\(\int \frac {x (a+b x+c x^2)^{3/2}}{(d+e x)^3} \, dx\) [80]

Optimal result

Integrand size = 23, antiderivative size = 288 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=-\frac {3 \left (2 \left (4 c d^2-e (2 b d-a e)\right )+e (4 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{4 e^4 (d+e x)}+\frac {(2 d+e x) \left (a+b x+c x^2\right )^{3/2}}{2 e^2 (d+e x)^2}-\frac {3 \left (4 c e (b d-a e)-(4 c d-b e)^2\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} e^5}-\frac {3 \left (16 c^2 d^3+b e^2 (5 b d-4 a e)-4 c d e (5 b d-3 a e)\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{8 e^5 \sqrt {c d^2-b d e+a e^2}} \] Output:

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

Mathematica [A] (verified)

Time = 12.04 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.76 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {\frac {d (a+x (b+c x))^{5/2}}{(d+e x)^2}-\frac {\left (6 c d^2+e (-5 b d+4 a e)\right ) (a+x (b+c x))^{5/2}}{2 \left (c d^2+e (-b d+a e)\right ) (d+e x)}-\frac {\frac {(a+x (b+c x))^{3/2} \left (b e^2 (-5 b d+4 a e)+c^2 \left (-8 d^3+6 d^2 e x\right )+c e (b d (13 d-5 e x)+2 a e (-3 d+2 e x))\right )}{2 e^2}+\frac {3 \left (-2 c^2 e \left (c d^2+e (-b d+a e)\right ) \sqrt {a+x (b+c x)} \left (b e^2 (4 b d-3 a e)+4 c^2 d^2 (2 d-e x)+c e (3 b d (-4 d+e x)-2 a e (-3 d+e x))\right )+c^{3/2} \left (16 c^2 d^2+b^2 e^2+4 c e (-3 b d+a e)\right ) \left (c d^2+e (-b d+a e)\right )^2 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )+c^2 \left (16 c^2 d^3+b e^2 (5 b d-4 a e)-4 c d e (5 b d-3 a e)\right ) \left (c d^2+e (-b d+a e)\right )^{3/2} \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )\right )}{4 c^2 e^5}}{-c d^2+e (b d-a e)}}{2 \left (c d^2+e (-b d+a e)\right )} \] Input:

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

Output:

((d*(a + x*(b + c*x))^(5/2))/(d + e*x)^2 - ((6*c*d^2 + e*(-5*b*d + 4*a*e)) 
*(a + x*(b + c*x))^(5/2))/(2*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)) - (((a 
+ x*(b + c*x))^(3/2)*(b*e^2*(-5*b*d + 4*a*e) + c^2*(-8*d^3 + 6*d^2*e*x) + 
c*e*(b*d*(13*d - 5*e*x) + 2*a*e*(-3*d + 2*e*x))))/(2*e^2) + (3*(-2*c^2*e*( 
c*d^2 + e*(-(b*d) + a*e))*Sqrt[a + x*(b + c*x)]*(b*e^2*(4*b*d - 3*a*e) + 4 
*c^2*d^2*(2*d - e*x) + c*e*(3*b*d*(-4*d + e*x) - 2*a*e*(-3*d + e*x))) + c^ 
(3/2)*(16*c^2*d^2 + b^2*e^2 + 4*c*e*(-3*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)])] + c^2*(16*c^ 
2*d^3 + b*e^2*(5*b*d - 4*a*e) - 4*c*d*e*(5*b*d - 3*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)))/(2*(c*d^2 + e*(-(b*d) + a*e)))
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {1230, 27, 1230, 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.

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

\(\Big \downarrow \) 1230

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1230

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

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 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. \(1234\) vs. \(2(257)=514\).

