\(\int \frac {(a+b x+c x^2)^{3/2} (d+e x+f x^2)}{(g+h x)^7} \, dx\) [43]

Optimal result

Integrand size = 32, antiderivative size = 660 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^7} \, dx=-\frac {\left (b^2-4 a c\right ) \left (24 c^2 d g^2+24 a^2 f h^2-12 a b h (2 f g+e h)-4 c \left (a f g^2-a h (7 e g-d h)+3 b g (e g+2 d h)\right )+b^2 \left (7 f g^2+h (5 e g+7 d h)\right )\right ) (b g-2 a h+(2 c g-b h) x) \sqrt {a+b x+c x^2}}{512 \left (c g^2-b g h+a h^2\right )^4 (g+h x)^2}+\frac {\left (24 c^2 d g^2+24 a^2 f h^2-12 a b h (2 f g+e h)-4 c \left (a f g^2-a h (7 e g-d h)+3 b g (e g+2 d h)\right )+b^2 \left (7 f g^2+h (5 e g+7 d h)\right )\right ) (b g-2 a h+(2 c g-b h) x) \left (a+b x+c x^2\right )^{3/2}}{192 \left (c g^2-b g h+a h^2\right )^3 (g+h x)^4}-\frac {\left (f g^2-h (e g-d h)\right ) \left (a+b x+c x^2\right )^{5/2}}{6 h \left (c g^2-b g h+a h^2\right ) (g+h x)^6}+\frac {\left (2 c \left (5 f g^3+g h (e g-7 d h)\right )-h \left (17 b f g^2-b h (5 e g+7 d h)-12 a h (2 f g-e h)\right )\right ) \left (a+b x+c x^2\right )^{5/2}}{60 h \left (c g^2-b g h+a h^2\right )^2 (g+h x)^5}+\frac {\left (b^2-4 a c\right )^2 \left (24 c^2 d g^2+24 a^2 f h^2-12 a b h (2 f g+e h)-4 c \left (a f g^2-a h (7 e g-d h)+3 b g (e g+2 d h)\right )+b^2 \left (7 f g^2+h (5 e g+7 d h)\right )\right ) \text {arctanh}\left (\frac {b g-2 a h+(2 c g-b h) x}{2 \sqrt {c g^2-b g h+a h^2} \sqrt {a+b x+c x^2}}\right )}{1024 \left (c g^2-b g h+a h^2\right )^{9/2}} \] Output:

-1/512*(-4*a*c+b^2)*(24*c^2*d*g^2+24*a^2*f*h^2-12*a*b*h*(e*h+2*f*g)-4*c*(a 
*f*g^2-a*h*(-d*h+7*e*g)+3*b*g*(2*d*h+e*g))+b^2*(7*f*g^2+h*(7*d*h+5*e*g)))* 
(b*g-2*a*h+(-b*h+2*c*g)*x)*(c*x^2+b*x+a)^(1/2)/(a*h^2-b*g*h+c*g^2)^4/(h*x+ 
g)^2+1/192*(24*c^2*d*g^2+24*a^2*f*h^2-12*a*b*h*(e*h+2*f*g)-4*c*(a*f*g^2-a* 
h*(-d*h+7*e*g)+3*b*g*(2*d*h+e*g))+b^2*(7*f*g^2+h*(7*d*h+5*e*g)))*(b*g-2*a* 
h+(-b*h+2*c*g)*x)*(c*x^2+b*x+a)^(3/2)/(a*h^2-b*g*h+c*g^2)^3/(h*x+g)^4-1/6* 
(f*g^2-h*(-d*h+e*g))*(c*x^2+b*x+a)^(5/2)/h/(a*h^2-b*g*h+c*g^2)/(h*x+g)^6+1 
/60*(2*c*(5*f*g^3+g*h*(-7*d*h+e*g))-h*(17*b*f*g^2-b*h*(7*d*h+5*e*g)-12*a*h 
*(-e*h+2*f*g)))*(c*x^2+b*x+a)^(5/2)/h/(a*h^2-b*g*h+c*g^2)^2/(h*x+g)^5+1/10 
24*(-4*a*c+b^2)^2*(24*c^2*d*g^2+24*a^2*f*h^2-12*a*b*h*(e*h+2*f*g)-4*c*(a*f 
*g^2-a*h*(-d*h+7*e*g)+3*b*g*(2*d*h+e*g))+b^2*(7*f*g^2+h*(7*d*h+5*e*g)))*ar 
ctanh(1/2*(b*g-2*a*h+(-b*h+2*c*g)*x)/(a*h^2-b*g*h+c*g^2)^(1/2)/(c*x^2+b*x+ 
a)^(1/2))/(a*h^2-b*g*h+c*g^2)^(9/2)
 

