\(\int (d+e x) (f+g x) (a+b x+c x^2)^{7/2} \, dx\) [900]

Optimal result

Integrand size = 25, antiderivative size = 394 \[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^{7/2} \, dx=-\frac {7 \left (b^2-4 a c\right )^3 \left (40 c^2 d f+11 b^2 e g-4 c (a e g+5 b (e f+d g))\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{131072 c^6}+\frac {7 \left (b^2-4 a c\right )^2 \left (40 c^2 d f+11 b^2 e g-4 c (a e g+5 b (e f+d g))\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{49152 c^5}-\frac {7 \left (b^2-4 a c\right ) \left (40 c^2 d f+11 b^2 e g-4 c (a e g+5 b (e f+d g))\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{15360 c^4}+\frac {\left (40 c^2 d f+11 b^2 e g-4 c (a e g+5 b (e f+d g))\right ) (b+2 c x) \left (a+b x+c x^2\right )^{7/2}}{640 c^3}-\frac {(11 b e g-20 c (e f+d g)-18 c e g x) \left (a+b x+c x^2\right )^{9/2}}{180 c^2}+\frac {7 \left (b^2-4 a c\right )^4 \left (40 c^2 d f+11 b^2 e g-4 c (a e g+5 b (e f+d g))\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{262144 c^{13/2}} \] Output:

-7/131072*(-4*a*c+b^2)^3*(40*c^2*d*f+11*b^2*e*g-4*c*(a*e*g+5*b*(d*g+e*f))) 
*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^6+7/49152*(-4*a*c+b^2)^2*(40*c^2*d*f+11*b 
^2*e*g-4*c*(a*e*g+5*b*(d*g+e*f)))*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c^5-7/1536 
0*(-4*a*c+b^2)*(40*c^2*d*f+11*b^2*e*g-4*c*(a*e*g+5*b*(d*g+e*f)))*(2*c*x+b) 
*(c*x^2+b*x+a)^(5/2)/c^4+1/640*(40*c^2*d*f+11*b^2*e*g-4*c*(a*e*g+5*b*(d*g+ 
e*f)))*(2*c*x+b)*(c*x^2+b*x+a)^(7/2)/c^3-1/180*(11*b*e*g-20*c*(d*g+e*f)-18 
*c*e*g*x)*(c*x^2+b*x+a)^(9/2)/c^2+7/262144*(-4*a*c+b^2)^4*(40*c^2*d*f+11*b 
^2*e*g-4*c*(a*e*g+5*b*(d*g+e*f)))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x 
+a)^(1/2))/c^(13/2)
 

Mathematica [A] (verified)

Time = 11.16 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.68 \[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^{7/2} \, dx=\frac {(a+x (b+c x))^{9/2} (-11 b e g+2 c (10 e f+10 d g+9 e g x))+\frac {3 \left (20 c^2 d f+\frac {11}{2} b^2 e g-2 c (a e g+5 b (e f+d g))\right ) \left (6144 c^{7/2} (b+2 c x) (a+x (b+c x))^{7/2}-7 \left (b^2-4 a c\right ) \left (256 c^{5/2} (b+2 c x) (a+x (b+c x))^{5/2}-5 \left (b^2-4 a c\right ) \left (16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left (b^2-4 a c\right ) \left (2 \sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)}-\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )\right )\right )\right )}{32768 c^{9/2}}}{180 c^2} \] Input:

Integrate[(d + e*x)*(f + g*x)*(a + b*x + c*x^2)^(7/2),x]
 

Output:

((a + x*(b + c*x))^(9/2)*(-11*b*e*g + 2*c*(10*e*f + 10*d*g + 9*e*g*x)) + ( 
3*(20*c^2*d*f + (11*b^2*e*g)/2 - 2*c*(a*e*g + 5*b*(e*f + d*g)))*(6144*c^(7 
/2)*(b + 2*c*x)*(a + x*(b + c*x))^(7/2) - 7*(b^2 - 4*a*c)*(256*c^(5/2)*(b 
+ 2*c*x)*(a + x*(b + c*x))^(5/2) - 5*(b^2 - 4*a*c)*(16*c^(3/2)*(b + 2*c*x) 
*(a + x*(b + c*x))^(3/2) - 3*(b^2 - 4*a*c)*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + 
 x*(b + c*x)] - (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b 
 + c*x)])])))))/(32768*c^(9/2)))/(180*c^2)
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.73, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1225, 1087, 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.

