3.3007 \(\int \frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x)^4} \, dx\)

Optimal. Leaf size=465 \[ -\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (b c-a d) (-4 a d f-5 b c f+9 b d e)}{54 (e+f x) (b e-a f)^2 (d e-c f)^2}+\frac{\sqrt [3]{a+b x} (c+d x)^{5/3} (-4 a d f-5 b c f+9 b d e)}{18 (e+f x)^2 (b e-a f) (d e-c f)^2}-\frac{f (a+b x)^{4/3} (c+d x)^{5/3}}{3 (e+f x)^3 (b e-a f) (d e-c f)}-\frac{(b c-a d)^2 \log (e+f x) (-4 a d f-5 b c f+9 b d e)}{162 (b e-a f)^{8/3} (d e-c f)^{7/3}}+\frac{(b c-a d)^2 (-4 a d f-5 b c f+9 b d e) \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{54 (b e-a f)^{8/3} (d e-c f)^{7/3}}+\frac{(b c-a d)^2 (-4 a d f-5 b c f+9 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3} (b e-a f)^{8/3} (d e-c f)^{7/3}} \]

[Out]

-(f*(a + b*x)^(4/3)*(c + d*x)^(5/3))/(3*(b*e - a*f)*(d*e - c*f)*(e + f*x)^3) + ((9*b*d*e - 5*b*c*f - 4*a*d*f)*
(a + b*x)^(1/3)*(c + d*x)^(5/3))/(18*(b*e - a*f)*(d*e - c*f)^2*(e + f*x)^2) - ((b*c - a*d)*(9*b*d*e - 5*b*c*f
- 4*a*d*f)*(a + b*x)^(1/3)*(c + d*x)^(2/3))/(54*(b*e - a*f)^2*(d*e - c*f)^2*(e + f*x)) + ((b*c - a*d)^2*(9*b*d
*e - 5*b*c*f - 4*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))])/(27*Sqrt[3]*(b*e - a*f)^(8/3)*(d*e - c*f)^(7/3)) - ((b*c - a*d)^2*(9*b*d*e - 5*b*c*f - 4*a*d*
f)*Log[e + f*x])/(162*(b*e - a*f)^(8/3)*(d*e - c*f)^(7/3)) + ((b*c - a*d)^2*(9*b*d*e - 5*b*c*f - 4*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)])/(54*(b*e - a*f)^(8/3)*(d*e - c*f)^(
7/3))

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Rubi [A]  time = 0.376137, antiderivative size = 465, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {96, 94, 91} \[ -\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (b c-a d) (-4 a d f-5 b c f+9 b d e)}{54 (e+f x) (b e-a f)^2 (d e-c f)^2}+\frac{\sqrt [3]{a+b x} (c+d x)^{5/3} (-4 a d f-5 b c f+9 b d e)}{18 (e+f x)^2 (b e-a f) (d e-c f)^2}-\frac{f (a+b x)^{4/3} (c+d x)^{5/3}}{3 (e+f x)^3 (b e-a f) (d e-c f)}-\frac{(b c-a d)^2 \log (e+f x) (-4 a d f-5 b c f+9 b d e)}{162 (b e-a f)^{8/3} (d e-c f)^{7/3}}+\frac{(b c-a d)^2 (-4 a d f-5 b c f+9 b d e) \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{54 (b e-a f)^{8/3} (d e-c f)^{7/3}}+\frac{(b c-a d)^2 (-4 a d f-5 b c f+9 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3} (b e-a f)^{8/3} (d e-c f)^{7/3}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(1/3)*(c + d*x)^(2/3))/(e + f*x)^4,x]

[Out]

