3.3042 \(\int \frac{(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^3} \, dx\)

Optimal. Leaf size=434 \[ \frac{3 d (a+b x)^{7/3}}{\sqrt [3]{c+d x} (e+f x)^2 (b c-a d) (d e-c f)}-\frac{(a+b x)^{4/3} (c+d x)^{2/3} (-7 a d f+b c f+6 b d e)}{2 (e+f x)^2 (b c-a d) (d e-c f)^2}+\frac{2 \sqrt [3]{a+b x} (c+d x)^{2/3} (-7 a d f+b c f+6 b d e)}{3 (e+f x) (d e-c f)^3}-\frac{(b c-a d) \log (e+f x) (-7 a d f+b c f+6 b d e)}{9 (b e-a f)^{2/3} (d e-c f)^{10/3}}+\frac{(b c-a d) (-7 a d f+b c f+6 b d e) \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{3 (b e-a f)^{2/3} (d e-c f)^{10/3}}+\frac{2 (b c-a d) (-7 a d f+b c f+6 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} (b e-a f)^{2/3} (d e-c f)^{10/3}} \]

[Out]

(3*d*(a + b*x)^(7/3))/((b*c - a*d)*(d*e - c*f)*(c + d*x)^(1/3)*(e + f*x)^2) - ((6*b*d*e + b*c*f - 7*a*d*f)*(a
+ b*x)^(4/3)*(c + d*x)^(2/3))/(2*(b*c - a*d)*(d*e - c*f)^2*(e + f*x)^2) + (2*(6*b*d*e + b*c*f - 7*a*d*f)*(a +
b*x)^(1/3)*(c + d*x)^(2/3))/(3*(d*e - c*f)^3*(e + f*x)) + (2*(b*c - a*d)*(6*b*d*e + b*c*f - 7*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)^(2/3)*(d*e - c*f)^(10/3)) - ((b*c - a*d)*(6*b*d*e + b*c*f - 7*a*d*f)*Log[e + f*x])/(9*(b*e - a*f)^(2/3)
*(d*e - c*f)^(10/3)) + ((b*c - a*d)*(6*b*d*e + b*c*f - 7*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)])/(3*(b*e - a*f)^(2/3)*(d*e - c*f)^(10/3))

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Rubi [A]  time = 0.302037, antiderivative size = 434, 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} \[ \frac{3 d (a+b x)^{7/3}}{\sqrt [3]{c+d x} (e+f x)^2 (b c-a d) (d e-c f)}-\frac{(a+b x)^{4/3} (c+d x)^{2/3} (-7 a d f+b c f+6 b d e)}{2 (e+f x)^2 (b c-a d) (d e-c f)^2}+\frac{2 \sqrt [3]{a+b x} (c+d x)^{2/3} (-7 a d f+b c f+6 b d e)}{3 (e+f x) (d e-c f)^3}-\frac{(b c-a d) \log (e+f x) (-7 a d f+b c f+6 b d e)}{9 (b e-a f)^{2/3} (d e-c f)^{10/3}}+\frac{(b c-a d) (-7 a d f+b c f+6 b d e) \log \left (\frac{\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{3 (b e-a f)^{2/3} (d e-c f)^{10/3}}+\frac{2 (b c-a d) (-7 a d f+b c f+6 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt{3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} (b e-a f)^{2/3} (d e-c f)^{10/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*d*(a + b*x)^(7/3))/((b*c - a*d)*(d*e - c*f)*(c + d*x)^(1/3)*(e + f*x)^2) - ((6*b*d*e + b*c*f - 7*a*d*f)*(a
+ b*x)^(4/3)*(c + d*x)^(2/3))/(2*(b*c - a*d)*(d*e - c*f)^2*(e + f*x)^2) + (2*(6*b*d*e + b*c*f - 7*a*d*f)*(a +
b*x)^(1/3)*(c + d*x)^(2/3))/(3*(d*e - c*f)^3*(e + f*x)) + (2*(b*c - a*d)*(6*b*d*e + b*c*f - 7*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)^(2/3)*(d*e - c*f)^(10/3)) - ((b*c - a*d)*(6*b*d*e + b*c*f - 7*a*d*f)*Log[e + f*x])/(9*(b*e - a*f)^(2/3)
*(d*e - c*f)^(10/3)) + ((b*c - a*d)*(6*b*d*e + b*c*f - 7*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)])/(3*(b*e - a*f)^(2/3)*(d*e - c*f)^(10/3))

