\(\int \frac {x (a+b x+c x^2)^{3/2}}{d+e x} \, dx\) [22]

Optimal result

Integrand size = 23, antiderivative size = 339 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=-\frac {\left (64 c^3 d^3+3 b^3 e^3-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (2 b d-3 a e)+2 c e \left (3 \left (b^2-4 a c\right ) e^2-8 c d (2 c d-b e)\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c^2 e^4}-\frac {(8 c d-3 b e-6 c e x) \left (a+b x+c x^2\right )^{3/2}}{24 c e^2}+\frac {\left (128 c^4 d^4+3 b^4 e^4+8 b^2 c e^3 (b d-3 a e)-192 c^3 d^2 e (b d-a e)+48 c^2 e^2 (b d-a e)^2\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{5/2} e^5}-\frac {d \left (c d^2-b d e+a e^2\right )^{3/2} \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e^5} \] Output:

-1/64*(64*c^3*d^3+3*b^3*e^3-16*c^2*d*e*(-4*a*e+5*b*d)+4*b*c*e^2*(-3*a*e+2* 
b*d)+2*c*e*(3*(-4*a*c+b^2)*e^2-8*c*d*(-b*e+2*c*d))*x)*(c*x^2+b*x+a)^(1/2)/ 
c^2/e^4-1/24*(-6*c*e*x-3*b*e+8*c*d)*(c*x^2+b*x+a)^(3/2)/c/e^2+1/128*(128*c 
^4*d^4+3*b^4*e^4+8*b^2*c*e^3*(-3*a*e+b*d)-192*c^3*d^2*e*(-a*e+b*d)+48*c^2* 
e^2*(-a*e+b*d)^2)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(5/ 
2)/e^5-d*(a*e^2-b*d*e+c*d^2)^(3/2)*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 = 2.87 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.96 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=\frac {\frac {2 e \sqrt {a+x (b+c x)} \left (-9 b^3 e^3+6 b c e^2 (-4 b d+10 a e+b e x)-16 c^3 \left (12 d^3-6 d^2 e x+4 d e^2 x^2-3 e^3 x^3\right )+8 c^2 e \left (a e (-32 d+15 e x)+b \left (30 d^2-14 d e x+9 e^2 x^2\right )\right )\right )}{c^2}-768 d \sqrt {-c d^2+b d e-a e^2} \left (c d^2+e (-b d+a e)\right ) \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )-\frac {3 \left (128 c^4 d^4+3 b^4 e^4+8 b^2 c e^3 (b d-3 a e)-192 c^3 d^2 e (b d-a e)+48 c^2 e^2 (b d-a e)^2\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{c^{5/2}}}{384 e^5} \] Input:

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

Output:

((2*e*Sqrt[a + x*(b + c*x)]*(-9*b^3*e^3 + 6*b*c*e^2*(-4*b*d + 10*a*e + b*e 
*x) - 16*c^3*(12*d^3 - 6*d^2*e*x + 4*d*e^2*x^2 - 3*e^3*x^3) + 8*c^2*e*(a*e 
*(-32*d + 15*e*x) + b*(30*d^2 - 14*d*e*x + 9*e^2*x^2))))/c^2 - 768*d*Sqrt[ 
-(c*d^2) + b*d*e - a*e^2]*(c*d^2 + e*(-(b*d) + a*e))*ArcTan[(Sqrt[c]*(d + 
e*x) - e*Sqrt[a + x*(b + c*x)])/Sqrt[-(c*d^2) + e*(b*d - a*e)]] - (3*(128* 
c^4*d^4 + 3*b^4*e^4 + 8*b^2*c*e^3*(b*d - 3*a*e) - 192*c^3*d^2*e*(b*d - a*e 
) + 48*c^2*e^2*(b*d - a*e)^2)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c* 
x)]])/c^(5/2))/(384*e^5)
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {1231, 27, 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.

