\(\int \frac {(a+b x)^m (c+d x)^n (g+h x)}{\sqrt {e+f x}} \, dx\) [231]

Optimal result

Integrand size = 29, antiderivative size = 264 \[ \int \frac {(a+b x)^m (c+d x)^n (g+h x)}{\sqrt {e+f x}} \, dx=-\frac {2 (b e-a f) h (a+b x)^m \left (-\frac {f (a+b x)}{b e-a f}\right )^{-m} (c+d x)^n \left (-\frac {f (c+d x)}{d e-c f}\right )^{-n} \sqrt {e+f x} \operatorname {AppellF1}\left (\frac {1}{2},-1-m,-n,\frac {3}{2},\frac {b (e+f x)}{b e-a f},\frac {d (e+f x)}{d e-c f}\right )}{b f^2}+\frac {2 (b g-a h) (a+b x)^m \left (-\frac {f (a+b x)}{b e-a f}\right )^{-m} (c+d x)^n \left (-\frac {f (c+d x)}{d e-c f}\right )^{-n} \sqrt {e+f x} \operatorname {AppellF1}\left (\frac {1}{2},-m,-n,\frac {3}{2},\frac {b (e+f x)}{b e-a f},\frac {d (e+f x)}{d e-c f}\right )}{b f} \] Output:

-2*(-a*f+b*e)*h*(b*x+a)^m*(d*x+c)^n*(f*x+e)^(1/2)*AppellF1(1/2,-1-m,-n,3/2 
,b*(f*x+e)/(-a*f+b*e),d*(f*x+e)/(-c*f+d*e))/b/f^2/((-f*(b*x+a)/(-a*f+b*e)) 
^m)/((-f*(d*x+c)/(-c*f+d*e))^n)+2*(-a*h+b*g)*(b*x+a)^m*(d*x+c)^n*(f*x+e)^( 
1/2)*AppellF1(1/2,-m,-n,3/2,b*(f*x+e)/(-a*f+b*e),d*(f*x+e)/(-c*f+d*e))/b/f 
/((-f*(b*x+a)/(-a*f+b*e))^m)/((-f*(d*x+c)/(-c*f+d*e))^n)
 

Mathematica [A] (warning: unable to verify)

Time = 0.33 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.78 \[ \int \frac {(a+b x)^m (c+d x)^n (g+h x)}{\sqrt {e+f x}} \, dx=-\frac {(a+b x)^{1+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \sqrt {e+f x} \left ((b e-a f) h \operatorname {AppellF1}\left (1+m,-n,-\frac {1}{2},2+m,\frac {d (a+b x)}{-b c+a d},\frac {f (a+b x)}{-b e+a f}\right )+b (f g-e h) \operatorname {AppellF1}\left (1+m,-n,\frac {1}{2},2+m,\frac {d (a+b x)}{-b c+a d},\frac {f (a+b x)}{-b e+a f}\right )\right )}{b f (-b e+a f) (1+m) \sqrt {\frac {b (e+f x)}{b e-a f}}} \] Input:

Integrate[((a + b*x)^m*(c + d*x)^n*(g + h*x))/Sqrt[e + f*x],x]
 

Output:

-(((a + b*x)^(1 + m)*(c + d*x)^n*Sqrt[e + f*x]*((b*e - a*f)*h*AppellF1[1 + 
 m, -n, -1/2, 2 + m, (d*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) + 
 a*f)] + b*(f*g - e*h)*AppellF1[1 + m, -n, 1/2, 2 + m, (d*(a + b*x))/(-(b* 
c) + a*d), (f*(a + b*x))/(-(b*e) + a*f)]))/(b*f*(-(b*e) + a*f)*(1 + m)*((b 
*(c + d*x))/(b*c - a*d))^n*Sqrt[(b*(e + f*x))/(b*e - a*f)]))
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {177, 157, 156, 155}

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.

\(\displaystyle \int \frac {(g+h x) (a+b x)^m (c+d x)^n}{\sqrt {e+f x}} \, dx\)

\(\Big \downarrow \) 177

\(\displaystyle \frac {(b g-a h) \int \frac {(a+b x)^m (c+d x)^n}{\sqrt {e+f x}}dx}{b}+\frac {h \int \frac {(a+b x)^{m+1} (c+d x)^n}{\sqrt {e+f x}}dx}{b}\)

\(\Big \downarrow \) 157

\(\displaystyle \frac {(b g-a h) (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \int \frac {(a+b x)^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n}{\sqrt {e+f x}}dx}{b}+\frac {h (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \int \frac {(a+b x)^{m+1} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n}{\sqrt {e+f x}}dx}{b}\)

\(\Big \downarrow \) 156

\(\displaystyle \frac {(b g-a h) (c+d x)^n \sqrt {\frac {b (e+f x)}{b e-a f}} \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \int \frac {(a+b x)^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n}{\sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}dx}{b \sqrt {e+f x}}+\frac {h (c+d x)^n \sqrt {\frac {b (e+f x)}{b e-a f}} \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \int \frac {(a+b x)^{m+1} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n}{\sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}dx}{b \sqrt {e+f x}}\)

\(\Big \downarrow \) 155

\(\displaystyle \frac {(b g-a h) (a+b x)^{m+1} (c+d x)^n \sqrt {\frac {b (e+f x)}{b e-a f}} \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \operatorname {AppellF1}\left (m+1,-n,\frac {1}{2},m+2,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^2 (m+1) \sqrt {e+f x}}+\frac {h (a+b x)^{m+2} (c+d x)^n \sqrt {\frac {b (e+f x)}{b e-a f}} \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \operatorname {AppellF1}\left (m+2,-n,\frac {1}{2},m+3,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b^2 (m+2) \sqrt {e+f x}}\)

Input:

Int[((a + b*x)^m*(c + d*x)^n*(g + h*x))/Sqrt[e + f*x],x]
 

Output:

((b*g - a*h)*(a + b*x)^(1 + m)*(c + d*x)^n*Sqrt[(b*(e + f*x))/(b*e - a*f)] 
*AppellF1[1 + m, -n, 1/2, 2 + m, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b 
*x))/(b*e - a*f))])/(b^2*(1 + m)*((b*(c + d*x))/(b*c - a*d))^n*Sqrt[e + f* 
x]) + (h*(a + b*x)^(2 + m)*(c + d*x)^n*Sqrt[(b*(e + f*x))/(b*e - a*f)]*App 
ellF1[2 + m, -n, 1/2, 3 + m, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x)) 
/(b*e - a*f))])/(b^2*(2 + m)*((b*(c + d*x))/(b*c - a*d))^n*Sqrt[e + f*x])
 

Defintions of rubi rules used

rule 155

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

rule 156

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) 
^(p_), x_] :> Simp[(e + f*x)^FracPart[p]/(Simplify[b/(b*e - a*f)]^IntPart[p 
]*(b*((e + f*x)/(b*e - a*f)))^FracPart[p])   Int[(a + b*x)^m*(c + d*x)^n*Si 
mp[b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)), x]^p, x], x] /; FreeQ[{a, b, c, 
 d, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !IntegerQ[p] & 
& GtQ[Simplify[b/(b*c - a*d)], 0] &&  !GtQ[Simplify[b/(b*e - a*f)], 0]
 

rule 157

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) 
^(p_), x_] :> Simp[(c + d*x)^FracPart[n]/(Simplify[b/(b*c - a*d)]^IntPart[n 
]*(b*((c + d*x)/(b*c - a*d)))^FracPart[n])   Int[(a + b*x)^m*Simp[b*(c/(b*c 
 - a*d)) + b*d*(x/(b*c - a*d)), x]^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, 
 d, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !IntegerQ[p] & 
&  !GtQ[Simplify[b/(b*c - a*d)], 0] &&  !SimplerQ[c + d*x, a + b*x] &&  !Si 
mplerQ[e + f*x, a + b*x]
 

rule 177

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h/b   Int[(a + b*x)^(m + 1)*(c + d 
*x)^n*(e + f*x)^p, x], x] + Simp[(b*g - a*h)/b   Int[(a + b*x)^m*(c + d*x)^ 
n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p}, x] && (Su 
mSimplerQ[m, 1] || ( !SumSimplerQ[n, 1] &&  !SumSimplerQ[p, 1]))
 

Maple [F]

\[\int \frac {\left (b x +a \right )^{m} \left (x d +c \right )^{n} \left (h x +g \right )}{\sqrt {f x +e}}d x\]

Input:

int((b*x+a)^m*(d*x+c)^n*(h*x+g)/(f*x+e)^(1/2),x)
 

Output:

int((b*x+a)^m*(d*x+c)^n*(h*x+g)/(f*x+e)^(1/2),x)
 

Fricas [F]

\[ \int \frac {(a+b x)^m (c+d x)^n (g+h x)}{\sqrt {e+f x}} \, dx=\int { \frac {{\left (h x + g\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n}}{\sqrt {f x + e}} \,d x } \] Input:

integrate((b*x+a)^m*(d*x+c)^n*(h*x+g)/(f*x+e)^(1/2),x, algorithm="fricas")
 

Output:

integral((h*x + g)*(b*x + a)^m*(d*x + c)^n/sqrt(f*x + e), x)
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^m (c+d x)^n (g+h x)}{\sqrt {e+f x}} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate((b*x+a)**m*(d*x+c)**n*(h*x+g)/(f*x+e)**(1/2),x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

\[ \int \frac {(a+b x)^m (c+d x)^n (g+h x)}{\sqrt {e+f x}} \, dx=\int { \frac {{\left (h x + g\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n}}{\sqrt {f x + e}} \,d x } \] Input:

integrate((b*x+a)^m*(d*x+c)^n*(h*x+g)/(f*x+e)^(1/2),x, algorithm="maxima")
 

Output:

integrate((h*x + g)*(b*x + a)^m*(d*x + c)^n/sqrt(f*x + e), x)
 

Giac [F]

\[ \int \frac {(a+b x)^m (c+d x)^n (g+h x)}{\sqrt {e+f x}} \, dx=\int { \frac {{\left (h x + g\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n}}{\sqrt {f x + e}} \,d x } \] Input:

integrate((b*x+a)^m*(d*x+c)^n*(h*x+g)/(f*x+e)^(1/2),x, algorithm="giac")
 

Output:

integrate((h*x + g)*(b*x + a)^m*(d*x + c)^n/sqrt(f*x + e), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^m (c+d x)^n (g+h x)}{\sqrt {e+f x}} \, dx=\int \frac {\left (g+h\,x\right )\,{\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^n}{\sqrt {e+f\,x}} \,d x \] Input:

int(((g + h*x)*(a + b*x)^m*(c + d*x)^n)/(e + f*x)^(1/2),x)
 

Output:

int(((g + h*x)*(a + b*x)^m*(c + d*x)^n)/(e + f*x)^(1/2), x)
 

Reduce [F]

\[ \int \frac {(a+b x)^m (c+d x)^n (g+h x)}{\sqrt {e+f x}} \, dx=\int \frac {\left (b x +a \right )^{m} \left (d x +c \right )^{n} \left (h x +g \right )}{\sqrt {f x +e}}d x \] Input:

int((b*x+a)^m*(d*x+c)^n*(h*x+g)/(f*x+e)^(1/2),x)
 

Output:

int((b*x+a)^m*(d*x+c)^n*(h*x+g)/(f*x+e)^(1/2),x)