-(f*(a + b*x)^(4/3)*(c + d*x)^(5/3))/(3*(b*e - a*f)*(d*e - c*f)*(e + f*x)^3) + ((9*b*d*e - 5*b*c*f - 4*a*d*f)*
(a + b*x)^(1/3)*(c + d*x)^(5/3))/(18*(b*e - a*f)*(d*e - c*f)^2*(e + f*x)^2) - ((b*c - a*d)*(9*b*d*e - 5*b*c*f
- 4*a*d*f)*(a + b*x)^(1/3)*(c + d*x)^(2/3))/(54*(b*e - a*f)^2*(d*e - c*f)^2*(e + f*x)) + ((b*c - a*d)^2*(9*b*d
*e - 5*b*c*f - 4*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))])/(27*Sqrt[3]*(b*e - a*f)^(8/3)*(d*e - c*f)^(7/3)) - ((b*c - a*d)^2*(9*b*d*e - 5*b*c*f - 4*a*d*
f)*Log[e + f*x])/(162*(b*e - a*f)^(8/3)*(d*e - c*f)^(7/3)) + ((b*c - a*d)^2*(9*b*d*e - 5*b*c*f - 4*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)])/(54*(b*e - a*f)^(8/3)*(d*e - c*f)^(
7/3))

Rule 96

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])

Rule 94

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[p, 1] &&  !SumSimplerQ[m, 1])

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} (c+d x)^{2/3}}{(e+f x)^4} \, dx &=-\frac{f (a+b x)^{4/3} (c+d x)^{5/3}}{3 (b e-a f) (d e-c f) (e+f x)^3}+\frac{(9 b d e-5 b c f-4 a d f) \int \frac{\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x)^3} \, dx}{9 (b e-a f) (d e-c f)}\\ &=-\frac{f (a+b x)^{4/3} (c+d x)^{5/3}}{3 (b e-a f) (d e-c f) (e+f x)^3}+\frac{(9 b d e-5 b c f-4 a d f) \sqrt [3]{a+b x} (c+d x)^{5/3}}{18 (b e-a f) (d e-c f)^2 (e+f x)^2}-\frac{((b c-a d) (9 b d e-5 b c f-4 a d f)) \int \frac{(c+d x)^{2/3}}{(a+b x)^{2/3} (e+f x)^2} \, dx}{54 (b e-a f) (d e-c f)^2}\\ &=-\frac{f (a+b x)^{4/3} (c+d x)^{5/3}}{3 (b e-a f) (d e-c f) (e+f x)^3}+\frac{(9 b d e-5 b c f-4 a d f) \sqrt [3]{a+b x} (c+d x)^{5/3}}{18 (b e-a f) (d e-c f)^2 (e+f x)^2}-\frac{(b c-a d) (9 b d e-5 b c f-4 a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{54 (b e-a f)^2 (d e-c f)^2 (e+f x)}-\frac{\left ((b c-a d)^2 (9 b d e-5 b c f-4 a d f)\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)^2}\\ &=-\frac{f (a+b x)^{4/3} (c+d x)^{5/3}}{3 (b e-a f) (d e-c f) (e+f x)^3}+\frac{(9 b d e-5 b c f-4 a d f) \sqrt [3]{a+b x} (c+d x)^{5/3}}{18 (b e-a f) (d e-c f)^2 (e+f x)^2}-\frac{(b c-a d) (9 b d e-5 b c f-4 a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{54 (b e-a f)^2 (d e-c f)^2 (e+f x)}+\frac{(b c-a d)^2 (9 b d e-5 b c f-4 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 )}{27 \sqrt{3} (b e-a f)^{8/3} (d e-c f)^{7/3}}-\frac{(b c-a d)^2 (9 b d e-5 b c f-4 a d f) \log (e+f x)}{162 (b e-a f)^{8/3} (d e-c f)^{7/3}}+\frac{(b c-a d)^2 (9 b d e-5 b c f-4 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 )}{54 (b e-a f)^{8/3} (d e-c f)^{7/3}}\\ \end{align*}

Mathematica [C]  time = 0.45879, size = 214, normalized size = 0.46 \[ \frac{\sqrt [3]{a+b x} \left (\frac{(e+f x) (-4 a d f-5 b c f+9 b d e) \left (3 (c+d x)^2 (b e-a f)^2-(e+f x) (b c-a d) \left (2 (e+f x) (b c-a d) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )+(c+d x) (b e-a f)\right )\right )}{3 (b e-a f)^2 (d e-c f)}-6 f (a+b x) (c+d x)^2\right )}{18 \sqrt [3]{c+d x} (e+f x)^3 (b e-a f) (d e-c f)} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(1/3)*(c + d*x)^(2/3))/(e + f*x)^4,x]

[Out]

