3.31.21 \(\int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)^3} \, dx\) [3021]

Optimal. Leaf size=477 \[ -\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x}}{2 (b e-a f) (d e-c f) (e+f x)^2}-\frac {f (9 b d e-4 b c f-5 a d f) (a+b x)^{2/3} \sqrt [3]{c+d x}}{6 (b e-a f)^2 (d e-c f)^2 (e+f x)}-\frac {\left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (9 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right ) \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}\right )}{3 \sqrt {3} (b e-a f)^{7/3} (d e-c f)^{8/3}}+\frac {\left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (9 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right ) \log (e+f x)}{18 (b e-a f)^{7/3} (d e-c f)^{8/3}}-\frac {\left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (9 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right ) \log \left (\frac {\sqrt [3]{d e-c f} \sqrt [3]{a+b x}}{\sqrt [3]{b e-a f}}-\sqrt [3]{c+d x}\right )}{6 (b e-a f)^{7/3} (d e-c f)^{8/3}} \]

[Out]

-1/2*f*(b*x+a)^(2/3)*(d*x+c)^(1/3)/(-a*f+b*e)/(-c*f+d*e)/(f*x+e)^2-1/6*f*(-5*a*d*f-4*b*c*f+9*b*d*e)*(b*x+a)^(2
/3)*(d*x+c)^(1/3)/(-a*f+b*e)^2/(-c*f+d*e)^2/(f*x+e)+1/18*(5*a^2*d^2*f^2-2*a*b*d*f*(-c*f+6*d*e)+b^2*(2*c^2*f^2-
6*c*d*e*f+9*d^2*e^2))*ln(f*x+e)/(-a*f+b*e)^(7/3)/(-c*f+d*e)^(8/3)-1/6*(5*a^2*d^2*f^2-2*a*b*d*f*(-c*f+6*d*e)+b^
2*(2*c^2*f^2-6*c*d*e*f+9*d^2*e^2))*ln((-c*f+d*e)^(1/3)*(b*x+a)^(1/3)/(-a*f+b*e)^(1/3)-(d*x+c)^(1/3))/(-a*f+b*e
)^(7/3)/(-c*f+d*e)^(8/3)-1/9*(5*a^2*d^2*f^2-2*a*b*d*f*(-c*f+6*d*e)+b^2*(2*c^2*f^2-6*c*d*e*f+9*d^2*e^2))*arctan
(1/3*3^(1/2)+2/3*(-c*f+d*e)^(1/3)*(b*x+a)^(1/3)/(-a*f+b*e)^(1/3)/(d*x+c)^(1/3)*3^(1/2))/(-a*f+b*e)^(7/3)/(-c*f
+d*e)^(8/3)*3^(1/2)

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Rubi [A]
time = 0.43, antiderivative size = 477, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {105, 156, 12, 93} \begin {gather*} -\frac {\left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (2 c^2 f^2-6 c d e f+9 d^2 e^2\right )\right ) \text {ArcTan}\left (\frac {2 \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}{\sqrt {3} \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} (b e-a f)^{7/3} (d e-c f)^{8/3}}+\frac {\log (e+f x) \left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (2 c^2 f^2-6 c d e f+9 d^2 e^2\right )\right )}{18 (b e-a f)^{7/3} (d e-c f)^{8/3}}-\frac {\left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (2 c^2 f^2-6 c d e f+9 d^2 e^2\right )\right ) \log \left (\frac {\sqrt [3]{a+b x} \sqrt [3]{d e-c f}}{\sqrt [3]{b e-a f}}-\sqrt [3]{c+d x}\right )}{6 (b e-a f)^{7/3} (d e-c f)^{8/3}}-\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x} (-5 a d f-4 b c f+9 b d e)}{6 (e+f x) (b e-a f)^2 (d e-c f)^2}-\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x}}{2 (e+f x)^2 (b e-a f) (d e-c f)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^(1/3)*(c + d*x)^(2/3)*(e + f*x)^3),x]

[Out]