\(\displaystyle \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^{7/2} \, dx\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {\left (-4 c (a e g+5 b (d g+e f))+11 b^2 e g+40 c^2 d f\right ) \int \left (c x^2+b x+a\right )^{7/2}dx}{40 c^2}-\frac {\left (a+b x+c x^2\right )^{9/2} (11 b e g-20 c (d g+e f)-18 c e g x)}{180 c^2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\left (-4 c (a e g+5 b (d g+e f))+11 b^2 e g+40 c^2 d f\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{7/2}}{16 c}-\frac {7 \left (b^2-4 a c\right ) \int \left (c x^2+b x+a\right )^{5/2}dx}{32 c}\right )}{40 c^2}-\frac {\left (a+b x+c x^2\right )^{9/2} (11 b e g-20 c (d g+e f)-18 c e g x)}{180 c^2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\left (-4 c (a e g+5 b (d g+e f))+11 b^2 e g+40 c^2 d f\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{7/2}}{16 c}-\frac {7 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \int \left (c x^2+b x+a\right )^{3/2}dx}{24 c}\right )}{32 c}\right )}{40 c^2}-\frac {\left (a+b x+c x^2\right )^{9/2} (11 b e g-20 c (d g+e f)-18 c e g x)}{180 c^2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\left (-4 c (a e g+5 b (d g+e f))+11 b^2 e g+40 c^2 d f\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{7/2}}{16 c}-\frac {7 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {c x^2+b x+a}dx}{16 c}\right )}{24 c}\right )}{32 c}\right )}{40 c^2}-\frac {\left (a+b x+c x^2\right )^{9/2} (11 b e g-20 c (d g+e f)-18 c e g x)}{180 c^2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\left (-4 c (a e g+5 b (d g+e f))+11 b^2 e g+40 c^2 d f\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{7/2}}{16 c}-\frac {7 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c}\right )}{16 c}\right )}{24 c}\right )}{32 c}\right )}{40 c^2}-\frac {\left (a+b x+c x^2\right )^{9/2} (11 b e g-20 c (d g+e f)-18 c e g x)}{180 c^2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\left (-4 c (a e g+5 b (d g+e f))+11 b^2 e g+40 c^2 d f\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{7/2}}{16 c}-\frac {7 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c}\right )}{16 c}\right )}{24 c}\right )}{32 c}\right )}{40 c^2}-\frac {\left (a+b x+c x^2\right )^{9/2} (11 b e g-20 c (d g+e f)-18 c e g x)}{180 c^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{7/2}}{16 c}-\frac {7 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}\right )}{16 c}\right )}{24 c}\right )}{32 c}\right ) \left (-4 c (a e g+5 b (d g+e f))+11 b^2 e g+40 c^2 d f\right )}{40 c^2}-\frac {\left (a+b x+c x^2\right )^{9/2} (11 b e g-20 c (d g+e f)-18 c e g x)}{180 c^2}\)

Input:

Int[(d + e*x)*(f + g*x)*(a + b*x + c*x^2)^(7/2),x]
 

Output:

-1/180*((11*b*e*g - 20*c*(e*f + d*g) - 18*c*e*g*x)*(a + b*x + c*x^2)^(9/2) 
)/c^2 + ((40*c^2*d*f + 11*b^2*e*g - 4*c*(a*e*g + 5*b*(e*f + d*g)))*(((b + 
2*c*x)*(a + b*x + c*x^2)^(7/2))/(16*c) - (7*(b^2 - 4*a*c)*(((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)))/(32*c)))/(40*c^2)
 

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 1225

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( 
x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 
 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), 
 x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p 
+ 3))/(2*c^2*(2*p + 3))   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, p}, x] &&  !LeQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(818\) vs. \(2(364)=728\).