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

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( fx+e \right ) ^{4}}\sqrt [3]{bx+a} \left ( dx+c \right ) ^{{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(1/3)*(d*x+c)^(2/3)/(f*x+e)^4,x)

[Out]

int((b*x+a)^(1/3)*(d*x+c)^(2/3)/(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{2}{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)^(2/3)/(f*x+e)^4,x, algorithm="maxima")

[Out]

integrate((b*x + a)^(1/3)*(d*x + c)^(2/3)/(f*x + e)^4, x)

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Fricas [B]  time = 14.8324, size = 16349, normalized size = 35.16 \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)^(2/3)/(f*x+e)^4,x, algorithm="fricas")

[Out]

[1/162*(3*sqrt(1/3)*(9*(b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*e^6 - (14*b^4*c^3*d - 15*a*b^3*c^2*d^2 - 12
*a^2*b^2*c*d^3 + 13*a^3*b*d^4)*e^5*f + (5*b^4*c^4 + 8*a*b^3*c^3*d - 27*a^2*b^2*c^2*d^2 + 10*a^3*b*c*d^3 + 4*a^
4*d^4)*e^4*f^2 - (5*a*b^3*c^4 - 6*a^2*b^2*c^3*d - 3*a^3*b*c^2*d^2 + 4*a^4*c*d^3)*e^3*f^3 + (9*(b^4*c^2*d^2 - 2
*a*b^3*c*d^3 + a^2*b^2*d^4)*e^3*f^3 - (14*b^4*c^3*d - 15*a*b^3*c^2*d^2 - 12*a^2*b^2*c*d^3 + 13*a^3*b*d^4)*e^2*
f^4 + (5*b^4*c^4 + 8*a*b^3*c^3*d - 27*a^2*b^2*c^2*d^2 + 10*a^3*b*c*d^3 + 4*a^4*d^4)*e*f^5 - (5*a*b^3*c^4 - 6*a
^2*b^2*c^3*d - 3*a^3*b*c^2*d^2 + 4*a^4*c*d^3)*f^6)*x^3 + 3*(9*(b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*e^4*
f^2 - (14*b^4*c^3*d - 15*a*b^3*c^2*d^2 - 12*a^2*b^2*c*d^3 + 13*a^3*b*d^4)*e^3*f^3 + (5*b^4*c^4 + 8*a*b^3*c^3*d
- 27*a^2*b^2*c^2*d^2 + 10*a^3*b*c*d^3 + 4*a^4*d^4)*e^2*f^4 - (5*a*b^3*c^4 - 6*a^2*b^2*c^3*d - 3*a^3*b*c^2*d^2
+ 4*a^4*c*d^3)*e*f^5)*x^2 + 3*(9*(b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*e^5*f - (14*b^4*c^3*d - 15*a*b^3
*c^2*d^2 - 12*a^2*b^2*c*d^3 + 13*a^3*b*d^4)*e^4*f^2 + (5*b^4*c^4 + 8*a*b^3*c^3*d - 27*a^2*b^2*c^2*d^2 + 10*a^3
*b*c*d^3 + 4*a^4*d^4)*e^3*f^3 - (5*a*b^3*c^4 - 6*a^2*b^2*c^3*d - 3*a^3*b*c^2*d^2 + 4*a^4*c*d^3)*e^2*f^4)*x)*sq
rt((-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)*(9*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*e^4 - (5*b^3*c^3 - 6*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 4*a^3*
d^3)*e^3*f + (9*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*e*f^3 - (5*b^3*c^3 - 6*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 4
