\(\int \frac {x^3}{(d+e x) (a+b x+c x^2)^{5/2}} \, dx\) [46]

Optimal result

Integrand size = 25, antiderivative size = 417 \[ \int \frac {x^3}{(d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \left (a \left (b^2 d-2 a c d-a b e\right )+\left (b^3 d-3 a b c d-a b^2 e+2 a^2 c e\right ) x\right )}{3 c \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (24 a^2 c^3 d^3-b^5 d^2 e-2 a b^2 c d \left (3 c d^2-2 a e^2\right )+a b^3 e \left (2 c d^2-a e^2\right )-4 a^2 b c e \left (4 c d^2+a e^2\right )+b^4 \left (c d^3+2 a d e^2\right )-c \left (2 b^4 d^2 e-2 a b^2 e \left (2 c d^2-a e^2\right )+8 a^2 c e \left (4 c d^2+a e^2\right )-4 a b c d \left (3 c d^2+2 a e^2\right )+b^3 \left (c d^3-4 a d e^2\right )\right ) x\right )}{3 c \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x+c x^2}}-\frac {d^3 e \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{\left (c d^2-b d e+a e^2\right )^{5/2}} \] Output:

2/3*(a*(-a*b*e-2*a*c*d+b^2*d)+(2*a^2*c*e-a*b^2*e-3*a*b*c*d+b^3*d)*x)/c/(-4 
*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(c*x^2+b*x+a)^(3/2)-2/3*(24*a^2*c^3*d^3-b^5* 
d^2*e-2*a*b^2*c*d*(-2*a*e^2+3*c*d^2)+a*b^3*e*(-a*e^2+2*c*d^2)-4*a^2*b*c*e* 
(a*e^2+4*c*d^2)+b^4*(2*a*d*e^2+c*d^3)-c*(2*b^4*d^2*e-2*a*b^2*e*(-a*e^2+2*c 
*d^2)+8*a^2*c*e*(a*e^2+4*c*d^2)-4*a*b*c*d*(2*a*e^2+3*c*d^2)+b^3*(-4*a*d*e^ 
2+c*d^3))*x)/c/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)^2/(c*x^2+b*x+a)^(1/2)-d^ 
3*e*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))/(a*e^2-b*d*e+c*d^2)^(5/2)
 

Mathematica [A] (verified)

Time = 12.20 (sec) , antiderivative size = 412, normalized size of antiderivative = 0.99 \[ \int \frac {x^3}{(d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \left (-b^3 d x+a b (-b d+3 c d x+b e x)+a^2 (b e+2 c (d-e x))\right )}{3 c \left (-b^2+4 a c\right ) \left (c d^2+e (-b d+a e)\right ) (a+x (b+c x))^{3/2}}+\frac {2 \left (b^5 d^2 e-b^4 d \left (2 a e^2+c d (d-2 e x)\right )+4 a b c \left (a^2 e^3-3 c^2 d^3 x+2 a c d e (2 d-e x)\right )+2 a b^2 c \left (c d^2 (3 d-2 e x)+a e^2 (-2 d+e x)\right )+b^3 \left (a^2 e^3+c^2 d^3 x-2 a c d e (d+2 e x)\right )+8 a^2 c^2 \left (a e^3 x+c d^2 (-3 d+4 e x)\right )\right )}{3 c \left (b^2-4 a c\right )^2 \left (c d^2+e (-b d+a e)\right )^2 \sqrt {a+x (b+c x)}}-\frac {d^3 e \log (d+e x)}{\left (c d^2+e (-b d+a e)\right )^{5/2}}+\frac {d^3 e \log \left (-b d+2 a e-2 c d x+b e x+2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}\right )}{\left (c d^2+e (-b d+a e)\right )^{5/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 446, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1264, 27, 1235, 27, 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.

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

\(\Big \downarrow \) 1264

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1235

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(2*(a*(b^2*d - 2*a*c*d - a*b*e) + (b^3*d - 3*a*b*c*d - a*b^2*e + 2*a^2*c*e 
)*x))/(3*c*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^(3/2)) 
+ ((-2*(d*(b*c*d - b^2*e + 2*a*c*e)*(b^3*d - a*b^2*e - 4*a^2*c*e) - a*(2*c 
*d - b*e)*(b^3*d*e + b^2*(3*c*d^2 - a*e^2) - 4*a*c*(3*c*d^2 + a*e^2)) + c* 
(d*(2*c*d - b*e)*(b^3*d - a*b^2*e - 4*a^2*c*e) - (b*d - 2*a*e)*(b^3*d*e + 
b^2*(3*c*d^2 - a*e^2) - 4*a*c*(3*c*d^2 + a*e^2)))*x))/((b^2 - 4*a*c)*(c*d^ 
2 - b*d*e + a*e^2)*Sqrt[a + b*x + c*x^2]) - (3*c*(b^2 - 4*a*c)*d^3*e*ArcTa 
nh[(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])])/(c*d^2 - b*d*e + a*e^2)^(3/2))/(3*c*(b^2 - 4*a*c)*(c*d^2 
- b*d*e + a*e^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 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 1235

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 
*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a 
+ b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] 
 + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^m 
*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 
 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* 
m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - 
f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] 
)
 

rule 1264

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d + e*x) 
^m*(f + g*x)^n, a + b*x + c*x^2, x], R = Coeff[PolynomialRemainder[(d + e*x 
)^m*(f + g*x)^n, a + b*x + c*x^2, x], x, 0], S = Coeff[PolynomialRemainder[ 
(d + e*x)^m*(f + g*x)^n, a + b*x + c*x^2, x], x, 1]}, Simp[(b*R - 2*a*S + ( 
2*c*R - b*S)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + S 
imp[1/((p + 1)*(b^2 - 4*a*c))   Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*E 
xpandToSum[((p + 1)*(b^2 - 4*a*c)*Q)/(d + e*x)^m - ((2*p + 3)*(2*c*R - b*S) 
)/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[n, 1] 
 && LtQ[p, -1] && ILtQ[m, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1130\) vs. \(2(401)=802\).