-1/2*(f*(a + b*x)^(2/3)*(c + d*x)^(1/3))/((b*e - a*f)*(d*e - c*f)*(e + f*x)^2) - (f*(9*b*d*e - 4*b*c*f - 5*a*d
*f)*(a + b*x)^(2/3)*(c + d*x)^(1/3))/(6*(b*e - a*f)^2*(d*e - c*f)^2*(e + f*x)) - ((5*a^2*d^2*f^2 - 2*a*b*d*f*(
6*d*e - c*f) + b^2*(9*d^2*e^2 - 6*c*d*e*f + 2*c^2*f^2))*ArcTan[1/Sqrt[3] + (2*(d*e - c*f)^(1/3)*(a + b*x)^(1/3
))/(Sqrt[3]*(b*e - a*f)^(1/3)*(c + d*x)^(1/3))])/(3*Sqrt[3]*(b*e - a*f)^(7/3)*(d*e - c*f)^(8/3)) + ((5*a^2*d^2
*f^2 - 2*a*b*d*f*(6*d*e - c*f) + b^2*(9*d^2*e^2 - 6*c*d*e*f + 2*c^2*f^2))*Log[e + f*x])/(18*(b*e - a*f)^(7/3)*
(d*e - c*f)^(8/3)) - ((5*a^2*d^2*f^2 - 2*a*b*d*f*(6*d*e - c*f) + b^2*(9*d^2*e^2 - 6*c*d*e*f + 2*c^2*f^2))*Log[
((d*e - c*f)^(1/3)*(a + b*x)^(1/3))/(b*e - a*f)^(1/3) - (c + d*x)^(1/3)])/(6*(b*e - a*f)^(7/3)*(d*e - c*f)^(8/
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 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> 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] + 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*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

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 {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)^3} \, dx &=-\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x}}{2 (b e-a f) (d e-c f) (e+f x)^2}-\frac {\int \frac {\frac {1}{3} (-6 b d e+4 b c f+5 a d f)+b d f x}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)^2} \, dx}{2 (b e-a f) (d e-c f)}\\ &=-\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x}}{2 (b e-a f) (d e-c f) (e+f x)^2}-\frac {f (9 b d e-4 b c f-5 a d f) (a+b x)^{2/3} \sqrt [3]{c+d x}}{6 (b e-a f)^2 (d e-c f)^2 (e+f x)}+\frac {\int \frac {2 \left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (9 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right )}{9 \sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)} \, dx}{2 (b e-a f)^2 (d e-c f)^2}\\ &=-\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x}}{2 (b e-a f) (d e-c f) (e+f x)^2}-\frac {f (9 b d e-4 b c f-5 a d f) (a+b x)^{2/3} \sqrt [3]{c+d x}}{6 (b e-a f)^2 (d e-c f)^2 (e+f x)}+\frac {\left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (9 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right ) \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)} \, dx}{9 (b e-a f)^2 (d e-c f)^2}\\ &=-\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x}}{2 (b e-a f) (d e-c f) (e+f x)^2}-\frac {f (9 b d e-4 b c f-5 a d f) (a+b x)^{2/3} \sqrt [3]{c+d x}}{6 (b e-a f)^2 (d e-c f)^2 (e+f x)}-\frac {\left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (9 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right ) \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}\right )}{3 \sqrt {3} (b e-a f)^{7/3} (d e-c f)^{8/3}}+\frac {\left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (9 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right ) \log (e+f x)}{18 (b e-a f)^{7/3} (d e-c f)^{8/3}}-\frac {\left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (9 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right ) \log \left (\frac {\sqrt [3]{d e-c f} \sqrt [3]{a+b x}}{\sqrt [3]{b e-a f}}-\sqrt [3]{c+d x}\right )}{6 (b e-a f)^{7/3} (d e-c f)^{8/3}}\\ \end {align*}

