\(\int \frac {x (a+b x+c x^2)^{3/2}}{(d+e x)^2} \, dx\) [79]

Optimal result

Integrand size = 23, antiderivative size = 296 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^2} \, dx=\frac {\left (32 c^2 d^2+b^2 e^2-4 c e (7 b d-2 a e)-2 c e (8 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{8 c e^4}+\frac {(4 d+e x) \left (a+b x+c x^2\right )^{3/2}}{3 e^2 (d+e x)}-\frac {\left (64 c^3 d^3+b^3 e^3-24 c^2 d e (3 b d-2 a e)+12 b c e^2 (b d-a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} e^5}+\frac {\sqrt {c d^2-b d e+a e^2} \left (8 c d^2-e (5 b d-2 a e)\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{2 e^5} \] Output:

1/8*(32*c^2*d^2+b^2*e^2-4*c*e*(-2*a*e+7*b*d)-2*c*e*(-b*e+8*c*d)*x)*(c*x^2+ 
b*x+a)^(1/2)/c/e^4+1/3*(e*x+4*d)*(c*x^2+b*x+a)^(3/2)/e^2/(e*x+d)-1/16*(64* 
c^3*d^3+b^3*e^3-24*c^2*d*e*(-2*a*e+3*b*d)+12*b*c*e^2*(-a*e+b*d))*arctanh(1 
/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(3/2)/e^5+1/2*(a*e^2-b*d*e+c*d 
^2)^(1/2)*(8*c*d^2-e*(-2*a*e+5*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^5
 

Mathematica [A] (verified)

Time = 11.01 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.03 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^2} \, dx=\frac {-3 \left (64 c^3 d^3+b^3 e^3+12 b c e^2 (b d-a e)+24 c^2 d e (-3 b d+2 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )+2 \sqrt {c} \left (\frac {e \sqrt {a+x (b+c x)} \left (3 b^2 e^2 (d+e x)+8 c^2 \left (12 d^3+6 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+2 c e \left (4 a e (7 d+4 e x)+b \left (-42 d^2-23 d e x+7 e^2 x^2\right )\right )\right )}{d+e x}-12 c \sqrt {c d^2+e (-b d+a e)} \left (8 c d^2+e (-5 b d+2 a e)\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )\right )}{48 c^{3/2} e^5} \] Input:

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

Output:

(-3*(64*c^3*d^3 + b^3*e^3 + 12*b*c*e^2*(b*d - a*e) + 24*c^2*d*e*(-3*b*d + 
2*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])] + 2*Sqrt[c] 
*((e*Sqrt[a + x*(b + c*x)]*(3*b^2*e^2*(d + e*x) + 8*c^2*(12*d^3 + 6*d^2*e* 
x - 2*d*e^2*x^2 + e^3*x^3) + 2*c*e*(4*a*e*(7*d + 4*e*x) + b*(-42*d^2 - 23* 
d*e*x + 7*e^2*x^2))))/(d + e*x) - 12*c*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*(8*c 
*d^2 + e*(-5*b*d + 2*a*e))*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*S 
qrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])]))/(48*c^(3/2)*e^5)
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {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.

