\(\int \frac {(a x^2+b x^3+c x^4)^{3/2}}{x} \, dx\) [57]

Optimal result

Integrand size = 24, antiderivative size = 260 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x} \, dx=\frac {\left (35 b^4-216 a b^2 c+240 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{3840 c^3}-\frac {b \left (105 b^4-760 a b^2 c+1296 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{7680 c^4 x}-\frac {x \left (b \left (7 b^2+12 a c\right )+6 c \left (7 b^2-20 a c\right ) x\right ) \sqrt {a x^2+b x^3+c x^4}}{960 c^2}+\frac {(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x}+\frac {\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \text {arctanh}\left (\frac {x (b+2 c x)}{2 \sqrt {c} \sqrt {a x^2+b x^3+c x^4}}\right )}{1024 c^{9/2}} \] Output:

1/3840*(240*a^2*c^2-216*a*b^2*c+35*b^4)*(c*x^4+b*x^3+a*x^2)^(1/2)/c^3-1/76 
80*b*(1296*a^2*c^2-760*a*b^2*c+105*b^4)*(c*x^4+b*x^3+a*x^2)^(1/2)/c^4/x-1/ 
960*x*(b*(12*a*c+7*b^2)+6*c*(-20*a*c+7*b^2)*x)*(c*x^4+b*x^3+a*x^2)^(1/2)/c 
^2+1/60*(10*c*x+3*b)*(c*x^4+b*x^3+a*x^2)^(3/2)/c/x+1/1024*(-4*a*c+b^2)^2*( 
-4*a*c+7*b^2)*arctanh(1/2*x*(2*c*x+b)/c^(1/2)/(c*x^4+b*x^3+a*x^2)^(1/2))/c 
^(9/2)
 

Mathematica [A] (verified)

Time = 0.95 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x} \, dx=\frac {x \sqrt {a+x (b+c x)} \left (2 \sqrt {c} \sqrt {a+x (b+c x)} \left (-105 b^5+70 b^4 c x+8 b^3 c \left (95 a-7 c x^2\right )+48 b^2 c^2 x \left (-9 a+c x^2\right )+160 c^3 x \left (3 a^2+14 a c x^2+8 c^2 x^4\right )+16 b c^2 \left (-81 a^2+18 a c x^2+104 c^2 x^4\right )\right )-15 \left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{15360 c^{9/2} \sqrt {x^2 (a+x (b+c x))}} \] Input:

Integrate[(a*x^2 + b*x^3 + c*x^4)^(3/2)/x,x]
 

Output:

(x*Sqrt[a + x*(b + c*x)]*(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-105*b^5 + 70*b 
^4*c*x + 8*b^3*c*(95*a - 7*c*x^2) + 48*b^2*c^2*x*(-9*a + c*x^2) + 160*c^3* 
x*(3*a^2 + 14*a*c*x^2 + 8*c^2*x^4) + 16*b*c^2*(-81*a^2 + 18*a*c*x^2 + 104* 
c^2*x^4)) - 15*(b^2 - 4*a*c)^2*(7*b^2 - 4*a*c)*Log[b + 2*c*x - 2*Sqrt[c]*S 
qrt[a + x*(b + c*x)]]))/(15360*c^(9/2)*Sqrt[x^2*(a + x*(b + c*x))])
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.19, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {1966, 27, 1981, 27, 1996, 27, 1996, 27, 1961, 1092, 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.

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

\(\Big \downarrow \) 1966

\(\displaystyle \frac {\int -\frac {1}{2} \left (4 a b+\left (7 b^2-20 a c\right ) x\right ) \sqrt {c x^4+b x^3+a x^2}dx}{20 c}+\frac {(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x}-\frac {\int \left (4 a b+\left (7 b^2-20 a c\right ) x\right ) \sqrt {c x^4+b x^3+a x^2}dx}{40 c}\)