*a^3*d^3)*f^4)*x^3 + 3*(9*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*e^2*f^2 - (5*b^3*c^3 - 6*a*b^2*c^2*d - 3*a^2
*b*c*d^2 + 4*a^3*d^3)*e*f^3)*x^2 + 3*(9*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*e^3*f - (5*b^3*c^3 - 6*a*b^2*c
^2*d - 3*a^2*b*c*d^2 + 4*a^3*d^3)*e^2*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)*(9*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*e^4 - (5*b^3*c^3 - 6*a*b^2*c^2*d - 3*a^2*b*c*d^2
+ 4*a^3*d^3)*e^3*f + (9*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*e*f^3 - (5*b^3*c^3 - 6*a*b^2*c^2*d - 3*a^2*b*c
*d^2 + 4*a^3*d^3)*f^4)*x^3 + 3*(9*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*e^2*f^2 - (5*b^3*c^3 - 6*a*b^2*c^2*d
- 3*a^2*b*c*d^2 + 4*a^3*d^3)*e*f^3)*x^2 + 3*(9*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*e^3*f - (5*b^3*c^3 - 6
*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 4*a^3*d^3)*e^2*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*(18*a^4*c^3*f^6 + 9*(2*b^4*c*d^2 + a*b^3*d^3)*e^6 - (28*b^4*c^2*d + 103*a*b^3*c*d^2 + 22*a
^2*b^2*d^3)*e^5*f + (10*b^4*c^3 + 147*a*b^3*c^2*d + 186*a^2*b^2*c*d^2 + 17*a^3*b*d^3)*e^4*f^2 - (53*a*b^3*c^3
+ 258*a^2*b^2*c^2*d + 135*a^3*b*c*d^2 + 4*a^4*d^3)*e^3*f^3 + (94*a^2*b^2*c^3 + 187*a^3*b*c^2*d + 34*a^4*c*d^2)
*e^2*f^4 - 3*(23*a^3*b*c^3 + 16*a^4*c^2*d)*e*f^5 + (9*b^4*d^3*e^5*f - 15*(b^4*c*d^2 + 2*a*b^3*d^3)*e^4*f^2 + (
11*b^4*c^2*d + 38*a*b^3*c*d^2 + 41*a^2*b^2*d^3)*e^3*f^3 - (5*b^4*c^3 + 18*a*b^3*c^2*d + 39*a^2*b^2*c*d^2 + 28*
a^3*b*d^3)*e^2*f^4 + (10*a*b^3*c^3 + 3*a^2*b^2*c^2*d + 24*a^3*b*c*d^2 + 8*a^4*d^3)*e*f^5 - (5*a^2*b^2*c^3 - 4*
a^3*b*c^2*d + 8*a^4*c*d^2)*f^6)*x^2 + (27*b^4*d^3*e^6 - 3*(17*b^4*c*d^2 + 31*a*b^3*d^3)*e^5*f + (37*b^4*c^2*d
+ 151*a*b^3*c*d^2 + 127*a^2*b^2*d^3)*e^4*f^2 - (13*b^4*c^3 + 87*a*b^3*c^2*d + 177*a^2*b^2*c*d^2 + 83*a^3*b*d^3
)*e^3*f^3 + (29*a*b^3*c^3 + 69*a^2*b^2*c^2*d + 105*a^3*b*c*d^2 + 22*a^4*d^3)*e^2*f^4 - (19*a^2*b^2*c^3 + 25*a^
3*b*c^2*d + 28*a^4*c*d^2)*e*f^5 + 3*(a^3*b*c^3 + 2*a^4*c^2*d)*f^6)*x)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(b^4*d^
3*e^10 - a^4*c^3*e^3*f^7 - (3*b^4*c*d^2 + 4*a*b^3*d^3)*e^9*f + 3*(b^4*c^2*d + 4*a*b^3*c*d^2 + 2*a^2*b^2*d^3)*e
^8*f^2 - (b^4*c^3 + 12*a*b^3*c^2*d + 18*a^2*b^2*c*d^2 + 4*a^3*b*d^3)*e^7*f^3 + (4*a*b^3*c^3 + 18*a^2*b^2*c^2*d