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

\(\Big \downarrow \) 1230

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

\(\Big \downarrow \) 1231

\(\displaystyle \frac {(4 d+e x) \left (a+b x+c x^2\right )^{3/2}}{3 e^2 (d+e x)}-\frac {-\frac {\int \frac {8 c e (b d-2 a e) (2 b d-a e)-d (8 c d-b e) \left (-e b^2+4 c d b-4 a c e\right )-\left (64 c^3 d^3-24 c^2 e (3 b d-2 a e) d+b^3 e^3+12 b c e^2 (b d-a e)\right ) x}{2 (d+e x) \sqrt {c x^2+b x+a}}dx}{4 c e^2}-\frac {\sqrt {a+b x+c x^2} \left (-4 c e (7 b d-2 a e)+b^2 e^2-2 c e x (8 c d-b e)+32 c^2 d^2\right )}{4 c e^2}}{2 e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(4 d+e x) \left (a+b x+c x^2\right )^{3/2}}{3 e^2 (d+e x)}-\frac {-\frac {\int \frac {8 c e (b d-2 a e) (2 b d-a e)-d (8 c d-b e) \left (-e b^2+4 c d b-4 a c e\right )-\left (64 c^3 d^3-24 c^2 e (3 b d-2 a e) d+b^3 e^3+12 b c e^2 (b d-a e)\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 c e^2}-\frac {\sqrt {a+b x+c x^2} \left (-4 c e (7 b d-2 a e)+b^2 e^2-2 c e x (8 c d-b e)+32 c^2 d^2\right )}{4 c e^2}}{2 e^2}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {(4 d+e x) \left (a+b x+c x^2\right )^{3/2}}{3 e^2 (d+e x)}-\frac {-\frac {\frac {8 c \left (a e^2-b d e+c d^2\right ) \left (2 a e^2-5 b d e+8 c d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {\left (-24 c^2 d e (3 b d-2 a e)+12 b c e^2 (b d-a e)+b^3 e^3+64 c^3 d^3\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}}{8 c e^2}-\frac {\sqrt {a+b x+c x^2} \left (-4 c e (7 b d-2 a e)+b^2 e^2-2 c e x (8 c d-b e)+32 c^2 d^2\right )}{4 c e^2}}{2 e^2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {(4 d+e x) \left (a+b x+c x^2\right )^{3/2}}{3 e^2 (d+e x)}-\frac {-\frac {\frac {8 c \left (a e^2-b d e+c d^2\right ) \left (2 a e^2-5 b d e+8 c d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {2 \left (-24 c^2 d e (3 b d-2 a e)+12 b c e^2 (b d-a e)+b^3 e^3+64 c^3 d^3\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}}{8 c e^2}-\frac {\sqrt {a+b x+c x^2} \left (-4 c e (7 b d-2 a e)+b^2 e^2-2 c e x (8 c d-b e)+32 c^2 d^2\right )}{4 c e^2}}{2 e^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {(4 d+e x) \left (a+b x+c x^2\right )^{3/2}}{3 e^2 (d+e x)}-\frac {-\frac {\frac {8 c \left (a e^2-b d e+c d^2\right ) \left (2 a e^2-5 b d e+8 c d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-24 c^2 d e (3 b d-2 a e)+12 b c e^2 (b d-a e)+b^3 e^3+64 c^3 d^3\right )}{\sqrt {c} e}}{8 c e^2}-\frac {\sqrt {a+b x+c x^2} \left (-4 c e (7 b d-2 a e)+b^2 e^2-2 c e x (8 c d-b e)+32 c^2 d^2\right )}{4 c e^2}}{2 e^2}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {(4 d+e x) \left (a+b x+c x^2\right )^{3/2}}{3 e^2 (d+e x)}-\frac {-\frac {-\frac {16 c \left (a e^2-b d e+c d^2\right ) \left (2 a e^2-5 b d e+8 c d^2\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-24 c^2 d e (3 b d-2 a e)+12 b c e^2 (b d-a e)+b^3 e^3+64 c^3 d^3\right )}{\sqrt {c} e}}{8 c e^2}-\frac {\sqrt {a+b x+c x^2} \left (-4 c e (7 b d-2 a e)+b^2 e^2-2 c e x (8 c d-b e)+32 c^2 d^2\right )}{4 c e^2}}{2 e^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {(4 d+e x) \left (a+b x+c x^2\right )^{3/2}}{3 e^2 (d+e x)}-\frac {-\frac {\frac {8 c \sqrt {a e^2-b d e+c d^2} \left (2 a e^2-5 b d e+8 c d^2\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e}-\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-24 c^2 d e (3 b d-2 a e)+12 b c e^2 (b d-a e)+b^3 e^3+64 c^3 d^3\right )}{\sqrt {c} e}}{8 c e^2}-\frac {\sqrt {a+b x+c x^2} \left (-4 c e (7 b d-2 a e)+b^2 e^2-2 c e x (8 c d-b e)+32 c^2 d^2\right )}{4 c e^2}}{2 e^2}\)

Input:

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

Output:

((4*d + e*x)*(a + b*x + c*x^2)^(3/2))/(3*e^2*(d + e*x)) - (-1/4*((32*c^2*d 
^2 + b^2*e^2 - 4*c*e*(7*b*d - 2*a*e) - 2*c*e*(8*c*d - b*e)*x)*Sqrt[a + b*x 
 + c*x^2])/(c*e^2) - (-(((64*c^3*d^3 + b^3*e^3 - 24*c^2*d*e*(3*b*d - 2*a*e 
) + 12*b*c*e^2*(b*d - a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + 
c*x^2])])/(Sqrt[c]*e)) + (8*c*Sqrt[c*d^2 - b*d*e + a*e^2]*(8*c*d^2 - 5*b*d 
*e + 2*a*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]*Sqrt[a + b*x + c*x^2])])/e)/(8*c*e^2))/(2*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 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. \(708\) vs. \(2(268)=536\).

Time = 1.45 (sec) , antiderivative size = 709, normalized size of antiderivative = 2.40

method	result	size
risch	\(\frac {\left (8 c^{2} e^{2} x^{2}+14 e^{2} x b c -24 c^{2} d e x +32 a c \,e^{2}+3 b^{2} e^{2}-60 b c d e +72 c^{2} d^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{24 c \,e^{4}}+\frac {\frac {\left (12 a b c \,e^{3}-48 d \,e^{2} a \,c^{2}-b^{3} e^{3}-12 d \,e^{2} b^{2} c +72 d^{2} e b \,c^{2}-64 d^{3} c^{3}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{e \sqrt {c}}-\frac {16 c \left (a^{2} e^{4}-4 d \,e^{3} a b +6 a c \,d^{2} e^{2}+3 d^{2} e^{2} b^{2}-8 b c \,d^{3} e +5 c^{2} d^{4}\right ) \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}-\frac {16 c d \left (a^{2} e^{4}-2 d \,e^{3} a b +2 a c \,d^{2} e^{2}+d^{2} e^{2} b^{2}-2 b c \,d^{3} e +c^{2} d^{4}\right ) \left (-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}+\frac {\left (b e -2 c d \right ) e \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}\right )}{e^{3}}}{16 e^{4} c}\)	\(709\)
default	\(\text {Expression too large to display}\)	\(1723\)