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{(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)^3} \, dx &=\frac{3 d (a+b x)^{7/3}}{(b c-a d) (d e-c f) \sqrt [3]{c+d x} (e+f x)^2}-\frac{(6 b d e+b c f-7 a d f) \int \frac{(a+b x)^{4/3}}{\sqrt [3]{c+d x} (e+f x)^3} \, dx}{(b c-a d) (d e-c f)}\\ &=\frac{3 d (a+b x)^{7/3}}{(b c-a d) (d e-c f) \sqrt [3]{c+d x} (e+f x)^2}-\frac{(6 b d e+b c f-7 a d f) (a+b x)^{4/3} (c+d x)^{2/3}}{2 (b c-a d) (d e-c f)^2 (e+f x)^2}+\frac{(2 (6 b d e+b c f-7 a d f)) \int \frac{\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)^2} \, dx}{3 (d e-c f)^2}\\ &=\frac{3 d (a+b x)^{7/3}}{(b c-a d) (d e-c f) \sqrt [3]{c+d x} (e+f x)^2}-\frac{(6 b d e+b c f-7 a d f) (a+b x)^{4/3} (c+d x)^{2/3}}{2 (b c-a d) (d e-c f)^2 (e+f x)^2}+\frac{2 (6 b d e+b c f-7 a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{3 (d e-c f)^3 (e+f x)}-\frac{(2 (b c-a d) (6 b d e+b c f-7 a d f)) \int \frac{1}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)} \, dx}{9 (d e-c f)^3}\\ &=\frac{3 d (a+b x)^{7/3}}{(b c-a d) (d e-c f) \sqrt [3]{c+d x} (e+f x)^2}-\frac{(6 b d e+b c f-7 a d f) (a+b x)^{4/3} (c+d x)^{2/3}}{2 (b c-a d) (d e-c f)^2 (e+f x)^2}+\frac{2 (6 b d e+b c f-7 a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{3 (d e-c f)^3 (e+f x)}+\frac{2 (b c-a d) (6 b d e+b c f-7 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)^{2/3} (d e-c f)^{10/3}}-\frac{(b c-a d) (6 b d e+b c f-7 a d f) \log (e+f x)}{9 (b e-a f)^{2/3} (d e-c f)^{10/3}}+\frac{(b c-a d) (6 b d e+b c f-7 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 )}{3 (b e-a f)^{2/3} (d e-c f)^{10/3}}\\ \end{align*}

Mathematica [C]  time = 0.488637, size = 214, normalized size = 0.49 \[ \frac{\frac{\sqrt [3]{a+b x} (-7 a d f+b c f+6 b d e) \left (3 (a+b x) (c+d x) (b e-a f) (d e-c f)-4 (e+f x) (b c-a d) \left ((c+d x) (b e-a f)-(e+f x) (b c-a d) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )\right )\right )}{3 (b e-a f) (d e-c f)^2}-6 d (a+b x)^{7/3}}{2 \sqrt [3]{c+d x} (e+f x)^2 (b c-a d) (c f-d e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*d*(a + b*x)^(7/3) + ((6*b*d*e + b*c*f - 7*a*d*f)*(a + b*x)^(1/3)*(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))])))/(3*(b*e - a*f)*(d*e - c*f)^2))/(2*(b*c - a*d)*(-(d*e) + c*
f)*(c + d*x)^(1/3)*(e + f*x)^2)

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( fx+e \right ) ^{3}} \left ( bx+a \right ) ^{{\frac{4}{3}}} \left ( dx+c \right ) ^{-{\frac{4}{3}}}}\, dx \end{align*}

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)^3,x)

[Out]

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

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

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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 8.00224, size = 14086, normalized size = 32.46 \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)^(4/3)/(d*x+c)^(4/3)/(f*x+e)^3,x, algorithm="fricas")

[Out]