+ 12*a^3*b*c*d^2 + a^4*d^3)*e^6*f^4 - 3*(2*a^2*b^2*c^3 + 4*a^3*b*c^2*d + a^4*c*d^2)*e^5*f^5 + (4*a^3*b*c^3 +
3*a^4*c^2*d)*e^4*f^6 + (b^4*d^3*e^7*f^3 - a^4*c^3*f^10 - (3*b^4*c*d^2 + 4*a*b^3*d^3)*e^6*f^4 + 3*(b^4*c^2*d +
4*a*b^3*c*d^2 + 2*a^2*b^2*d^3)*e^5*f^5 - (b^4*c^3 + 12*a*b^3*c^2*d + 18*a^2*b^2*c*d^2 + 4*a^3*b*d^3)*e^4*f^6 +
(4*a*b^3*c^3 + 18*a^2*b^2*c^2*d + 12*a^3*b*c*d^2 + a^4*d^3)*e^3*f^7 - 3*(2*a^2*b^2*c^3 + 4*a^3*b*c^2*d + a^4*
c*d^2)*e^2*f^8 + (4*a^3*b*c^3 + 3*a^4*c^2*d)*e*f^9)*x^3 + 3*(b^4*d^3*e^8*f^2 - a^4*c^3*e*f^9 - (3*b^4*c*d^2 +
4*a*b^3*d^3)*e^7*f^3 + 3*(b^4*c^2*d + 4*a*b^3*c*d^2 + 2*a^2*b^2*d^3)*e^6*f^4 - (b^4*c^3 + 12*a*b^3*c^2*d + 18*
a^2*b^2*c*d^2 + 4*a^3*b*d^3)*e^5*f^5 + (4*a*b^3*c^3 + 18*a^2*b^2*c^2*d + 12*a^3*b*c*d^2 + a^4*d^3)*e^4*f^6 - 3
*(2*a^2*b^2*c^3 + 4*a^3*b*c^2*d + a^4*c*d^2)*e^3*f^7 + (4*a^3*b*c^3 + 3*a^4*c^2*d)*e^2*f^8)*x^2 + 3*(b^4*d^3*e
^9*f - a^4*c^3*e^2*f^8 - (3*b^4*c*d^2 + 4*a*b^3*d^3)*e^8*f^2 + 3*(b^4*c^2*d + 4*a*b^3*c*d^2 + 2*a^2*b^2*d^3)*e
^7*f^3 - (b^4*c^3 + 12*a*b^3*c^2*d + 18*a^2*b^2*c*d^2 + 4*a^3*b*d^3)*e^6*f^4 + (4*a*b^3*c^3 + 18*a^2*b^2*c^2*d
+ 12*a^3*b*c*d^2 + a^4*d^3)*e^5*f^5 - 3*(2*a^2*b^2*c^3 + 4*a^3*b*c^2*d + a^4*c*d^2)*e^4*f^6 + (4*a^3*b*c^3 +
3*a^4*c^2*d)*e^3*f^7)*x), -1/162*(6*sqrt(1/3)*(9*(b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*e^6 - (14*b^4*c^3
*d - 15*a*b^3*c^2*d^2 - 12*a^2*b^2*c*d^3 + 13*a^3*b*d^4)*e^5*f + (5*b^4*c^4 + 8*a*b^3*c^3*d - 27*a^2*b^2*c^2*d
^2 + 10*a^3*b*c*d^3 + 4*a^4*d^4)*e^4*f^2 - (5*a*b^3*c^4 - 6*a^2*b^2*c^3*d - 3*a^3*b*c^2*d^2 + 4*a^4*c*d^3)*e^3
*f^3 + (9*(b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*e^3*f^3 - (14*b^4*c^3*d - 15*a*b^3*c^2*d^2 - 12*a^2*b^2*
c*d^3 + 13*a^3*b*d^4)*e^2*f^4 + (5*b^4*c^4 + 8*a*b^3*c^3*d - 27*a^2*b^2*c^2*d^2 + 10*a^3*b*c*d^3 + 4*a^4*d^4)*
e*f^5 - (5*a*b^3*c^4 - 6*a^2*b^2*c^3*d - 3*a^3*b*c^2*d^2 + 4*a^4*c*d^3)*f^6)*x^3 + 3*(9*(b^4*c^2*d^2 - 2*a*b^3
*c*d^3 + a^2*b^2*d^4)*e^4*f^2 - (14*b^4*c^3*d - 15*a*b^3*c^2*d^2 - 12*a^2*b^2*c*d^3 + 13*a^3*b*d^4)*e^3*f^3 +
(5*b^4*c^4 + 8*a*b^3*c^3*d - 27*a^2*b^2*c^2*d^2 + 10*a^3*b*c*d^3 + 4*a^4*d^4)*e^2*f^4 - (5*a*b^3*c^4 - 6*a^2*b
^2*c^3*d - 3*a^3*b*c^2*d^2 + 4*a^4*c*d^3)*e*f^5)*x^2 + 3*(9*(b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*e^5*f
- (14*b^4*c^3*d - 15*a*b^3*c^2*d^2 - 12*a^2*b^2*c*d^3 + 13*a^3*b*d^4)*e^4*f^2 + (5*b^4*c^4 + 8*a*b^3*c^3*d - 2
