\(\int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^3} \, dx\) [3013]

Optimal result

Integrand size = 26, antiderivative size = 386 \[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^3} \, dx=-\frac {f (a+b x)^{4/3} (c+d x)^{2/3}}{2 (b e-a f) (d e-c f) (e+f x)^2}+\frac {(3 b d e-b c f-2 a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{3 (b e-a f) (d e-c f)^2 (e+f x)}+\frac {(b c-a d) (3 b d e-b c f-2 a d f) \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}\right )}{3 \sqrt {3} (b e-a f)^{5/3} (d e-c f)^{7/3}}-\frac {(b c-a d) (3 b d e-b c f-2 a d f) \log (e+f x)}{18 (b e-a f)^{5/3} (d e-c f)^{7/3}}+\frac {(b c-a d) (3 b d e-b c f-2 a d f) \log \left (-\sqrt [3]{a+b x}+\frac {\sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt [3]{d e-c f}}\right )}{6 (b e-a f)^{5/3} (d e-c f)^{7/3}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {98, 96, 93} \[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^3} \, dx=\frac {(b c-a d) (-2 a d f-b c f+3 b d e) \arctan \left (\frac {2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt {3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} (b e-a f)^{5/3} (d e-c f)^{7/3}}-\frac {f (a+b x)^{4/3} (c+d x)^{2/3}}{2 (e+f x)^2 (b e-a f) (d e-c f)}+\frac {\sqrt [3]{a+b x} (c+d x)^{2/3} (-2 a d f-b c f+3 b d e)}{3 (e+f x) (b e-a f) (d e-c f)^2}-\frac {(b c-a d) \log (e+f x) (-2 a d f-b c f+3 b d e)}{18 (b e-a f)^{5/3} (d e-c f)^{7/3}}+\frac {(b c-a d) (-2 a d f-b c f+3 b d e) \log \left (\frac {\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{6 (b e-a f)^{5/3} (d e-c f)^{7/3}} \]

[In]

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

[Out]

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

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 96

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[n*((d*e - c*f)/((m + 1)*(b*e - a*f
))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

Rule 98

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[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 10.17 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.45 \[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^3} \, dx=\frac {\sqrt [3]{a+b x} \left (-3 f (a+b x) (c+d x)+\frac {2 (3 b d e-b c f-2 a d f) (e+f x) \left ((b e-a f) (c+d x)-(b c-a d) (e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {4}{3},\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )\right )}{(b e-a f) (d e-c f)}\right )}{6 (b e-a f) (d e-c f) \sqrt [3]{c+d x} (e+f x)^2} \]

[In]

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

[Out]

((a + b*x)^(1/3)*(-3*f*(a + b*x)*(c + d*x) + (2*(3*b*d*e - b*c*f - 2*a*d*f)*(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))]))/((b*e
- a*f)*(d*e - c*f))))/(6*(b*e - a*f)*(d*e - c*f)*(c + d*x)^(1/3)*(e + f*x)^2)

Maple [F]

\[\int \frac {\left (b x +a \right )^{\frac {1}{3}}}{\left (d x +c \right )^{\frac {1}{3}} \left (f x +e \right )^{3}}d x\]

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2307 vs. \(2 (336) = 672\).