\(\Big \downarrow \) 1981

\(\displaystyle \frac {(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x}-\frac {\frac {\int -\frac {x^2 \left (4 a b \left (7 b^2-36 a c\right )+\left (35 b^4-216 a c b^2+240 a^2 c^2\right ) x\right )}{2 \sqrt {c x^4+b x^3+a x^2}}dx}{24 c}+\frac {x \sqrt {a x^2+b x^3+c x^4} \left (6 c x \left (7 b^2-20 a c\right )+b \left (12 a c+7 b^2\right )\right )}{24 c}}{40 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x}-\frac {\frac {x \left (6 c x \left (7 b^2-20 a c\right )+b \left (12 a c+7 b^2\right )\right ) \sqrt {a x^2+b x^3+c x^4}}{24 c}-\frac {\int \frac {x^2 \left (4 a b \left (7 b^2-36 a c\right )+\left (35 b^4-216 a c b^2+240 a^2 c^2\right ) x\right )}{\sqrt {c x^4+b x^3+a x^2}}dx}{48 c}}{40 c}\)

\(\Big \downarrow \) 1996

\(\displaystyle \frac {(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x}-\frac {\frac {x \left (6 c x \left (7 b^2-20 a c\right )+b \left (12 a c+7 b^2\right )\right ) \sqrt {a x^2+b x^3+c x^4}}{24 c}-\frac {\frac {\left (240 a^2 c^2-216 a b^2 c+35 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {\int \frac {x \left (2 a \left (35 b^4-216 a c b^2+240 a^2 c^2\right )+b \left (105 b^4-760 a c b^2+1296 a^2 c^2\right ) x\right )}{2 \sqrt {c x^4+b x^3+a x^2}}dx}{2 c}}{48 c}}{40 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x}-\frac {\frac {x \left (6 c x \left (7 b^2-20 a c\right )+b \left (12 a c+7 b^2\right )\right ) \sqrt {a x^2+b x^3+c x^4}}{24 c}-\frac {\frac {\left (240 a^2 c^2-216 a b^2 c+35 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {\int \frac {x \left (2 a \left (35 b^4-216 a c b^2+240 a^2 c^2\right )+b \left (105 b^4-760 a c b^2+1296 a^2 c^2\right ) x\right )}{\sqrt {c x^4+b x^3+a x^2}}dx}{4 c}}{48 c}}{40 c}\)

\(\Big \downarrow \) 1996

\(\displaystyle \frac {(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x}-\frac {\frac {x \left (6 c x \left (7 b^2-20 a c\right )+b \left (12 a c+7 b^2\right )\right ) \sqrt {a x^2+b x^3+c x^4}}{24 c}-\frac {\frac {\left (240 a^2 c^2-216 a b^2 c+35 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {\frac {b \left (1296 a^2 c^2-760 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{c x}-\frac {\int \frac {15 \left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) x}{2 \sqrt {c x^4+b x^3+a x^2}}dx}{c}}{4 c}}{48 c}}{40 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x}-\frac {\frac {x \left (6 c x \left (7 b^2-20 a c\right )+b \left (12 a c+7 b^2\right )\right ) \sqrt {a x^2+b x^3+c x^4}}{24 c}-\frac {\frac {\left (240 a^2 c^2-216 a b^2 c+35 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {\frac {b \left (1296 a^2 c^2-760 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{c x}-\frac {15 \left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \int \frac {x}{\sqrt {c x^4+b x^3+a x^2}}dx}{2 c}}{4 c}}{48 c}}{40 c}\)

\(\Big \downarrow \) 1961

\(\displaystyle \frac {(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x}-\frac {\frac {x \left (6 c x \left (7 b^2-20 a c\right )+b \left (12 a c+7 b^2\right )\right ) \sqrt {a x^2+b x^3+c x^4}}{24 c}-\frac {\frac {\left (240 a^2 c^2-216 a b^2 c+35 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {\frac {b \left (1296 a^2 c^2-760 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{c x}-\frac {15 x \left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \sqrt {a+b x+c x^2} \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{2 c \sqrt {a x^2+b x^3+c x^4}}}{4 c}}{48 c}}{40 c}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x}-\frac {\frac {x \left (6 c x \left (7 b^2-20 a c\right )+b \left (12 a c+7 b^2\right )\right ) \sqrt {a x^2+b x^3+c x^4}}{24 c}-\frac {\frac {\left (240 a^2 c^2-216 a b^2 c+35 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {\frac {b \left (1296 a^2 c^2-760 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{c x}-\frac {15 x \left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \sqrt {a+b x+c x^2} \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{c \sqrt {a x^2+b x^3+c x^4}}}{4 c}}{48 c}}{40 c}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x}-\frac {\frac {x \left (6 c x \left (7 b^2-20 a c\right )+b \left (12 a c+7 b^2\right )\right ) \sqrt {a x^2+b x^3+c x^4}}{24 c}-\frac {\frac {\left (240 a^2 c^2-216 a b^2 c+35 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {\frac {b \left (1296 a^2 c^2-760 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{c x}-\frac {15 x \left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{3/2} \sqrt {a x^2+b x^3+c x^4}}}{4 c}}{48 c}}{40 c}\)

