a^3*b*d^4)*e^4*f^3 - (5*b^4*c^4 + 26*a*b^3*c^3*d + 27*a^2*b^2*c^2*d^2 - 44*a^
3*b*c*d^3 - 14*a^4*d^4)*e^3*f^4 + (5*a*b^3*c^4 + 3*a^2*b^2*c^3*d + 6*a^3*b*c^2*d^2 - 14*a^4*c*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(sq
rt(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)) + (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*d^2 - a*b^2*d^3)*e^5 - 18*(b
^3*c^2*d + a*b^2*c*d^2 - 2*a^2*b*d^3)*e^4*f + (5*b^3*c^3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 14*a^3*d^3)*e^3*f^2
+ (27*(b^3*c*d^2 - a*b^2*d^3)*e^2*f^3 - 18*(b^3*c^2*d + a*b^2*c*d^2 - 2*a^2*b*d^3)*e*f^4 + (5*b^3*c^3 + 3*a*b
^2*c^2*d + 6*a^2*b*c*d^2 - 14*a^3*d^3)*f^5)*x^3 + 3*(27*(b^3*c*d^2 - a*b^2*d^3)*e^3*f^2 - 18*(b^3*c^2*d + a*b^
2*c*d^2 - 2*a^2*b*d^3)*e^2*f^3 + (5*b^3*c^3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 14*a^3*d^3)*e*f^4)*x^2 + 3*(27*(
b^3*c*d^2 - a*b^2*d^3)*e^4*f - 18*(b^3*c^2*d + a*b^2*c*d^2 - 2*a^2*b*d^3)*e^3*f^2 + (5*b^3*c^3 + 3*a*b^2*c^2*d
+ 6*a^2*b*c*d^2 - 14*a^3*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)) - 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*d^2 - a*b^2*d^3)*e^5 - 18*(b^3*c^2*d + a*b^2*c*d^2 - 2*a^2*b*d^3)*e^4*f + (5*b^3*c^3 +
3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 14*a^3*d^3)*e^3*f^2 + (27*(b^3*c*d^2 - a*b^2*d^3)*e^2*f^3 - 18*(b^3*c^2*d + a*
b^2*c*d^2 - 2*a^2*b*d^3)*e*f^4 + (5*b^3*c^3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 14*a^3*d^3)*f^5)*x^3 + 3*(27*(b^
3*c*d^2 - a*b^2*d^3)*e^3*f^2 - 18*(b^3*c^2*d + a*b^2*c*d^2 - 2*a^2*b*d^3)*e^2*f^3 + (5*b^3*c^3 + 3*a*b^2*c^2*d
+ 6*a^2*b*c*d^2 - 14*a^3*d^3)*e*f^4)*x^2 + 3*(27*(b^3*c*d^2 - a*b^2*d^3)*e^4*f - 18*(b^3*c^2*d + a*b^2*c*d^2
- 2*a^2*b*d^3)*e^3*f^2 + (5*b^3*c^3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 14*a^3*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*(54*b^4*d^3*e^7 - 18*a^4*c^3*f^7 - 18*(5*b^4*c*d^2
+ 13*a*b^3*d^3)*e^6*f + (46*b^4*c^2*d + 409*a*b^3*c*d^2 + 373*a^2*b^2*d^3)*e^5*f^2 - 2*(5*b^4*c^3 + 114*a*b^3*
c^2*d + 336*a^2*b^2*c*d^2 + 130*a^3*b*d^3)*e^4*f^3 + (53*a*b^3*c^3 + 393*a^2*b^2*c^2*d + 477*a^3*b*c*d^2 + 67*
a^4*d^3)*e^3*f^4 - 2*(47*a^2*b^2*c^3 + 143*a^3*b*c^2*d + 62*a^4*c*d^2)*e^2*f^5 + 3*(23*a^3*b*c^3 + 25*a^4*c^2*
d)*e*f^6 + (18*b^4*d^3*e^5*f^2 - 3*(b^4*c*d^2 + 29*a*b^3*d^3)*e^4*f^3 - 4*(5*b^4*c^2*d - 13*a*b^3*c*d^2 - 37*a
^2*b^2*d^3)*e^3*f^4 + (5*b^4*c^3 + 45*a*b^3*c^2*d - 123*a^2*b^2*c*d^2 - 107*a^3*b*d^3)*e^2*f^5 - 2*(5*a*b^3*c^
3 + 15*a^2*b^2*c^2*d - 51*a^3*b*c*d^2 - 14*a^4*d^3)*e*f^6 + (5*a^2*b^2*c^3 + 5*a^3*b*c^2*d - 28*a^4*c*d^2)*f^7
