3.31.43 \(\int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^4} \, dx\) [3043]

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

[Out]

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+129*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)

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Rubi [A]
time = 0.99, antiderivative size = 645, normalized size of antiderivative = 1.00, 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 = {100, 156, 12, 93} \begin {gather*} \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 ) \text {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 )}{27 \sqrt {3} (b e-a f)^{5/3} (d e-c f)^{13/3}}+\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 {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} \end {gather*}

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 12

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

Rule 93

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]

Rule 100

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 156

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] && ILtQ[m, -1]

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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 10.57, size = 320, normalized size = 0.50 \begin {gather*} \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) \, _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 )\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} \end {gather*}

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 = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{\frac {4}{3}}}{\left (d x +c \right )^{\frac {4}{3}} \left (f x +e \right )^{4}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 6231 vs. \(2 (611) = 1222\).
time = 20.83, size = 12621, normalized size = 19.57 \begin {gather*} \text {Too large to display} \end {gather*}

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)*((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 + (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 - 27*(b^4*c^2*d^3 - a*b^3*c*d^4 + (b^4*c*d^4 - a
*b^3*d^5)*x)*e^7 - 9*(9*(b^4*c*d^4 - a*b^3*d^5)*f*x^2 + (7*b^4*c^2*d^3 - 17*a*b^3*c*d^4 + 10*a^2*b^2*d^5)*f*x
- 2*(b^4*c^3*d^2 + 4*a*b^3*c^2*d^3 - 5*a^2*b^2*c*d^4)*f)*e^6 - (81*(b^4*c*d^4 - a*b^3*d^5)*f^2*x^3 + 27*(b^4*c
^2*d^3 - 11*a*b^3*c*d^4 + 10*a^2*b^2*d^5)*f^2*x^2 - 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)*f^2*x - 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)*f^2)*e^5 - (27
*(b^4*c*d^4 - a*b^3*d^5)*f^3*x^4 - 27*(b^4*c^2*d^3 + 9*a*b^3*c*d^4 - 10*a^2*b^2*d^5)*f^3*x^3 - 6*(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)*f^3*x^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)*f^3*x + (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)*f^3)*e^4 + (18*(b^4*c^2*d^3 + 4*a*b^3*c*d^4 - 5*a^2*b^2*d^5)*f^4*x^4 + 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)*f^4*x^3 + 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)*f^4*x^2 - (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)*f^4*x + (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^4)*e^
3 + (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)*f^5*x^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)*f^5*x^3 - 3*(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)*f^5*x^2 + 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)
