\(\int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x)^2} \, dx\) [3005]

Optimal result

Integrand size = 26, antiderivative size = 417 \[ \int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x)^2} \, dx=-\frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{f (e+f x)}-\frac {\sqrt {3} \sqrt [3]{b} d^{2/3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{f^2}+\frac {(3 b d e-b c f-2 a d f) \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}\right )}{\sqrt {3} f^2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}-\frac {\sqrt [3]{b} d^{2/3} \log (a+b x)}{2 f^2}-\frac {(3 b d e-b c f-2 a d f) \log (e+f x)}{6 f^2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}+\frac {(3 b d e-b c f-2 a d f) \log \left (-\sqrt [3]{a+b x}+\frac {\sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt [3]{d e-c f}}\right )}{2 f^2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}-\frac {3 \sqrt [3]{b} d^{2/3} \log \left (-1+\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{2 f^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {99, 163, 61, 93} \[ \int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x)^2} \, dx=-\frac {\sqrt {3} \sqrt [3]{b} d^{2/3} \arctan \left (\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac {1}{\sqrt {3}}\right )}{f^2}+\frac {(-2 a d f-b c f+3 b d e) \arctan \left (\frac {2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt {3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} f^2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}-\frac {3 \sqrt [3]{b} d^{2/3} \log \left (\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 f^2}-\frac {\log (e+f x) (-2 a d f-b c f+3 b d e)}{6 f^2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}+\frac {(-2 a d f-b c f+3 b d e) \log \left (\frac {\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 f^2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}-\frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{f (e+f x)}-\frac {\sqrt [3]{b} d^{2/3} \log (a+b x)}{2 f^2} \]

[In]

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

[Out]

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

Rule 61

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[d/b, 3]}, Simp[(-Sqrt
[3])*(q/d)*ArcTan[2*q*((a + b*x)^(1/3)/(Sqrt[3]*(c + d*x)^(1/3))) + 1/Sqrt[3]], x] + (-Simp[3*(q/(2*d))*Log[q*
((a + b*x)^(1/3)/(c + d*x)^(1/3)) - 1], x] - Simp[(q/(2*d))*Log[c + d*x], x])] /; FreeQ[{a, b, c, d}, x] && Ne
Q[b*c - a*d, 0] && PosQ[d/b]

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 99

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/(b*(m + 1))), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 163

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

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

Mathematica [A] (verified)

Time = 1.90 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.26 \[ \int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x)^2} \, dx=\frac {-\frac {6 f \sqrt [3]{a+b x} (c+d x)^{2/3}}{e+f x}+6 \sqrt {3} \sqrt [3]{b} d^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}}{\sqrt {3}}\right )+\frac {2 \sqrt {3} (3 b d e-b c f-2 a d f) \arctan \left (\frac {1-\frac {2 \sqrt [3]{-d e+c f} \sqrt [3]{a+b x}}{\sqrt [3]{b e-a f} \sqrt [3]{c+d x}}}{\sqrt {3}}\right )}{(b e-a f)^{2/3} \sqrt [3]{-d e+c f}}-6 \sqrt [3]{b} d^{2/3} \log \left (\sqrt [3]{b}-\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}\right )+3 \sqrt [3]{b} d^{2/3} \log \left (b^{2/3}+\frac {d^{2/3} (a+b x)^{2/3}}{(c+d x)^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}\right )+\frac {2 (-3 b d e+b c f+2 a d f) \log \left (\sqrt [3]{b e-a f}+\frac {\sqrt [3]{-d e+c f} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}\right )}{(b e-a f)^{2/3} \sqrt [3]{-d e+c f}}+\frac {(3 b d e-b c f-2 a d f) \log \left ((b e-a f)^{2/3}+\frac {(-d e+c f)^{2/3} (a+b x)^{2/3}}{(c+d x)^{2/3}}-\frac {\sqrt [3]{b e-a f} \sqrt [3]{-d e+c f} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}\right )}{(b e-a f)^{2/3} \sqrt [3]{-d e+c f}}}{6 f^2} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {\left (b x +a \right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {2}{3}}}{\left (f x +e \right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1726 vs. \(2 (342) = 684\).