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Mathematica [A]
time = 4.57, size = 525, normalized size = 1.10 \begin {gather*} \frac {1}{18} \left (\frac {3 f (a+b x)^{2/3} \sqrt [3]{c+d x} (-3 b d e (4 e+3 f x)+b c f (7 e+4 f x)+a f (8 d e-3 c f+5 d f x))}{(b e-a f)^2 (d e-c f)^2 (e+f x)^2}-\frac {2 \sqrt {3} \left (5 a^2 d^2 f^2+2 a b d f (-6 d e+c f)+b^2 \left (9 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{-b e+a f} \sqrt [3]{c+d x}}{\sqrt [3]{d e-c f} \sqrt [3]{a+b x}}}{\sqrt {3}}\right )}{(-b e+a f)^{7/3} (d e-c f)^{8/3}}+\frac {2 \left (5 a^2 d^2 f^2+2 a b d f (-6 d e+c f)+b^2 \left (9 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right ) \log \left (\sqrt [3]{d e-c f}+\frac {\sqrt [3]{-b e+a f} \sqrt [3]{c+d x}}{\sqrt [3]{a+b x}}\right )}{(-b e+a f)^{7/3} (d e-c f)^{8/3}}-\frac {\left (5 a^2 d^2 f^2+2 a b d f (-6 d e+c f)+b^2 \left (9 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right ) \log \left ((d e-c f)^{2/3}-\frac {\sqrt [3]{-b e+a f} \sqrt [3]{d e-c f} \sqrt [3]{c+d x}}{\sqrt [3]{a+b x}}+\frac {(-b e+a f)^{2/3} (c+d x)^{2/3}}{(a+b x)^{2/3}}\right )}{(-b e+a f)^{7/3} (d e-c f)^{8/3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^(1/3)*(c + d*x)^(2/3)*(e + f*x)^3),x]

[Out]

((3*f*(a + b*x)^(2/3)*(c + d*x)^(1/3)*(-3*b*d*e*(4*e + 3*f*x) + b*c*f*(7*e + 4*f*x) + a*f*(8*d*e - 3*c*f + 5*d
*f*x)))/((b*e - a*f)^2*(d*e - c*f)^2*(e + f*x)^2) - (2*Sqrt[3]*(5*a^2*d^2*f^2 + 2*a*b*d*f*(-6*d*e + c*f) + b^2
*(9*d^2*e^2 - 6*c*d*e*f + 2*c^2*f^2))*ArcTan[(1 - (2*(-(b*e) + a*f)^(1/3)*(c + d*x)^(1/3))/((d*e - c*f)^(1/3)*
(a + b*x)^(1/3)))/Sqrt[3]])/((-(b*e) + a*f)^(7/3)*(d*e - c*f)^(8/3)) + (2*(5*a^2*d^2*f^2 + 2*a*b*d*f*(-6*d*e +
c*f) + b^2*(9*d^2*e^2 - 6*c*d*e*f + 2*c^2*f^2))*Log[(d*e - c*f)^(1/3) + ((-(b*e) + a*f)^(1/3)*(c + d*x)^(1/3)
)/(a + b*x)^(1/3)])/((-(b*e) + a*f)^(7/3)*(d*e - c*f)^(8/3)) - ((5*a^2*d^2*f^2 + 2*a*b*d*f*(-6*d*e + c*f) + b^
2*(9*d^2*e^2 - 6*c*d*e*f + 2*c^2*f^2))*Log[(d*e - c*f)^(2/3) - ((-(b*e) + a*f)^(1/3)*(d*e - c*f)^(1/3)*(c + d*
x)^(1/3))/(a + b*x)^(1/3) + ((-(b*e) + a*f)^(2/3)*(c + d*x)^(2/3))/(a + b*x)^(2/3)])/((-(b*e) + a*f)^(7/3)*(d*
e - c*f)^(8/3)))/18

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b x +a \right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {2}{3}} \left (f x +e \right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^(1/3)/(d*x+c)^(2/3)/(f*x+e)^3,x)

[Out]

int(1/(b*x+a)^(1/3)/(d*x+c)^(2/3)/(f*x+e)^3,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(1/(b*x+a)^(1/3)/(d*x+c)^(2/3)/(f*x+e)^3,x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(1/3)*(d*x + c)^(2/3)*(f*x + e)^3), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2627 vs. \(2 (445) = 890\).
time = 15.73, size = 5414, normalized size = 11.35 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^(1/3)/(d*x+c)^(2/3)/(f*x+e)^3,x, algorithm="fricas")

