\(\int \frac {(a+b x+c x^2)^{5/2}}{(d+e x)^3} \, dx\) [599]

Optimal result

Integrand size = 22, antiderivative size = 331 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\frac {5 \left (16 c^2 d^2+5 b^2 e^2-4 c e (5 b d-a e)-4 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{8 e^5}+\frac {5 (8 c d-3 b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{12 e^3 (d+e x)}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2}-\frac {5 (2 c d-b e) \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 \sqrt {c} e^6}+\frac {5 \sqrt {c d^2-b d e+a e^2} \left (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{8 e^6} \] Output:

5/8*(16*c^2*d^2+5*b^2*e^2-4*c*e*(-a*e+5*b*d)-4*c*e*(-b*e+2*c*d)*x)*(c*x^2+ 
b*x+a)^(1/2)/e^5+5/12*(2*c*e*x-3*b*e+8*c*d)*(c*x^2+b*x+a)^(3/2)/e^3/(e*x+d 
)-1/2*(c*x^2+b*x+a)^(5/2)/e/(e*x+d)^2-5/16*(-b*e+2*c*d)*(16*c^2*d^2+b^2*e^ 
2-4*c*e*(-3*a*e+4*b*d))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2)) 
/c^(1/2)/e^6+5/8*(a*e^2-b*d*e+c*d^2)^(1/2)*(16*c^2*d^2+3*b^2*e^2-4*c*e*(-a 
*e+4*b*d))*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2 
)/(c*x^2+b*x+a)^(1/2))/e^6
 

Mathematica [A] (verified)

Time = 11.25 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\frac {-\frac {2 (a+x (b+c x))^{5/2}}{(d+e x)^2}+\frac {5 (2 c d-b e) (a+x (b+c x))^{5/2}}{\left (c d^2+e (-b d+a e)\right ) (d+e x)}+\frac {5 \left (-\frac {(a+x (b+c x))^{3/2} \left (3 b^2 e^2+2 c^2 d (4 d-3 e x)+c e (-11 b d+2 a e+3 b e x)\right )}{3 e^2}+\frac {-2 c^2 e \left (c d^2+e (-b d+a e)\right ) \sqrt {a+x (b+c x)} \left (5 b^2 e^2+8 c^2 d (2 d-e x)+4 c e (-5 b d+a e+b e x)\right )+c^{3/2} (2 c d-b e) \left (c d^2+e (-b d+a e)\right ) \left (16 c^2 d^2+b^2 e^2+4 c e (-4 b d+3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )+2 c^2 \left (16 c^2 d^2+3 b^2 e^2+4 c e (-4 b d+a e)\right ) \left (c d^2+e (-b d+a e)\right )^{3/2} \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{4 c^2 e^5}\right )}{-c d^2+e (b d-a e)}}{4 e} \] Input:

Integrate[(a + b*x + c*x^2)^(5/2)/(d + e*x)^3,x]
 

Output:

((-2*(a + x*(b + c*x))^(5/2))/(d + e*x)^2 + (5*(2*c*d - b*e)*(a + x*(b + c 
*x))^(5/2))/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x)) + (5*(-1/3*((a + x*(b + 
 c*x))^(3/2)*(3*b^2*e^2 + 2*c^2*d*(4*d - 3*e*x) + c*e*(-11*b*d + 2*a*e + 3 
*b*e*x)))/e^2 + (-2*c^2*e*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[a + x*(b + c*x)] 
*(5*b^2*e^2 + 8*c^2*d*(2*d - e*x) + 4*c*e*(-5*b*d + a*e + b*e*x)) + c^(3/2 
)*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))*(16*c^2*d^2 + b^2*e^2 + 4*c*e*( 
-4*b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + 
2*c^2*(16*c^2*d^2 + 3*b^2*e^2 + 4*c*e*(-4*b*d + a*e))*(c*d^2 + e*(-(b*d) + 
 a*e))^(3/2)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e* 
(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(4*c^2*e^5)))/(-(c*d^2) + e*(b*d 
- a*e)))/(4*e)
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {1161, 1230, 1231, 27, 1269, 1092, 219, 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.

