\(\int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x)^4} \, dx\) [3007]

Optimal result

Integrand size = 26, antiderivative size = 465 \[ \int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x)^4} \, dx=-\frac {f (a+b x)^{4/3} (c+d x)^{5/3}}{3 (b e-a f) (d e-c f) (e+f x)^3}+\frac {(9 b d e-5 b c f-4 a d f) \sqrt [3]{a+b x} (c+d x)^{5/3}}{18 (b e-a f) (d e-c f)^2 (e+f x)^2}-\frac {(b c-a d) (9 b d e-5 b c f-4 a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{54 (b e-a f)^2 (d e-c f)^2 (e+f x)}+\frac {(b c-a d)^2 (9 b d e-5 b c f-4 a d f) \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}\right )}{27 \sqrt {3} (b e-a f)^{8/3} (d e-c f)^{7/3}}-\frac {(b c-a d)^2 (9 b d e-5 b c f-4 a d f) \log (e+f x)}{162 (b e-a f)^{8/3} (d e-c f)^{7/3}}+\frac {(b c-a d)^2 (9 b d e-5 b c f-4 a d f) \log \left (-\sqrt [3]{a+b x}+\frac {\sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt [3]{d e-c f}}\right )}{54 (b e-a f)^{8/3} (d e-c f)^{7/3}} \]

[Out]

-1/3*f*(b*x+a)^(4/3)*(d*x+c)^(5/3)/(-a*f+b*e)/(-c*f+d*e)/(f*x+e)^3+1/18*(-4*a*d*f-5*b*c*f+9*b*d*e)*(b*x+a)^(1/
3)*(d*x+c)^(5/3)/(-a*f+b*e)/(-c*f+d*e)^2/(f*x+e)^2-1/54*(-a*d+b*c)*(-4*a*d*f-5*b*c*f+9*b*d*e)*(b*x+a)^(1/3)*(d
*x+c)^(2/3)/(-a*f+b*e)^2/(-c*f+d*e)^2/(f*x+e)-1/162*(-a*d+b*c)^2*(-4*a*d*f-5*b*c*f+9*b*d*e)*ln(f*x+e)/(-a*f+b*
e)^(8/3)/(-c*f+d*e)^(7/3)+1/54*(-a*d+b*c)^2*(-4*a*d*f-5*b*c*f+9*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)^(8/3)/(-c*f+d*e)^(7/3)+1/81*(-a*d+b*c)^2*(-4*a*d*f-5*b*c*f+9*b*d*e)*arc
tan(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)^(8/3)/(-
c*f+d*e)^(7/3)*3^(1/2)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {98, 96, 93} \[ \int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x)^4} \, dx=\frac {(b c-a d)^2 (-4 a d f-5 b c f+9 b d e) \arctan \left (\frac {2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt {3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac {1}{\sqrt {3}}\right )}{27 \sqrt {3} (b e-a f)^{8/3} (d e-c f)^{7/3}}-\frac {\sqrt [3]{a+b x} (c+d x)^{2/3} (b c-a d) (-4 a d f-5 b c f+9 b d e)}{54 (e+f x) (b e-a f)^2 (d e-c f)^2}+\frac {\sqrt [3]{a+b x} (c+d x)^{5/3} (-4 a d f-5 b c f+9 b d e)}{18 (e+f x)^2 (b e-a f) (d e-c f)^2}-\frac {f (a+b x)^{4/3} (c+d x)^{5/3}}{3 (e+f x)^3 (b e-a f) (d e-c f)}-\frac {(b c-a d)^2 \log (e+f x) (-4 a d f-5 b c f+9 b d e)}{162 (b e-a f)^{8/3} (d e-c f)^{7/3}}+\frac {(b c-a d)^2 (-4 a d f-5 b c f+9 b d e) \log \left (\frac {\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{54 (b e-a f)^{8/3} (d e-c f)^{7/3}} \]

[In]

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

[Out]

-1/3*(f*(a + b*x)^(4/3)*(c + d*x)^(5/3))/((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)*Lo
g[-(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 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*} \text {integral}& = -\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 [A] (verified)