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

\(\Big \downarrow \) 1231

\(\displaystyle -\frac {\int -\frac {\left (d \left (-3 e b^2+8 c d b-4 a c e\right )+\left (16 c^2 d^2-3 b^2 e^2-4 c e (2 b d-3 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{2 (d+e x)}dx}{8 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d-6 c e x)}{24 c e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\left (d \left (-3 e b^2+8 c d b-4 a c e\right )+\left (16 c^2 d^2-3 b^2 e^2-4 c e (2 b d-3 a e)\right ) x\right ) \sqrt {c x^2+b x+a}}{d+e x}dx}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d-6 c e x)}{24 c e^2}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {-\frac {\int -\frac {d \left (3 e^3 b^4+8 c d e^2 b^3-8 c e \left (10 c d^2+3 a e^2\right ) b^2+32 c^2 d \left (2 c d^2+5 a e^2\right ) b-16 a c^2 e \left (4 c d^2+5 a e^2\right )\right )+\left (128 c^4 d^4-192 c^3 e (b d-a e) d^2+3 b^4 e^4+48 c^2 e^2 (b d-a e)^2+8 b^2 c e^3 (b d-3 a e)\right ) x}{2 (d+e x) \sqrt {c x^2+b x+a}}dx}{4 c e^2}-\frac {\sqrt {a+b x+c x^2} \left (-2 c e x \left (-4 c e (2 b d-3 a e)-3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (2 b d-3 a e)+3 b^3 e^3+64 c^3 d^3\right )}{4 c e^2}}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d-6 c e x)}{24 c e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {d \left (3 e^3 b^4+8 c d e^2 b^3-8 c e \left (10 c d^2+3 a e^2\right ) b^2+32 c^2 d \left (2 c d^2+5 a e^2\right ) b-16 a c^2 e \left (4 c d^2+5 a e^2\right )\right )+\left (128 c^4 d^4-192 c^3 e (b d-a e) d^2+3 b^4 e^4+48 c^2 e^2 (b d-a e)^2+8 b^2 c e^3 (b d-3 a e)\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 c e^2}-\frac {\sqrt {a+b x+c x^2} \left (-2 c e x \left (-4 c e (2 b d-3 a e)-3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (2 b d-3 a e)+3 b^3 e^3+64 c^3 d^3\right )}{4 c e^2}}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d-6 c e x)}{24 c e^2}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\frac {\frac {\left (8 b^2 c e^3 (b d-3 a e)-192 c^3 d^2 e (b d-a e)+48 c^2 e^2 (b d-a e)^2+3 b^4 e^4+128 c^4 d^4\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {128 c^2 d \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 c e^2}-\frac {\sqrt {a+b x+c x^2} \left (-2 c e x \left (-4 c e (2 b d-3 a e)-3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (2 b d-3 a e)+3 b^3 e^3+64 c^3 d^3\right )}{4 c e^2}}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d-6 c e x)}{24 c e^2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {\frac {2 \left (8 b^2 c e^3 (b d-3 a e)-192 c^3 d^2 e (b d-a e)+48 c^2 e^2 (b d-a e)^2+3 b^4 e^4+128 c^4 d^4\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}-\frac {128 c^2 d \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 c e^2}-\frac {\sqrt {a+b x+c x^2} \left (-2 c e x \left (-4 c e (2 b d-3 a e)-3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (2 b d-3 a e)+3 b^3 e^3+64 c^3 d^3\right )}{4 c e^2}}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d-6 c e x)}{24 c e^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (8 b^2 c e^3 (b d-3 a e)-192 c^3 d^2 e (b d-a e)+48 c^2 e^2 (b d-a e)^2+3 b^4 e^4+128 c^4 d^4\right )}{\sqrt {c} e}-\frac {128 c^2 d \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 c e^2}-\frac {\sqrt {a+b x+c x^2} \left (-2 c e x \left (-4 c e (2 b d-3 a e)-3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (2 b d-3 a e)+3 b^3 e^3+64 c^3 d^3\right )}{4 c e^2}}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d-6 c e x)}{24 c e^2}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {\frac {256 c^2 d \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}+\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (8 b^2 c e^3 (b d-3 a e)-192 c^3 d^2 e (b d-a e)+48 c^2 e^2 (b d-a e)^2+3 b^4 e^4+128 c^4 d^4\right )}{\sqrt {c} e}}{8 c e^2}-\frac {\sqrt {a+b x+c x^2} \left (-2 c e x \left (-4 c e (2 b d-3 a e)-3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (2 b d-3 a e)+3 b^3 e^3+64 c^3 d^3\right )}{4 c e^2}}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d-6 c e x)}{24 c e^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (8 b^2 c e^3 (b d-3 a e)-192 c^3 d^2 e (b d-a e)+48 c^2 e^2 (b d-a e)^2+3 b^4 e^4+128 c^4 d^4\right )}{\sqrt {c} e}-\frac {128 c^2 d \left (a e^2-b d e+c d^2\right )^{3/2} \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e}}{8 c e^2}-\frac {\sqrt {a+b x+c x^2} \left (-2 c e x \left (-4 c e (2 b d-3 a e)-3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (2 b d-3 a e)+3 b^3 e^3+64 c^3 d^3\right )}{4 c e^2}}{16 c e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d-6 c e x)}{24 c e^2}\)