Input:

Int[(a*x^2 + b*x^3 + c*x^4)^(3/2)/x,x]
 

Output:

((3*b + 10*c*x)*(a*x^2 + b*x^3 + c*x^4)^(3/2))/(60*c*x) - ((x*(b*(7*b^2 + 
12*a*c) + 6*c*(7*b^2 - 20*a*c)*x)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(24*c) - (( 
(35*b^4 - 216*a*b^2*c + 240*a^2*c^2)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(2*c) - 
((b*(105*b^4 - 760*a*b^2*c + 1296*a^2*c^2)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(c 
*x) - (15*(b^2 - 4*a*c)^2*(7*b^2 - 4*a*c)*x*Sqrt[a + b*x + c*x^2]*ArcTanh[ 
(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*c^(3/2)*Sqrt[a*x^2 + b* 
x^3 + c*x^4]))/(4*c))/(48*c))/(40*c)
 

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 1961

Int[(x_)^(m_.)/Sqrt[(b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.)] 
, x_Symbol] :> Simp[x^(q/2)*(Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))]/Sqrt[a 
*x^q + b*x^n + c*x^(2*n - q)])   Int[x^(m - q/2)/Sqrt[a + b*x^(n - q) + c*x 
^(2*(n - q))], x], x] /; FreeQ[{a, b, c, m, n, q}, x] && EqQ[r, 2*n - q] && 
 PosQ[n - q] && ((EqQ[m, 1] && EqQ[n, 3] && EqQ[q, 2]) || ((EqQ[m + 1/2] || 
 EqQ[m, 3/2] || EqQ[m, 1/2] || EqQ[m, 5/2]) && EqQ[n, 3] && EqQ[q, 1]))
 

rule 1966

Int[(x_)^(m_.)*((b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(p_ 
), x_Symbol] :> Simp[x^(m - n + q + 1)*(b*(n - q)*p + c*(m + p*q + (n - q)* 
(2*p - 1) + 1)*x^(n - q))*((a*x^q + b*x^n + c*x^(2*n - q))^p/(c*(m + p*(2*n 
 - q) + 1)*(m + p*q + (n - q)*(2*p - 1) + 1))), x] + Simp[(n - q)*(p/(c*(m 
+ p*(2*n - q) + 1)*(m + p*q + (n - q)*(2*p - 1) + 1)))   Int[x^(m - (n - 2* 
q))*Simp[(-a)*b*(m + p*q - n + q + 1) + (2*a*c*(m + p*q + (n - q)*(2*p - 1) 
 + 1) - b^2*(m + p*q + (n - q)*(p - 1) + 1))*x^(n - q), x]*(a*x^q + b*x^n + 
 c*x^(2*n - q))^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && EqQ[r, 2*n - q] & 
& PosQ[n - q] &&  !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[ 
p, 0] && RationalQ[m, q] && GtQ[m + p*q + 1, n - q] && NeQ[m + p*(2*n - q) 
+ 1, 0] && NeQ[m + p*q + (n - q)*(2*p - 1) + 1, 0]
 