Time = 1.93 (sec) , antiderivative size = 819, normalized size of antiderivative = 2.08

method	result	size
default	\(d f \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{16 c}+\frac {7 \left (4 a c -b^{2}\right ) \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 )}{32 c}\right )+\left (d g +e f \right ) \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {9}{2}}}{9 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{16 c}+\frac {7 \left (4 a c -b^{2}\right ) \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 )}{32 c}\right )}{2 c}\right )+e g \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {9}{2}}}{10 c}-\frac {11 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {9}{2}}}{9 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{16 c}+\frac {7 \left (4 a c -b^{2}\right ) \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 )}{32 c}\right )}{2 c}\right )}{20 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{16 c}+\frac {7 \left (4 a c -b^{2}\right ) \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 )}{32 c}\right )}{10 c}\right )\)	\(819\)
risch	\(\text {Expression too large to display}\)	\(1528\)

Input:

int((e*x+d)*(g*x+f)*(c*x^2+b*x+a)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

d*f*(1/16*(2*c*x+b)/c*(c*x^2+b*x+a)^(7/2)+7/32*(4*a*c-b^2)/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))))))+(d*g+e*f)*(1/ 
9*(c*x^2+b*x+a)^(9/2)/c-1/2*b/c*(1/16*(2*c*x+b)/c*(c*x^2+b*x+a)^(7/2)+7/32 
*(4*a*c-b^2)/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)))))))+e*g*(1/10*x*(c*x^2+b*x+a)^(9/2)/c-11/20*b/c*(1/9*(c*x^2+b* 
x+a)^(9/2)/c-1/2*b/c*(1/16*(2*c*x+b)/c*(c*x^2+b*x+a)^(7/2)+7/32*(4*a*c-b^2 
)/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))) 
))))-1/10*a/c*(1/16*(2*c*x+b)/c*(c*x^2+b*x+a)^(7/2)+7/32*(4*a*c-b^2)/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. 1353 vs. \(2 (364) = 728\).

Time = 0.61 (sec) , antiderivative size = 2709, normalized size of antiderivative = 6.88 \[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^{7/2} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)*(g*x+f)*(c*x^2+b*x+a)^(7/2),x, algorithm="fricas")
 

Output:

[-1/23592960*(315*(20*(2*(b^8*c^2 - 16*a*b^6*c^3 + 96*a^2*b^4*c^4 - 256*a^ 
3*b^2*c^5 + 256*a^4*c^6)*d - (b^9*c - 16*a*b^7*c^2 + 96*a^2*b^5*c^3 - 256* 
a^3*b^3*c^4 + 256*a^4*b*c^5)*e)*f - (20*(b^9*c - 16*a*b^7*c^2 + 96*a^2*b^5 
*c^3 - 256*a^3*b^3*c^4 + 256*a^4*b*c^5)*d - (11*b^10 - 180*a*b^8*c + 1120* 
a^2*b^6*c^2 - 3200*a^3*b^4*c^3 + 3840*a^4*b^2*c^4 - 1024*a^5*c^5)*e)*g)*sq 
rt(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*(589824*c^10*e*g*x^9 + 32768*(20*c^10*e*f + (20*c^10 
*d + 61*b*c^9*e)*g)*x^8 + 2048*(20*(18*c^10*d + 55*b*c^9*e)*f + (1100*b*c^ 
9*d + (1123*b^2*c^8 + 1116*a*c^9)*e)*g)*x^7 + 1024*(20*(126*b*c^9*d + (129 
*b^2*c^8 + 128*a*c^9)*e)*f + (20*(129*b^2*c^8 + 128*a*c^9)*d + (885*b^3*c^ 
7 + 5252*a*b*c^8)*e)*g)*x^6 + 256*(20*(6*(101*b^2*c^8 + 100*a*c^9)*d + (20 
9*b^3*c^7 + 1236*a*b*c^8)*e)*f + (20*(209*b^3*c^7 + 1236*a*b*c^8)*d + (5*b 
^4*c^6 + 12840*a*b^2*c^7 + 12624*a^2*c^8)*e)*g)*x^5 + 128*(20*(30*(17*b^3* 
c^7 + 100*a*b*c^8)*d + (b^4*c^6 + 1572*a*b^2*c^7 + 1536*a^2*c^8)*e)*f + (2 
0*(b^4*c^6 + 1572*a*b^2*c^7 + 1536*a^2*c^8)*d - (11*b^5*c^5 - 120*a*b^3*c^ 
6 - 32400*a^2*b*c^7)*e)*g)*x^4 + 16*(20*(6*(3*b^4*c^6 + 2696*a*b^2*c^7 + 2 
608*a^2*c^8)*d - (9*b^5*c^5 - 104*a*b^3*c^6 - 16752*a^2*b*c^7)*e)*f - (20* 
(9*b^5*c^5 - 104*a*b^3*c^6 - 16752*a^2*b*c^7)*d - (99*b^6*c^4 - 1180*a*b^4 
*c^5 + 4560*a^2*b^2*c^6 + 116160*a^3*c^7)*e)*g)*x^3 - 8*(20*(6*(7*b^5*c^5 
- 88*a*b^3*c^6 - 7824*a^2*b*c^7)*d - (21*b^6*c^4 - 264*a*b^4*c^5 + 1104...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 22197 vs. \(2 (405) = 810\).