[-1/18*(6*sqrt(1/3)*(6*(b^3*c^2*d^2 - a*b^2*c*d^3)*e^5 - (5*b^3*c^3*d + 8*a*b^2*c^2*d^2 - 13*a^2*b*c*d^3)*e^4*
f - (b^3*c^4 - 13*a*b^2*c^3*d + 5*a^2*b*c^2*d^2 + 7*a^3*c*d^3)*e^3*f^2 + (a*b^2*c^4 - 8*a^2*b*c^3*d + 7*a^3*c^
2*d^2)*e^2*f^3 + (6*(b^3*c*d^3 - a*b^2*d^4)*e^3*f^2 - (5*b^3*c^2*d^2 + 8*a*b^2*c*d^3 - 13*a^2*b*d^4)*e^2*f^3 -
(b^3*c^3*d - 13*a*b^2*c^2*d^2 + 5*a^2*b*c*d^3 + 7*a^3*d^4)*e*f^4 + (a*b^2*c^3*d - 8*a^2*b*c^2*d^2 + 7*a^3*c*d
^3)*f^5)*x^3 + (12*(b^3*c*d^3 - a*b^2*d^4)*e^4*f - 2*(2*b^3*c^2*d^2 + 11*a*b^2*c*d^3 - 13*a^2*b*d^4)*e^3*f^2 -
(7*b^3*c^3*d - 18*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3 + 14*a^3*d^4)*e^2*f^3 - (b^3*c^4 - 15*a*b^2*c^3*d + 21*a^2*b*
c^2*d^2 - 7*a^3*c*d^3)*e*f^4 + (a*b^2*c^4 - 8*a^2*b*c^3*d + 7*a^3*c^2*d^2)*f^5)*x^2 + (6*(b^3*c*d^3 - a*b^2*d^
4)*e^5 + (7*b^3*c^2*d^2 - 20*a*b^2*c*d^3 + 13*a^2*b*d^4)*e^4*f - (11*b^3*c^3*d + 3*a*b^2*c^2*d^2 - 21*a^2*b*c*
d^3 + 7*a^3*d^4)*e^3*f^2 - (2*b^3*c^4 - 27*a*b^2*c^3*d + 18*a^2*b*c^2*d^2 + 7*a^3*c*d^3)*e^2*f^3 + 2*(a*b^2*c^
4 - 8*a^2*b*c^3*d + 7*a^3*c^2*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)) + 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)*(6*(b^2*c^2*d - a*b*c*d^2)*e^3 + (b^2*c^3 - 8*a*b*c
^2*d + 7*a^2*c*d^2)*e^2*f + (6*(b^2*c*d^2 - a*b*d^3)*e*f^2 + (b^2*c^2*d - 8*a*b*c*d^2 + 7*a^2*d^3)*f^3)*x^3 +
(12*(b^2*c*d^2 - a*b*d^3)*e^2*f + 2*(4*b^2*c^2*d - 11*a*b*c*d^2 + 7*a^2*d^3)*e*f^2 + (b^2*c^3 - 8*a*b*c^2*d +
7*a^2*c*d^2)*f^3)*x^2 + (6*(b^2*c*d^2 - a*b*d^3)*e^3 + (13*b^2*c^2*d - 20*a*b*c*d^2 + 7*a^2*d^3)*e^2*f + 2*(b^
2*c^3 - 8*a*b*c^2*d + 7*a^2*c*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)) - 4*(-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)*(6*(b^2*c^2*d - a*b*c*d^2)*e^3 + (b^2*c^3 - 8*a*b*c^2*d + 7*a^2*c*d^2)*e^2*f + (6*(b^2*c*d^2
- a*b*d^3)*e*f^2 + (b^2*c^2*d - 8*a*b*c*d^2 + 7*a^2*d^3)*f^3)*x^3 + (12*(b^2*c*d^2 - a*b*d^3)*e^2*f + 2*(4*b^2
*c^2*d - 11*a*b*c*d^2 + 7*a^2*d^3)*e*f^2 + (b^2*c^3 - 8*a*b*c^2*d + 7*a^2*c*d^2)*f^3)*x^2 + (6*(b^2*c*d^2 - a*
b*d^3)*e^3 + (13*b^2*c^2*d - 20*a*b*c*d^2 + 7*a^2*d^3)*e^2*f + 2*(b^2*c^3 - 8*a*b*c^2*d + 7*a^2*c*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*(3*a^3*c^3*f^5 - 6*(4*b^3*c*d^2