Time = 2.75 (sec) , antiderivative size = 3608, normalized size of antiderivative = 8.65 \[ \int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x)^2} \, dx=\text {Too large to display} \]

[In]

integrate((b*x+a)^(1/3)*(d*x+c)^(2/3)/(f*x+e)^2,x, algorithm="fricas")

[Out]

[1/6*(3*sqrt(1/3)*(3*b^2*d^2*e^4 - (4*b^2*c*d + 5*a*b*d^2)*e^3*f + (b^2*c^2 + 6*a*b*c*d + 2*a^2*d^2)*e^2*f^2 -
(a*b*c^2 + 2*a^2*c*d)*e*f^3 + (3*b^2*d^2*e^3*f - (4*b^2*c*d + 5*a*b*d^2)*e^2*f^2 + (b^2*c^2 + 6*a*b*c*d + 2*a
^2*d^2)*e*f^3 - (a*b*c^2 + 2*a^2*c*d)*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)) - 6*sqrt(3)*(b^2*d*e^4 - a^
2*c*e*f^3 - (b^2*c + 2*a*b*d)*e^3*f + (2*a*b*c + a^2*d)*e^2*f^2 + (b^2*d*e^3*f - a^2*c*f^4 - (b^2*c + 2*a*b*d)
*e^2*f^2 + (2*a*b*c + a^2*d)*e*f^3)*x)*(-b*d^2)^(1/3)*arctan(1/3*(2*sqrt(3)*(-b*d^2)^(2/3)*(b*x + a)^(1/3)*(d*
x + c)^(2/3) + sqrt(3)*(b*d^2*x + b*c*d))/(b*d^2*x + b*c*d)) - (-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*d*e^2 - (b*c + 2*a*d)*e*f + (3*b*d*e*f - (b*c + 2*a*d)*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*d*e^2 - (b*c + 2*a*d)*e*f + (
3*b*d*e*f - (b*c + 2*a*d)*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*
(b^2*d*e^4 - a^2*c*e*f^3 - (b^2*c + 2*a*b*d)*e^3*f + (2*a*b*c + a^2*d)*e^2*f^2 + (b^2*d*e^3*f - a^2*c*f^4 - (b
^2*c + 2*a*b*d)*e^2*f^2 + (2*a*b*c + a^2*d)*e*f^3)*x)*(-b*d^2)^(1/3)*log(((b*x + a)^(2/3)*(d*x + c)^(1/3)*d^2
- (-b*d^2)^(1/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3)*d + (-b*d^2)^(2/3)*(d*x + c))/(d*x + c)) + 6*(b^2*d*e^4 - a^2
*c*e*f^3 - (b^2*c + 2*a*b*d)*e^3*f + (2*a*b*c + a^2*d)*e^2*f^2 + (b^2*d*e^3*f - a^2*c*f^4 - (b^2*c + 2*a*b*d)*
e^2*f^2 + (2*a*b*c + a^2*d)*e*f^3)*x)*(-b*d^2)^(1/3)*log(((b*x + a)^(1/3)*(d*x + c)^(2/3)*d + (-b*d^2)^(1/3)*(
d*x + c))/(d*x + c)) - 6*(b^2*d*e^3*f - a^2*c*f^4 - (b^2*c + 2*a*b*d)*e^2*f^2 + (2*a*b*c + a^2*d)*e*f^3)*(b*x
+ a)^(1/3)*(d*x + c)^(2/3))/(b^2*d*e^4*f^2 - a^2*c*e*f^5 - (b^2*c + 2*a*b*d)*e^3*f^3 + (2*a*b*c + a^2*d)*e^2*f
^4 + (b^2*d*e^3*f^3 - a^2*c*f^6 - (b^2*c + 2*a*b*d)*e^2*f^4 + (2*a*b*c + a^2*d)*e*f^5)*x), -1/6*(6*sqrt(1/3)*(
