\(\int (e x)^m (A+B x) (a+b x+c x^2)^{5/2} \, dx\) [233]

Optimal result

Integrand size = 25, antiderivative size = 281 \[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {A (e x)^{1+m} \left (a+b x+c x^2\right )^{5/2} \operatorname {AppellF1}\left (1+m,-\frac {5}{2},-\frac {5}{2},2+m,-\frac {2 c x}{b-\sqrt {b^2-4 a c}},-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{e (1+m) \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )^{5/2} \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )^{5/2}}+\frac {B (e x)^{2+m} \left (a+b x+c x^2\right )^{5/2} \operatorname {AppellF1}\left (2+m,-\frac {5}{2},-\frac {5}{2},3+m,-\frac {2 c x}{b-\sqrt {b^2-4 a c}},-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{e^2 (2+m) \left (1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}\right )^{5/2} \left (1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )^{5/2}} \] Output:

A*(e*x)^(1+m)*(c*x^2+b*x+a)^(5/2)*AppellF1(1+m,-5/2,-5/2,2+m,-2*c*x/(b-(-4 
*a*c+b^2)^(1/2)),-2*c*x/(b+(-4*a*c+b^2)^(1/2)))/e/(1+m)/(1+2*c*x/(b-(-4*a* 
c+b^2)^(1/2)))^(5/2)/(1+2*c*x/(b+(-4*a*c+b^2)^(1/2)))^(5/2)+B*(e*x)^(2+m)* 
(c*x^2+b*x+a)^(5/2)*AppellF1(2+m,-5/2,-5/2,3+m,-2*c*x/(b-(-4*a*c+b^2)^(1/2 
)),-2*c*x/(b+(-4*a*c+b^2)^(1/2)))/e^2/(2+m)/(1+2*c*x/(b-(-4*a*c+b^2)^(1/2) 
))^(5/2)/(1+2*c*x/(b+(-4*a*c+b^2)^(1/2)))^(5/2)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(618\) vs. \(2(281)=562\).

Time = 3.61 (sec) , antiderivative size = 618, normalized size of antiderivative = 2.20 \[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {x (e x)^m \sqrt {a+x (b+c x)} \left (a^2 A \left (720+1044 m+580 m^2+155 m^3+20 m^4+m^5\right ) \operatorname {AppellF1}\left (1+m,-\frac {1}{2},-\frac {1}{2},2+m,-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )+(1+m) x \left (a (2 A b+a B) \left (360+342 m+119 m^2+18 m^3+m^4\right ) \operatorname {AppellF1}\left (2+m,-\frac {1}{2},-\frac {1}{2},3+m,-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )+(2+m) x \left (\left (2 a b B+A \left (b^2+2 a c\right )\right ) \left (120+74 m+15 m^2+m^3\right ) \operatorname {AppellF1}\left (3+m,-\frac {1}{2},-\frac {1}{2},4+m,-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )+(3+m) x \left (\left (b^2 B+2 A b c+2 a B c\right ) \left (30+11 m+m^2\right ) \operatorname {AppellF1}\left (4+m,-\frac {1}{2},-\frac {1}{2},5+m,-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )+c (4+m) x \left ((2 b B+A c) (6+m) \operatorname {AppellF1}\left (5+m,-\frac {1}{2},-\frac {1}{2},6+m,-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )+B c (5+m) x \operatorname {AppellF1}\left (6+m,-\frac {1}{2},-\frac {1}{2},7+m,-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )\right )\right )\right )\right )\right )}{(1+m) (2+m) (3+m) (4+m) (5+m) (6+m) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{b+\sqrt {b^2-4 a c}}}} \] Input:

Integrate[(e*x)^m*(A + B*x)*(a + b*x + c*x^2)^(5/2),x]
 

Output:

(x*(e*x)^m*Sqrt[a + x*(b + c*x)]*(a^2*A*(720 + 1044*m + 580*m^2 + 155*m^3 
+ 20*m^4 + m^5)*AppellF1[1 + m, -1/2, -1/2, 2 + m, (-2*c*x)/(b + Sqrt[b^2 
- 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])] + (1 + m)*x*(a*(2*A*b + a*B)* 
(360 + 342*m + 119*m^2 + 18*m^3 + m^4)*AppellF1[2 + m, -1/2, -1/2, 3 + m, 
(-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])] + (2 + 
 m)*x*((2*a*b*B + A*(b^2 + 2*a*c))*(120 + 74*m + 15*m^2 + m^3)*AppellF1[3 
+ m, -1/2, -1/2, 4 + m, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sq 
rt[b^2 - 4*a*c])] + (3 + m)*x*((b^2*B + 2*A*b*c + 2*a*B*c)*(30 + 11*m + m^ 
2)*AppellF1[4 + m, -1/2, -1/2, 5 + m, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2 
*c*x)/(-b + Sqrt[b^2 - 4*a*c])] + c*(4 + m)*x*((2*b*B + A*c)*(6 + m)*Appel 
lF1[5 + m, -1/2, -1/2, 6 + m, (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(- 
b + Sqrt[b^2 - 4*a*c])] + B*c*(5 + m)*x*AppellF1[6 + m, -1/2, -1/2, 7 + m, 
 (-2*c*x)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])])))))) 
/((1 + m)*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m)*Sqrt[(b - Sqrt[b^2 - 4*a 
*c] + 2*c*x)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x) 
/(b + Sqrt[b^2 - 4*a*c])])
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1269, 1179, 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.

\(\displaystyle \int (A+B x) (e x)^m \left (a+b x+c x^2\right )^{5/2} \, dx\)

\(\Big \downarrow \) 1269

\(\displaystyle A \int (e x)^m \left (c x^2+b x+a\right )^{5/2}dx+\frac {B \int (e x)^{m+1} \left (c x^2+b x+a\right )^{5/2}dx}{e}\)

\(\Big \downarrow \) 1179

\(\displaystyle \frac {A \left (a+b x+c x^2\right )^{5/2} \int (e x)^m \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )^{5/2} \left (\frac {2 c x}{b+\sqrt {b^2-4 a c}}+1\right )^{5/2}d(e x)}{e \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )^{5/2} \left (\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1\right )^{5/2}}+\frac {B \left (a+b x+c x^2\right )^{5/2} \int (e x)^{m+1} \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )^{5/2} \left (\frac {2 c x}{b+\sqrt {b^2-4 a c}}+1\right )^{5/2}d(e x)}{e^2 \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )^{5/2} \left (\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1\right )^{5/2}}\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {A (e x)^{m+1} \left (a+b x+c x^2\right )^{5/2} \operatorname {AppellF1}\left (m+1,-\frac {5}{2},-\frac {5}{2},m+2,-\frac {2 c x}{b-\sqrt {b^2-4 a c}},-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{e (m+1) \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )^{5/2} \left (\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1\right )^{5/2}}+\frac {B (e x)^{m+2} \left (a+b x+c x^2\right )^{5/2} \operatorname {AppellF1}\left (m+2,-\frac {5}{2},-\frac {5}{2},m+3,-\frac {2 c x}{b-\sqrt {b^2-4 a c}},-\frac {2 c x}{b+\sqrt {b^2-4 a c}}\right )}{e^2 (m+2) \left (\frac {2 c x}{b-\sqrt {b^2-4 a c}}+1\right )^{5/2} \left (\frac {2 c x}{\sqrt {b^2-4 a c}+b}+1\right )^{5/2}}\)

Input:

Int[(e*x)^m*(A + B*x)*(a + b*x + c*x^2)^(5/2),x]
 

Output:

(A*(e*x)^(1 + m)*(a + b*x + c*x^2)^(5/2)*AppellF1[1 + m, -5/2, -5/2, 2 + m 
, (-2*c*x)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x)/(b + Sqrt[b^2 - 4*a*c])])/(e* 
(1 + m)*(1 + (2*c*x)/(b - Sqrt[b^2 - 4*a*c]))^(5/2)*(1 + (2*c*x)/(b + Sqrt 
[b^2 - 4*a*c]))^(5/2)) + (B*(e*x)^(2 + m)*(a + b*x + c*x^2)^(5/2)*AppellF1 
[2 + m, -5/2, -5/2, 3 + m, (-2*c*x)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x)/(b + 
 Sqrt[b^2 - 4*a*c])])/(e^2*(2 + m)*(1 + (2*c*x)/(b - Sqrt[b^2 - 4*a*c]))^( 
5/2)*(1 + (2*c*x)/(b + Sqrt[b^2 - 4*a*c]))^(5/2))
 

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 1179

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[(a + b*x + c*x^2)^p/(e*(1 - ( 
d + e*x)/(d - e*((b - q)/(2*c))))^p*(1 - (d + e*x)/(d - e*((b + q)/(2*c)))) 
^p)   Subst[Int[x^m*Simp[1 - x/(d - e*((b - q)/(2*c))), x]^p*Simp[1 - x/(d 
- e*((b + q)/(2*c))), x]^p, x], x, d + e*x], x]] /; FreeQ[{a, b, c, d, e, m 
, p}, x]
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

Maple [F]

\[\int \left (e x \right )^{m} \left (B x +A \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}d x\]

Input:

int((e*x)^m*(B*x+A)*(c*x^2+b*x+a)^(5/2),x)
 

Output:

int((e*x)^m*(B*x+A)*(c*x^2+b*x+a)^(5/2),x)
 

Fricas [F]

\[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} {\left (B x + A\right )} \left (e x\right )^{m} \,d x } \] Input:

integrate((e*x)^m*(B*x+A)*(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
 

Output:

integral((B*c^2*x^5 + (2*B*b*c + A*c^2)*x^4 + (B*b^2 + 2*(B*a + A*b)*c)*x^ 
3 + A*a^2 + (2*B*a*b + A*b^2 + 2*A*a*c)*x^2 + (B*a^2 + 2*A*a*b)*x)*sqrt(c* 
x^2 + b*x + a)*(e*x)^m, x)
 

Sympy [F]

\[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\int \left (e x\right )^{m} \left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {5}{2}}\, dx \] Input:

integrate((e*x)**m*(B*x+A)*(c*x**2+b*x+a)**(5/2),x)
 

Output:

Integral((e*x)**m*(A + B*x)*(a + b*x + c*x**2)**(5/2), x)
 

Maxima [F]

\[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} {\left (B x + A\right )} \left (e x\right )^{m} \,d x } \] Input:

integrate((e*x)^m*(B*x+A)*(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")
 

Output:

integrate((c*x^2 + b*x + a)^(5/2)*(B*x + A)*(e*x)^m, x)
 

Giac [F]

\[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\int { {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} {\left (B x + A\right )} \left (e x\right )^{m} \,d x } \] Input:

integrate((e*x)^m*(B*x+A)*(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
 

Output:

integrate((c*x^2 + b*x + a)^(5/2)*(B*x + A)*(e*x)^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\int {\left (e\,x\right )}^m\,\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x \] Input:

int((e*x)^m*(A + B*x)*(a + b*x + c*x^2)^(5/2),x)
 

Output:

int((e*x)^m*(A + B*x)*(a + b*x + c*x^2)^(5/2), x)
 

Reduce [F]

\[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx=\int \left (e x \right )^{m} \left (B x +A \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}d x \] Input:

int((e*x)^m*(B*x+A)*(c*x^2+b*x+a)^(5/2),x)
 

Output:

int((e*x)^m*(B*x+A)*(c*x^2+b*x+a)^(5/2),x)