Time = 0.73 (sec) , antiderivative size = 4776, normalized size of antiderivative = 12.37 \[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/18*(3*sqrt(1/3)*(3*(b^3*c*d^2 - a*b^2*d^3)*e^5 - (4*b^3*c^2*d + a*b^2*c*d^2 - 5*a^2*b*d^3)*e^4*f + (b^3*c^3
+ 5*a*b^2*c^2*d - 4*a^2*b*c*d^2 - 2*a^3*d^3)*e^3*f^2 - (a*b^2*c^3 + a^2*b*c^2*d - 2*a^3*c*d^2)*e^2*f^3 + (3*(
b^3*c*d^2 - a*b^2*d^3)*e^3*f^2 - (4*b^3*c^2*d + a*b^2*c*d^2 - 5*a^2*b*d^3)*e^2*f^3 + (b^3*c^3 + 5*a*b^2*c^2*d
- 4*a^2*b*c*d^2 - 2*a^3*d^3)*e*f^4 - (a*b^2*c^3 + a^2*b*c^2*d - 2*a^3*c*d^2)*f^5)*x^2 + 2*(3*(b^3*c*d^2 - a*b^
2*d^3)*e^4*f - (4*b^3*c^2*d + a*b^2*c*d^2 - 5*a^2*b*d^3)*e^3*f^2 + (b^3*c^3 + 5*a*b^2*c^2*d - 4*a^2*b*c*d^2 -
2*a^3*d^3)*e^2*f^3 - (a*b^2*c^3 + a^2*b*c^2*d - 2*a^3*c*d^2)*e*f^4)*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)*(3*(b^2*c*d - a*b*d^2)*e^3 - (b
^2*c^2 + a*b*c*d - 2*a^2*d^2)*e^2*f + (3*(b^2*c*d - a*b*d^2)*e*f^2 - (b^2*c^2 + a*b*c*d - 2*a^2*d^2)*f^3)*x^2
+ 2*(3*(b^2*c*d - a*b*d^2)*e^2*f - (b^2*c^2 + a*b*c*d - 2*a^2*d^2)*e*f^2)*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)*(3*(b^2*c*d - a*b*d^2)*e^3 - (b^2*c^2 + a*b*c*d - 2*a^2*d^2)*e^2
*f + (3*(b^2*c*d - a*b*d^2)*e*f^2 - (b^2*c^2 + a*b*c*d - 2*a^2*d^2)*f^3)*x^2 + 2*(3*(b^2*c*d - a*b*d^2)*e^2*f
- (b^2*c^2 + a*b*c*d - 2*a^2*d^2)*e*f^2)*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*(6*b^3*d^2*e^5 - 3*a^3*c^2*f^5 - (8*b^3*c*d + 19*a*b^2*d^2)*e^4*f + 2*(b^3*c^2 + 13*a*b^2*c*d + 10*a^
2*b*d^2)*e^3*f^2 - 7*(a*b^2*c^2 + 4*a^2*b*c*d + a^3*d^2)*e^2*f^3 + 2*(4*a^2*b*c^2 + 5*a^3*c*d)*e*f^4 + (3*b^3*
d^2*e^4*f - 2*(b^3*c*d + 5*a*b^2*d^2)*e^3*f^2 - (b^3*c^2 - 8*a*b^2*c*d - 11*a^2*b*d^2)*e^2*f^3 + 2*(a*b^2*c^2
- 5*a^2*b*c*d - 2*a^3*d^2)*e*f^4 - (a^2*b*c^2 - 4*a^3*c*d)*f^5)*x)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(b^3*d^3*e
^8 + a^3*c^3*e^2*f^6 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^7*f + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*e^6*f^2 - (
b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*e^5*f^3 + 3*(a*b^2*c^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*e^4*f^4
- 3*(a^2*b*c^3 + a^3*c^2*d)*e^3*f^5 + (b^3*d^3*e^6*f^2 + a^3*c^3*f^8 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^5*f^3 + 3*
(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*e^4*f^4 - (b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*e^3*f^5
+ 3*(a*b^2*c^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*e^2*f^6 - 3*(a^2*b*c^3 + a^3*c^2*d)*e*f^7)*x^2 + 2*(b^3*d^3*e^7*f
+ a^3*c^3*e*f^7 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^6*f^2 + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*e^5*f^3 - (b^3
*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*e^4*f^4 + 3*(a*b^2*c^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*e^3*f^5 -
3*(a^2*b*c^3 + a^3*c^2*d)*e^2*f^6)*x), -1/18*(6*sqrt(1/3)*(3*(b^3*c*d^2 - a*b^2*d^3)*e^5 - (4*b^3*c^2*d + a*b^
2*c*d^2 - 5*a^2*b*d^3)*e^4*f + (b^3*c^3 + 5*a*b^2*c^2*d - 4*a^2*b*c*d^2 - 2*a^3*d^3)*e^3*f^2 - (a*b^2*c^3 + a^
2*b*c^2*d - 2*a^3*c*d^2)*e^2*f^3 + (3*(b^3*c*d^2 - a*b^2*d^3)*e^3*f^2 - (4*b^3*c^2*d + a*b^2*c*d^2 - 5*a^2*b*d
^3)*e^2*f^3 + (b^3*c^3 + 5*a*b^2*c^2*d - 4*a^2*b*c*d^2 - 2*a^3*d^3)*e*f^4 - (a*b^2*c^3 + a^2*b*c^2*d - 2*a^3*c