\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx\)

\(\Big \downarrow \) 1161

\(\displaystyle \frac {5 \int \frac {(b+2 c x) \left (c x^2+b x+a\right )^{3/2}}{(d+e x)^2}dx}{4 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 1230

\(\displaystyle \frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d+2 c e x)}{3 e^2 (d+e x)}-\frac {\int \frac {\left (-3 e b^2+8 c d b-4 a c e+8 c (2 c d-b e) x\right ) \sqrt {c x^2+b x+a}}{d+e x}dx}{2 e^2}\right )}{4 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d+2 c e x)}{3 e^2 (d+e x)}-\frac {-\frac {\int \frac {2 c \left (e (b d-2 a e) \left (-3 e b^2+8 c d b-4 a c e\right )-2 d (2 c d-b e) \left (-e b^2+4 c d b-4 a c e\right )-(2 c d-b e) \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right )}{(d+e x) \sqrt {c x^2+b x+a}}dx}{4 c e^2}-\frac {\sqrt {a+b x+c x^2} \left (-4 c e (5 b d-a e)+5 b^2 e^2-4 c e x (2 c d-b e)+16 c^2 d^2\right )}{e^2}}{2 e^2}\right )}{4 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d+2 c e x)}{3 e^2 (d+e x)}-\frac {-\frac {\int \frac {e (b d-2 a e) \left (-3 e b^2+8 c d b-4 a c e\right )-2 d (2 c d-b e) \left (-e b^2+4 c d b-4 a c e\right )-(2 c d-b e) \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (-4 c e (5 b d-a e)+5 b^2 e^2-4 c e x (2 c d-b e)+16 c^2 d^2\right )}{e^2}}{2 e^2}\right )}{4 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d+2 c e x)}{3 e^2 (d+e x)}-\frac {-\frac {\frac {2 \left (a e^2-b d e+c d^2\right ) \left (4 a c e^2+3 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {(2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (-4 c e (5 b d-a e)+5 b^2 e^2-4 c e x (2 c d-b e)+16 c^2 d^2\right )}{e^2}}{2 e^2}\right )}{4 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d+2 c e x)}{3 e^2 (d+e x)}-\frac {-\frac {\frac {2 \left (a e^2-b d e+c d^2\right ) \left (4 a c e^2+3 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {2 (2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (-4 c e (5 b d-a e)+5 b^2 e^2-4 c e x (2 c d-b e)+16 c^2 d^2\right )}{e^2}}{2 e^2}\right )}{4 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d+2 c e x)}{3 e^2 (d+e x)}-\frac {-\frac {\frac {2 \left (a e^2-b d e+c d^2\right ) \left (4 a c e^2+3 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )}{\sqrt {c} e}}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (-4 c e (5 b d-a e)+5 b^2 e^2-4 c e x (2 c d-b e)+16 c^2 d^2\right )}{e^2}}{2 e^2}\right )}{4 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d+2 c e x)}{3 e^2 (d+e x)}-\frac {-\frac {-\frac {4 \left (a e^2-b d e+c d^2\right ) \left (4 a c e^2+3 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}-\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )}{\sqrt {c} e}}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (-4 c e (5 b d-a e)+5 b^2 e^2-4 c e x (2 c d-b e)+16 c^2 d^2\right )}{e^2}}{2 e^2}\right )}{4 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d+2 c e x)}{3 e^2 (d+e x)}-\frac {-\frac {\frac {2 \sqrt {a e^2-b d e+c d^2} \left (4 a c e^2+3 b^2 e^2-16 b c d e+16 c^2 d^2\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e}-\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )}{\sqrt {c} e}}{2 e^2}-\frac {\sqrt {a+b x+c x^2} \left (-4 c e (5 b d-a e)+5 b^2 e^2-4 c e x (2 c d-b e)+16 c^2 d^2\right )}{e^2}}{2 e^2}\right )}{4 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{2 e (d+e x)^2}\)

Input:

Int[(a + b*x + c*x^2)^(5/2)/(d + e*x)^3,x]
 

Output:

-1/2*(a + b*x + c*x^2)^(5/2)/(e*(d + e*x)^2) + (5*(((8*c*d - 3*b*e + 2*c*e 
*x)*(a + b*x + c*x^2)^(3/2))/(3*e^2*(d + e*x)) - (-(((16*c^2*d^2 + 5*b^2*e 
^2 - 4*c*e*(5*b*d - a*e) - 4*c*e*(2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/e 
^2) - (-(((2*c*d - b*e)*(16*c^2*d^2 + b^2*e^2 - 4*c*e*(4*b*d - 3*a*e))*Arc 
Tanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt[c]*e)) + (2*Sqr 
t[c*d^2 - b*d*e + a*e^2]*(16*c^2*d^2 - 16*b*c*d*e + 3*b^2*e^2 + 4*a*c*e^2) 
*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sq 
rt[a + b*x + c*x^2])])/e)/(2*e^2))/(2*e^2)))/(4*e)
 

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 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 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 1161

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 1))), x] - Si 
mp[p/(e*(m + 1))   Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^(p - 
 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && GtQ[p, 0] && (IntegerQ[p] || 
 LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, 
 c, d, e, m, p, x]
 

rule 1230

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - 
 d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p 
+ 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a 
+ b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m 
+ 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, 
 x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 
1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ 
[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1231

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) 
 - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ 
(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 
 2*p + 2))   Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* 
a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* 
c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c 
^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x 
] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  !R 
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Integer 
Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1430\) vs. \(2(299)=598\).

Time = 1.42 (sec) , antiderivative size = 1431, normalized size of antiderivative = 4.32

method	result	size
risch	\(\text {Expression too large to display}\)	\(1431\)
default	\(\text {Expression too large to display}\)	\(2822\)

Input:

int((c*x^2+b*x+a)^(5/2)/(e*x+d)^3,x,method=_RETURNVERBOSE)
 

Output:

1/24*(8*c^2*e^2*x^2+26*b*c*e^2*x-36*c^2*d*e*x+56*a*c*e^2+33*b^2*e^2-162*b* 
c*d*e+144*c^2*d^2)*(c*x^2+b*x+a)^(1/2)/e^5+1/16/e^5*(-48/e^2*(a^2*c*e^4+a* 
b^2*e^4-6*a*b*c*d*e^3+6*a*c^2*d^2*e^2-b^3*d*e^3+6*b^2*c*d^2*e^2-10*b*c^2*d 
^3*e+5*c^3*d^4)/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/ 
e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2+( 
b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))+48/e^3*(a^2* 
b*e^5-2*a^2*c*d*e^4-2*a*b^2*d*e^4+6*a*b*c*d^2*e^3-4*a*c^2*d^3*e^2+b^3*d^2* 
e^3-4*b^2*c*d^3*e^2+5*b*c^2*d^4*e-2*c^3*d^5)*(-1/(a*e^2-b*d*e+c*d^2)*e^2/( 
x+d/e)*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1 
/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2 
*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2) 
^(1/2)*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/ 
(x+d/e)))+16*(a^3*e^6-3*a^2*b*d*e^5+3*a^2*c*d^2*e^4+3*a*b^2*d^2*e^4-6*a*b* 
c*d^3*e^3+3*a*c^2*d^4*e^2-b^3*d^3*e^3+3*b^2*c*d^4*e^2-3*b*c^2*d^5*e+c^3*d^ 
6)/e^4*(-1/2/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^2*(c*(x+d/e)^2+(b*e-2*c*d)/e* 
(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-3/4*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^ 
2)*(-1/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+ 
(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)/((a*e 
^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+ 
d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 733 vs. \(2 (299) = 598\).

Time = 65.41 (sec) , antiderivative size = 3019, normalized size of antiderivative = 9.12 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(e*x+d)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{3}}\, dx \] Input:

integrate((c*x**2+b*x+a)**(5/2)/(e*x+d)**3,x)
 

Output:

Integral((a + b*x + c*x**2)**(5/2)/(d + e*x)**3, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(e*x+d)^3,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*e^2-b*d*e>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. 1500 vs. \(2 (299) = 598\).