Input:

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

Output:

1/24/c*(8*c^2*e^2*x^2+14*b*c*e^2*x-24*c^2*d*e*x+32*a*c*e^2+3*b^2*e^2-60*b* 
c*d*e+72*c^2*d^2)*(c*x^2+b*x+a)^(1/2)/e^4+1/16/e^4/c*((12*a*b*c*e^3-48*a*c 
^2*d*e^2-b^3*e^3-12*b^2*c*d*e^2+72*b*c^2*d^2*e-64*c^3*d^3)/e*ln((1/2*b+c*x 
)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-16*c/e^2*(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)/((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*c*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^3*(-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))))
 

Fricas [A] (verification not implemented)

Time = 112.28 (sec) , antiderivative size = 2135, normalized size of antiderivative = 7.21 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/96*(3*(64*c^3*d^4 - 72*b*c^2*d^3*e + 12*(b^2*c + 4*a*c^2)*d^2*e^2 + (b 
^3 - 12*a*b*c)*d*e^3 + (64*c^3*d^3*e - 72*b*c^2*d^2*e^2 + 12*(b^2*c + 4*a* 
c^2)*d*e^3 + (b^3 - 12*a*b*c)*e^4)*x)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b 
^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 24*(8*c^3*d^3 
- 5*b*c^2*d^2*e + 2*a*c^2*d*e^2 + (8*c^3*d^2*e - 5*b*c^2*d*e^2 + 2*a*c^2*e 
^3)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a 
*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 - 4*sqrt(c*d^2 - 
 b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x) - 2* 
(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2) 
) - 4*(8*c^3*e^4*x^3 + 96*c^3*d^3*e - 84*b*c^2*d^2*e^2 + (3*b^2*c + 56*a*c 
^2)*d*e^3 - 2*(8*c^3*d*e^3 - 7*b*c^2*e^4)*x^2 + (48*c^3*d^2*e^2 - 46*b*c^2 
*d*e^3 + (3*b^2*c + 32*a*c^2)*e^4)*x)*sqrt(c*x^2 + b*x + a))/(c^2*e^6*x + 
c^2*d*e^5), 1/48*(3*(64*c^3*d^4 - 72*b*c^2*d^3*e + 12*(b^2*c + 4*a*c^2)*d^ 
2*e^2 + (b^3 - 12*a*b*c)*d*e^3 + (64*c^3*d^3*e - 72*b*c^2*d^2*e^2 + 12*(b^ 
2*c + 4*a*c^2)*d*e^3 + (b^3 - 12*a*b*c)*e^4)*x)*sqrt(-c)*arctan(1/2*sqrt(c 
*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) + 12*(8*c^3* 
d^3 - 5*b*c^2*d^2*e + 2*a*c^2*d*e^2 + (8*c^3*d^2*e - 5*b*c^2*d*e^2 + 2*a*c 
^2*e^3)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 
 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 - 4*sqrt(c*d 
^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.59 (sec) , antiderivative size = 1343, normalized size of antiderivative = 4.54 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(48*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**2 + 48*sqr 
t(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**3*x - 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)*b*c**2*d**2*e - 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)*b*c**2*d*e**2*x + 192*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**3 + 192*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**2*e*x - 48*sqrt(a*e**2 - b*d*e + c*d**2)*log(d 
+ e*x)*a*c**2*d*e**2 - 48*sqrt(a*e**2 - b*d*e + c*d**2)*log(d + e*x)*a*c** 
2*e**3*x + 120*sqrt(a*e**2 - b*d*e + c*d**2)*log(d + e*x)*b*c**2*d**2*e + 
120*sqrt(a*e**2 - b*d*e + c*d**2)*log(d + e*x)*b*c**2*d*e**2*x - 192*sqrt( 
a*e**2 - b*d*e + c*d**2)*log(d + e*x)*c**3*d**3 - 192*sqrt(a*e**2 - b*d*e 
+ c*d**2)*log(d + e*x)*c**3*d**2*e*x + 112*sqrt(a + b*x + c*x**2)*a*c**2*d 
*e**3 + 64*sqrt(a + b*x + c*x**2)*a*c**2*e**4*x + 6*sqrt(a + b*x + c*x**2) 
*b**2*c*d*e**3 + 6*sqrt(a + b*x + c*x**2)*b**2*c*e**4*x - 168*sqrt(a + b*x 
 + c*x**2)*b*c**2*d**2*e**2 - 92*sqrt(a + b*x + c*x**2)*b*c**2*d*e**3*x...