Time = 1.49 (sec) , antiderivative size = 22197, normalized size of antiderivative = 56.34 \[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^{7/2} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)*(g*x+f)*(c*x**2+b*x+a)**(7/2),x)
 

Output:

Piecewise((sqrt(a + b*x + c*x**2)*(c**3*e*g*x**9/10 + x**8*(61*b*c**3*e*g/ 
20 + c**4*d*g + c**4*e*f)/(9*c) + x**7*(31*a*c**3*e*g/10 + 6*b**2*c**2*e*g 
 + 4*b*c**3*d*g + 4*b*c**3*e*f - 17*b*(61*b*c**3*e*g/20 + c**4*d*g + c**4* 
e*f)/(18*c) + c**4*d*f)/(8*c) + x**6*(12*a*b*c**2*e*g + 4*a*c**3*d*g + 4*a 
*c**3*e*f - 8*a*(61*b*c**3*e*g/20 + c**4*d*g + c**4*e*f)/(9*c) + 4*b**3*c* 
e*g + 6*b**2*c**2*d*g + 6*b**2*c**2*e*f + 4*b*c**3*d*f - 15*b*(31*a*c**3*e 
*g/10 + 6*b**2*c**2*e*g + 4*b*c**3*d*g + 4*b*c**3*e*f - 17*b*(61*b*c**3*e* 
g/20 + c**4*d*g + c**4*e*f)/(18*c) + c**4*d*f)/(16*c))/(7*c) + x**5*(6*a** 
2*c**2*e*g + 12*a*b**2*c*e*g + 12*a*b*c**2*d*g + 12*a*b*c**2*e*f + 4*a*c** 
3*d*f - 7*a*(31*a*c**3*e*g/10 + 6*b**2*c**2*e*g + 4*b*c**3*d*g + 4*b*c**3* 
e*f - 17*b*(61*b*c**3*e*g/20 + c**4*d*g + c**4*e*f)/(18*c) + c**4*d*f)/(8* 
c) + b**4*e*g + 4*b**3*c*d*g + 4*b**3*c*e*f + 6*b**2*c**2*d*f - 13*b*(12*a 
*b*c**2*e*g + 4*a*c**3*d*g + 4*a*c**3*e*f - 8*a*(61*b*c**3*e*g/20 + c**4*d 
*g + c**4*e*f)/(9*c) + 4*b**3*c*e*g + 6*b**2*c**2*d*g + 6*b**2*c**2*e*f + 
4*b*c**3*d*f - 15*b*(31*a*c**3*e*g/10 + 6*b**2*c**2*e*g + 4*b*c**3*d*g + 4 
*b*c**3*e*f - 17*b*(61*b*c**3*e*g/20 + c**4*d*g + c**4*e*f)/(18*c) + c**4* 
d*f)/(16*c))/(14*c))/(6*c) + x**4*(12*a**2*b*c*e*g + 6*a**2*c**2*d*g + 6*a 
**2*c**2*e*f + 4*a*b**3*e*g + 12*a*b**2*c*d*g + 12*a*b**2*c*e*f + 12*a*b*c 
**2*d*f - 6*a*(12*a*b*c**2*e*g + 4*a*c**3*d*g + 4*a*c**3*e*f - 8*a*(61*b*c 
**3*e*g/20 + c**4*d*g + c**4*e*f)/(9*c) + 4*b**3*c*e*g + 6*b**2*c**2*d*...
 

