\(\int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x)^4} \, dx\) [1632]

Optimal result

Integrand size = 26, antiderivative size = 482 \[ \int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x)^4} \, dx=-\frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{3 f (e+f x)^3}+\frac {(3 b d e-b c f-2 a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{18 f (b e-a f) (d e-c f) (e+f x)^2}+\frac {\left (8 a^2 d^2 f^2-4 a b d f (3 d e+c f)+b^2 \left (9 d^2 e^2-6 c d e f+5 c^2 f^2\right )\right ) \sqrt [3]{a+b x} (c+d x)^{2/3}}{54 f (b e-a f)^2 (d e-c f)^2 (e+f x)}+\frac {(b c-a d)^2 (9 b d e-5 b c f-4 a d f) \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}\right )}{27 \sqrt {3} (b e-a f)^{8/3} (d e-c f)^{7/3}}-\frac {(b c-a d)^2 (9 b d e-5 b c f-4 a d f) \log (e+f x)}{162 (b e-a f)^{8/3} (d e-c f)^{7/3}}+\frac {(b c-a d)^2 (9 b d e-5 b c f-4 a d f) \log \left (-\sqrt [3]{a+b x}+\frac {\sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt [3]{d e-c f}}\right )}{54 (b e-a f)^{8/3} (d e-c f)^{7/3}} \] Output:

-1/3*(b*x+a)^(1/3)*(d*x+c)^(2/3)/f/(f*x+e)^3+1/18*(-2*a*d*f-b*c*f+3*b*d*e) 
*(b*x+a)^(1/3)*(d*x+c)^(2/3)/f/(-a*f+b*e)/(-c*f+d*e)/(f*x+e)^2+1/54*(8*a^2 
*d^2*f^2-4*a*b*d*f*(c*f+3*d*e)+b^2*(5*c^2*f^2-6*c*d*e*f+9*d^2*e^2))*(b*x+a 
)^(1/3)*(d*x+c)^(2/3)/f/(-a*f+b*e)^2/(-c*f+d*e)^2/(f*x+e)+1/81*(-a*d+b*c)^ 
2*(-4*a*d*f-5*b*c*f+9*b*d*e)*arctan(1/3*3^(1/2)+2/3*(-a*f+b*e)^(1/3)*(d*x+ 
c)^(1/3)*3^(1/2)/(-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)-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)
 

Mathematica [A] (verified)

Time = 6.93 (sec) , antiderivative size = 577, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x)^4} \, dx=\frac {1}{162} (b c-a d)^2 \left (\frac {3 \sqrt [3]{a+b x} (c+d x)^{2/3} \left (a b \left (-3 c^2 f^2 (-11 e+f x)+3 d^2 e \left (3 e^2-13 e f x-4 f^2 x^2\right )-2 c d f \left (29 e^2-5 e f x+2 f^2 x^2\right )\right )+2 a^2 f \left (-9 c^2 f^2-3 c d f (-5 e+f x)+d^2 \left (-2 e^2+11 e f x+4 f^2 x^2\right )\right )+b^2 \left (9 d^2 e^2 x (3 e+f x)+6 c d e \left (3 e^2-4 e f x-f^2 x^2\right )+c^2 f \left (-10 e^2+13 e f x+5 f^2 x^2\right )\right )\right )}{(b c-a d)^2 (b e-a f)^2 (d e-c f)^2 (e+f x)^3}+\frac {2 \sqrt {3} (9 b d e-5 b c f-4 a d f) \arctan \left (\frac {1-\frac {2 \sqrt [3]{-d e+c f} \sqrt [3]{a+b x}}{\sqrt [3]{b e-a f} \sqrt [3]{c+d x}}}{\sqrt {3}}\right )}{(b e-a f)^{8/3} (-d e+c f)^{7/3}}-\frac {2 (9 b d e-5 b c f-4 a d f) \log \left (\sqrt [3]{b e-a f}+\frac {\sqrt [3]{-d e+c f} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}\right )}{(b e-a f)^{8/3} (-d e+c f)^{7/3}}+\frac {(9 b d e-5 b c f-4 a d f) \log \left ((b e-a f)^{2/3}+\frac {(-d e+c f)^{2/3} (a+b x)^{2/3}}{(c+d x)^{2/3}}-\frac {\sqrt [3]{b e-a f} \sqrt [3]{-d e+c f} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}\right )}{(b e-a f)^{8/3} (-d e+c f)^{7/3}}\right ) \] Input:

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

Output:

((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
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 409, normalized size of antiderivative = 0.85, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {107, 105, 105, 102}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 107

\(\displaystyle \frac {(-4 a d f-5 b c f+9 b d e) \int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x)^3}dx}{9 (b e-a f) (d e-c f)}-\frac {f (a+b x)^{4/3} (c+d x)^{5/3}}{3 (e+f x)^3 (b e-a f) (d e-c f)}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {(-4 a d f-5 b c f+9 b d e) \left (\frac {\sqrt [3]{a+b x} (c+d x)^{5/3}}{2 (e+f x)^2 (d e-c f)}-\frac {(b c-a d) \int \frac {(c+d x)^{2/3}}{(a+b x)^{2/3} (e+f x)^2}dx}{6 (d e-c f)}\right )}{9 (b e-a f) (d e-c f)}-\frac {f (a+b x)^{4/3} (c+d x)^{5/3}}{3 (e+f x)^3 (b e-a f) (d e-c f)}\)

\(\Big \downarrow \) 105

\(\displaystyle \frac {(-4 a d f-5 b c f+9 b d e) \left (\frac {\sqrt [3]{a+b x} (c+d x)^{5/3}}{2 (e+f x)^2 (d e-c f)}-\frac {(b c-a d) \left (\frac {2 (b c-a d) \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)}dx}{3 (b e-a f)}+\frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x) (b e-a f)}\right )}{6 (d e-c f)}\right )}{9 (b e-a f) (d e-c f)}-\frac {f (a+b x)^{4/3} (c+d x)^{5/3}}{3 (e+f x)^3 (b e-a f) (d e-c f)}\)

\(\Big \downarrow \) 102

\(\displaystyle \frac {(-4 a d f-5 b c f+9 b d e) \left (\frac {\sqrt [3]{a+b x} (c+d x)^{5/3}}{2 (e+f x)^2 (d e-c f)}-\frac {(b c-a d) \left (\frac {2 (b c-a d) \left (-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt {3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac {1}{\sqrt {3}}\right )}{(b e-a f)^{2/3} \sqrt [3]{d e-c f}}+\frac {\log (e+f x)}{2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}-\frac {3 \log \left (\frac {\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 (b e-a f)^{2/3} \sqrt [3]{d e-c f}}\right )}{3 (b e-a f)}+\frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x) (b e-a f)}\right )}{6 (d e-c f)}\right )}{9 (b e-a f) (d e-c f)}-\frac {f (a+b x)^{4/3} (c+d x)^{5/3}}{3 (e+f x)^3 (b e-a f) (d e-c f)}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 102

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.) 
*(x_))), x_] :> 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 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[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 107

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> 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] + Simp[(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] || SumSimplerQ[m, 1])
 

Maple [F]

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

Input:

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

Output:

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. \(2 (426) = 852\).

Time = 1.10 (sec) , antiderivative size = 7932, normalized size of antiderivative = 16.46 \[ \int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{(e+f x)^4} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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