7*a^2*b^2*c^2*d^2 + 10*a^3*b*c*d^3 + 4*a^4*d^4)*e^3*f^3 - (5*a*b^3*c^4 - 6*a^2*b^2*c^3*d - 3*a^3*b*c^2*d^2 + 4
*a^4*c*d^3)*e^2*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(sqrt(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)*(9*(b^3*c
^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*e^4 - (5*b^3*c^3 - 6*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 4*a^3*d^3)*e^3*f + (9*(b^
3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*e*f^3 - (5*b^3*c^3 - 6*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 4*a^3*d^3)*f^4)*x^3
+ 3*(9*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*e^2*f^2 - (5*b^3*c^3 - 6*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 4*a^3*d^
3)*e*f^3)*x^2 + 3*(9*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*e^3*f - (5*b^3*c^3 - 6*a*b^2*c^2*d - 3*a^2*b*c*d^
2 + 4*a^3*d^3)*e^2*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)*(
9*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*e^4 - (5*b^3*c^3 - 6*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 4*a^3*d^3)*e^3*f
+ (9*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*e*f^3 - (5*b^3*c^3 - 6*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 4*a^3*d^3)*f
^4)*x^3 + 3*(9*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*e^2*f^2 - (5*b^3*c^3 - 6*a*b^2*c^2*d - 3*a^2*b*c*d^2 +
4*a^3*d^3)*e*f^3)*x^2 + 3*(9*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*e^3*f - (5*b^3*c^3 - 6*a*b^2*c^2*d - 3*a^
2*b*c*d^2 + 4*a^3*d^3)*e^2*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
*(18*a^4*c^3*f^6 + 9*(2*b^4*c*d^2 + a*b^3*d^3)*e^6 - (28*b^4*c^2*d + 103*a*b^3*c*d^2 + 22*a^2*b^2*d^3)*e^5*f +
(10*b^4*c^3 + 147*a*b^3*c^2*d + 186*a^2*b^2*c*d^2 + 17*a^3*b*d^3)*e^4*f^2 - (53*a*b^3*c^3 + 258*a^2*b^2*c^2*d
+ 135*a^3*b*c*d^2 + 4*a^4*d^3)*e^3*f^3 + (94*a^2*b^2*c^3 + 187*a^3*b*c^2*d + 34*a^4*c*d^2)*e^2*f^4 - 3*(23*a^
3*b*c^3 + 16*a^4*c^2*d)*e*f^5 + (9*b^4*d^3*e^5*f - 15*(b^4*c*d^2 + 2*a*b^3*d^3)*e^4*f^2 + (11*b^4*c^2*d + 38*a
*b^3*c*d^2 + 41*a^2*b^2*d^3)*e^3*f^3 - (5*b^4*c^3 + 18*a*b^3*c^2*d + 39*a^2*b^2*c*d^2 + 28*a^3*b*d^3)*e^2*f^4
+ (10*a*b^3*c^3 + 3*a^2*b^2*c^2*d + 24*a^3*b*c*d^2 + 8*a^4*d^3)*e*f^5 - (5*a^2*b^2*c^3 - 4*a^3*b*c^2*d + 8*a^4
*c*d^2)*f^6)*x^2 + (27*b^4*d^3*e^6 - 3*(17*b^4*c*d^2 + 31*a*b^3*d^3)*e^5*f + (37*b^4*c^2*d + 151*a*b^3*c*d^2 +
