\(\int \frac {(a+b x+c x^2)^{4/3}}{(d+e x)^2} \, dx\) [684]

Optimal result

Integrand size = 22, antiderivative size = 189 \[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(d+e x)^2} \, dx=\frac {12\ 2^{2/3} \left (a+b x+c x^2\right )^{4/3} \operatorname {AppellF1}\left (-\frac {5}{3},-\frac {4}{3},-\frac {4}{3},-\frac {2}{3},\frac {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{5 e \left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{4/3} \left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{4/3} (d+e x)} \] Output:

12/5*2^(2/3)*(c*x^2+b*x+a)^(4/3)*AppellF1(-5/3,-4/3,-4/3,-2/3,(2*d-(b+(-4* 
a*c+b^2)^(1/2))*e/c)/(2*e*x+2*d),1/2*(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)/c/(e 
*x+d))/e/(e*(b-(-4*a*c+b^2)^(1/2)+2*c*x)/c/(e*x+d))^(4/3)/(e*(2*c*x+(-4*a* 
c+b^2)^(1/2)+b)/c/(e*x+d))^(4/3)/(e*x+d)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1178, 27, 150}

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.

\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(d+e x)^2} \, dx\)

\(\Big \downarrow \) 1178

\(\displaystyle -\frac {4\ 2^{2/3} \left (\frac {1}{d+e x}\right )^{8/3} \left (a+b x+c x^2\right )^{4/3} \int \frac {\left (2-\frac {2 d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{c}}{d+e x}\right )^{4/3} \left (2-\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{d+e x}\right )^{4/3}}{4\ 2^{2/3} \left (\frac {1}{d+e x}\right )^{8/3}}d\frac {1}{d+e x}}{e \left (\frac {e \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{4/3} \left (\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{4/3}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\left (\frac {1}{d+e x}\right )^{8/3} \left (a+b x+c x^2\right )^{4/3} \int \frac {\left (2-\frac {2 d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{c}}{d+e x}\right )^{4/3} \left (2-\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{d+e x}\right )^{4/3}}{\left (\frac {1}{d+e x}\right )^{8/3}}d\frac {1}{d+e x}}{e \left (\frac {e \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{4/3} \left (\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{4/3}}\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {12\ 2^{2/3} \left (a+b x+c x^2\right )^{4/3} \operatorname {AppellF1}\left (-\frac {5}{3},-\frac {4}{3},-\frac {4}{3},-\frac {2}{3},\frac {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{5 e (d+e x) \left (\frac {e \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{4/3} \left (\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{4/3}}\)

Input:

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

Output:

(12*2^(2/3)*(a + b*x + c*x^2)^(4/3)*AppellF1[-5/3, -4/3, -4/3, -2/3, (2*c* 
d - (b - Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*d - ((b + Sqrt[b^2 - 4* 
a*c])*e)/c)/(2*(d + e*x))])/(5*e*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*( 
d + e*x)))^(4/3)*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(4/3) 
*(d + e*x))
 

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 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 1178

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(-(1/(d + e*x))^(2*p))*((a + 
b*x + c*x^2)^p/(e*(e*((b - q + 2*c*x)/(2*c*(d + e*x))))^p*(e*((b + q + 2*c* 
x)/(2*c*(d + e*x))))^p))   Subst[Int[x^(-m - 2*(p + 1))*Simp[1 - (d - e*((b 
 - q)/(2*c)))*x, x]^p*Simp[1 - (d - e*((b + q)/(2*c)))*x, x]^p, x], x, 1/(d 
 + e*x)], x]] /; FreeQ[{a, b, c, d, e, p}, x] && ILtQ[m, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{(d+e x)^2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{1,[0,1,1,2,0,0,0]%%%}+%%%{-1,[0,1,0,1,1,1,0]%%%}+%%%{1,[0, 
1,0,0,2,0
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 15*(a + b*x + c*x**2)**(1/3)*a*b*e**2 + 9*(a + b*x + c*x**2)**(1/3)*a* 
c*d*e + 18*(a + b*x + c*x**2)**(1/3)*b**2*d*e + 9*(a + b*x + c*x**2)**(1/3 
)*b**2*e**2*x - 24*(a + b*x + c*x**2)**(1/3)*b*c*d**2 - 21*(a + b*x + c*x* 
*2)**(1/3)*b*c*d*e*x + 3*(a + b*x + c*x**2)**(1/3)*b*c*e**2*x**2 + 12*(a + 
 b*x + c*x**2)**(1/3)*c**2*d**2*x - 3*(a + b*x + c*x**2)**(1/3)*c**2*d*e*x 
**2 - 10*int((a + b*x + c*x**2)**(1/3)/(a*b*d**2*e + 2*a*b*d*e**2*x + a*b* 
e**3*x**2 - a*c*d**3 - 2*a*c*d**2*e*x - a*c*d*e**2*x**2 + b**2*d**2*e*x + 
2*b**2*d*e**2*x**2 + b**2*e**3*x**3 - b*c*d**3*x - b*c*d**2*e*x**2 + b*c*d 
*e**2*x**3 + b*c*e**3*x**4 - c**2*d**3*x**2 - 2*c**2*d**2*e*x**3 - c**2*d* 
e**2*x**4),x)*a**2*b**2*d*e**4 - 10*int((a + b*x + c*x**2)**(1/3)/(a*b*d** 
2*e + 2*a*b*d*e**2*x + a*b*e**3*x**2 - a*c*d**3 - 2*a*c*d**2*e*x - a*c*d*e 
**2*x**2 + b**2*d**2*e*x + 2*b**2*d*e**2*x**2 + b**2*e**3*x**3 - b*c*d**3* 
x - b*c*d**2*e*x**2 + b*c*d*e**2*x**3 + b*c*e**3*x**4 - c**2*d**3*x**2 - 2 
*c**2*d**2*e*x**3 - c**2*d*e**2*x**4),x)*a**2*b**2*e**5*x + 14*int((a + b* 
x + c*x**2)**(1/3)/(a*b*d**2*e + 2*a*b*d*e**2*x + a*b*e**3*x**2 - a*c*d**3 
 - 2*a*c*d**2*e*x - a*c*d*e**2*x**2 + b**2*d**2*e*x + 2*b**2*d*e**2*x**2 + 
 b**2*e**3*x**3 - b*c*d**3*x - b*c*d**2*e*x**2 + b*c*d*e**2*x**3 + b*c*e** 
3*x**4 - c**2*d**3*x**2 - 2*c**2*d**2*e*x**3 - c**2*d*e**2*x**4),x)*a**2*b 
*c*d**2*e**3 + 14*int((a + b*x + c*x**2)**(1/3)/(a*b*d**2*e + 2*a*b*d*e**2 
*x + a*b*e**3*x**2 - a*c*d**3 - 2*a*c*d**2*e*x - a*c*d*e**2*x**2 + b**2...