Input:

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

Output:

-1/24*((8*c*d - 3*b*e - 6*c*e*x)*(a + b*x + c*x^2)^(3/2))/(c*e^2) + (-1/4* 
((64*c^3*d^3 + 3*b^3*e^3 - 16*c^2*d*e*(5*b*d - 4*a*e) + 4*b*c*e^2*(2*b*d - 
 3*a*e) - 2*c*e*(16*c^2*d^2 - 3*b^2*e^2 - 4*c*e*(2*b*d - 3*a*e))*x)*Sqrt[a 
 + b*x + c*x^2])/(c*e^2) + (((128*c^4*d^4 + 3*b^4*e^4 + 8*b^2*c*e^3*(b*d - 
 3*a*e) - 192*c^3*d^2*e*(b*d - a*e) + 48*c^2*e^2*(b*d - a*e)^2)*ArcTanh[(b 
 + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt[c]*e) - (128*c^2*d*(c* 
d^2 - b*d*e + a*e^2)^(3/2)*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))/(16*c*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 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 [A] (verified)

Time = 1.50 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.47

method	result	size
risch	\(\frac {\left (48 c^{3} x^{3} e^{3}+72 b \,c^{2} e^{3} x^{2}-64 c^{3} d \,e^{2} x^{2}+120 a \,c^{2} e^{3} x +6 x \,b^{2} c \,e^{3}-112 b \,c^{2} d \,e^{2} x +96 d^{2} e \,c^{3} x +60 a b c \,e^{3}-256 d \,e^{2} a \,c^{2}-9 b^{3} e^{3}-24 d \,e^{2} b^{2} c +240 d^{2} e b \,c^{2}-192 d^{3} c^{3}\right ) \sqrt {c \,x^{2}+b x +a}}{192 c^{2} e^{4}}+\frac {\frac {\left (48 e^{4} a^{2} c^{2}-24 a \,b^{2} c \,e^{4}-96 a b \,c^{2} d \,e^{3}+192 d^{2} e^{2} a \,c^{3}+3 b^{4} e^{4}+8 d \,e^{3} b^{3} c +48 d^{2} e^{2} b^{2} c^{2}-192 d^{3} e b \,c^{3}+128 d^{4} c^{4}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{e \sqrt {c}}+\frac {128 d \left (a^{2} e^{4}-2 d \,e^{3} a b +2 a c \,d^{2} e^{2}+d^{2} e^{2} b^{2}-2 b c \,d^{3} e +c^{2} d^{4}\right ) c^{2} \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}}{128 e^{4} c^{2}}\)	\(500\)
default	\(\frac {\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}}{e}-\frac {d \left (\frac {\left (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}}\right )^{\frac {3}{2}}}{3}+\frac {\left (b e -2 c d \right ) \left (\frac {\left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right ) \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}}}}{4 c}+\frac {\left (\frac {4 c \left (a \,e^{2}-b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\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}}}\right )}{8 c^{\frac {3}{2}}}\right )}{2 e}+\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (\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}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\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}}}\right )}{2 e \sqrt {c}}-\frac {\left (a \,e^{2}-b d e +c \,d^{2}\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}}}}\right )}{e^{2}}\right )}{e^{2}}\)	\(735\)