)*x^2 + (54*b^4*d^3*e^6*f - 3*(13*b^4*c*d^2 + 83*a*b^3*d^3)*e^5*f^2 - 7*(4*b^4*c^2*d - 35*a*b^3*c*d^2 - 59*a^2
*b^2*d^3)*e^4*f^3 + (13*b^4*c^3 + 33*a*b^3*c^2*d - 471*a^2*b^2*c*d^2 - 295*a^3*b*d^3)*e^3*f^4 - (29*a*b^3*c^3
- 39*a^2*b^2*c^2*d - 363*a^3*b*c*d^2 - 77*a^4*d^3)*e^2*f^5 + (19*a^2*b^2*c^3 - 65*a^3*b*c^2*d - 98*a^4*c*d^2)*
e*f^6 - 3*(a^3*b*c^3 - 7*a^4*c^2*d)*f^7)*x)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(b^4*d^4*e^11 + a^4*c^4*e^3*f^8 -
4*(b^4*c*d^3 + a*b^3*d^4)*e^10*f + 2*(3*b^4*c^2*d^2 + 8*a*b^3*c*d^3 + 3*a^2*b^2*d^4)*e^9*f^2 - 4*(b^4*c^3*d +
6*a*b^3*c^2*d^2 + 6*a^2*b^2*c*d^3 + a^3*b*d^4)*e^8*f^3 + (b^4*c^4 + 16*a*b^3*c^3*d + 36*a^2*b^2*c^2*d^2 + 16*
a^3*b*c*d^3 + a^4*d^4)*e^7*f^4 - 4*(a*b^3*c^4 + 6*a^2*b^2*c^3*d + 6*a^3*b*c^2*d^2 + a^4*c*d^3)*e^6*f^5 + 2*(3*
a^2*b^2*c^4 + 8*a^3*b*c^3*d + 3*a^4*c^2*d^2)*e^5*f^6 - 4*(a^3*b*c^4 + a^4*c^3*d)*e^4*f^7 + (b^4*d^4*e^8*f^3 +
a^4*c^4*f^11 - 4*(b^4*c*d^3 + a*b^3*d^4)*e^7*f^4 + 2*(3*b^4*c^2*d^2 + 8*a*b^3*c*d^3 + 3*a^2*b^2*d^4)*e^6*f^5 -
4*(b^4*c^3*d + 6*a*b^3*c^2*d^2 + 6*a^2*b^2*c*d^3 + a^3*b*d^4)*e^5*f^6 + (b^4*c^4 + 16*a*b^3*c^3*d + 36*a^2*b^
2*c^2*d^2 + 16*a^3*b*c*d^3 + a^4*d^4)*e^4*f^7 - 4*(a*b^3*c^4 + 6*a^2*b^2*c^3*d + 6*a^3*b*c^2*d^2 + a^4*c*d^3)*
e^3*f^8 + 2*(3*a^2*b^2*c^4 + 8*a^3*b*c^3*d + 3*a^4*c^2*d^2)*e^2*f^9 - 4*(a^3*b*c^4 + a^4*c^3*d)*e*f^10)*x^3 +
3*(b^4*d^4*e^9*f^2 + a^4*c^4*e*f^10 - 4*(b^4*c*d^3 + a*b^3*d^4)*e^8*f^3 + 2*(3*b^4*c^2*d^2 + 8*a*b^3*c*d^3 + 3
*a^2*b^2*d^4)*e^7*f^4 - 4*(b^4*c^3*d + 6*a*b^3*c^2*d^2 + 6*a^2*b^2*c*d^3 + a^3*b*d^4)*e^6*f^5 + (b^4*c^4 + 16*
a*b^3*c^3*d + 36*a^2*b^2*c^2*d^2 + 16*a^3*b*c*d^3 + a^4*d^4)*e^5*f^6 - 4*(a*b^3*c^4 + 6*a^2*b^2*c^3*d + 6*a^3*
b*c^2*d^2 + a^4*c*d^3)*e^4*f^7 + 2*(3*a^2*b^2*c^4 + 8*a^3*b*c^3*d + 3*a^4*c^2*d^2)*e^3*f^8 - 4*(a^3*b*c^4 + a^
4*c^3*d)*e^2*f^9)*x^2 + 3*(b^4*d^4*e^10*f + a^4*c^4*e^2*f^9 - 4*(b^4*c*d^3 + a*b^3*d^4)*e^9*f^2 + 2*(3*b^4*c^2
*d^2 + 8*a*b^3*c*d^3 + 3*a^2*b^2*d^4)*e^8*f^3 - 4*(b^4*c^3*d + 6*a*b^3*c^2*d^2 + 6*a^2*b^2*c*d^3 + a^3*b*d^4)*
e^7*f^4 + (b^4*c^4 + 16*a*b^3*c^3*d + 36*a^2*b^2*c^2*d^2 + 16*a^3*b*c*d^3 + a^4*d^4)*e^6*f^5 - 4*(a*b^3*c^4 +
6*a^2*b^2*c^3*d + 6*a^3*b*c^2*d^2 + a^4*c*d^3)*e^5*f^6 + 2*(3*a^2*b^2*c^4 + 8*a^3*b*c^3*d + 3*a^4*c^2*d^2)*e^4
*f^7 - 4*(a^3*b*c^4 + a^4*c^3*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)**(1/3)/(d*x+c)**(1/3)/(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{1}{3}}}{{\left (d x + c\right )}^{\frac{1}{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)^(1/3)/(d*x+c)^(1/3)/(f*x+e)^4,x, algorithm="giac")

[Out]

integrate((b*x + a)^(1/3)/((d*x + c)^(1/3)*(f*x + e)^4), x)