\(\int (f+g x) (a+b x+c x^2)^{5/2} \, dx\) [892]

Optimal result

Integrand size = 20, antiderivative size = 207 \[ \int (f+g x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {5 \left (b^2-4 a c\right )^2 (2 c f-b g) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^4}-\frac {5 \left (b^2-4 a c\right ) (2 c f-b g) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^3}+\frac {(2 c f-b g) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{24 c^2}+\frac {g \left (a+b x+c x^2\right )^{7/2}}{7 c}-\frac {5 \left (b^2-4 a c\right )^3 (2 c f-b g) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{9/2}} \] Output:

5/1024*(-4*a*c+b^2)^2*(-b*g+2*c*f)*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^4-5/384 
*(-4*a*c+b^2)*(-b*g+2*c*f)*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c^3+1/24*(-b*g+2* 
c*f)*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c^2+1/7*g*(c*x^2+b*x+a)^(7/2)/c-5/2048* 
(-4*a*c+b^2)^3*(-b*g+2*c*f)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1 
/2))/c^(9/2)
 

Mathematica [A] (verified)

Time = 4.22 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.50 \[ \int (f+g x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {\sqrt {c} \sqrt {a+x (b+c x)} \left (-105 b^6 g+70 b^5 c (3 f+g x)+28 b^4 c (40 a g-c x (5 f+2 g x))+16 b^3 c^2 \left (c x^2 (7 f+3 g x)-14 a (10 f+3 g x)\right )+64 c^3 \left (48 a^3 g+8 c^3 x^5 (7 f+6 g x)+3 a^2 c x (77 f+48 g x)+2 a c^2 x^3 (91 f+72 g x)\right )+16 b^2 c^2 \left (-231 a^2 g+6 a c x (14 f+5 g x)+2 c^2 x^3 (189 f+148 g x)\right )+32 b c^3 \left (3 a^2 (77 f+19 g x)+8 c^2 x^4 (35 f+29 g x)+2 a c x^2 (273 f+197 g x)\right )\right )+105 \left (b^2-4 a c\right )^3 (-2 c f+b g) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{21504 c^{9/2}} \] Input:

Integrate[(f + g*x)*(a + b*x + c*x^2)^(5/2),x]
 

Output:

(Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-105*b^6*g + 70*b^5*c*(3*f + g*x) + 28*b^4 
*c*(40*a*g - c*x*(5*f + 2*g*x)) + 16*b^3*c^2*(c*x^2*(7*f + 3*g*x) - 14*a*( 
10*f + 3*g*x)) + 64*c^3*(48*a^3*g + 8*c^3*x^5*(7*f + 6*g*x) + 3*a^2*c*x*(7 
7*f + 48*g*x) + 2*a*c^2*x^3*(91*f + 72*g*x)) + 16*b^2*c^2*(-231*a^2*g + 6* 
a*c*x*(14*f + 5*g*x) + 2*c^2*x^3*(189*f + 148*g*x)) + 32*b*c^3*(3*a^2*(77* 
f + 19*g*x) + 8*c^2*x^4*(35*f + 29*g*x) + 2*a*c*x^2*(273*f + 197*g*x))) + 
105*(b^2 - 4*a*c)^3*(-2*c*f + b*g)*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a 
+ x*(b + c*x)])])/(21504*c^(9/2))
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1160, 1087, 1087, 1087, 1092, 219}

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.

\(\displaystyle \int (f+g x) \left (a+b x+c x^2\right )^{5/2} \, dx\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {(2 c f-b g) \int \left (c x^2+b x+a\right )^{5/2}dx}{2 c}+\frac {g \left (a+b x+c x^2\right )^{7/2}}{7 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {(2 c f-b g) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \int \left (c x^2+b x+a\right )^{3/2}dx}{24 c}\right )}{2 c}+\frac {g \left (a+b x+c x^2\right )^{7/2}}{7 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {(2 c f-b g) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {c x^2+b x+a}dx}{16 c}\right )}{24 c}\right )}{2 c}+\frac {g \left (a+b x+c x^2\right )^{7/2}}{7 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {(2 c f-b g) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c}\right )}{16 c}\right )}{24 c}\right )}{2 c}+\frac {g \left (a+b x+c x^2\right )^{7/2}}{7 c}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {(2 c f-b g) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c}\right )}{16 c}\right )}{24 c}\right )}{2 c}+\frac {g \left (a+b x+c x^2\right )^{7/2}}{7 c}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {(2 c f-b g) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}\right )}{16 c}\right )}{24 c}\right )}{2 c}+\frac {g \left (a+b x+c x^2\right )^{7/2}}{7 c}\)

