\(\int \frac {(a+b \sqrt {x}+c x)^p}{x} \, dx\) [82]

Optimal result

Integrand size = 18, antiderivative size = 162 \[ \int \frac {\left (a+b \sqrt {x}+c x\right )^p}{x} \, dx=\frac {4^p \left (\frac {b-\sqrt {b^2-4 a c}+2 c \sqrt {x}}{c \sqrt {x}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c \sqrt {x}}{c \sqrt {x}}\right )^{-p} \left (a+b \sqrt {x}+c x\right )^p \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,-\frac {b-\sqrt {b^2-4 a c}}{2 c \sqrt {x}},-\frac {b+\sqrt {b^2-4 a c}}{2 c \sqrt {x}}\right )}{p} \] Output:

4^p*(a+b*x^(1/2)+c*x)^p*AppellF1(-2*p,-p,-p,1-2*p,-1/2*(b-(-4*a*c+b^2)^(1/ 
2))/c/x^(1/2),-1/2*(b+(-4*a*c+b^2)^(1/2))/c/x^(1/2))/p/(((b-(-4*a*c+b^2)^( 
1/2)+2*c*x^(1/2))/c/x^(1/2))^p)/(((b+(-4*a*c+b^2)^(1/2)+2*c*x^(1/2))/c/x^( 
1/2))^p)
 

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.02 \[ \int \frac {\left (a+b \sqrt {x}+c x\right )^p}{x} \, dx=\frac {4^p \left (\frac {b-\sqrt {b^2-4 a c}+2 c \sqrt {x}}{c \sqrt {x}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c \sqrt {x}}{c \sqrt {x}}\right )^{-p} \left (a+b \sqrt {x}+c x\right )^p \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {-b-\sqrt {b^2-4 a c}}{2 c \sqrt {x}},\frac {-b+\sqrt {b^2-4 a c}}{2 c \sqrt {x}}\right )}{p} \] Input:

Integrate[(a + b*Sqrt[x] + c*x)^p/x,x]
 

Output:

(4^p*(a + b*Sqrt[x] + c*x)^p*AppellF1[-2*p, -p, -p, 1 - 2*p, (-b - Sqrt[b^ 
2 - 4*a*c])/(2*c*Sqrt[x]), (-b + Sqrt[b^2 - 4*a*c])/(2*c*Sqrt[x])])/(p*((b 
 - Sqrt[b^2 - 4*a*c] + 2*c*Sqrt[x])/(c*Sqrt[x]))^p*((b + Sqrt[b^2 - 4*a*c] 
 + 2*c*Sqrt[x])/(c*Sqrt[x]))^p)
 

Rubi [A] (warning: unable to verify)

Time = 0.31 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1693, 1178, 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.

\(\displaystyle \int \frac {\left (a+b \sqrt {x}+c x\right )^p}{x} \, dx\)

\(\Big \downarrow \) 1693

\(\displaystyle 2 \int \frac {\left (a+c x+b \sqrt {x}\right )^p}{\sqrt {x}}d\sqrt {x}\)

\(\Big \downarrow \) 1178

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

\(\Big \downarrow \) 150

\(\displaystyle \frac {2^{2 p} x^{-2 p} \left (\frac {-\sqrt {b^2-4 a c}+b+2 c \sqrt {x}}{c \sqrt {x}}\right )^{-p} \left (\frac {\sqrt {b^2-4 a c}+b+2 c \sqrt {x}}{c \sqrt {x}}\right )^{-p} \left (a+b \sqrt {x}+c x\right )^p \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,-\frac {b-\sqrt {b^2-4 a c}}{2 c \sqrt {x}},-\frac {b+\sqrt {b^2-4 a c}}{2 c \sqrt {x}}\right )}{p}\)

Input:

Int[(a + b*Sqrt[x] + c*x)^p/x,x]
 

Output:

(2^(2*p)*(a + b*Sqrt[x] + c*x)^p*AppellF1[-2*p, -p, -p, 1 - 2*p, -1/2*(b - 
 Sqrt[b^2 - 4*a*c])/(c*Sqrt[x]), -1/2*(b + Sqrt[b^2 - 4*a*c])/(c*Sqrt[x])] 
)/(p*((b - Sqrt[b^2 - 4*a*c] + 2*c*Sqrt[x])/(c*Sqrt[x]))^p*((b + Sqrt[b^2 
- 4*a*c] + 2*c*Sqrt[x])/(c*Sqrt[x]))^p*x^(2*p))
 

Defintions of rubi rules used

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]
 

rule 1693

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol 
] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, 
x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && IntegerQ 
[Simplify[(m + 1)/n]]
 

Maple [F]

\[\int \frac {\left (a +b \sqrt {x}+c x \right )^{p}}{x}d x\]

Input:

int((a+b*x^(1/2)+c*x)^p/x,x)
 

Output:

int((a+b*x^(1/2)+c*x)^p/x,x)
 

Fricas [F]

\[ \int \frac {\left (a+b \sqrt {x}+c x\right )^p}{x} \, dx=\int { \frac {{\left (c x + b \sqrt {x} + a\right )}^{p}}{x} \,d x } \] Input:

integrate((a+b*x^(1/2)+c*x)^p/x,x, algorithm="fricas")
 

Output:

integral((c*x + b*sqrt(x) + a)^p/x, x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b \sqrt {x}+c x\right )^p}{x} \, dx=\text {Timed out} \] Input:

integrate((a+b*x**(1/2)+c*x)**p/x,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b \sqrt {x}+c x\right )^p}{x} \, dx=\int { \frac {{\left (c x + b \sqrt {x} + a\right )}^{p}}{x} \,d x } \] Input:

integrate((a+b*x^(1/2)+c*x)^p/x,x, algorithm="maxima")
 

Output:

integrate((c*x + b*sqrt(x) + a)^p/x, x)
 

Giac [F]

\[ \int \frac {\left (a+b \sqrt {x}+c x\right )^p}{x} \, dx=\int { \frac {{\left (c x + b \sqrt {x} + a\right )}^{p}}{x} \,d x } \] Input:

integrate((a+b*x^(1/2)+c*x)^p/x,x, algorithm="giac")
 

Output:

integrate((c*x + b*sqrt(x) + a)^p/x, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \sqrt {x}+c x\right )^p}{x} \, dx=\int \frac {{\left (a+c\,x+b\,\sqrt {x}\right )}^p}{x} \,d x \] Input:

int((a + c*x + b*x^(1/2))^p/x,x)
 

Output:

int((a + c*x + b*x^(1/2))^p/x, x)
 

Reduce [F]

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

int((a+b*x^(1/2)+c*x)^p/x,x)
 

Output:

(2*(sqrt(x)*b + a + c*x)**p + int((sqrt(x)*b + a + c*x)**p/(a**2 + 2*a*c*x 
 - b**2*x + c**2*x**2),x)*a*c*p + int((sqrt(x)*b + a + c*x)**p/(a**2*x + 2 
*a*c*x**2 - b**2*x**2 + c**2*x**3),x)*a**2*p - int((sqrt(x)*b + a + c*x)** 
p/(sqrt(x)*b + a + c*x),x)*c*p - int((sqrt(x)*(sqrt(x)*b + a + c*x)**p)/(a 
**2*x + 2*a*c*x**2 - b**2*x**2 + c**2*x**3),x)*a*b*p)/p