\(\int \frac {(a+b x+c x^2)^{5/2}}{(d+e x)^4} \, dx\) [600]

Optimal result

Integrand size = 22, antiderivative size = 337 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^4} \, dx=-\frac {5 \left (16 c^2 d^2+b^2 e^2-4 c e (3 b d-a e)+4 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{8 e^5 (d+e x)}+\frac {5 (4 c d-b e+2 c e x) \left (a+b x+c x^2\right )^{3/2}}{12 e^3 (d+e x)^2}-\frac {\left (a+b x+c x^2\right )^{5/2}}{3 e (d+e x)^3}+\frac {5 \sqrt {c} \left (16 c^2 d^2+3 b^2 e^2-4 c e (4 b d-a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 e^6}-\frac {5 (2 c d-b e) \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{16 e^6 \sqrt {c d^2-b d e+a e^2}} \] Output:

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

Mathematica [A] (verified)

Time = 12.12 (sec) , antiderivative size = 559, normalized size of antiderivative = 1.66 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^4} \, dx=\frac {-\frac {2 (a+x (b+c x))^{5/2}}{(d+e x)^3}+\frac {5 (2 c d-b e) (a+x (b+c x))^{5/2}}{2 \left (c d^2+e (-b d+a e)\right ) (d+e x)^2}+\frac {5 \left (-\frac {\left (12 c^2 d^2+b^2 e^2+4 c e (-3 b d+2 a e)\right ) (a+x (b+c x))^{5/2}}{2 (d+e x)}+\frac {(a+x (b+c x))^{3/2} \left (b^3 e^3-4 c^3 d^2 (4 d-3 e x)+b c e^2 (-15 b d+10 a e+b e x)+2 c^2 e (3 b d (5 d-2 e x)+2 a e (-3 d+2 e x))\right )}{2 e^2}+\frac {3 \left (2 e \left (c d^2+e (-b d+a e)\right ) \sqrt {a+x (b+c x)} \left (b^3 e^3+8 c^3 d^2 (-2 d+e x)+b c e^2 (-13 b d+8 a e+b e x)+4 c^2 e (b d (7 d-2 e x)+a e (-3 d+e x))\right )+2 \sqrt {c} \left (16 c^2 d^2+3 b^2 e^2+4 c e (-4 b d+a e)\right ) \left (c d^2+e (-b d+a e)\right )^2 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )+(2 c d-b e) \left (c d^2+e (-b d+a e)\right )^{3/2} \left (16 c^2 d^2+b^2 e^2+4 c e (-4 b d+3 a e)\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )\right )}{4 e^5}\right )}{2 \left (c d^2+e (-b d+a e)\right )^2}}{6 e} \] Input:

Integrate[(a + b*x + c*x^2)^(5/2)/(d + e*x)^4,x]
 

Output:

((-2*(a + x*(b + c*x))^(5/2))/(d + e*x)^3 + (5*(2*c*d - b*e)*(a + x*(b + c 
*x))^(5/2))/(2*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + (5*(-1/2*((12*c^2 
*d^2 + b^2*e^2 + 4*c*e*(-3*b*d + 2*a*e))*(a + x*(b + c*x))^(5/2))/(d + e*x 
) + ((a + x*(b + c*x))^(3/2)*(b^3*e^3 - 4*c^3*d^2*(4*d - 3*e*x) + b*c*e^2* 
(-15*b*d + 10*a*e + b*e*x) + 2*c^2*e*(3*b*d*(5*d - 2*e*x) + 2*a*e*(-3*d + 
2*e*x))))/(2*e^2) + (3*(2*e*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[a + x*(b + c*x 
)]*(b^3*e^3 + 8*c^3*d^2*(-2*d + e*x) + b*c*e^2*(-13*b*d + 8*a*e + b*e*x) + 
 4*c^2*e*(b*d*(7*d - 2*e*x) + a*e*(-3*d + e*x))) + 2*Sqrt[c]*(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))^2*ArcTanh[(b 
 + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + (2*c*d - b*e)*(c*d^2 + e*(- 
(b*d) + a*e))^(3/2)*(16*c^2*d^2 + b^2*e^2 + 4*c*e*(-4*b*d + 3*a*e))*ArcTan 
h[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqr 
t[a + x*(b + c*x)])]))/(4*e^5)))/(2*(c*d^2 + e*(-(b*d) + a*e))^2))/(6*e)
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {1161, 1230, 27, 1230, 25, 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.