127*a^2*b^2*d^3)*e^4*f^2 - (13*b^4*c^3 + 87*a*b^3*c^2*d + 177*a^2*b^2*c*d^2 + 83*a^3*b*d^3)*e^3*f^3 + (29*a*b
^3*c^3 + 69*a^2*b^2*c^2*d + 105*a^3*b*c*d^2 + 22*a^4*d^3)*e^2*f^4 - (19*a^2*b^2*c^3 + 25*a^3*b*c^2*d + 28*a^4*
c*d^2)*e*f^5 + 3*(a^3*b*c^3 + 2*a^4*c^2*d)*f^6)*x)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(b^4*d^3*e^10 - a^4*c^3*e^
3*f^7 - (3*b^4*c*d^2 + 4*a*b^3*d^3)*e^9*f + 3*(b^4*c^2*d + 4*a*b^3*c*d^2 + 2*a^2*b^2*d^3)*e^8*f^2 - (b^4*c^3 +
12*a*b^3*c^2*d + 18*a^2*b^2*c*d^2 + 4*a^3*b*d^3)*e^7*f^3 + (4*a*b^3*c^3 + 18*a^2*b^2*c^2*d + 12*a^3*b*c*d^2 +
a^4*d^3)*e^6*f^4 - 3*(2*a^2*b^2*c^3 + 4*a^3*b*c^2*d + a^4*c*d^2)*e^5*f^5 + (4*a^3*b*c^3 + 3*a^4*c^2*d)*e^4*f^
6 + (b^4*d^3*e^7*f^3 - a^4*c^3*f^10 - (3*b^4*c*d^2 + 4*a*b^3*d^3)*e^6*f^4 + 3*(b^4*c^2*d + 4*a*b^3*c*d^2 + 2*a
^2*b^2*d^3)*e^5*f^5 - (b^4*c^3 + 12*a*b^3*c^2*d + 18*a^2*b^2*c*d^2 + 4*a^3*b*d^3)*e^4*f^6 + (4*a*b^3*c^3 + 18*
a^2*b^2*c^2*d + 12*a^3*b*c*d^2 + a^4*d^3)*e^3*f^7 - 3*(2*a^2*b^2*c^3 + 4*a^3*b*c^2*d + a^4*c*d^2)*e^2*f^8 + (4
*a^3*b*c^3 + 3*a^4*c^2*d)*e*f^9)*x^3 + 3*(b^4*d^3*e^8*f^2 - a^4*c^3*e*f^9 - (3*b^4*c*d^2 + 4*a*b^3*d^3)*e^7*f^
3 + 3*(b^4*c^2*d + 4*a*b^3*c*d^2 + 2*a^2*b^2*d^3)*e^6*f^4 - (b^4*c^3 + 12*a*b^3*c^2*d + 18*a^2*b^2*c*d^2 + 4*a
^3*b*d^3)*e^5*f^5 + (4*a*b^3*c^3 + 18*a^2*b^2*c^2*d + 12*a^3*b*c*d^2 + a^4*d^3)*e^4*f^6 - 3*(2*a^2*b^2*c^3 + 4
*a^3*b*c^2*d + a^4*c*d^2)*e^3*f^7 + (4*a^3*b*c^3 + 3*a^4*c^2*d)*e^2*f^8)*x^2 + 3*(b^4*d^3*e^9*f - a^4*c^3*e^2*
f^8 - (3*b^4*c*d^2 + 4*a*b^3*d^3)*e^8*f^2 + 3*(b^4*c^2*d + 4*a*b^3*c*d^2 + 2*a^2*b^2*d^3)*e^7*f^3 - (b^4*c^3 +
12*a*b^3*c^2*d + 18*a^2*b^2*c*d^2 + 4*a^3*b*d^3)*e^6*f^4 + (4*a*b^3*c^3 + 18*a^2*b^2*c^2*d + 12*a^3*b*c*d^2 +
a^4*d^3)*e^5*f^5 - 3*(2*a^2*b^2*c^3 + 4*a^3*b*c^2*d + a^4*c*d^2)*e^4*f^6 + (4*a^3*b*c^3 + 3*a^4*c^2*d)*e^3*f^
7)*x)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac{2}{3}}}{\left (e + f x\right )^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(1/3)*(d*x+c)**(2/3)/(f*x+e)**4,x)

[Out]

Integral((a + b*x)**(1/3)*(c + d*x)**(2/3)/(e + f*x)**4, x)

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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{2}{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)^(2/3)/(f*x+e)^4,x, algorithm="giac")

[Out]

integrate((b*x + a)^(1/3)*(d*x + c)^(2/3)/(f*x + e)^4, x)