Mathematica [A] (verified)

Time = 16.44 (sec) , antiderivative size = 1222, normalized size of antiderivative = 1.85 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^7} \, dx =\text {Too large to display} \] Input:

Integrate[((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^7,x]
 

Output:

(f*(b*g - 2*a*h + (2*c*g - b*h)*x)*(a + x*(b + c*x))^(3/2))/(8*h^2*(c*g^2 
- h*(b*g - a*h))*(g + h*x)^4) - ((f*g^2 - h*(e*g - d*h))*(a + b*x + c*x^2) 
*(a + x*(b + c*x))^(3/2))/(6*h*(c*g^2 - h*(b*g - a*h))*(g + h*x)^6) + ((2* 
f*g - e*h)*(a + b*x + c*x^2)*(a + x*(b + c*x))^(3/2))/(5*h*(c*g^2 - h*(b*g 
 - a*h))*(g + h*x)^5) + ((2*c*g - b*h)*(-2*f*g + e*h)*(a + x*(b + c*x))^(3 
/2)*(((b*g - 2*a*h + (2*c*g - b*h)*x)*(a + b*x + c*x^2)^(3/2))/(8*(c*g^2 - 
 b*g*h + a*h^2)*(g + h*x)^4) - (3*(b^2 - 4*a*c)*(((b*g - 2*a*h + (2*c*g - 
b*h)*x)*Sqrt[a + b*x + c*x^2])/(4*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^2) + ( 
(b^2 - 4*a*c)*ArcTanh[(-(b*g) + 2*a*h - (2*c*g - b*h)*x)/(2*Sqrt[c*g^2 - b 
*g*h + a*h^2]*Sqrt[a + b*x + c*x^2])])/(2*Sqrt[c*g^2 - b*g*h + a*h^2]*(4*c 
*g^2 - 4*b*g*h + 4*a*h^2))))/(16*(c*g^2 - b*g*h + a*h^2))))/(2*h^2*(c*g^2 
- b*g*h + a*h^2)*(a + b*x + c*x^2)^(3/2)) - ((f*g^2 - e*g*h + d*h^2)*(a + 
x*(b + c*x))^(3/2)*(((c*g*h - (h*(-12*c*g + 7*b*h))/2)*(a + b*x + c*x^2)^( 
5/2))/(5*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^5) - ((-2*(a*c*h^2 + (c*g*(-12* 
c*g + 7*b*h))/2) + b*(c*g*h + (h*(-12*c*g + 7*b*h))/2))*(((b*g - 2*a*h + ( 
2*c*g - b*h)*x)*(a + b*x + c*x^2)^(3/2))/(8*(c*g^2 - b*g*h + a*h^2)*(g + h 
*x)^4) - (3*(b^2 - 4*a*c)*(((b*g - 2*a*h + (2*c*g - b*h)*x)*Sqrt[a + b*x + 
 c*x^2])/(4*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^2) + ((b^2 - 4*a*c)*ArcTanh[ 
(-(b*g) + 2*a*h - (2*c*g - b*h)*x)/(2*Sqrt[c*g^2 - b*g*h + a*h^2]*Sqrt[a + 
 b*x + c*x^2])])/(2*Sqrt[c*g^2 - b*g*h + a*h^2]*(4*c*g^2 - 4*b*g*h + 4*...
 