Time = 1.54 (sec) , antiderivative size = 1235, normalized size of antiderivative = 4.29

method	result	size
risch	\(\text {Expression too large to display}\)	\(1235\)
default	\(\text {Expression too large to display}\)	\(2953\)

Input:

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

Output:

1/4*(2*c*e*x+5*b*e-12*c*d)*(c*x^2+b*x+a)^(1/2)/e^4+1/8/e^4*(-8/e^2*(2*a*b* 
e^3-6*a*c*d*e^2-3*b^2*d*e^2+12*b*c*d^2*e-10*c^2*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))+8/e^3*(a^2*e^4-4*a*b*d*e^3+6*a*c*d^2*e^2+3*b^2*d^ 
2*e^2-8*b*c*d^3*e+5*c^2*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)))+3*(4*a 
*c*e^2+b^2*e^2-12*b*c*d*e+16*c^2*d^2)/e*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+ 
a)^(1/2))/c^(1/2)-8*d*(a^2*e^4-2*a*b*d*e^3+2*a*c*d^2*e^2+b^2*d^2*e^2-2*b*c 
*d^3*e+c^2*d^4)/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/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+c*d^2)/e^...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 908 vs. \(2 (257) = 514\).

Time = 104.09 (sec) , antiderivative size = 3719, normalized size of antiderivative = 12.91 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x*(c*x^2+b*x+a)^(3/2)/(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. 973 vs. \(2 (257) = 514\).

Time = 0.43 (sec) , antiderivative size = 973, normalized size of antiderivative = 3.38 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

1/4*sqrt(c*x^2 + b*x + a)*(2*c*x/e^3 - (12*c^2*d*e^8 - 5*b*c*e^9)/(c*e^12) 
) - 3/4*(16*c^2*d^3 - 20*b*c*d^2*e + 5*b^2*d*e^2 + 12*a*c*d*e^2 - 4*a*b*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^5) - 3/8*(16*c^2*d^2 - 
12*b*c*d*e + b^2*e^2 + 4*a*c*e^2)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x 
+ a))*sqrt(c) + b))/(sqrt(c)*e^5) - 1/4*(32*(sqrt(c)*x - sqrt(c*x^2 + b*x 
+ a))^3*c^2*d^3*e - 36*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b*c*d^2*e^2 + 
 9*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^2*d*e^3 + 12*(sqrt(c)*x - sqrt( 
c*x^2 + b*x + a))^3*a*c*d*e^3 - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a* 
b*e^4 + 56*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^(5/2)*d^4 - 44*(sqrt(c) 
*x - sqrt(c*x^2 + b*x + a))^2*b*c^(3/2)*d^3*e + 3*(sqrt(c)*x - sqrt(c*x^2 
+ b*x + a))^2*b^2*sqrt(c)*d^2*e^2 - 12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)) 
^2*a*c^(3/2)*d^2*e^2 + 20*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b*sqrt(c 
)*d*e^3 - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*sqrt(c)*e^4 + 56*(sq 
rt(c)*x - sqrt(c*x^2 + b*x + a))*b*c^2*d^4 - 48*(sqrt(c)*x - sqrt(c*x^2 + 
b*x + a))*b^2*c*d^3*e - 80*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*c^2*d^3*e 
 + 7*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^3*d^2*e^2 + 80*(sqrt(c)*x - sqr 
t(c*x^2 + b*x + a))*a*b*c*d^2*e^2 - 11*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)) 
*a*b^2*d*e^3 - 20*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*c*d*e^3 + 4*(sqr 
t(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b*e^4 + 14*b^2*c^(3/2)*d^4 - 9*b^3*...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.60 (sec) , antiderivative size = 3154, normalized size of antiderivative = 10.95 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 12*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**3 
- 24*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**4*x - 1 
2*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*e**5*x**2 + 36* 
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**3*e**2 + 72*s 
qrt(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**3*x + 36* 
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**4*x**2 + 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**2*c*d**3*e**2 + 30* 
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**3*x + 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**2*c*d*e**4*x**2 - 6 
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)*b*c**2*d**4*e - 12...