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

\(\Big \downarrow \) 1161

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

\(\Big \downarrow \) 1230

\(\displaystyle \frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-b e+4 c d+2 c e x)}{2 e^2 (d+e x)^2}-\frac {3 \int \frac {2 \left (-e b^2+4 c d b-4 a c e+4 c (2 c d-b e) x\right ) \sqrt {c x^2+b x+a}}{(d+e x)^2}dx}{8 e^2}\right )}{6 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{3 e (d+e x)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-b e+4 c d+2 c e x)}{2 e^2 (d+e x)^2}-\frac {3 \int \frac {\left (-e b^2+4 c d b-4 a c e+4 c (2 c d-b e) x\right ) \sqrt {c x^2+b x+a}}{(d+e x)^2}dx}{4 e^2}\right )}{6 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{3 e (d+e x)^3}\)

\(\Big \downarrow \) 1230

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1269

\(\displaystyle \frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-b e+4 c d+2 c e x)}{2 e^2 (d+e x)^2}-\frac {3 \left (\frac {\frac {(2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {2 c \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}}{2 e^2}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right )}{e^2 (d+e x)}\right )}{4 e^2}\right )}{6 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{3 e (d+e x)^3}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-b e+4 c d+2 c e x)}{2 e^2 (d+e x)^2}-\frac {3 \left (\frac {\frac {(2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {4 c \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}}{2 e^2}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right )}{e^2 (d+e x)}\right )}{4 e^2}\right )}{6 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{3 e (d+e x)^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-b e+4 c d+2 c e x)}{2 e^2 (d+e x)^2}-\frac {3 \left (\frac {\frac {(2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {2 \sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right )}{e}}{2 e^2}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right )}{e^2 (d+e x)}\right )}{4 e^2}\right )}{6 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{3 e (d+e x)^3}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-b e+4 c d+2 c e x)}{2 e^2 (d+e x)^2}-\frac {3 \left (\frac {-\frac {2 (2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}-\frac {2 \sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right )}{e}}{2 e^2}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right )}{e^2 (d+e x)}\right )}{4 e^2}\right )}{6 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{3 e (d+e x)^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {5 \left (\frac {\left (a+b x+c x^2\right )^{3/2} (-b e+4 c d+2 c e x)}{2 e^2 (d+e x)^2}-\frac {3 \left (\frac {\frac {(2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e \sqrt {a e^2-b d e+c d^2}}-\frac {2 \sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right )}{e}}{2 e^2}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right )}{e^2 (d+e x)}\right )}{4 e^2}\right )}{6 e}-\frac {\left (a+b x+c x^2\right )^{5/2}}{3 e (d+e x)^3}\)

Input:

Int[(a + b*x + c*x^2)^(5/2)/(d + e*x)^4,x]
 

Output:

-1/3*(a + b*x + c*x^2)^(5/2)/(e*(d + e*x)^3) + (5*(((4*c*d - b*e + 2*c*e*x 
)*(a + b*x + c*x^2)^(3/2))/(2*e^2*(d + e*x)^2) - (3*(((16*c^2*d^2 + b^2*e^ 
2 - 4*c*e*(3*b*d - a*e) + 4*c*e*(2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(e 
^2*(d + e*x)) + ((-2*Sqrt[c]*(16*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(4*b*d - a*e) 
)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/e + ((2*c*d - b* 
e)*(16*c^2*d^2 + b^2*e^2 - 4*c*e*(4*b*d - 3*a*e))*ArcTanh[(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])])/(e 
*Sqrt[c*d^2 - b*d*e + a*e^2]))/(2*e^2)))/(4*e^2)))/(6*e)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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. \(2384\) vs. \(2(305)=610\).