Rubi [A] (verified)

Time = 1.75 (sec) , antiderivative size = 549, normalized size of antiderivative = 0.83, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {2181, 27, 1228, 1152, 1152, 1154, 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.

\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^7} \, dx\)

\(\Big \downarrow \) 2181

\(\displaystyle -\frac {\int -\frac {\left (\frac {5 b f g^2}{h}+12 c d g-5 b e g-12 a f g-7 b d h+12 a e h+2 \left (\frac {5 c f g^2}{h}+c e g-6 b f g-c d h+6 a f h\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{2 (g+h x)^6}dx}{6 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (f g^2-h (e g-d h)\right )}{6 h (g+h x)^6 \left (a h^2-b g h+c g^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\left (12 c d g-b \left (-\frac {5 f g^2}{h}+5 e g+7 d h\right )-12 a (f g-e h)-2 \left (6 b f g-6 a f h-c \left (\frac {5 f g^2}{h}+e g-d h\right )\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{(g+h x)^6}dx}{12 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (f g^2-h (e g-d h)\right )}{6 h (g+h x)^6 \left (a h^2-b g h+c g^2\right )}\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {\frac {\left (24 a^2 f h^2-4 c \left (-a h (7 e g-d h)+a f g^2+3 b g (2 d h+e g)\right )-12 a b h (e h+2 f g)+b^2 \left (h (7 d h+5 e g)+7 f g^2\right )+24 c^2 d g^2\right ) \int \frac {\left (c x^2+b x+a\right )^{3/2}}{(g+h x)^5}dx}{2 \left (a h^2-b g h+c g^2\right )}+\frac {\left (a+b x+c x^2\right )^{5/2} \left (2 c \left (g h (e g-7 d h)+5 f g^3\right )-h \left (-12 a h (2 f g-e h)-b h (7 d h+5 e g)+17 b f g^2\right )\right )}{5 h (g+h x)^5 \left (a h^2-b g h+c g^2\right )}}{12 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (f g^2-h (e g-d h)\right )}{6 h (g+h x)^6 \left (a h^2-b g h+c g^2\right )}\)

\(\Big \downarrow \) 1152

\(\displaystyle \frac {\frac {\left (24 a^2 f h^2-4 c \left (-a h (7 e g-d h)+a f g^2+3 b g (2 d h+e g)\right )-12 a b h (e h+2 f g)+b^2 \left (h (7 d h+5 e g)+7 f g^2\right )+24 c^2 d g^2\right ) \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-2 a h+x (2 c g-b h)+b g)}{8 (g+h x)^4 \left (a h^2-b g h+c g^2\right )}-\frac {3 \left (b^2-4 a c\right ) \int \frac {\sqrt {c x^2+b x+a}}{(g+h x)^3}dx}{16 \left (a h^2-b g h+c g^2\right )}\right )}{2 \left (a h^2-b g h+c g^2\right )}+\frac {\left (a+b x+c x^2\right )^{5/2} \left (2 c \left (g h (e g-7 d h)+5 f g^3\right )-h \left (-12 a h (2 f g-e h)-b h (7 d h+5 e g)+17 b f g^2\right )\right )}{5 h (g+h x)^5 \left (a h^2-b g h+c g^2\right )}}{12 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (f g^2-h (e g-d h)\right )}{6 h (g+h x)^6 \left (a h^2-b g h+c g^2\right )}\)