- 3*a*b^2*d^3)*e^5 + (20*b^3*c^2*d + 43*a*b^2*c*d^2 - 36*a^2*b*d^3)*e^4*f + 2*(2*b^3*c^3 - 28*a*b^2*c^2*d - 7
*a^2*b*c*d^2 + 9*a^3*d^3)*e^3*f^2 - (5*a*b^2*c^3 - 52*a^2*b*c^2*d + 5*a^3*c*d^2)*e^2*f^3 - 2*(a^2*b*c^3 + 8*a^
3*c^2*d)*e*f^4 - (3*b^3*d^3*e^4*f + 2*(11*b^3*c*d^2 - 17*a*b^2*d^3)*e^3*f^2 - (25*b^3*c^2*d + 16*a*b^2*c*d^2 -
59*a^2*b*d^3)*e^2*f^3 + 2*(25*a*b^2*c^2*d - 17*a^2*b*c*d^2 - 14*a^3*d^3)*e*f^4 - (25*a^2*b*c^2*d - 28*a^3*c*d
^2)*f^5)*x^2 - (6*b^3*d^3*e^5 + (37*b^3*c*d^2 - 61*a*b^2*d^3)*e^4*f - 4*(9*b^3*c^2*d + 8*a*b^2*c*d^2 - 26*a^2*
b*d^3)*e^3*f^2 - (7*b^3*c^3 - 79*a*b^2*c^2*d + 47*a^2*b*c*d^2 + 49*a^3*d^3)*e^2*f^3 + 2*(7*a*b^2*c^3 - 25*a^2*
b*c^2*d + 21*a^3*c*d^2)*e*f^4 - 7*(a^2*b*c^3 - a^3*c^2*d)*f^5)*x)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(b^2*c*d^4*
e^8 + a^2*c^5*e^2*f^6 - 2*(2*b^2*c^2*d^3 + a*b*c*d^4)*e^7*f + (6*b^2*c^3*d^2 + 8*a*b*c^2*d^3 + a^2*c*d^4)*e^6*
f^2 - 4*(b^2*c^4*d + 3*a*b*c^3*d^2 + a^2*c^2*d^3)*e^5*f^3 + (b^2*c^5 + 8*a*b*c^4*d + 6*a^2*c^3*d^2)*e^4*f^4 -
2*(a*b*c^5 + 2*a^2*c^4*d)*e^3*f^5 + (b^2*d^5*e^6*f^2 + a^2*c^4*d*f^8 - 2*(2*b^2*c*d^4 + a*b*d^5)*e^5*f^3 + (6*
b^2*c^2*d^3 + 8*a*b*c*d^4 + a^2*d^5)*e^4*f^4 - 4*(b^2*c^3*d^2 + 3*a*b*c^2*d^3 + a^2*c*d^4)*e^3*f^5 + (b^2*c^4*
d + 8*a*b*c^3*d^2 + 6*a^2*c^2*d^3)*e^2*f^6 - 2*(a*b*c^4*d + 2*a^2*c^3*d^2)*e*f^7)*x^3 + (2*b^2*d^5*e^7*f + a^2
*c^5*f^8 - (7*b^2*c*d^4 + 4*a*b*d^5)*e^6*f^2 + 2*(4*b^2*c^2*d^3 + 7*a*b*c*d^4 + a^2*d^5)*e^5*f^3 - (2*b^2*c^3*
d^2 + 16*a*b*c^2*d^3 + 7*a^2*c*d^4)*e^4*f^4 - 2*(b^2*c^4*d - 2*a*b*c^3*d^2 - 4*a^2*c^2*d^3)*e^3*f^5 + (b^2*c^5
+ 4*a*b*c^4*d - 2*a^2*c^3*d^2)*e^2*f^6 - 2*(a*b*c^5 + a^2*c^4*d)*e*f^7)*x^2 + (b^2*d^5*e^8 + 2*a^2*c^5*e*f^7
- 2*(b^2*c*d^4 + a*b*d^5)*e^7*f - (2*b^2*c^2*d^3 - 4*a*b*c*d^4 - a^2*d^5)*e^6*f^2 + 2*(4*b^2*c^3*d^2 + 2*a*b*c
^2*d^3 - a^2*c*d^4)*e^5*f^3 - (7*b^2*c^4*d + 16*a*b*c^3*d^2 + 2*a^2*c^2*d^3)*e^4*f^4 + 2*(b^2*c^5 + 7*a*b*c^4*
d + 4*a^2*c^3*d^2)*e^3*f^5 - (4*a*b*c^5 + 7*a^2*c^4*d)*e^2*f^6)*x), -1/18*(12*sqrt(1/3)*(6*(b^3*c^2*d^2 - a*b^
2*c*d^3)*e^5 - (5*b^3*c^3*d + 8*a*b^2*c^2*d^2 - 13*a^2*b*c*d^3)*e^4*f - (b^3*c^4 - 13*a*b^2*c^3*d + 5*a^2*b*c^