*d^2)*f^5)*x^2 + 2*(3*(b^3*c*d^2 - a*b^2*d^3)*e^4*f - (4*b^3*c^2*d + a*b^2*c*d^2 - 5*a^2*b*d^3)*e^3*f^2 + (b^3
*c^3 + 5*a*b^2*c^2*d - 4*a^2*b*c*d^2 - 2*a^3*d^3)*e^2*f^3 - (a*b^2*c^3 + a^2*b*c^2*d - 2*a^3*c*d^2)*e*f^4)*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(sqr
t(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)*(3*(b^2*c*d - a*b*d^2)*e^3 - (b^2*c^2 +
a*b*c*d - 2*a^2*d^2)*e^2*f + (3*(b^2*c*d - a*b*d^2)*e*f^2 - (b^2*c^2 + a*b*c*d - 2*a^2*d^2)*f^3)*x^2 + 2*(3*(b
^2*c*d - a*b*d^2)*e^2*f - (b^2*c^2 + a*b*c*d - 2*a^2*d^2)*e*f^2)*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)*(3*(b^2*c*d - a*b*d^2)*e^3 - (b^2*c^2 + a*b*c*d - 2*a^2*d^2)*e^2*f + (3*(
b^2*c*d - a*b*d^2)*e*f^2 - (b^2*c^2 + a*b*c*d - 2*a^2*d^2)*f^3)*x^2 + 2*(3*(b^2*c*d - a*b*d^2)*e^2*f - (b^2*c^
2 + a*b*c*d - 2*a^2*d^2)*e*f^2)*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*
(6*b^3*d^2*e^5 - 3*a^3*c^2*f^5 - (8*b^3*c*d + 19*a*b^2*d^2)*e^4*f + 2*(b^3*c^2 + 13*a*b^2*c*d + 10*a^2*b*d^2)*
e^3*f^2 - 7*(a*b^2*c^2 + 4*a^2*b*c*d + a^3*d^2)*e^2*f^3 + 2*(4*a^2*b*c^2 + 5*a^3*c*d)*e*f^4 + (3*b^3*d^2*e^4*f
- 2*(b^3*c*d + 5*a*b^2*d^2)*e^3*f^2 - (b^3*c^2 - 8*a*b^2*c*d - 11*a^2*b*d^2)*e^2*f^3 + 2*(a*b^2*c^2 - 5*a^2*b
*c*d - 2*a^3*d^2)*e*f^4 - (a^2*b*c^2 - 4*a^3*c*d)*f^5)*x)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(b^3*d^3*e^8 + a^3*
c^3*e^2*f^6 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^7*f + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*e^6*f^2 - (b^3*c^3 +
9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*e^5*f^3 + 3*(a*b^2*c^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*e^4*f^4 - 3*(a^2
*b*c^3 + a^3*c^2*d)*e^3*f^5 + (b^3*d^3*e^6*f^2 + a^3*c^3*f^8 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^5*f^3 + 3*(b^3*c^2*
d + 3*a*b^2*c*d^2 + a^2*b*d^3)*e^4*f^4 - (b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*e^3*f^5 + 3*(a*b^
2*c^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*e^2*f^6 - 3*(a^2*b*c^3 + a^3*c^2*d)*e*f^7)*x^2 + 2*(b^3*d^3*e^7*f + a^3*c^3
*e*f^7 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^6*f^2 + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*e^5*f^3 - (b^3*c^3 + 9*
a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*e^4*f^4 + 3*(a*b^2*c^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*e^3*f^5 - 3*(a^2*b*
c^3 + a^3*c^2*d)*e^2*f^6)*x)]

Sympy [F]

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

[In]

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

[Out]

Integral((a + b*x)**(1/3)/((c + d*x)**(1/3)*(e + f*x)**3), x)

Maxima [F]

\[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^3} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{3}}}{{\left (d x + c\right )}^{\frac {1}{3}} {\left (f x + e\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^3} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{3}}}{{\left (d x + c\right )}^{\frac {1}{3}} {\left (f x + e\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^3} \, dx=\int \frac {{\left (a+b\,x\right )}^{1/3}}{{\left (e+f\,x\right )}^3\,{\left (c+d\,x\right )}^{1/3}} \,d x \]

[In]

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

[Out]

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