Maxima [F(-2)]

Exception generated. \[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^{7/2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)*(g*x+f)*(c*x^2+b*x+a)^(7/2),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. 1410 vs. \(2 (364) = 728\).

Time = 0.47 (sec) , antiderivative size = 1410, normalized size of antiderivative = 3.58 \[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^{7/2} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)*(g*x+f)*(c*x^2+b*x+a)^(7/2),x, algorithm="giac")
 

Output:

1/5898240*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(4*(2*(16*(18*c^3*e*g*x + ( 
20*c^12*e*f + 20*c^12*d*g + 61*b*c^11*e*g)/c^9)*x + (360*c^12*d*f + 1100*b 
*c^11*e*f + 1100*b*c^11*d*g + 1123*b^2*c^10*e*g + 1116*a*c^11*e*g)/c^9)*x 
+ (2520*b*c^11*d*f + 2580*b^2*c^10*e*f + 2560*a*c^11*e*f + 2580*b^2*c^10*d 
*g + 2560*a*c^11*d*g + 885*b^3*c^9*e*g + 5252*a*b*c^10*e*g)/c^9)*x + (1212 
0*b^2*c^10*d*f + 12000*a*c^11*d*f + 4180*b^3*c^9*e*f + 24720*a*b*c^10*e*f 
+ 4180*b^3*c^9*d*g + 24720*a*b*c^10*d*g + 5*b^4*c^8*e*g + 12840*a*b^2*c^9* 
e*g + 12624*a^2*c^10*e*g)/c^9)*x + (10200*b^3*c^9*d*f + 60000*a*b*c^10*d*f 
 + 20*b^4*c^8*e*f + 31440*a*b^2*c^9*e*f + 30720*a^2*c^10*e*f + 20*b^4*c^8* 
d*g + 31440*a*b^2*c^9*d*g + 30720*a^2*c^10*d*g - 11*b^5*c^7*e*g + 120*a*b^ 
3*c^8*e*g + 32400*a^2*b*c^9*e*g)/c^9)*x + (360*b^4*c^8*d*f + 323520*a*b^2* 
c^9*d*f + 312960*a^2*c^10*d*f - 180*b^5*c^7*e*f + 2080*a*b^3*c^8*e*f + 335 
040*a^2*b*c^9*e*f - 180*b^5*c^7*d*g + 2080*a*b^3*c^8*d*g + 335040*a^2*b*c^ 
9*d*g + 99*b^6*c^6*e*g - 1180*a*b^4*c^7*e*g + 4560*a^2*b^2*c^8*e*g + 11616 
0*a^3*c^9*e*g)/c^9)*x - (840*b^5*c^7*d*f - 10560*a*b^3*c^8*d*f - 938880*a^ 
2*b*c^9*d*f - 420*b^6*c^6*e*f + 5280*a*b^4*c^7*e*f - 22080*a^2*b^2*c^8*e*f 
 - 327680*a^3*c^9*e*f - 420*b^6*c^6*d*g + 5280*a*b^4*c^7*d*g - 22080*a^2*b 
^2*c^8*d*g - 327680*a^3*c^9*d*g + 231*b^7*c^5*e*g - 2988*a*b^5*c^6*e*g + 1 
3200*a^2*b^3*c^7*e*g - 20800*a^3*b*c^8*e*g)/c^9)*x + (4200*b^6*c^6*d*f - 5 
7120*a*b^4*c^7*d*f + 270720*a^2*b^2*c^8*d*f + 2142720*a^3*c^9*d*f - 210...
 

Mupad [F(-1)]

Timed out. \[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^{7/2} \, dx=\int \left (f+g\,x\right )\,\left (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{7/2} \,d x \] Input:

int((f + g*x)*(d + e*x)*(a + b*x + c*x^2)^(7/2),x)
 

Output:

int((f + g*x)*(d + e*x)*(a + b*x + c*x^2)^(7/2), x)
 

Reduce [F]

\[ \int (d+e x) (f+g x) \left (a+b x+c x^2\right )^{7/2} \, dx=\int \left (e x +d \right ) \left (g x +f \right ) \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}d x \] Input:

int((e*x+d)*(g*x+f)*(c*x^2+b*x+a)^(7/2),x)
 

Output:

int((e*x+d)*(g*x+f)*(c*x^2+b*x+a)^(7/2),x)