[Out]

[-1/18*(3*sqrt(1/3)*((2*a*b^2*c^3 + 2*a^2*b*c^2*d + 5*a^3*c*d^2)*f^6*x^2 + 9*b^3*d^3*e^6 + 3*(6*b^3*d^3*f*x -
(5*b^3*c*d^2 + 7*a*b^2*d^3)*f)*e^5 + (9*b^3*d^3*f^2*x^2 - 6*(5*b^3*c*d^2 + 7*a*b^2*d^3)*f^2*x + (8*b^3*c^2*d +
29*a*b^2*c*d^2 + 17*a^2*b*d^3)*f^2)*e^4 - (3*(5*b^3*c*d^2 + 7*a*b^2*d^3)*f^3*x^2 - 2*(8*b^3*c^2*d + 29*a*b^2*
c*d^2 + 17*a^2*b*d^3)*f^3*x + (2*b^3*c^3 + 10*a*b^2*c^2*d + 19*a^2*b*c*d^2 + 5*a^3*d^3)*f^3)*e^3 + ((8*b^3*c^2
*d + 29*a*b^2*c*d^2 + 17*a^2*b*d^3)*f^4*x^2 - 2*(2*b^3*c^3 + 10*a*b^2*c^2*d + 19*a^2*b*c*d^2 + 5*a^3*d^3)*f^4*
x + (2*a*b^2*c^3 + 2*a^2*b*c^2*d + 5*a^3*c*d^2)*f^4)*e^2 - ((2*b^3*c^3 + 10*a*b^2*c^2*d + 19*a^2*b*c*d^2 + 5*a
^3*d^3)*f^5*x^2 - 2*(2*a*b^2*c^3 + 2*a^2*b*c^2*d + 5*a^3*c*d^2)*f^5*x)*e)*sqrt(-(a*c^2*f^3 - b*d^2*e^3 - (b*c^
2 + 2*a*c*d)*f^2*e + (2*b*c*d + a*d^2)*f*e^2)^(1/3)/(a*f - b*e))*log(-(3*a*c^2*f^2 + (b*c^2 + 2*a*c*d)*f^2*x -
3*(a*c^2*f^3 - b*d^2*e^3 - (b*c^2 + 2*a*c*d)*f^2*e + (2*b*c*d + a*d^2)*f*e^2)^(1/3)*(c*f - d*e)*(b*x + a)^(2/
3)*(d*x + c)^(1/3) + 3*sqrt(1/3)*(2*(a*c*f^2 + b*d*e^2 - (b*c + a*d)*f*e)*(b*x + a)^(1/3)*(d*x + c)^(2/3) - (a
*c^2*f^3 - b*d^2*e^3 - (b*c^2 + 2*a*c*d)*f^2*e + (2*b*c*d + a*d^2)*f*e^2)^(2/3)*(b*x + a)^(2/3)*(d*x + c)^(1/3
) - (a*c^2*f^3 - b*d^2*e^3 - (b*c^2 + 2*a*c*d)*f^2*e + (2*b*c*d + a*d^2)*f*e^2)^(1/3)*(b*c*f*x + a*c*f - (b*d*
x + a*d)*e))*sqrt(-(a*c^2*f^3 - b*d^2*e^3 - (b*c^2 + 2*a*c*d)*f^2*e + (2*b*c*d + a*d^2)*f*e^2)^(1/3)/(a*f - b*
e)) + (3*b*d^2*x + 2*b*c*d + a*d^2)*e^2 - 2*((2*b*c*d + a*d^2)*f*x + (b*c^2 + 2*a*c*d)*f)*e)/(f*x + e)) - 2*((
2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*f^4*x^2 + 9*b^2*d^2*e^4 + 6*(3*b^2*d^2*f*x - (b^2*c*d + 2*a*b*d^2)*f)*e^3 +
(9*b^2*d^2*f^2*x^2 - 12*(b^2*c*d + 2*a*b*d^2)*f^2*x + (2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*f^2)*e^2 - 2*(3*(b^
2*c*d + 2*a*b*d^2)*f^3*x^2 - (2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*f^3*x)*e)*(a*c^2*f^3 - b*d^2*e^3 - (b*c^2 + 2