2*d^2 + 7*a^3*c*d^3)*e^3*f^2 + (a*b^2*c^4 - 8*a^2*b*c^3*d + 7*a^3*c^2*d^2)*e^2*f^3 + (6*(b^3*c*d^3 - a*b^2*d^4
)*e^3*f^2 - (5*b^3*c^2*d^2 + 8*a*b^2*c*d^3 - 13*a^2*b*d^4)*e^2*f^3 - (b^3*c^3*d - 13*a*b^2*c^2*d^2 + 5*a^2*b*c
*d^3 + 7*a^3*d^4)*e*f^4 + (a*b^2*c^3*d - 8*a^2*b*c^2*d^2 + 7*a^3*c*d^3)*f^5)*x^3 + (12*(b^3*c*d^3 - a*b^2*d^4)
*e^4*f - 2*(2*b^3*c^2*d^2 + 11*a*b^2*c*d^3 - 13*a^2*b*d^4)*e^3*f^2 - (7*b^3*c^3*d - 18*a*b^2*c^2*d^2 - 3*a^2*b
*c*d^3 + 14*a^3*d^4)*e^2*f^3 - (b^3*c^4 - 15*a*b^2*c^3*d + 21*a^2*b*c^2*d^2 - 7*a^3*c*d^3)*e*f^4 + (a*b^2*c^4
- 8*a^2*b*c^3*d + 7*a^3*c^2*d^2)*f^5)*x^2 + (6*(b^3*c*d^3 - a*b^2*d^4)*e^5 + (7*b^3*c^2*d^2 - 20*a*b^2*c*d^3 +
13*a^2*b*d^4)*e^4*f - (11*b^3*c^3*d + 3*a*b^2*c^2*d^2 - 21*a^2*b*c*d^3 + 7*a^3*d^4)*e^3*f^2 - (2*b^3*c^4 - 27
*a*b^2*c^3*d + 18*a^2*b*c^2*d^2 + 7*a^3*c*d^3)*e^2*f^3 + 2*(a*b^2*c^4 - 8*a^2*b*c^3*d + 7*a^3*c^2*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))*arcta
n(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)) + 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)*(6*(b^2*c^2*d - a*b*c*d^2)*e
^3 + (b^2*c^3 - 8*a*b*c^2*d + 7*a^2*c*d^2)*e^2*f + (6*(b^2*c*d^2 - a*b*d^3)*e*f^2 + (b^2*c^2*d - 8*a*b*c*d^2 +
7*a^2*d^3)*f^3)*x^3 + (12*(b^2*c*d^2 - a*b*d^3)*e^2*f + 2*(4*b^2*c^2*d - 11*a*b*c*d^2 + 7*a^2*d^3)*e*f^2 + (b
^2*c^3 - 8*a*b*c^2*d + 7*a^2*c*d^2)*f^3)*x^2 + (6*(b^2*c*d^2 - a*b*d^3)*e^3 + (13*b^2*c^2*d - 20*a*b*c*d^2 + 7
*a^2*d^3)*e^2*f + 2*(b^2*c^3 - 8*a*b*c^2*d + 7*a^2*c*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)) - 4*(-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)*(6*(b^2*c^2*d - a*b*c*d^2)*e^3 + (b^2*c^3 - 8*a*b*c^2*d + 7*a^2*c*d^2)
*e^2*f + (6*(b^2*c*d^2 - a*b*d^3)*e*f^2 + (b^2*c^2*d - 8*a*b*c*d^2 + 7*a^2*d^3)*f^3)*x^3 + (12*(b^2*c*d^2 - a*
b*d^3)*e^2*f + 2*(4*b^2*c^2*d - 11*a*b*c*d^2 + 7*a^2*d^3)*e*f^2 + (b^2*c^3 - 8*a*b*c^2*d + 7*a^2*c*d^2)*f^3)*x
^2 + (6*(b^2*c*d^2 - a*b*d^3)*e^3 + (13*b^2*c^2*d - 20*a*b*c*d^2 + 7*a^2*d^3)*e^2*f + 2*(b^2*c^3 - 8*a*b*c^2*d
+ 7*a^2*c*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*(3*a^3*c
^3*f^5 - 6*(4*b^3*c*d^2 - 3*a*b^2*d^3)*e^5 + (20*b^3*c^2*d + 43*a*b^2*c*d^2 - 36*a^2*b*d^3)*e^4*f + 2*(2*b^3*c