Input:

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

Output:

1/192/c^2*(48*c^3*e^3*x^3+72*b*c^2*e^3*x^2-64*c^3*d*e^2*x^2+120*a*c^2*e^3* 
x+6*b^2*c*e^3*x-112*b*c^2*d*e^2*x+96*c^3*d^2*e*x+60*a*b*c*e^3-256*a*c^2*d* 
e^2-9*b^3*e^3-24*b^2*c*d*e^2+240*b*c^2*d^2*e-192*c^3*d^3)*(c*x^2+b*x+a)^(1 
/2)/e^4+1/128/e^4/c^2*((48*a^2*c^2*e^4-24*a*b^2*c*e^4-96*a*b*c^2*d*e^3+192 
*a*c^3*d^2*e^2+3*b^4*e^4+8*b^3*c*d*e^3+48*b^2*c^2*d^2*e^2-192*b*c^3*d^3*e+ 
128*c^4*d^4)/e*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)+128*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)*c^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)^(1/2))/(x+d/e)))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 25.09 (sec) , antiderivative size = 10332, normalized size of antiderivative = 30.48 \[ \int \frac {x \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx =\text {Too large to display} \] Input:

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

Output:

(384*sqrt(4*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*b*e - 8*sqrt(c)*sqrt(a*e 
**2 - b*d*e + c*d**2)*c*d - 4*a*c*e**2 - b**2*e**2 + 8*b*c*d*e - 8*c**2*d* 
*2)*sqrt(a*e**2 - b*d*e + c*d**2)*atan((2*sqrt(c)*sqrt(a + b*x + c*x**2)*e 
 + b*e + 2*c*e*x)/sqrt(4*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*b*e - 8*sqr 
t(c)*sqrt(a*e**2 - b*d*e + c*d**2)*c*d - 4*a*c*e**2 - b**2*e**2 + 8*b*c*d* 
e - 8*c**2*d**2))*a*b*c**3*d*e**3 - 768*sqrt(4*sqrt(c)*sqrt(a*e**2 - b*d*e 
 + c*d**2)*b*e - 8*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*c*d - 4*a*c*e**2 
- b**2*e**2 + 8*b*c*d*e - 8*c**2*d**2)*sqrt(a*e**2 - b*d*e + c*d**2)*atan( 
(2*sqrt(c)*sqrt(a + b*x + c*x**2)*e + b*e + 2*c*e*x)/sqrt(4*sqrt(c)*sqrt(a 
*e**2 - b*d*e + c*d**2)*b*e - 8*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*c*d 
- 4*a*c*e**2 - b**2*e**2 + 8*b*c*d*e - 8*c**2*d**2))*a*c**4*d**2*e**2 - 38 
4*sqrt(4*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*b*e - 8*sqrt(c)*sqrt(a*e**2 
 - b*d*e + c*d**2)*c*d - 4*a*c*e**2 - b**2*e**2 + 8*b*c*d*e - 8*c**2*d**2) 
*sqrt(a*e**2 - b*d*e + c*d**2)*atan((2*sqrt(c)*sqrt(a + b*x + c*x**2)*e + 
b*e + 2*c*e*x)/sqrt(4*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*b*e - 8*sqrt(c 
)*sqrt(a*e**2 - b*d*e + c*d**2)*c*d - 4*a*c*e**2 - b**2*e**2 + 8*b*c*d*e - 
 8*c**2*d**2))*b**2*c**3*d**2*e**2 + 1152*sqrt(4*sqrt(c)*sqrt(a*e**2 - b*d 
*e + c*d**2)*b*e - 8*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*c*d - 4*a*c*e** 
2 - b**2*e**2 + 8*b*c*d*e - 8*c**2*d**2)*sqrt(a*e**2 - b*d*e + c*d**2)*ata 
n((2*sqrt(c)*sqrt(a + b*x + c*x**2)*e + b*e + 2*c*e*x)/sqrt(4*sqrt(c)*s...