*a*c*d)*f^2*e + (2*b*c*d + a*d^2)*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*c^2*f^3 - b*d^2*e^3 - (b*c^2 + 2*a*c*d)*f^2*e + (2*b*c*d + a*d^2)*f*e^2)^(2/3)*(b*x + a))/(b*
x + a)) + ((2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*f^4*x^2 + 9*b^2*d^2*e^4 + 6*(3*b^2*d^2*f*x - (b^2*c*d + 2*a*b*d
^2)*f)*e^3 + (9*b^2*d^2*f^2*x^2 - 12*(b^2*c*d + 2*a*b*d^2)*f^2*x + (2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*f^2)*e^
2 - 2*(3*(b^2*c*d + 2*a*b*d^2)*f^3*x^2 - (2*b^2*c^2 + 2*a*b*c*d + 5*a^2*d^2)*f^3*x)*e)*(a*c^2*f^3 - b*d^2*e^3
- (b*c^2 + 2*a*c*d)*f^2*e + (2*b*c*d + a*d^2)*f*e^2)^(2/3)*log(((a*c*f^2 + b*d*e^2 - (b*c + a*d)*f*e)*(b*x + a
)^(1/3)*(d*x + c)^(2/3) + (a*c^2*f^3 - b*d^2*e^3 - (b*c^2 + 2*a*c*d)*f^2*e + (2*b*c*d + a*d^2)*f*e^2)^(2/3)*(b
*x + a)^(2/3)*(d*x + c)^(1/3) + (a*c^2*f^3 - b*d^2*e^3 - (b*c^2 + 2*a*c*d)*f^2*e + (2*b*c*d + a*d^2)*f*e^2)^(1
/3)*(b*c*f*x + a*c*f - (b*d*x + a*d)*e))/(b*x + a)) + 3*(3*a^2*c^3*f^6 - (4*a*b*c^3 + 5*a^2*c^2*d)*f^6*x - 12*
b^2*d^3*f*e^5 - (9*b^2*d^3*f^2*x - (31*b^2*c*d^2 + 20*a*b*d^3)*f^2)*e^4 + 2*((11*b^2*c*d^2 + 7*a*b*d^3)*f^3*x
- (13*b^2*c^2*d + 25*a*b*c*d^2 + 4*a^2*d^3)*f^3)*e^3 - ((17*b^2*c^2*d + 32*a*b*c*d^2 + 5*a^2*d^3)*f^4*x - (7*b
^2*c^3 + 40*a*b*c^2*d + 19*a^2*c*d^2)*f^4)*e^2 + 2*((2*b^2*c^3 + 11*a*b*c^2*d + 5*a^2*c*d^2)*f^5*x - (5*a*b*c^
3 + 7*a^2*c^2*d)*f^5)*e)*(b*x + a)^(2/3)*(d*x + c)^(1/3))/(a^3*c^4*f^9*x^2 - b^3*d^4*e^9 - (2*b^3*d^4*f*x - (4
*b^3*c*d^3 + 3*a*b^2*d^4)*f)*e^8 - (b^3*d^4*f^2*x^2 - 2*(4*b^3*c*d^3 + 3*a*b^2*d^4)*f^2*x + 3*(2*b^3*c^2*d^2 +
4*a*b^2*c*d^3 + a^2*b*d^4)*f^2)*e^7 + ((4*b^3*c*d^3 + 3*a*b^2*d^4)*f^3*x^2 - 6*(2*b^3*c^2*d^2 + 4*a*b^2*c*d^3
+ a^2*b*d^4)*f^3*x + (4*b^3*c^3*d + 18*a*b^2*c^2*d^2 + 12*a^2*b*c*d^3 + a^3*d^4)*f^3)*e^6 - (3*(2*b^3*c^2*d^2
+ 4*a*b^2*c*d^3 + a^2*b*d^4)*f^4*x^2 - 2*(4*b^3*c^3*d + 18*a*b^2*c^2*d^2 + 12*a^2*b*c*d^3 + a^3*d^4)*f^4*x +