Input:

Int[(f + g*x)*(a + b*x + c*x^2)^(5/2),x]
 

Output:

(g*(a + b*x + c*x^2)^(7/2))/(7*c) + ((2*c*f - b*g)*(((b + 2*c*x)*(a + b*x 
+ c*x^2)^(5/2))/(12*c) - (5*(b^2 - 4*a*c)*(((b + 2*c*x)*(a + b*x + c*x^2)^ 
(3/2))/(8*c) - (3*(b^2 - 4*a*c)*(((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*c) 
 - ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/ 
(8*c^(3/2))))/(16*c)))/(24*c)))/(2*c)
 

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1160

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b 
*e)/(2*c)   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && NeQ[p, -1]
 

Maple [A] (verified)

Time = 1.88 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.52

method	result	size
default	\(f \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )+g \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )\)	\(314\)
risch	\(\frac {\left (3072 c^{6} g \,x^{6}+7424 b \,c^{5} g \,x^{5}+3584 c^{6} f \,x^{5}+9216 a \,c^{5} g \,x^{4}+4736 b^{2} c^{4} g \,x^{4}+8960 b \,c^{5} f \,x^{4}+12608 a b \,c^{4} g \,x^{3}+11648 a \,c^{5} f \,x^{3}+48 b^{3} c^{3} g \,x^{3}+6048 b^{2} c^{4} f \,x^{3}+9216 a^{2} c^{4} g \,x^{2}+480 a \,b^{2} c^{3} g \,x^{2}+17472 a b \,c^{4} f \,x^{2}-56 b^{4} c^{2} g \,x^{2}+112 b^{3} c^{3} f \,x^{2}+1824 a^{2} b \,c^{3} g x +14784 a^{2} c^{4} f x -672 a \,b^{3} c^{2} g x +1344 a \,b^{2} c^{3} f x +70 b^{5} c g x -140 b^{4} c^{2} f x +3072 a^{3} c^{3} g -3696 a^{2} b^{2} c^{2} g +7392 a^{2} b \,c^{3} f +1120 a \,b^{4} c g -2240 a \,b^{3} c^{2} f -105 b^{6} g +210 b^{5} c f \right ) \sqrt {c \,x^{2}+b x +a}}{21504 c^{4}}-\frac {5 \left (64 a^{3} b \,c^{3} g -128 a^{3} c^{4} f -48 a^{2} b^{3} c^{2} g +96 a^{2} b^{2} c^{3} f +12 a \,b^{5} c g -24 a \,b^{4} c^{2} f -b^{7} g +2 b^{6} c f \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2048 c^{\frac {9}{2}}}\)	\(413\)

Input:

int((g*x+f)*(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

f*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x+b)* 
(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2 
)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))) 
+g*(1/7*(c*x^2+b*x+a)^(7/2)/c-1/2*b/c*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/ 
c+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2) 
/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+ 
c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (181) = 362\).

Time = 0.14 (sec) , antiderivative size = 857, normalized size of antiderivative = 4.14 \[ \int (f+g x) \left (a+b x+c x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:

integrate((g*x+f)*(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
 

Output:

[1/86016*(105*(2*(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*f - 
(b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*g)*sqrt(c)*log(-8*c^2*x 
^2 - 8*b*c*x - b^2 + 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) 
+ 4*(3072*c^7*g*x^6 + 256*(14*c^7*f + 29*b*c^6*g)*x^5 + 128*(70*b*c^6*f + 
(37*b^2*c^5 + 72*a*c^6)*g)*x^4 + 16*(14*(27*b^2*c^5 + 52*a*c^6)*f + (3*b^3 
*c^4 + 788*a*b*c^5)*g)*x^3 + 8*(14*(b^3*c^4 + 156*a*b*c^5)*f - (7*b^4*c^3 
- 60*a*b^2*c^4 - 1152*a^2*c^5)*g)*x^2 + 14*(15*b^5*c^2 - 160*a*b^3*c^3 + 5 
28*a^2*b*c^4)*f - (105*b^6*c - 1120*a*b^4*c^2 + 3696*a^2*b^2*c^3 - 3072*a^ 
3*c^4)*g - 2*(14*(5*b^4*c^3 - 48*a*b^2*c^4 - 528*a^2*c^5)*f - (35*b^5*c^2 
- 336*a*b^3*c^3 + 912*a^2*b*c^4)*g)*x)*sqrt(c*x^2 + b*x + a))/c^5, 1/43008 
*(105*(2*(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)*f - (b^7 - 1 
2*a*b^5*c + 48*a^2*b^3*c^2 - 64*a^3*b*c^3)*g)*sqrt(-c)*arctan(1/2*sqrt(c*x 
^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) + 2*(3072*c^7* 
g*x^6 + 256*(14*c^7*f + 29*b*c^6*g)*x^5 + 128*(70*b*c^6*f + (37*b^2*c^5 + 
72*a*c^6)*g)*x^4 + 16*(14*(27*b^2*c^5 + 52*a*c^6)*f + (3*b^3*c^4 + 788*a*b 
*c^5)*g)*x^3 + 8*(14*(b^3*c^4 + 156*a*b*c^5)*f - (7*b^4*c^3 - 60*a*b^2*c^4 
 - 1152*a^2*c^5)*g)*x^2 + 14*(15*b^5*c^2 - 160*a*b^3*c^3 + 528*a^2*b*c^4)* 
f - (105*b^6*c - 1120*a*b^4*c^2 + 3696*a^2*b^2*c^3 - 3072*a^3*c^4)*g - 2*( 
14*(5*b^4*c^3 - 48*a*b^2*c^4 - 528*a^2*c^5)*f - (35*b^5*c^2 - 336*a*b^3*c^ 
3 + 912*a^2*b*c^4)*g)*x)*sqrt(c*x^2 + b*x + a))/c^5]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2428 vs. \(2 (197) = 394\).

Time = 0.64 (sec) , antiderivative size = 2428, normalized size of antiderivative = 11.73 \[ \int (f+g x) \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((g*x+f)*(c*x**2+b*x+a)**(5/2),x)
 

Output:

Piecewise((sqrt(a + b*x + c*x**2)*(c**2*g*x**6/7 + x**5*(29*b*c**2*g/14 + 
c**3*f)/(6*c) + x**4*(15*a*c**2*g/7 + 3*b**2*c*g + 3*b*c**2*f - 11*b*(29*b 
*c**2*g/14 + c**3*f)/(12*c))/(5*c) + x**3*(6*a*b*c*g + 3*a*c**2*f - 5*a*(2 
9*b*c**2*g/14 + c**3*f)/(6*c) + b**3*g + 3*b**2*c*f - 9*b*(15*a*c**2*g/7 + 
 3*b**2*c*g + 3*b*c**2*f - 11*b*(29*b*c**2*g/14 + c**3*f)/(12*c))/(10*c))/ 
(4*c) + x**2*(3*a**2*c*g + 3*a*b**2*g + 6*a*b*c*f - 4*a*(15*a*c**2*g/7 + 3 
*b**2*c*g + 3*b*c**2*f - 11*b*(29*b*c**2*g/14 + c**3*f)/(12*c))/(5*c) + b* 
*3*f - 7*b*(6*a*b*c*g + 3*a*c**2*f - 5*a*(29*b*c**2*g/14 + c**3*f)/(6*c) + 
 b**3*g + 3*b**2*c*f - 9*b*(15*a*c**2*g/7 + 3*b**2*c*g + 3*b*c**2*f - 11*b 
*(29*b*c**2*g/14 + c**3*f)/(12*c))/(10*c))/(8*c))/(3*c) + x*(3*a**2*b*g + 
3*a**2*c*f + 3*a*b**2*f - 3*a*(6*a*b*c*g + 3*a*c**2*f - 5*a*(29*b*c**2*g/1 
4 + c**3*f)/(6*c) + b**3*g + 3*b**2*c*f - 9*b*(15*a*c**2*g/7 + 3*b**2*c*g 
+ 3*b*c**2*f - 11*b*(29*b*c**2*g/14 + c**3*f)/(12*c))/(10*c))/(4*c) - 5*b* 
(3*a**2*c*g + 3*a*b**2*g + 6*a*b*c*f - 4*a*(15*a*c**2*g/7 + 3*b**2*c*g + 3 
*b*c**2*f - 11*b*(29*b*c**2*g/14 + c**3*f)/(12*c))/(5*c) + b**3*f - 7*b*(6 
*a*b*c*g + 3*a*c**2*f - 5*a*(29*b*c**2*g/14 + c**3*f)/(6*c) + b**3*g + 3*b 
**2*c*f - 9*b*(15*a*c**2*g/7 + 3*b**2*c*g + 3*b*c**2*f - 11*b*(29*b*c**2*g 
/14 + c**3*f)/(12*c))/(10*c))/(8*c))/(6*c))/(2*c) + (a**3*g + 3*a**2*b*f - 
 2*a*(3*a**2*c*g + 3*a*b**2*g + 6*a*b*c*f - 4*a*(15*a*c**2*g/7 + 3*b**2*c* 
g + 3*b*c**2*f - 11*b*(29*b*c**2*g/14 + c**3*f)/(12*c))/(5*c) + b**3*f ...
 