^3 - 28*a*b^2*c^2*d - 7*a^2*b*c*d^2 + 9*a^3*d^3)*e^3*f^2 - (5*a*b^2*c^3 - 52*a^2*b*c^2*d + 5*a^3*c*d^2)*e^2*f^
3 - 2*(a^2*b*c^3 + 8*a^3*c^2*d)*e*f^4 - (3*b^3*d^3*e^4*f + 2*(11*b^3*c*d^2 - 17*a*b^2*d^3)*e^3*f^2 - (25*b^3*c
^2*d + 16*a*b^2*c*d^2 - 59*a^2*b*d^3)*e^2*f^3 + 2*(25*a*b^2*c^2*d - 17*a^2*b*c*d^2 - 14*a^3*d^3)*e*f^4 - (25*a
^2*b*c^2*d - 28*a^3*c*d^2)*f^5)*x^2 - (6*b^3*d^3*e^5 + (37*b^3*c*d^2 - 61*a*b^2*d^3)*e^4*f - 4*(9*b^3*c^2*d +
8*a*b^2*c*d^2 - 26*a^2*b*d^3)*e^3*f^2 - (7*b^3*c^3 - 79*a*b^2*c^2*d + 47*a^2*b*c*d^2 + 49*a^3*d^3)*e^2*f^3 + 2
*(7*a*b^2*c^3 - 25*a^2*b*c^2*d + 21*a^3*c*d^2)*e*f^4 - 7*(a^2*b*c^3 - a^3*c^2*d)*f^5)*x)*(b*x + a)^(1/3)*(d*x
+ c)^(2/3))/(b^2*c*d^4*e^8 + a^2*c^5*e^2*f^6 - 2*(2*b^2*c^2*d^3 + a*b*c*d^4)*e^7*f + (6*b^2*c^3*d^2 + 8*a*b*c^
2*d^3 + a^2*c*d^4)*e^6*f^2 - 4*(b^2*c^4*d + 3*a*b*c^3*d^2 + a^2*c^2*d^3)*e^5*f^3 + (b^2*c^5 + 8*a*b*c^4*d + 6*
a^2*c^3*d^2)*e^4*f^4 - 2*(a*b*c^5 + 2*a^2*c^4*d)*e^3*f^5 + (b^2*d^5*e^6*f^2 + a^2*c^4*d*f^8 - 2*(2*b^2*c*d^4 +
a*b*d^5)*e^5*f^3 + (6*b^2*c^2*d^3 + 8*a*b*c*d^4 + a^2*d^5)*e^4*f^4 - 4*(b^2*c^3*d^2 + 3*a*b*c^2*d^3 + a^2*c*d
^4)*e^3*f^5 + (b^2*c^4*d + 8*a*b*c^3*d^2 + 6*a^2*c^2*d^3)*e^2*f^6 - 2*(a*b*c^4*d + 2*a^2*c^3*d^2)*e*f^7)*x^3 +
(2*b^2*d^5*e^7*f + a^2*c^5*f^8 - (7*b^2*c*d^4 + 4*a*b*d^5)*e^6*f^2 + 2*(4*b^2*c^2*d^3 + 7*a*b*c*d^4 + a^2*d^5
)*e^5*f^3 - (2*b^2*c^3*d^2 + 16*a*b*c^2*d^3 + 7*a^2*c*d^4)*e^4*f^4 - 2*(b^2*c^4*d - 2*a*b*c^3*d^2 - 4*a^2*c^2*
d^3)*e^3*f^5 + (b^2*c^5 + 4*a*b*c^4*d - 2*a^2*c^3*d^2)*e^2*f^6 - 2*(a*b*c^5 + a^2*c^4*d)*e*f^7)*x^2 + (b^2*d^5
*e^8 + 2*a^2*c^5*e*f^7 - 2*(b^2*c*d^4 + a*b*d^5)*e^7*f - (2*b^2*c^2*d^3 - 4*a*b*c*d^4 - a^2*d^5)*e^6*f^2 + 2*(
4*b^2*c^3*d^2 + 2*a*b*c^2*d^3 - a^2*c*d^4)*e^5*f^3 - (7*b^2*c^4*d + 16*a*b*c^3*d^2 + 2*a^2*c^2*d^3)*e^4*f^4 +
2*(b^2*c^5 + 7*a*b*c^4*d + 4*a^2*c^3*d^2)*e^3*f^5 - (4*a*b*c^5 + 7*a^2*c^4*d)*e^2*f^6)*x)]

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

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)**3,x)

[Out]

Timed out

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

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)^3,x, algorithm="giac")

[Out]

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