Time = 1.43 (sec) , antiderivative size = 2385, normalized size of antiderivative = 7.08

method	result	size
risch	\(\text {Expression too large to display}\)	\(2385\)
default	\(\text {Expression too large to display}\)	\(4598\)

Input:

int((c*x^2+b*x+a)^(5/2)/(e*x+d)^4,x,method=_RETURNVERBOSE)
 

Output:

1/4*c*(2*c*e*x+9*b*e-16*c*d)*(c*x^2+b*x+a)^(1/2)/e^5+1/8/e^5*(-8/e^2*(6*a* 
b*c*e^3-12*a*c^2*d*e^2+b^3*e^3-12*b^2*c*d*e^2+30*b*c^2*d^2*e-20*c^3*d^3)/( 
(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))+24/e^3*(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)*(-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)))+24/e^4*(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)*(-1/2/(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/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e)))+1/2*c/(a*e 
^2-b*d*e+c*d^2)*e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1261 vs. \(2 (305) = 610\).

Time = 101.44 (sec) , antiderivative size = 5131, normalized size of antiderivative = 15.23 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^4} \, dx=\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(e*x+d)^4,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^4} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{4}}\, dx \] Input:

integrate((c*x**2+b*x+a)**(5/2)/(e*x+d)**4,x)
 

Output:

Integral((a + b*x + c*x**2)**(5/2)/(d + e*x)**4, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^4} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(e*x+d)^4,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. 2152 vs. \(2 (305) = 610\).

Time = 3.94 (sec) , antiderivative size = 2152, normalized size of antiderivative = 6.39 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^4} \, dx=\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(5/2)/(e*x+d)^4,x, algorithm="giac")
 

Output:

1/4*sqrt(c*x^2 + b*x + a)*(2*c^2*x/e^4 - (16*c^3*d*e^10 - 9*b*c^2*e^11)/(c 
*e^15)) - 5/8*(32*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 - 12*a*b*c*e^3)*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) - 5/8*(16*c^3*d^2 - 16*b*c^2*d*e + 3*b^2*c*e^2 + 4*a*c^2*e^2)*log(ab 
s(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/(sqrt(c)*e^6) - 1/2 
4*(480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*c^3*d^3*e^2 - 720*(sqrt(c)*x 
- sqrt(c*x^2 + b*x + a))^5*b*c^2*d^2*e^3 + 306*(sqrt(c)*x - sqrt(c*x^2 + b 
*x + a))^5*b^2*c*d*e^4 + 216*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*c^2*d 
*e^4 - 33*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^3*e^5 - 108*(sqrt(c)*x - 
 sqrt(c*x^2 + b*x + a))^5*a*b*c*e^5 + 1680*(sqrt(c)*x - sqrt(c*x^2 + b*x + 
 a))^4*c^(7/2)*d^4*e - 2160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b*c^(5/2 
)*d^3*e^2 + 666*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^2*c^(3/2)*d^2*e^3 
+ 216*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*c^(5/2)*d^2*e^3 - 21*(sqrt(c 
)*x - sqrt(c*x^2 + b*x + a))^4*b^3*sqrt(c)*d*e^4 + 324*(sqrt(c)*x - sqrt(c 
*x^2 + b*x + a))^4*a*b*c^(3/2)*d*e^4 - 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + 
 a))^4*a*b^2*sqrt(c)*e^5 - 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*c 
^(3/2)*e^5 + 1504*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*c^4*d^5 - 400*(sqr 
t(c)*x - sqrt(c*x^2 + b*x + a))^3*b*c^3*d^4*e - 1308*(sqrt(c)*x - sqrt(c*x 
^2 + b*x + a))^3*b^2*c^2*d^3*e^2 - 1808*(sqrt(c)*x - sqrt(c*x^2 + b*x +...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^4} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (d+e\,x\right )}^4} \,d x \] Input:

int((a + b*x + c*x^2)^(5/2)/(d + e*x)^4,x)
 

Output:

int((a + b*x + c*x^2)^(5/2)/(d + e*x)^4, x)
 

Reduce [B] (verification not implemented)

Time = 2.53 (sec) , antiderivative size = 4902, normalized size of antiderivative = 14.55 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^4} \, dx =\text {Too large to display} \] Input:

int((c*x^2+b*x+a)^(5/2)/(e*x+d)^4,x)
 

Output:

( - 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)*a*b*c*d**3*e**3 
 - 540*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*b*c*d**2*e**4* 
x - 540*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*b*c*d*e**5*x* 
*2 - 180*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqr 
t(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a*b*c*e**6*x** 
3 + 360*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**4*e** 
2 + 1080*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqr 
t(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**3*e* 
*3*x + 1080*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**4*x**2 + 360*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**5*x**3 - 15*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**3* 
d**3*e**3 - 45*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**3*...