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

Optimal. Leaf size=386 \[ -\frac {f (a+b x)^{4/3} (c+d x)^{2/3}}{2 (b e-a f) (d e-c f) (e+f x)^2}+\frac {(3 b d e-b c f-2 a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{3 (b e-a f) (d e-c f)^2 (e+f x)}+\frac {(b c-a d) (3 b d e-b c f-2 a d f) \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}\right )}{3 \sqrt {3} (b e-a f)^{5/3} (d e-c f)^{7/3}}-\frac {(b c-a d) (3 b d e-b c f-2 a d f) \log (e+f x)}{18 (b e-a f)^{5/3} (d e-c f)^{7/3}}+\frac {(b c-a d) (3 b d e-b c f-2 a d f) \log \left (-\sqrt [3]{a+b x}+\frac {\sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt [3]{d e-c f}}\right )}{6 (b e-a f)^{5/3} (d e-c f)^{7/3}} \]

[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)

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Rubi [A]
time = 0.14, antiderivative size = 386, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {98, 96, 93} \begin {gather*} \frac {(b c-a d) (-2 a d f-b c f+3 b d e) \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 )}{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}} \end {gather*}

Antiderivative was successfully verified.

[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*} \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) \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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 10.15, size = 175, normalized size = 0.45 \begin {gather*} \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) \, _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)}\right )}{6 (b e-a f) (d e-c f) \sqrt [3]{c+d x} (e+f x)^2} \end {gather*}

Antiderivative was successfully verified.

[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)

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

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)^3,x)

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2284 vs. \(2 (353) = 706\).
time = 3.25, size = 4727, normalized size = 12.25 \begin {gather*} \text {Too large to display} \end {gather*}

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)^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{a + b x}}{\sqrt [3]{c + d x} \left (e + f x\right )^{3}}\, dx \end {gather*}

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)**3,x)

[Out]

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

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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)^(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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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