3*b^2*d^2*e^4 - (4*b^2*c*d + 5*a*b*d^2)*e^3*f + (b^2*c^2 + 6*a*b*c*d + 2*a^2*d^2)*e^2*f^2 - (a*b*c^2 + 2*a^2*c
*d)*e*f^3 + (3*b^2*d^2*e^3*f - (4*b^2*c*d + 5*a*b*d^2)*e^2*f^2 + (b^2*c^2 + 6*a*b*c*d + 2*a^2*d^2)*e*f^3 - (a*
b*c^2 + 2*a^2*c*d)*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)) + 6*sqrt(3)*(b^2*d*e^4 - a^2*c*e*f^3 - (b^2*c + 2*a*b*d)*e^3*f + (2*a*b*c + a^2*d)*e^2*f^2
+ (b^2*d*e^3*f - a^2*c*f^4 - (b^2*c + 2*a*b*d)*e^2*f^2 + (2*a*b*c + a^2*d)*e*f^3)*x)*(-b*d^2)^(1/3)*arctan(1/3
*(2*sqrt(3)*(-b*d^2)^(2/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3) + sqrt(3)*(b*d^2*x + b*c*d))/(b*d^2*x + b*c*d)) + (
-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*d*e^2 - (b*c + 2*a*d)*e
*f + (3*b*d*e*f - (b*c + 2*a*d)*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*d*e^2 - (b*c + 2*a*d)*e*f + (3*b*d*e*f - (b*c + 2*a*d)*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*(b^2*d*e^4 - a^2*c*e*f^3 - (b^2*c + 2*a*b*d)*e^3*f + (2*a*b*c +
a^2*d)*e^2*f^2 + (b^2*d*e^3*f - a^2*c*f^4 - (b^2*c + 2*a*b*d)*e^2*f^2 + (2*a*b*c + a^2*d)*e*f^3)*x)*(-b*d^2)^(
1/3)*log(((b*x + a)^(2/3)*(d*x + c)^(1/3)*d^2 - (-b*d^2)^(1/3)*(b*x + a)^(1/3)*(d*x + c)^(2/3)*d + (-b*d^2)^(2
/3)*(d*x + c))/(d*x + c)) - 6*(b^2*d*e^4 - a^2*c*e*f^3 - (b^2*c + 2*a*b*d)*e^3*f + (2*a*b*c + a^2*d)*e^2*f^2 +
(b^2*d*e^3*f - a^2*c*f^4 - (b^2*c + 2*a*b*d)*e^2*f^2 + (2*a*b*c + a^2*d)*e*f^3)*x)*(-b*d^2)^(1/3)*log(((b*x +
a)^(1/3)*(d*x + c)^(2/3)*d + (-b*d^2)^(1/3)*(d*x + c))/(d*x + c)) + 6*(b^2*d*e^3*f - a^2*c*f^4 - (b^2*c + 2*a
*b*d)*e^2*f^2 + (2*a*b*c + a^2*d)*e*f^3)*(b*x + a)^(1/3)*(d*x + c)^(2/3))/(b^2*d*e^4*f^2 - a^2*c*e*f^5 - (b^2*
c + 2*a*b*d)*e^3*f^3 + (2*a*b*c + a^2*d)*e^2*f^4 + (b^2*d*e^3*f^3 - a^2*c*f^6 - (b^2*c + 2*a*b*d)*e^2*f^4 + (2
*a*b*c + a^2*d)*e*f^5)*x)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

integrate((b*x+a)^(1/3)*(d*x+c)^(2/3)/(f*x+e)^2,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x)^2} \, dx=\int \frac {{\left (a+b\,x\right )}^{1/3}\,{\left (c+d\,x\right )}^{2/3}}{{\left (e+f\,x\right )}^2} \,d x \]

[In]

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

[Out]

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