\(\int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)} \, dx\) [1665]

Optimal result

Integrand size = 26, antiderivative size = 381 \[ \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)} \, dx=\frac {3 (b c-a d) \sqrt [3]{a+b x}}{d (d e-c f) \sqrt [3]{c+d x}}-\frac {\sqrt {3} b^{4/3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{d^{4/3} f}+\frac {\sqrt {3} (b e-a f)^{4/3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}\right )}{f (d e-c f)^{4/3}}-\frac {b^{4/3} \log (a+b x)}{2 d^{4/3} f}-\frac {(b e-a f)^{4/3} \log (e+f x)}{2 f (d e-c f)^{4/3}}+\frac {3 (b e-a f)^{4/3} \log \left (-\sqrt [3]{a+b x}+\frac {\sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt [3]{d e-c f}}\right )}{2 f (d e-c f)^{4/3}}-\frac {3 b^{4/3} \log \left (1-\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{2 d^{4/3} f} \] Output:

3*(-a*d+b*c)*(b*x+a)^(1/3)/d/(-c*f+d*e)/(d*x+c)^(1/3)-3^(1/2)*b^(4/3)*arct 
an(1/3*3^(1/2)+2/3*b^(1/3)*(d*x+c)^(1/3)*3^(1/2)/d^(1/3)/(b*x+a)^(1/3))/d^ 
(4/3)/f+3^(1/2)*(-a*f+b*e)^(4/3)*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))/f/(-c*f+d*e)^(4/3)-1/ 
2*b^(4/3)*ln(b*x+a)/d^(4/3)/f-1/2*(-a*f+b*e)^(4/3)*ln(f*x+e)/f/(-c*f+d*e)^ 
(4/3)+3/2*(-a*f+b*e)^(4/3)*ln(-(b*x+a)^(1/3)+(-a*f+b*e)^(1/3)*(d*x+c)^(1/3 
)/(-c*f+d*e)^(1/3))/f/(-c*f+d*e)^(4/3)-3/2*b^(4/3)*ln(1-b^(1/3)*(d*x+c)^(1 
/3)/d^(1/3)/(b*x+a)^(1/3))/d^(4/3)/f
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 10.14 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.39 \[ \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)} \, dx=\frac {3 \sqrt [3]{a+b x} \left (-\frac {4 (b e-a f) \left (-1+\operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {4}{3},\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )\right )}{d e-c f}+\frac {(a+b x) \left (\frac {b (c+d x)}{b c-a d}\right )^{4/3} \operatorname {Hypergeometric2F1}\left (\frac {4}{3},\frac {4}{3},\frac {7}{3},\frac {d (a+b x)}{-b c+a d}\right )}{c+d x}\right )}{4 f \sqrt [3]{c+d x}} \] Input:

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

Output:

(3*(a + b*x)^(1/3)*((-4*(b*e - a*f)*(-1 + Hypergeometric2F1[1/3, 1, 4/3, ( 
(d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))]))/(d*e - c*f) + ((a + b*x) 
*((b*(c + d*x))/(b*c - a*d))^(4/3)*Hypergeometric2F1[4/3, 4/3, 7/3, (d*(a 
+ b*x))/(-(b*c) + a*d)])/(c + d*x)))/(4*f*(c + d*x)^(1/3))
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {109, 27, 175, 71, 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.

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 175

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

\(\Big \downarrow \) 71

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

\(\Big \downarrow \) 102

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 71

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] && PosQ[d/b]
 

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 109

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f 
*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) 
 Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) 
+ c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) 
 + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, 
d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || 
IntegersQ[m, n + p] || IntegersQ[p, m + n])
 

rule 175

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ 
)))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b   Int[(c + d*x)^n*(e + f*x)^p, x] 
, x] + Simp[(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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 738 vs. \(2 (305) = 610\).

Time = 0.28 (sec) , antiderivative size = 738, normalized size of antiderivative = 1.94 \[ \int \frac {(a+b x)^{4/3}}{(c+d x)^{4/3} (e+f x)} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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