*f^5*x)*e^2 - ((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)*f^6*x^4 + (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)*f^6*x^3 - 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)*f^6*x^2)*e)*sqrt(-(a^2*c*f^3 - b^2*d*e^3 - (2*a*b*c + a^2*d)
*f^2*e + (b^2*c + 2*a*b*d)*f*e^2)^(1/3)/(c*f - d*e))*log((3*a^2*c*f^2 + (2*a*b*c + a^2*d)*f^2*x - 3*(a^2*c*f^3
- b^2*d*e^3 - (2*a*b*c + a^2*d)*f^2*e + (b^2*c + 2*a*b*d)*f*e^2)^(1/3)*(a*f - b*e)*(b*x + a)^(1/3)*(d*x + c)^
(2/3) + 3*sqrt(1/3)*(2*(a*c*f^2 + b*d*e^2 - (b*c + a*d)*f*e)*(b*x + a)^(2/3)*(d*x + c)^(1/3) - (a^2*c*f^3 - b^
2*d*e^3 - (2*a*b*c + a^2*d)*f^2*e + (b^2*c + 2*a*b*d)*f*e^2)^(2/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3) - (a^2*c*f^
3 - b^2*d*e^3 - (2*a*b*c + a^2*d)*f^2*e + (b^2*c + 2*a*b*d)*f*e^2)^(1/3)*(a*d*f*x + a*c*f - (b*d*x + b*c)*e))*
sqrt(-(a^2*c*f^3 - b^2*d*e^3 - (2*a*b*c + a^2*d)*f^2*e + (b^2*c + 2*a*b*d)*f*e^2)^(1/3)/(c*f - d*e)) + (3*b^2*
d*x + b^2*c + 2*a*b*d)*e^2 - 2*((b^2*c + 2*a*b*d)*f*x + (2*a*b*c + a^2*d)*f)*e)/(f*x + e)) + 2*((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 + (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 - 27*(b^3*c^2*d^2 - a*b^2*c*d^3 + (b^3*c*d^3 - a*b^2*d^4)*x)*e^5 - 9*(9*(b^3*c*d^3 - a*b^2*d^4)*
f*x^2 + (10*b^3*c^2*d^2 - 17*a*b^2*c*d^3 + 7*a^2*b*d^4)*f*x + (b^3*c^3*d - 8*a*b^2*c^2*d^2 + 7*a^2*b*c*d^3)*f)
*e^4 - (81*(b^3*c*d^3 - a*b^2*d^4)*f^2*x^3 + 27*(4*b^3*c^2*d^2 - 11*a*b^2*c*d^3 + 7*a^2*b*d^4)*f^2*x^2 + (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)*f^2*x - (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^2)*e^3 - 3*(9*(b^3*c*d^3 - a*b^2*d^4)*f^3*x^4 + 9*(2*b^3*c^2*d^2 - 9*a*b^2*c*d^3 + 7*a^2*
b*d^4)*f^3*x^3 + (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)*f^3*x^2 - (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^3*x)*e^2 - 3*(3*(b^3*c^2*d^2 - 8*a*b^2*c*d^3 + 7*a^2*b*d^4)*f^4*x^4
+ (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)*f^4*x^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)*f^4*x^2)*e)*(a^2*c*f^3 - b^2*d*e^3 - (2*a*b*c + a^2*d)*f^2*e + (b^2*c + 2*a*b*d)*f*e
^2)^(2/3)*log(((a*c*f^2 + b*d*e^2 - (b*c + a*d)*f*e)*(b*x + a)^(2/3)*(d*x + c)^(1/3) + (a^2*c*f^3 - b^2*d*e^3
- (2*a*b*c + a^2*d)*f^2*e + (b^2*c + 2*a*b*d)*f*e^2)^(2/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3) + (a^2*c*f^3 - b^2*
d*e^3 - (2*a*b*c + a^2*d)*f^2*e + (b^2*c + 2*a*b*d)*f*e^2)^(1/3)*(a*d*f*x + a*c*f - (b*d*x + b*c)*e))/(d*x + c
)) - 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 + (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 - 27*(b^3*c^2*d^2 - a*b^2*c*d^3 + (b^3*c*d^3 - a*b^2*d^4)*x)*e^5 - 9*(9*(b
^3*c*d^3 - a*b^2*d^4)*f*x^2 + (10*b^3*c^2*d^2 - 17*a*b^2*c*d^3 + 7*a^2*b*d^4)*f*x + (b^3*c^3*d - 8*a*b^2*c^2*d
^2 + 7*a^2*b*c*d^3)*f)*e^4 - (81*(b^3*c*d^3 - a*b^2*d^4)*f^2*x^3 + 27*(4*b^3*c^2*d^2 - 11*a*b^2*c*d^3 + 7*a^2*
b*d^4)*f^2*x^2 + (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)*f^2*x - (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^2)*e^3 - 3*(9*(b^3*c*d^3 - a*b^2*d^4)*f^3*x^4 + 9*(2*b^3*c^2*d^2 -
9*a*b^2*c*d^3 + 7*a^2*b*d^4)*f^3*x^3 + (8*b^3*c...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^{4/3}}{{\left (e+f\,x\right )}^4\,{\left (c+d\,x\right )}^{4/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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