\(\Big \downarrow \) 1152

\(\displaystyle \frac {\frac {\left (24 a^2 f h^2-4 c \left (-a h (7 e g-d h)+a f g^2+3 b g (2 d h+e g)\right )-12 a b h (e h+2 f g)+b^2 \left (h (7 d h+5 e g)+7 f g^2\right )+24 c^2 d g^2\right ) \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-2 a h+x (2 c g-b h)+b g)}{8 (g+h x)^4 \left (a h^2-b g h+c g^2\right )}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} (-2 a h+x (2 c g-b h)+b g)}{4 (g+h x)^2 \left (a h^2-b g h+c g^2\right )}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+b x+a}}dx}{8 \left (a h^2-b g h+c g^2\right )}\right )}{16 \left (a h^2-b g h+c g^2\right )}\right )}{2 \left (a h^2-b g h+c g^2\right )}+\frac {\left (a+b x+c x^2\right )^{5/2} \left (2 c \left (g h (e g-7 d h)+5 f g^3\right )-h \left (-12 a h (2 f g-e h)-b h (7 d h+5 e g)+17 b f g^2\right )\right )}{5 h (g+h x)^5 \left (a h^2-b g h+c g^2\right )}}{12 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (f g^2-h (e g-d h)\right )}{6 h (g+h x)^6 \left (a h^2-b g h+c g^2\right )}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {\left (24 a^2 f h^2-4 c \left (-a h (7 e g-d h)+a f g^2+3 b g (2 d h+e g)\right )-12 a b h (e h+2 f g)+b^2 \left (h (7 d h+5 e g)+7 f g^2\right )+24 c^2 d g^2\right ) \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-2 a h+x (2 c g-b h)+b g)}{8 (g+h x)^4 \left (a h^2-b g h+c g^2\right )}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 \left (c g^2-b h g+a h^2\right )-\frac {(b g-2 a h+(2 c g-b h) x)^2}{c x^2+b x+a}}d\left (-\frac {b g-2 a h+(2 c g-b h) x}{\sqrt {c x^2+b x+a}}\right )}{4 \left (a h^2-b g h+c g^2\right )}+\frac {\sqrt {a+b x+c x^2} (-2 a h+x (2 c g-b h)+b g)}{4 (g+h x)^2 \left (a h^2-b g h+c g^2\right )}\right )}{16 \left (a h^2-b g h+c g^2\right )}\right )}{2 \left (a h^2-b g h+c g^2\right )}+\frac {\left (a+b x+c x^2\right )^{5/2} \left (2 c \left (g h (e g-7 d h)+5 f g^3\right )-h \left (-12 a h (2 f g-e h)-b h (7 d h+5 e g)+17 b f g^2\right )\right )}{5 h (g+h x)^5 \left (a h^2-b g h+c g^2\right )}}{12 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (f g^2-h (e g-d h)\right )}{6 h (g+h x)^6 \left (a h^2-b g h+c g^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\left (\frac {\left (a+b x+c x^2\right )^{3/2} (-2 a h+x (2 c g-b h)+b g)}{8 (g+h x)^4 \left (a h^2-b g h+c g^2\right )}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {\sqrt {a+b x+c x^2} (-2 a h+x (2 c g-b h)+b g)}{4 (g+h x)^2 \left (a h^2-b g h+c g^2\right )}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {-2 a h+x (2 c g-b h)+b g}{2 \sqrt {a+b x+c x^2} \sqrt {a h^2-b g h+c g^2}}\right )}{8 \left (a h^2-b g h+c g^2\right )^{3/2}}\right )}{16 \left (a h^2-b g h+c g^2\right )}\right ) \left (24 a^2 f h^2-4 c \left (-a h (7 e g-d h)+a f g^2+3 b g (2 d h+e g)\right )-12 a b h (e h+2 f g)+b^2 \left (h (7 d h+5 e g)+7 f g^2\right )+24 c^2 d g^2\right )}{2 \left (a h^2-b g h+c g^2\right )}+\frac {\left (a+b x+c x^2\right )^{5/2} \left (2 c \left (g h (e g-7 d h)+5 f g^3\right )-h \left (-12 a h (2 f g-e h)-b h (7 d h+5 e g)+17 b f g^2\right )\right )}{5 h (g+h x)^5 \left (a h^2-b g h+c g^2\right )}}{12 \left (a h^2-b g h+c g^2\right )}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (f g^2-h (e g-d h)\right )}{6 h (g+h x)^6 \left (a h^2-b g h+c g^2\right )}\)