rule 1981

Int[((c_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.))^(p_)*((A_) + ( 
B_.)*(x_)^(r_.)), x_Symbol] :> Simp[x*(b*B*(n - q)*p + A*c*(p*q + (n - q)*( 
2*p + 1) + 1) + B*c*(p*(2*n - q) + 1)*x^(n - q))*((a*x^q + b*x^n + c*x^(2*n 
 - q))^p/(c*(p*(2*n - q) + 1)*(p*q + (n - q)*(2*p + 1) + 1))), x] + Simp[(n 
 - q)*(p/(c*(p*(2*n - q) + 1)*(p*q + (n - q)*(2*p + 1) + 1)))   Int[x^q*(2* 
a*A*c*(p*q + (n - q)*(2*p + 1) + 1) - a*b*B*(p*q + 1) + (2*a*B*c*(p*(2*n - 
q) + 1) + A*b*c*(p*q + (n - q)*(2*p + 1) + 1) - b^2*B*(p*q + (n - q)*p + 1) 
)*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p - 1), x], x] /; FreeQ[{a, b 
, c, A, B, n, q}, x] && EqQ[r, n - q] && EqQ[j, 2*n - q] &&  !IntegerQ[p] & 
& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && NeQ[p*(2*n - q) + 1, 0] && NeQ[p*q + 
(n - q)*(2*p + 1) + 1, 0]
 

rule 1996

Int[(x_)^(m_.)*((c_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.))^(p_ 
.)*((A_) + (B_.)*(x_)^(r_.)), x_Symbol] :> Simp[B*x^(m - n + 1)*((a*x^q + b 
*x^n + c*x^(2*n - q))^(p + 1)/(c*(m + p*q + (n - q)*(2*p + 1) + 1))), x] - 
Simp[1/(c*(m + p*q + (n - q)*(2*p + 1) + 1))   Int[x^(m - n + q)*Simp[a*B*( 
m + p*q - n + q + 1) + (b*B*(m + p*q + (n - q)*p + 1) - A*c*(m + p*q + (n - 
 q)*(2*p + 1) + 1))*x^(n - q), x]*(a*x^q + b*x^n + c*x^(2*n - q))^p, x], x] 
 /; FreeQ[{a, b, c, A, B}, x] && EqQ[r, n - q] && EqQ[j, 2*n - q] &&  !Inte 
gerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GeQ[p, -1] && LtQ[p, 0] && 
RationalQ[m, q] && GeQ[m + p*q, n - q - 1] && NeQ[m + p*q + (n - q)*(2*p + 
1) + 1, 0]
 

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.87

method	result	size
risch	\(-\frac {\left (-1280 c^{5} x^{5}-1664 b \,c^{4} x^{4}-2240 a \,c^{4} x^{3}-48 b^{2} c^{3} x^{3}-288 a b \,c^{3} x^{2}+56 b^{3} c^{2} x^{2}-480 a^{2} c^{3} x +432 a \,b^{2} c^{2} x -70 b^{4} c x +1296 a^{2} b \,c^{2}-760 a \,b^{3} c +105 b^{5}\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{7680 c^{4} x}-\frac {\left (64 a^{3} c^{3}-144 a^{2} b^{2} c^{2}+60 a \,b^{4} c -7 b^{6}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{1024 c^{\frac {9}{2}} x \sqrt {c \,x^{2}+b x +a}}\)	\(225\)
default	\(\frac {\left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (2560 x \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} c^{\frac {9}{2}}-640 c^{\frac {9}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a x -960 c^{\frac {9}{2}} \sqrt {c \,x^{2}+b x +a}\, a^{2} x -1792 c^{\frac {7}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b +1120 c^{\frac {7}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{2} x -320 c^{\frac {7}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a b +1920 c^{\frac {7}{2}} \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} x -480 c^{\frac {7}{2}} \sqrt {c \,x^{2}+b x +a}\, a^{2} b +560 c^{\frac {5}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{3}-420 c^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}\, b^{4} x +960 c^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}\, a \,b^{3}-210 c^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}\, b^{5}-960 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) a^{3} c^{4}+2160 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) a^{2} b^{2} c^{3}-900 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) a \,b^{4} c^{2}+105 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) b^{6} c \right )}{15360 x^{3} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{\frac {11}{2}}}\)	\(431\)

Input:

int((c*x^4+b*x^3+a*x^2)^(3/2)/x,x,method=_RETURNVERBOSE)
 

Output:

-1/7680*(-1280*c^5*x^5-1664*b*c^4*x^4-2240*a*c^4*x^3-48*b^2*c^3*x^3-288*a* 
b*c^3*x^2+56*b^3*c^2*x^2-480*a^2*c^3*x+432*a*b^2*c^2*x-70*b^4*c*x+1296*a^2 
*b*c^2-760*a*b^3*c+105*b^5)/c^4*(x^2*(c*x^2+b*x+a))^(1/2)/x-1/1024*(64*a^3 
*c^3-144*a^2*b^2*c^2+60*a*b^4*c-7*b^6)/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x 
^2+b*x+a)^(1/2))*(x^2*(c*x^2+b*x+a))^(1/2)/x/(c*x^2+b*x+a)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.82 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x} \, dx=\left [-\frac {15 \, {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {c} x \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) - 4 \, {\left (1280 \, c^{6} x^{5} + 1664 \, b c^{5} x^{4} - 105 \, b^{5} c + 760 \, a b^{3} c^{2} - 1296 \, a^{2} b c^{3} + 16 \, {\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} x^{3} - 8 \, {\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} x^{2} + 2 \, {\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{30720 \, c^{5} x}, -\frac {15 \, {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-c} x \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) - 2 \, {\left (1280 \, c^{6} x^{5} + 1664 \, b c^{5} x^{4} - 105 \, b^{5} c + 760 \, a b^{3} c^{2} - 1296 \, a^{2} b c^{3} + 16 \, {\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} x^{3} - 8 \, {\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} x^{2} + 2 \, {\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{15360 \, c^{5} x}\right ] \] Input:

integrate((c*x^4+b*x^3+a*x^2)^(3/2)/x,x, algorithm="fricas")
 

Output:

[-1/30720*(15*(7*b^6 - 60*a*b^4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(c)* 
x*log(-(8*c^2*x^3 + 8*b*c*x^2 - 4*sqrt(c*x^4 + b*x^3 + a*x^2)*(2*c*x + b)* 
sqrt(c) + (b^2 + 4*a*c)*x)/x) - 4*(1280*c^6*x^5 + 1664*b*c^5*x^4 - 105*b^5 
*c + 760*a*b^3*c^2 - 1296*a^2*b*c^3 + 16*(3*b^2*c^4 + 140*a*c^5)*x^3 - 8*( 
7*b^3*c^3 - 36*a*b*c^4)*x^2 + 2*(35*b^4*c^2 - 216*a*b^2*c^3 + 240*a^2*c^4) 
*x)*sqrt(c*x^4 + b*x^3 + a*x^2))/(c^5*x), -1/15360*(15*(7*b^6 - 60*a*b^4*c 
 + 144*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(-c)*x*arctan(1/2*sqrt(c*x^4 + b*x^3 
+ a*x^2)*(2*c*x + b)*sqrt(-c)/(c^2*x^3 + b*c*x^2 + a*c*x)) - 2*(1280*c^6*x 
^5 + 1664*b*c^5*x^4 - 105*b^5*c + 760*a*b^3*c^2 - 1296*a^2*b*c^3 + 16*(3*b 
^2*c^4 + 140*a*c^5)*x^3 - 8*(7*b^3*c^3 - 36*a*b*c^4)*x^2 + 2*(35*b^4*c^2 - 
 216*a*b^2*c^3 + 240*a^2*c^4)*x)*sqrt(c*x^4 + b*x^3 + a*x^2))/(c^5*x)]
 

Sympy [F]

\[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x} \, dx=\int \frac {\left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac {3}{2}}}{x}\, dx \] Input:

integrate((c*x**4+b*x**3+a*x**2)**(3/2)/x,x)
 

Output:

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

Maxima [F]

\[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x} \, dx=\int { \frac {{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}{x} \,d x } \] Input:

integrate((c*x^4+b*x^3+a*x^2)^(3/2)/x,x, algorithm="maxima")
 