(b^3*c^4 + 12*a*b^2*c^3*d + 18*a^2*b*c^2*d^2 + 4*a^3*c*d^3)*f^4)*e^5 + ((4*b^3*c^3*d + 18*a*b^2*c^2*d^2 + 12*a
^2*b*c*d^3 + a^3*d^4)*f^5*x^2 - 2*(b^3*c^4 + 12*a*b^2*c^3*d + 18*a^2*b*c^2*d^2 + 4*a^3*c*d^3)*f^5*x + 3*(a*b^2
*c^4 + 4*a^2*b*c^3*d + 2*a^3*c^2*d^2)*f^5)*e^4 - ((b^3*c^4 + 12*a*b^2*c^3*d + 18*a^2*b*c^2*d^2 + 4*a^3*c*d^3)*
f^6*x^2 - 6*(a*b^2*c^4 + 4*a^2*b*c^3*d + 2*a^3*c^2*d^2)*f^6*x + (3*a^2*b*c^4 + 4*a^3*c^3*d)*f^6)*e^3 + (a^3*c^
4*f^7 + 3*(a*b^2*c^4 + 4*a^2*b*c^3*d + 2*a^3*c^2*d^2)*f^7*x^2 - 2*(3*a^2*b*c^4 + 4*a^3*c^3*d)*f^7*x)*e^2 + (2*
a^3*c^4*f^8*x - (3*a^2*b*c^4 + 4*a^3*c^3*d)*f^8*x^2)*e), -1/18*(6*sqrt(1/3)*((2*a*b^2*c^3 + 2*a^2*b*c^2*d + 5*
a^3*c*d^2)*f^6*x^2 + 9*b^3*d^3*e^6 + 3*(6*b^3*d^3*f*x - (5*b^3*c*d^2 + 7*a*b^2*d^3)*f)*e^5 + (9*b^3*d^3*f^2*x^
2 - 6*(5*b^3*c*d^2 + 7*a*b^2*d^3)*f^2*x + (8*b^3*c^2*d + 29*a*b^2*c*d^2 + 17*a^2*b*d^3)*f^2)*e^4 - (3*(5*b^3*c
*d^2 + 7*a*b^2*d^3)*f^3*x^2 - 2*(8*b^3*c^2*d + 29*a*b^2*c*d^2 + 17*a^2*b*d^3)*f^3*x + (2*b^3*c^3 + 10*a*b^2*c^
2*d + 19*a^2*b*c*d^2 + 5*a^3*d^3)*f^3)*e^3 + ((8*b^3*c^2*d + 29*a*b^2*c*d^2 + 17*a^2*b*d^3)*f^4*x^2 - 2*(2*b^3
*c^3 + 10*a*b^2*c^2*d + 19*a^2*b*c*d^2 + 5*a^3*d^3)*f^4*x + (2*a*b^2*c^3 + 2*a^2*b*c^2*d + 5*a^3*c*d^2)*f^4)*e
^2 - ((2*b^3*c^3 + 10*a*b^2*c^2*d + 19*a^2*b*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(1/(b*x+a)**(1/3)/(d*x+c)**(2/3)/(f*x+e)**3,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(1/(b*x+a)^(1/3)/(d*x+c)^(2/3)/(f*x+e)^3,x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(1/3)*(d*x + c)^(2/3)*(f*x + e)^3), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (e+f\,x\right )}^3\,{\left (a+b\,x\right )}^{1/3}\,{\left (c+d\,x\right )}^{2/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((e + f*x)^3*(a + b*x)^(1/3)*(c + d*x)^(2/3)),x)

[Out]

int(1/((e + f*x)^3*(a + b*x)^(1/3)*(c + d*x)^(2/3)), x)

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