Time = 0.36 (sec) , antiderivative size = 1500, normalized size of antiderivative = 4.53 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(e*x+d)^3,x, algorithm="giac")
 

Output:

1/24*sqrt(c*x^2 + b*x + a)*(2*x*(4*c^2*x/e^3 - (18*c^4*d*e^14 - 13*b*c^3*e 
^15)/(c^2*e^18)) + (144*c^4*d^2*e^13 - 162*b*c^3*d*e^14 + 33*b^2*c^2*e^15 
+ 56*a*c^3*e^15)/(c^2*e^18)) + 5/4*(16*c^3*d^4 - 32*b*c^2*d^3*e + 19*b^2*c 
*d^2*e^2 + 20*a*c^2*d^2*e^2 - 3*b^3*d*e^3 - 20*a*b*c*d*e^3 + 3*a*b^2*e^4 + 
 4*a^2*c*e^4)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d)/ 
sqrt(-c*d^2 + b*d*e - a*e^2))/(sqrt(-c*d^2 + b*d*e - a*e^2)*e^6) + 5/16*(3 
2*c^3*d^3 - 48*b*c^2*d^2*e + 18*b^2*c*d*e^2 + 24*a*c^2*d*e^2 - b^3*e^3 - 1 
2*a*b*c*e^3)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/( 
sqrt(c)*e^6) + 1/4*(40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*c^3*d^4*e - 8 
0*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b*c^2*d^3*e^2 + 49*(sqrt(c)*x - sq 
rt(c*x^2 + b*x + a))^3*b^2*c*d^2*e^3 + 44*(sqrt(c)*x - sqrt(c*x^2 + b*x + 
a))^3*a*c^2*d^2*e^3 - 9*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^3*d*e^4 - 
44*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*c*d*e^4 + 9*(sqrt(c)*x - sqrt 
(c*x^2 + b*x + a))^3*a*b^2*e^5 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a 
^2*c*e^5 + 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^(7/2)*d^5 - 120*(sqr 
t(c)*x - sqrt(c*x^2 + b*x + a))^2*b*c^(5/2)*d^4*e + 51*(sqrt(c)*x - sqrt(c 
*x^2 + b*x + a))^2*b^2*c^(3/2)*d^3*e^2 + 36*(sqrt(c)*x - sqrt(c*x^2 + b*x 
+ a))^2*a*c^(5/2)*d^3*e^2 - 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^3*sq 
rt(c)*d^2*e^3 + 12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b*c^(3/2)*d^2*e 
^3 - 21*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^2*sqrt(c)*d*e^4 - 36*...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (d+e\,x\right )}^3} \,d x \] Input:

int((a + b*x + c*x^2)^(5/2)/(d + e*x)^3,x)
 

Output:

int((a + b*x + c*x^2)^(5/2)/(d + e*x)^3, x)
 

Reduce [B] (verification not implemented)

Time = 2.27 (sec) , antiderivative size = 2452, normalized size of antiderivative = 7.41 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:

int((c*x^2+b*x+a)^(5/2)/(e*x+d)^3,x)
 

Output:

(120*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e** 
2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a*c**2*d**2*e**2 + 24 
0*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - 
 b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a*c**2*d*e**3*x + 120*sq 
rt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d 
*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a*c**2*e**4*x**2 + 90*sqrt(a 
*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + 
 c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*b**2*c*d**2*e**2 + 180*sqrt(a*e* 
*2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c* 
d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*b**2*c*d*e**3*x + 90*sqrt(a*e**2 - 
b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) 
 - 2*a*e + b*d - b*e*x + 2*c*d*x)*b**2*c*e**4*x**2 - 480*sqrt(a*e**2 - b*d 
*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 
2*a*e + b*d - b*e*x + 2*c*d*x)*b*c**2*d**3*e - 960*sqrt(a*e**2 - b*d*e + c 
*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e 
+ b*d - b*e*x + 2*c*d*x)*b*c**2*d**2*e**2*x - 480*sqrt(a*e**2 - b*d*e + c* 
d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + 
 b*d - b*e*x + 2*c*d*x)*b*c**2*d*e**3*x**2 + 480*sqrt(a*e**2 - b*d*e + c*d 
**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + 
b*d - b*e*x + 2*c*d*x)*c**3*d**4 + 960*sqrt(a*e**2 - b*d*e + c*d**2)*lo...