Time = 7.42 (sec) , antiderivative size = 577, normalized size of antiderivative = 1.24 \[ \int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x)^4} \, dx=\frac {1}{162} (b c-a d)^2 \left (\frac {3 \sqrt [3]{a+b x} (c+d x)^{2/3} \left (a b \left (-3 c^2 f^2 (-11 e+f x)+3 d^2 e \left (3 e^2-13 e f x-4 f^2 x^2\right )-2 c d f \left (29 e^2-5 e f x+2 f^2 x^2\right )\right )+2 a^2 f \left (-9 c^2 f^2-3 c d f (-5 e+f x)+d^2 \left (-2 e^2+11 e f x+4 f^2 x^2\right )\right )+b^2 \left (9 d^2 e^2 x (3 e+f x)+6 c d e \left (3 e^2-4 e f x-f^2 x^2\right )+c^2 f \left (-10 e^2+13 e f x+5 f^2 x^2\right )\right )\right )}{(b c-a d)^2 (b e-a f)^2 (d e-c f)^2 (e+f x)^3}+\frac {2 \sqrt {3} (9 b d e-5 b c f-4 a d f) \arctan \left (\frac {1-\frac {2 \sqrt [3]{-d e+c f} \sqrt [3]{a+b x}}{\sqrt [3]{b e-a f} \sqrt [3]{c+d x}}}{\sqrt {3}}\right )}{(b e-a f)^{8/3} (-d e+c f)^{7/3}}-\frac {2 (9 b d e-5 b c f-4 a d f) \log \left (\sqrt [3]{b e-a f}+\frac {\sqrt [3]{-d e+c f} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}\right )}{(b e-a f)^{8/3} (-d e+c f)^{7/3}}+\frac {(9 b d e-5 b c f-4 a d f) \log \left ((b e-a f)^{2/3}+\frac {(-d e+c f)^{2/3} (a+b x)^{2/3}}{(c+d x)^{2/3}}-\frac {\sqrt [3]{b e-a f} \sqrt [3]{-d e+c f} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}\right )}{(b e-a f)^{8/3} (-d e+c f)^{7/3}}\right ) \]

[In]

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

[Out]

((b*c - a*d)^2*((3*(a + b*x)^(1/3)*(c + d*x)^(2/3)*(a*b*(-3*c^2*f^2*(-11*e + f*x) + 3*d^2*e*(3*e^2 - 13*e*f*x
- 4*f^2*x^2) - 2*c*d*f*(29*e^2 - 5*e*f*x + 2*f^2*x^2)) + 2*a^2*f*(-9*c^2*f^2 - 3*c*d*f*(-5*e + f*x) + d^2*(-2*
e^2 + 11*e*f*x + 4*f^2*x^2)) + b^2*(9*d^2*e^2*x*(3*e + f*x) + 6*c*d*e*(3*e^2 - 4*e*f*x - f^2*x^2) + c^2*f*(-10
*e^2 + 13*e*f*x + 5*f^2*x^2))))/((b*c - a*d)^2*(b*e - a*f)^2*(d*e - c*f)^2*(e + f*x)^3) + (2*Sqrt[3]*(9*b*d*e
- 5*b*c*f - 4*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)^(8/3)*(-(d*e) + c*f)^(7/3)) - (2*(9*b*d*e - 5*b*c*f - 4*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)^(8/3)*(-(d*e) + c*f)^(7/3)) + ((9*b*d*e
- 5*b*c*f - 4*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)^(8/3)*(-(d*e) + c*f)^(7/3))))
/162

Maple [F]

\[\int \frac {\left (b x +a \right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {2}{3}}}{\left (f x +e \right )^{4}}d x\]

[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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3888 vs. \(2 (409) = 818\).

Time = 1.17 (sec) , antiderivative size = 7932, normalized size of antiderivative = 17.06 \[ \int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x)^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x)^4} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{{\left (f x + e\right )}^{4}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x)^4} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{{\left (f x + e\right )}^{4}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x)^4} \, dx=\int \frac {{\left (a+b\,x\right )}^{1/3}\,{\left (c+d\,x\right )}^{2/3}}{{\left (e+f\,x\right )}^4} \,d x \]

[In]

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

[Out]

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