Time = 1.47 (sec) , antiderivative size = 1131, normalized size of antiderivative = 2.71

method	result	size
default	\(\text {Expression too large to display}\)	\(1131\)

Input:

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

Output:

d^2/e^3*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^ 
2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2))+1/e*(-1/2*x/c/(c*x^2+b*x+a)^(3/2)-1/4*b/c 
*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x 
+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2)))+1/2*a/c*(2/ 
3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b) 
/(c*x^2+b*x+a)^(1/2)))-d/e^2*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(2/3*(2*c 
*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^ 
2+b*x+a)^(1/2)))-d^3/e^4*(1/3/(a*e^2-b*d*e+c*d^2)*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)^(3/2)-1/2*(b*e-2*c*d)*e/(a*e^2-b*d 
*e+c*d^2)*(2/3*(2*c*(x+d/e)+(b*e-2*c*d)/e)/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b 
*e-2*c*d)^2/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)^(3/2)+16/3*c/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)^2*(2*c*(x+ 
d/e)+(b*e-2*c*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/(a*e^2-b*d*e+c*d^2)*e^2*(1/(a*e^2-b*d*e+c*d^2)*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)-(b*e-2*c*d)*e/(a 
*e^2-b*d*e+c*d^2)*(2*c*(x+d/e)+(b*e-2*c*d)/e)/(4*c*(a*e^2-b*d*e+c*d^2)/e^2 
-(b*e-2*c*d)^2/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)-1/(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+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/...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2279 vs. \(2 (401) = 802\).

Time = 1.30 (sec) , antiderivative size = 4600, normalized size of antiderivative = 11.03 \[ \int \frac {x^3}{(d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^3/(e*x+d)/(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((b/e-(2*c*d)/e^2)^2>0)', see `as 
sume?` for
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8382 vs. \(2 (401) = 802\).

Time = 0.48 (sec) , antiderivative size = 8382, normalized size of antiderivative = 20.10 \[ \int \frac {x^3}{(d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-2*d^3*e*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))/((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2 
*e^2 - 2*a*b*d*e^3 + a^2*e^4)*sqrt(-c*d^2 + b*d*e - a*e^2)) + 2/3*((((b^3* 
c^8*d^15 - 12*a*b*c^9*d^15 - 4*b^4*c^7*d^14*e + 68*a*b^2*c^8*d^14*e + 32*a 
^2*c^9*d^14*e + 3*b^5*c^6*d^13*e^2 - 154*a*b^3*c^7*d^13*e^2 - 272*a^2*b*c^ 
8*d^13*e^2 + 10*b^6*c^5*d^12*e^3 + 186*a*b^4*c^6*d^12*e^3 + 866*a^2*b^2*c^ 
7*d^12*e^3 + 200*a^3*c^8*d^12*e^3 - 25*b^7*c^4*d^11*e^4 - 160*a*b^5*c^5*d^ 
11*e^4 - 1381*a^2*b^3*c^6*d^11*e^4 - 1236*a^3*b*c^7*d^11*e^4 + 24*b^8*c^3* 
d^10*e^5 + 152*a*b^6*c^4*d^10*e^5 + 1240*a^2*b^4*c^5*d^10*e^5 + 2952*a^3*b 
^2*c^6*d^10*e^5 + 528*a^4*c^7*d^10*e^5 - 11*b^9*c^2*d^9*e^6 - 138*a*b^7*c^ 
3*d^9*e^6 - 742*a^2*b^5*c^4*d^9*e^6 - 3500*a^3*b^3*c^5*d^9*e^6 - 2520*a^4* 
b*c^6*d^9*e^6 + 2*b^10*c*d^8*e^7 + 74*a*b^8*c^2*d^8*e^7 + 422*a^2*b^6*c^3* 
d^8*e^7 + 2260*a^3*b^4*c^4*d^8*e^7 + 4510*a^4*b^2*c^5*d^8*e^7 + 760*a^5*c^ 
6*d^8*e^7 - 16*a*b^9*c*d^7*e^8 - 221*a^2*b^7*c^2*d^7*e^8 - 960*a^3*b^5*c^3 
*d^7*e^8 - 3785*a^4*b^3*c^4*d^7*e^8 - 2740*a^5*b*c^5*d^7*e^8 + 56*a^2*b^8* 
c*d^6*e^9 + 396*a^3*b^6*c^2*d^6*e^9 + 1620*a^4*b^4*c^3*d^6*e^9 + 3460*a^5* 
b^2*c^4*d^6*e^9 + 640*a^6*c^5*d^6*e^9 - 112*a^3*b^7*c*d^5*e^10 - 493*a^4*b 
^5*c^2*d^5*e^10 - 1834*a^5*b^3*c^3*d^5*e^10 - 1632*a^6*b*c^4*d^5*e^10 + 14 
0*a^4*b^6*c*d^4*e^11 + 466*a^5*b^4*c^2*d^4*e^11 + 1278*a^6*b^2*c^3*d^4*e^1 
1 + 312*a^7*c^4*d^4*e^11 - 112*a^5*b^5*c*d^3*e^12 - 339*a^6*b^3*c^2*d^3...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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