\(\int \frac {(a+b x+c x^2)^{4/3}}{d+e x} \, dx\) [683]

Optimal result

Integrand size = 22, antiderivative size = 180 \[ \int \frac {\left (a+b x+c x^2\right )^{4/3}}{d+e x} \, dx=\frac {3 \left (a+b x+c x^2\right )^{4/3} \operatorname {AppellF1}\left (-\frac {8}{3},-\frac {4}{3},-\frac {4}{3},-\frac {5}{3},\frac {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{\sqrt [3]{2} e \left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{4/3} \left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{4/3}} \] Output:

3/2*(c*x^2+b*x+a)^(4/3)*AppellF1(-8/3,-4/3,-4/3,-5/3,(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))*2^ 
(2/3)/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)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 180, 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.

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

\(\Big \downarrow \) 1178

\(\displaystyle -\frac {4\ 2^{2/3} \left (\frac {1}{d+e x}\right )^{8/3} \left (a+b x+c x^2\right )^{4/3} \int \frac {\left (2-\frac {2 d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{c}}{d+e x}\right )^{4/3} \left (2-\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{d+e x}\right )^{4/3}}{4\ 2^{2/3} \left (\frac {1}{d+e x}\right )^{11/3}}d\frac {1}{d+e x}}{e \left (\frac {e \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{4/3} \left (\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{4/3}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\left (\frac {1}{d+e x}\right )^{8/3} \left (a+b x+c x^2\right )^{4/3} \int \frac {\left (2-\frac {2 d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{c}}{d+e x}\right )^{4/3} \left (2-\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{d+e x}\right )^{4/3}}{\left (\frac {1}{d+e x}\right )^{11/3}}d\frac {1}{d+e x}}{e \left (\frac {e \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{4/3} \left (\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{4/3}}\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {3 \left (a+b x+c x^2\right )^{4/3} \operatorname {AppellF1}\left (-\frac {8}{3},-\frac {4}{3},-\frac {4}{3},-\frac {5}{3},\frac {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{\sqrt [3]{2} e \left (\frac {e \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{4/3} \left (\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{4/3}}\)

Input:

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

Output:

(3*(a + b*x + c*x^2)^(4/3)*AppellF1[-8/3, -4/3, -4/3, -5/3, (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))])/(2^(1/3)*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))
 

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]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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