\(\int \frac {(a^2+2 a b x+b^2 x^2)^p}{\sqrt {d+e x}} \, dx\) [442]

Optimal result

Integrand size = 28, antiderivative size = 81 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^p}{\sqrt {d+e x}} \, dx=\frac {2 \left (-\frac {e (a+b x)}{b d-a e}\right )^{-2 p} \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-2 p,\frac {3}{2},\frac {b (d+e x)}{b d-a e}\right )}{e} \] Output:

2*(e*x+d)^(1/2)*(b^2*x^2+2*a*b*x+a^2)^p*hypergeom([1/2, -2*p],[3/2],b*(e*x 
+d)/(-a*e+b*d))/e/((-e*(b*x+a)/(-a*e+b*d))^(2*p))
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^p}{\sqrt {d+e x}} \, dx=\frac {2 \left (\frac {e (a+b x)}{-b d+a e}\right )^{-2 p} \left ((a+b x)^2\right )^p \sqrt {d+e x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-2 p,\frac {3}{2},\frac {b (d+e x)}{b d-a e}\right )}{e} \] Input:

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^p/Sqrt[d + e*x],x]
 

Output:

(2*((a + b*x)^2)^p*Sqrt[d + e*x]*Hypergeometric2F1[1/2, -2*p, 3/2, (b*(d + 
 e*x))/(b*d - a*e)])/(e*((e*(a + b*x))/(-(b*d) + a*e))^(2*p))
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {1102, 80, 79}

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.

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

\(\Big \downarrow \) 1102

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

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

\(\displaystyle \frac {2 \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^p \left (-\frac {e (a+b x)}{b d-a e}\right )^{-2 p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-2 p,\frac {3}{2},\frac {b (d+e x)}{b d-a e}\right )}{e}\)

Input:

Int[(a^2 + 2*a*b*x + b^2*x^2)^p/Sqrt[d + e*x],x]
 

Output:

(2*Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^p*Hypergeometric2F1[1/2, -2*p, 
3/2, (b*(d + e*x))/(b*d - a*e)])/(e*(-((e*(a + b*x))/(b*d - a*e)))^(2*p))
 

Defintions of rubi rules used

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 80

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c 
 + d*x)^FracPart[n]/((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, x], x] /; FreeQ[{a, b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integ 
erQ[n] && (RationalQ[m] ||  !SimplerQ[n + 1, m + 1])
 

rule 1102

Int[((d_.) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*F 
racPart[p]))   Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, 
 d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0]
 

Maple [F]

\[\int \frac {\left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p}}{\sqrt {e x +d}}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((b^2*x^2 + 2*a*b*x + a^2)^p/sqrt(e*x + d), x)
 

Sympy [F]

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

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

Output:

Integral(((a + b*x)**2)**p/sqrt(d + e*x), x)
 

Maxima [F]

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

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

Output:

integrate((b^2*x^2 + 2*a*b*x + a^2)^p/sqrt(e*x + d), x)
 

Giac [F]

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

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

Output:

integrate((b^2*x^2 + 2*a*b*x + a^2)^p/sqrt(e*x + d), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^p}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {e x +d}\, \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a -4 \left (\int \frac {\sqrt {e x +d}\, \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} x}{4 b^{2} d e p \,x^{2}+4 a b d e p x +a b \,e^{2} x^{2}+4 b^{2} d^{2} p x +a^{2} e^{2} x +4 a b \,d^{2} p +a b d e x +a^{2} d e}d x \right ) a^{2} b \,e^{2} p -16 \left (\int \frac {\sqrt {e x +d}\, \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} x}{4 b^{2} d e p \,x^{2}+4 a b d e p x +a b \,e^{2} x^{2}+4 b^{2} d^{2} p x +a^{2} e^{2} x +4 a b \,d^{2} p +a b d e x +a^{2} d e}d x \right ) a \,b^{2} d e \,p^{2}+4 \left (\int \frac {\sqrt {e x +d}\, \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} x}{4 b^{2} d e p \,x^{2}+4 a b d e p x +a b \,e^{2} x^{2}+4 b^{2} d^{2} p x +a^{2} e^{2} x +4 a b \,d^{2} p +a b d e x +a^{2} d e}d x \right ) a \,b^{2} d e p +16 \left (\int \frac {\sqrt {e x +d}\, \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} x}{4 b^{2} d e p \,x^{2}+4 a b d e p x +a b \,e^{2} x^{2}+4 b^{2} d^{2} p x +a^{2} e^{2} x +4 a b \,d^{2} p +a b d e x +a^{2} d e}d x \right ) b^{3} d^{2} p^{2}}{4 b d p +a e} \] Input:

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

Output:

(2*(sqrt(d + e*x)*(a**2 + 2*a*b*x + b**2*x**2)**p*a - 2*int((sqrt(d + e*x) 
*(a**2 + 2*a*b*x + b**2*x**2)**p*x)/(a**2*d*e + a**2*e**2*x + 4*a*b*d**2*p 
 + 4*a*b*d*e*p*x + a*b*d*e*x + a*b*e**2*x**2 + 4*b**2*d**2*p*x + 4*b**2*d* 
e*p*x**2),x)*a**2*b*e**2*p - 8*int((sqrt(d + e*x)*(a**2 + 2*a*b*x + b**2*x 
**2)**p*x)/(a**2*d*e + a**2*e**2*x + 4*a*b*d**2*p + 4*a*b*d*e*p*x + a*b*d* 
e*x + a*b*e**2*x**2 + 4*b**2*d**2*p*x + 4*b**2*d*e*p*x**2),x)*a*b**2*d*e*p 
**2 + 2*int((sqrt(d + e*x)*(a**2 + 2*a*b*x + b**2*x**2)**p*x)/(a**2*d*e + 
a**2*e**2*x + 4*a*b*d**2*p + 4*a*b*d*e*p*x + a*b*d*e*x + a*b*e**2*x**2 + 4 
*b**2*d**2*p*x + 4*b**2*d*e*p*x**2),x)*a*b**2*d*e*p + 8*int((sqrt(d + e*x) 
*(a**2 + 2*a*b*x + b**2*x**2)**p*x)/(a**2*d*e + a**2*e**2*x + 4*a*b*d**2*p 
 + 4*a*b*d*e*p*x + a*b*d*e*x + a*b*e**2*x**2 + 4*b**2*d**2*p*x + 4*b**2*d* 
e*p*x**2),x)*b**3*d**2*p**2))/(a*e + 4*b*d*p)