Input:

Int[((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^7,x]
 

Output:

-1/6*((f*g^2 - h*(e*g - d*h))*(a + b*x + c*x^2)^(5/2))/(h*(c*g^2 - b*g*h + 
 a*h^2)*(g + h*x)^6) + (((2*c*(5*f*g^3 + g*h*(e*g - 7*d*h)) - h*(17*b*f*g^ 
2 - b*h*(5*e*g + 7*d*h) - 12*a*h*(2*f*g - e*h)))*(a + b*x + c*x^2)^(5/2))/ 
(5*h*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^5) + ((24*c^2*d*g^2 + 24*a^2*f*h^2 
- 12*a*b*h*(2*f*g + e*h) - 4*c*(a*f*g^2 - a*h*(7*e*g - d*h) + 3*b*g*(e*g + 
 2*d*h)) + b^2*(7*f*g^2 + h*(5*e*g + 7*d*h)))*(((b*g - 2*a*h + (2*c*g - b* 
h)*x)*(a + b*x + c*x^2)^(3/2))/(8*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^4) - ( 
3*(b^2 - 4*a*c)*(((b*g - 2*a*h + (2*c*g - b*h)*x)*Sqrt[a + b*x + c*x^2])/( 
4*(c*g^2 - b*g*h + a*h^2)*(g + h*x)^2) - ((b^2 - 4*a*c)*ArcTanh[(b*g - 2*a 
*h + (2*c*g - b*h)*x)/(2*Sqrt[c*g^2 - b*g*h + a*h^2]*Sqrt[a + b*x + c*x^2] 
)])/(8*(c*g^2 - b*g*h + a*h^2)^(3/2))))/(16*(c*g^2 - b*g*h + a*h^2))))/(2* 
(c*g^2 - b*g*h + a*h^2)))/(12*(c*g^2 - b*g*h + a*h^2))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 1152

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b 
*x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a 
*c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)))   Int[(d + e*x)^(m + 2)*(a + b*x + 
 c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] 
 && GtQ[p, 0]
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^ 
(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 2181

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_ 
), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = Polynomi 
alRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*x^2) 
^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Simp[1/((m + 1)*(c*d^2 - 
b*d*e + a*e^2))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m 
+ 1)*(c*d^2 - b*d*e + a*e^2)*Qx + c*d*R*(m + 1) - b*e*R*(m + p + 2) - c*e*R 
*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, 
x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(20683\) vs. \(2(634)=1268\).

Time = 1.14 (sec) , antiderivative size = 20684, normalized size of antiderivative = 31.34

method	result	size
default	\(\text {Expression too large to display}\)	\(20684\)

Input:

int((c*x^2+b*x+a)^(3/2)*(f*x^2+e*x+d)/(h*x+g)^7,x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^7} \, dx=\text {Timed out} \] Input:

integrate((c*x^2+b*x+a)^(3/2)*(f*x^2+e*x+d)/(h*x+g)^7,x, algorithm="fricas 
")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^7} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}} \left (d + e x + f x^{2}\right )}{\left (g + h x\right )^{7}}\, dx \] Input:

integrate((c*x**2+b*x+a)**(3/2)*(f*x**2+e*x+d)/(h*x+g)**7,x)
 

Output:

Integral((a + b*x + c*x**2)**(3/2)*(d + e*x + f*x**2)/(g + h*x)**7, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^7} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((c*x^2+b*x+a)^(3/2)*(f*x^2+e*x+d)/(h*x+g)^7,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(a*h^2-b*g*h>0)', see `assume?` f 
or more de
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48343 vs. \(2 (634) = 1268\).

