3.3021 \(\int \frac{1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.63327, antiderivative size = 477, normalized size of antiderivative = 1., 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 = {129, 151, 12, 91} \[ \frac{\log (e+f x) \left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (2 c^2 f^2-6 c d e f+9 d^2 e^2\right )\right )}{18 (b e-a f)^{7/3} (d e-c f)^{8/3}}-\frac{\left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (2 c^2 f^2-6 c d e f+9 d^2 e^2\right )\right ) \log \left (\frac{\sqrt [3]{a+b x} \sqrt [3]{d e-c f}}{\sqrt [3]{b e-a f}}-\sqrt [3]{c+d x}\right )}{6 (b e-a f)^{7/3} (d e-c f)^{8/3}}-\frac{\left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (2 c^2 f^2-6 c d e f+9 d^2 e^2\right )\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}{\sqrt{3} \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} (b e-a f)^{7/3} (d e-c f)^{8/3}}-\frac{f (a+b x)^{2/3} \sqrt [3]{c+d x} (-5 a d f-4 b c f+9 b d e)}{6 (e+f x) (b e-a f)^2 (d e-c f)^2}-\frac{f (a+b x)^{2/3} \sqrt [3]{c+d x}}{2 (e+f x)^2 (b e-a f) (d e-c f)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(f*(a + b*x)^(2/3)*(c + d*x)^(1/3))/(2*(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 129

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, m, n, p}, x] && ILtQ[m + n
+ p + 2, 0] && NeQ[m, -1] && (SumSimplerQ[m, 1] || ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && S
umSimplerQ[p, 1])))

Rule 151

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] && LtQ[m, -1] && IntegerQ[m]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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{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*}

Mathematica [C]  time = 0.275586, size = 221, normalized size = 0.46 \[ \frac{(a+b x)^{2/3} \left (\frac{\left (5 a^2 d^2 f^2+2 a b d f (c f-6 d e)+b^2 \left (2 c^2 f^2-6 c d e f+9 d^2 e^2\right )\right ) \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(b e-a f)^2 (d e-c f)}+\frac{f (c+d x) (5 a d f+4 b c f-9 b d e)}{(e+f x) (b e-a f) (d e-c f)}-\frac{3 f (c+d x)}{(e+f x)^2}\right )}{6 (c+d x)^{2/3} (b e-a f) (d e-c f)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)^(2/3)*((-3*f*(c + d*x))/(e + f*x)^2 + (f*(-9*b*d*e + 4*b*c*f + 5*a*d*f)*(c + d*x))/((b*e - a*f)*(d*
e - c*f)*(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))*Hy
pergeometric2F1[2/3, 1, 5/3, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))])/((b*e - a*f)^2*(d*e - c*f))))/(
6*(b*e - a*f)*(d*e - c*f)*(c + d*x)^(2/3))

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( fx+e \right ) ^{3}}{\frac{1}{\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(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., size = 0, normalized size = 0. \begin{align*} \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}}\,{d x} \end{align*}

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]  time = 50.2783, size = 11379, normalized size = 23.86 \begin{align*} \text{result too large to display} \end{align*}

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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}}\,{d x} \end{align*}

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)