Maxima [F(-2)]

Exception generated. \[ \int (f+g x) \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((g*x+f)*(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (181) = 362\).

Time = 0.20 (sec) , antiderivative size = 423, normalized size of antiderivative = 2.04 \[ \int (f+g x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {1}{21504} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (12 \, c^{2} g x + \frac {14 \, c^{8} f + 29 \, b c^{7} g}{c^{6}}\right )} x + \frac {70 \, b c^{7} f + 37 \, b^{2} c^{6} g + 72 \, a c^{7} g}{c^{6}}\right )} x + \frac {378 \, b^{2} c^{6} f + 728 \, a c^{7} f + 3 \, b^{3} c^{5} g + 788 \, a b c^{6} g}{c^{6}}\right )} x + \frac {14 \, b^{3} c^{5} f + 2184 \, a b c^{6} f - 7 \, b^{4} c^{4} g + 60 \, a b^{2} c^{5} g + 1152 \, a^{2} c^{6} g}{c^{6}}\right )} x - \frac {70 \, b^{4} c^{4} f - 672 \, a b^{2} c^{5} f - 7392 \, a^{2} c^{6} f - 35 \, b^{5} c^{3} g + 336 \, a b^{3} c^{4} g - 912 \, a^{2} b c^{5} g}{c^{6}}\right )} x + \frac {210 \, b^{5} c^{3} f - 2240 \, a b^{3} c^{4} f + 7392 \, a^{2} b c^{5} f - 105 \, b^{6} c^{2} g + 1120 \, a b^{4} c^{3} g - 3696 \, a^{2} b^{2} c^{4} g + 3072 \, a^{3} c^{5} g}{c^{6}}\right )} + \frac {5 \, {\left (2 \, b^{6} c f - 24 \, a b^{4} c^{2} f + 96 \, a^{2} b^{2} c^{3} f - 128 \, a^{3} c^{4} f - b^{7} g + 12 \, a b^{5} c g - 48 \, a^{2} b^{3} c^{2} g + 64 \, a^{3} b c^{3} g\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{2048 \, c^{\frac {9}{2}}} \] Input:

integrate((g*x+f)*(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
 

Output:

1/21504*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(12*c^2*g*x + (14*c^8*f + 29* 
b*c^7*g)/c^6)*x + (70*b*c^7*f + 37*b^2*c^6*g + 72*a*c^7*g)/c^6)*x + (378*b 
^2*c^6*f + 728*a*c^7*f + 3*b^3*c^5*g + 788*a*b*c^6*g)/c^6)*x + (14*b^3*c^5 
*f + 2184*a*b*c^6*f - 7*b^4*c^4*g + 60*a*b^2*c^5*g + 1152*a^2*c^6*g)/c^6)* 
x - (70*b^4*c^4*f - 672*a*b^2*c^5*f - 7392*a^2*c^6*f - 35*b^5*c^3*g + 336* 
a*b^3*c^4*g - 912*a^2*b*c^5*g)/c^6)*x + (210*b^5*c^3*f - 2240*a*b^3*c^4*f 
+ 7392*a^2*b*c^5*f - 105*b^6*c^2*g + 1120*a*b^4*c^3*g - 3696*a^2*b^2*c^4*g 
 + 3072*a^3*c^5*g)/c^6) + 5/2048*(2*b^6*c*f - 24*a*b^4*c^2*f + 96*a^2*b^2* 
c^3*f - 128*a^3*c^4*f - b^7*g + 12*a*b^5*c*g - 48*a^2*b^3*c^2*g + 64*a^3*b 
*c^3*g)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(9/2 
)
 

Mupad [F(-1)]

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

int((f + g*x)*(a + b*x + c*x^2)^(5/2),x)
 

Output:

int((f + g*x)*(a + b*x + c*x^2)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 1.81 (sec) , antiderivative size = 987, normalized size of antiderivative = 4.77 \[ \int (f+g x) \left (a+b x+c x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:

int((g*x+f)*(c*x^2+b*x+a)^(5/2),x)
 

Output:

(6144*sqrt(a + b*x + c*x**2)*a**3*c**4*g - 7392*sqrt(a + b*x + c*x**2)*a** 
2*b**2*c**3*g + 14784*sqrt(a + b*x + c*x**2)*a**2*b*c**4*f + 3648*sqrt(a + 
 b*x + c*x**2)*a**2*b*c**4*g*x + 29568*sqrt(a + b*x + c*x**2)*a**2*c**5*f* 
x + 18432*sqrt(a + b*x + c*x**2)*a**2*c**5*g*x**2 + 2240*sqrt(a + b*x + c* 
x**2)*a*b**4*c**2*g - 4480*sqrt(a + b*x + c*x**2)*a*b**3*c**3*f - 1344*sqr 
t(a + b*x + c*x**2)*a*b**3*c**3*g*x + 2688*sqrt(a + b*x + c*x**2)*a*b**2*c 
**4*f*x + 960*sqrt(a + b*x + c*x**2)*a*b**2*c**4*g*x**2 + 34944*sqrt(a + b 
*x + c*x**2)*a*b*c**5*f*x**2 + 25216*sqrt(a + b*x + c*x**2)*a*b*c**5*g*x** 
3 + 23296*sqrt(a + b*x + c*x**2)*a*c**6*f*x**3 + 18432*sqrt(a + b*x + c*x* 
*2)*a*c**6*g*x**4 - 210*sqrt(a + b*x + c*x**2)*b**6*c*g + 420*sqrt(a + b*x 
 + c*x**2)*b**5*c**2*f + 140*sqrt(a + b*x + c*x**2)*b**5*c**2*g*x - 280*sq 
rt(a + b*x + c*x**2)*b**4*c**3*f*x - 112*sqrt(a + b*x + c*x**2)*b**4*c**3* 
g*x**2 + 224*sqrt(a + b*x + c*x**2)*b**3*c**4*f*x**2 + 96*sqrt(a + b*x + c 
*x**2)*b**3*c**4*g*x**3 + 12096*sqrt(a + b*x + c*x**2)*b**2*c**5*f*x**3 + 
9472*sqrt(a + b*x + c*x**2)*b**2*c**5*g*x**4 + 17920*sqrt(a + b*x + c*x**2 
)*b*c**6*f*x**4 + 14848*sqrt(a + b*x + c*x**2)*b*c**6*g*x**5 + 7168*sqrt(a 
 + b*x + c*x**2)*c**7*f*x**5 + 6144*sqrt(a + b*x + c*x**2)*c**7*g*x**6 - 6 
720*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c 
- b**2))*a**3*b*c**3*g + 13440*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x** 
2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*c**4*f + 5040*sqrt(c)*log((2*s...