Time = 6.32 (sec) , antiderivative size = 48343, normalized size of antiderivative = 73.25 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^7} \, dx=\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(3/2)*(f*x^2+e*x+d)/(h*x+g)^7,x, algorithm="giac")
 

Output:

1/512*(24*b^4*c^2*d*g^2 - 192*a*b^2*c^3*d*g^2 + 384*a^2*c^4*d*g^2 - 12*b^5 
*c*e*g^2 + 96*a*b^3*c^2*e*g^2 - 192*a^2*b*c^3*e*g^2 + 7*b^6*f*g^2 - 60*a*b 
^4*c*f*g^2 + 144*a^2*b^2*c^2*f*g^2 - 64*a^3*c^3*f*g^2 - 24*b^5*c*d*g*h + 1 
92*a*b^3*c^2*d*g*h - 384*a^2*b*c^3*d*g*h + 5*b^6*e*g*h - 12*a*b^4*c*e*g*h 
- 144*a^2*b^2*c^2*e*g*h + 448*a^3*c^3*e*g*h - 24*a*b^5*f*g*h + 192*a^2*b^3 
*c*f*g*h - 384*a^3*b*c^2*f*g*h + 7*b^6*d*h^2 - 60*a*b^4*c*d*h^2 + 144*a^2* 
b^2*c^2*d*h^2 - 64*a^3*c^3*d*h^2 - 12*a*b^5*e*h^2 + 96*a^2*b^3*c*e*h^2 - 1 
92*a^3*b*c^2*e*h^2 + 24*a^2*b^4*f*h^2 - 192*a^3*b^2*c*f*h^2 + 384*a^4*c^2* 
f*h^2)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*h + sqrt(c)*g)/sqrt(-c 
*g^2 + b*g*h - a*h^2))/((c^4*g^8 - 4*b*c^3*g^7*h + 6*b^2*c^2*g^6*h^2 + 4*a 
*c^3*g^6*h^2 - 4*b^3*c*g^5*h^3 - 12*a*b*c^2*g^5*h^3 + b^4*g^4*h^4 + 12*a*b 
^2*c*g^4*h^4 + 6*a^2*c^2*g^4*h^4 - 4*a*b^3*g^3*h^5 - 12*a^2*b*c*g^3*h^5 + 
6*a^2*b^2*g^2*h^6 + 4*a^3*c*g^2*h^6 - 4*a^3*b*g*h^7 + a^4*h^8)*sqrt(-c*g^2 
 + b*g*h - a*h^2)) + 1/7680*(15360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11* 
c^6*f*g^8*h^5 - 61440*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*b*c^5*f*g^7*h 
^6 + 92160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*b^2*c^4*f*g^6*h^7 + 6144 
0*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*a*c^5*f*g^6*h^7 - 61440*(sqrt(c)* 
x - sqrt(c*x^2 + b*x + a))^11*b^3*c^3*f*g^5*h^8 - 184320*(sqrt(c)*x - sqrt 
(c*x^2 + b*x + a))^11*a*b*c^4*f*g^5*h^8 + 15360*(sqrt(c)*x - sqrt(c*x^2 + 
b*x + a))^11*b^4*c^2*f*g^4*h^9 + 184320*(sqrt(c)*x - sqrt(c*x^2 + b*x +...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^7} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}\,\left (f\,x^2+e\,x+d\right )}{{\left (g+h\,x\right )}^7} \,d x \] Input:

int(((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^7,x)
 

Output:

int(((a + b*x + c*x^2)^(3/2)*(d + e*x + f*x^2))/(g + h*x)^7, x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^7} \, dx=\int \frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (f \,x^{2}+e x +d \right )}{\left (h x +g \right )^{7}}d x \] Input:

int((c*x^2+b*x+a)^(3/2)*(f*x^2+e*x+d)/(h*x+g)^7,x)
 

Output:

int((c*x^2+b*x+a)^(3/2)*(f*x^2+e*x+d)/(h*x+g)^7,x)