Output:

integrate((c*x^4 + b*x^3 + a*x^2)^(3/2)/x, x)
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.37 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x} \, dx=\frac {1}{7680} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, c x \mathrm {sgn}\left (x\right ) + 13 \, b \mathrm {sgn}\left (x\right )\right )} x + \frac {3 \, b^{2} c^{4} \mathrm {sgn}\left (x\right ) + 140 \, a c^{5} \mathrm {sgn}\left (x\right )}{c^{5}}\right )} x - \frac {7 \, b^{3} c^{3} \mathrm {sgn}\left (x\right ) - 36 \, a b c^{4} \mathrm {sgn}\left (x\right )}{c^{5}}\right )} x + \frac {35 \, b^{4} c^{2} \mathrm {sgn}\left (x\right ) - 216 \, a b^{2} c^{3} \mathrm {sgn}\left (x\right ) + 240 \, a^{2} c^{4} \mathrm {sgn}\left (x\right )}{c^{5}}\right )} x - \frac {105 \, b^{5} c \mathrm {sgn}\left (x\right ) - 760 \, a b^{3} c^{2} \mathrm {sgn}\left (x\right ) + 1296 \, a^{2} b c^{3} \mathrm {sgn}\left (x\right )}{c^{5}}\right )} - \frac {{\left (7 \, b^{6} \mathrm {sgn}\left (x\right ) - 60 \, a b^{4} c \mathrm {sgn}\left (x\right ) + 144 \, a^{2} b^{2} c^{2} \mathrm {sgn}\left (x\right ) - 64 \, a^{3} c^{3} \mathrm {sgn}\left (x\right )\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{1024 \, c^{\frac {9}{2}}} + \frac {{\left (105 \, b^{6} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 900 \, a b^{4} c \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 2160 \, a^{2} b^{2} c^{2} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 960 \, a^{3} c^{3} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 210 \, \sqrt {a} b^{5} \sqrt {c} - 1520 \, a^{\frac {3}{2}} b^{3} c^{\frac {3}{2}} + 2592 \, a^{\frac {5}{2}} b c^{\frac {5}{2}}\right )} \mathrm {sgn}\left (x\right )}{15360 \, c^{\frac {9}{2}}} \] Input:

integrate((c*x^4+b*x^3+a*x^2)^(3/2)/x,x, algorithm="giac")
 

Output:

1/7680*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*c*x*sgn(x) + 13*b*sgn(x))*x + 
 (3*b^2*c^4*sgn(x) + 140*a*c^5*sgn(x))/c^5)*x - (7*b^3*c^3*sgn(x) - 36*a*b 
*c^4*sgn(x))/c^5)*x + (35*b^4*c^2*sgn(x) - 216*a*b^2*c^3*sgn(x) + 240*a^2* 
c^4*sgn(x))/c^5)*x - (105*b^5*c*sgn(x) - 760*a*b^3*c^2*sgn(x) + 1296*a^2*b 
*c^3*sgn(x))/c^5) - 1/1024*(7*b^6*sgn(x) - 60*a*b^4*c*sgn(x) + 144*a^2*b^2 
*c^2*sgn(x) - 64*a^3*c^3*sgn(x))*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + 
 a))*sqrt(c) + b))/c^(9/2) + 1/15360*(105*b^6*log(abs(b - 2*sqrt(a)*sqrt(c 
))) - 900*a*b^4*c*log(abs(b - 2*sqrt(a)*sqrt(c))) + 2160*a^2*b^2*c^2*log(a 
bs(b - 2*sqrt(a)*sqrt(c))) - 960*a^3*c^3*log(abs(b - 2*sqrt(a)*sqrt(c))) + 
 210*sqrt(a)*b^5*sqrt(c) - 1520*a^(3/2)*b^3*c^(3/2) + 2592*a^(5/2)*b*c^(5/ 
2))*sgn(x)/c^(9/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x} \, dx=\int \frac {{\left (c\,x^4+b\,x^3+a\,x^2\right )}^{3/2}}{x} \,d x \] Input:

int((a*x^2 + b*x^3 + c*x^4)^(3/2)/x,x)
 

Output:

int((a*x^2 + b*x^3 + c*x^4)^(3/2)/x, x)
 

Reduce [F]

\[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x} \, dx=\int \frac {\left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}}}{x}d x \] Input:

int((c*x^4+b*x^3+a*x^2)^(3/2)/x,x)
 

Output:

int((c*x^4+b*x^3+a*x^2)^(3/2)/x,x)