\(\int x (a x^2+b x^3+c x^4)^{3/2} \, dx\) [38]

Optimal result

Integrand size = 22, antiderivative size = 422 \[ \int x \left (a x^2+b x^3+c x^4\right )^{3/2} \, dx=\frac {\left (1155 b^6-8988 a b^4 c+18896 a^2 b^2 c^2-6720 a^3 c^3\right ) \sqrt {a x^2+b x^3+c x^4}}{286720 c^5}-\frac {b \left (3465 b^6-30660 a b^4 c+81648 a^2 b^2 c^2-58816 a^3 c^3\right ) \sqrt {a x^2+b x^3+c x^4}}{573440 c^6 x}-\frac {b \left (231 b^4-1560 a b^2 c+2416 a^2 c^2\right ) x \sqrt {a x^2+b x^3+c x^4}}{71680 c^4}+\frac {\left (99 b^4-568 a b^2 c+560 a^2 c^2\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}{35840 c^3}-\frac {x^3 \left (b \left (11 b^2+68 a c\right )+10 c \left (11 b^2-28 a c\right ) x\right ) \sqrt {a x^2+b x^3+c x^4}}{4480 c^2}+\frac {x (3 b+14 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{112 c}+\frac {3 \left (b^2-4 a c\right )^2 \left (33 b^4-72 a b^2 c+16 a^2 c^2\right ) x \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{13/2} \sqrt {a x^2+b x^3+c x^4}} \]

[Out]

1/112*x*(14*c*x+3*b)*(c*x^4+b*x^3+a*x^2)^(3/2)/c+3/32768*(-4*a*c+b^2)^2*(16*a^2*c^2-72*a*b^2*c+33*b^4)*x*arcta
nh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))*(c*x^2+b*x+a)^(1/2)/c^(13/2)/(c*x^4+b*x^3+a*x^2)^(1/2)+1/286720*
(-6720*a^3*c^3+18896*a^2*b^2*c^2-8988*a*b^4*c+1155*b^6)*(c*x^4+b*x^3+a*x^2)^(1/2)/c^5-1/573440*b*(-58816*a^3*c
^3+81648*a^2*b^2*c^2-30660*a*b^4*c+3465*b^6)*(c*x^4+b*x^3+a*x^2)^(1/2)/c^6/x-1/71680*b*(2416*a^2*c^2-1560*a*b^
2*c+231*b^4)*x*(c*x^4+b*x^3+a*x^2)^(1/2)/c^4+1/35840*(560*a^2*c^2-568*a*b^2*c+99*b^4)*x^2*(c*x^4+b*x^3+a*x^2)^
(1/2)/c^3-1/4480*x^3*(b*(68*a*c+11*b^2)+10*c*(-28*a*c+11*b^2)*x)*(c*x^4+b*x^3+a*x^2)^(1/2)/c^2

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1933, 1959, 1963, 12, 1928, 635, 212} \[ \int x \left (a x^2+b x^3+c x^4\right )^{3/2} \, dx=\frac {3 x \left (b^2-4 a c\right )^2 \left (16 a^2 c^2-72 a b^2 c+33 b^4\right ) \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{13/2} \sqrt {a x^2+b x^3+c x^4}}-\frac {b x \left (2416 a^2 c^2-1560 a b^2 c+231 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{71680 c^4}+\frac {x^2 \left (560 a^2 c^2-568 a b^2 c+99 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{35840 c^3}-\frac {b \left (-58816 a^3 c^3+81648 a^2 b^2 c^2-30660 a b^4 c+3465 b^6\right ) \sqrt {a x^2+b x^3+c x^4}}{573440 c^6 x}+\frac {\left (-6720 a^3 c^3+18896 a^2 b^2 c^2-8988 a b^4 c+1155 b^6\right ) \sqrt {a x^2+b x^3+c x^4}}{286720 c^5}-\frac {x^3 \left (10 c x \left (11 b^2-28 a c\right )+b \left (68 a c+11 b^2\right )\right ) \sqrt {a x^2+b x^3+c x^4}}{4480 c^2}+\frac {x (3 b+14 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{112 c} \]

[In]

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

[Out]

((1155*b^6 - 8988*a*b^4*c + 18896*a^2*b^2*c^2 - 6720*a^3*c^3)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(286720*c^5) - (b*(
3465*b^6 - 30660*a*b^4*c + 81648*a^2*b^2*c^2 - 58816*a^3*c^3)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(573440*c^6*x) - (b
*(231*b^4 - 1560*a*b^2*c + 2416*a^2*c^2)*x*Sqrt[a*x^2 + b*x^3 + c*x^4])/(71680*c^4) + ((99*b^4 - 568*a*b^2*c +
560*a^2*c^2)*x^2*Sqrt[a*x^2 + b*x^3 + c*x^4])/(35840*c^3) - (x^3*(b*(11*b^2 + 68*a*c) + 10*c*(11*b^2 - 28*a*c
)*x)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(4480*c^2) + (x*(3*b + 14*c*x)*(a*x^2 + b*x^3 + c*x^4)^(3/2))/(112*c) + (3*(
b^2 - 4*a*c)^2*(33*b^4 - 72*a*b^2*c + 16*a^2*c^2)*x*Sqrt[a + b*x + c*x^2]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[
a + b*x + c*x^2])])/(32768*c^(13/2)*Sqrt[a*x^2 + b*x^3 + c*x^4])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1928

Int[(x_)^(m_.)/Sqrt[(b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.)], x_Symbol] :> Dist[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 1933

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] + Dist[(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 1959

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

Rule 1963

Int[(x_)^(m_.)*((c_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.))^(p_.)*((A_) + (B_.)*(x_)^(r_.)), x_Sym
bol] :> 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] - Dist[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] &&  !IntegerQ[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]

Rubi steps \begin{align*} \text {integral}& = \frac {x (3 b+14 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{112 c}+\frac {3 \int x^2 \left (-4 a b-\frac {1}{2} \left (11 b^2-28 a c\right ) x\right ) \sqrt {a x^2+b x^3+c x^4} \, dx}{112 c} \\ & = -\frac {x^3 \left (b \left (11 b^2+68 a c\right )+10 c \left (11 b^2-28 a c\right ) x\right ) \sqrt {a x^2+b x^3+c x^4}}{4480 c^2}+\frac {x (3 b+14 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{112 c}+\frac {\int \frac {x^4 \left (2 a b \left (11 b^2-52 a c\right )+\frac {1}{4} \left (99 b^4-568 a b^2 c+560 a^2 c^2\right ) x\right )}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{2240 c^2} \\ & = \frac {\left (99 b^4-568 a b^2 c+560 a^2 c^2\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}{35840 c^3}-\frac {x^3 \left (b \left (11 b^2+68 a c\right )+10 c \left (11 b^2-28 a c\right ) x\right ) \sqrt {a x^2+b x^3+c x^4}}{4480 c^2}+\frac {x (3 b+14 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{112 c}-\frac {\int \frac {x^3 \left (\frac {3}{4} a \left (99 b^4-568 a b^2 c+560 a^2 c^2\right )+\frac {3}{8} b \left (231 b^4-1560 a b^2 c+2416 a^2 c^2\right ) x\right )}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{8960 c^3} \\ & = -\frac {b \left (231 b^4-1560 a b^2 c+2416 a^2 c^2\right ) x \sqrt {a x^2+b x^3+c x^4}}{71680 c^4}+\frac {\left (99 b^4-568 a b^2 c+560 a^2 c^2\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}{35840 c^3}-\frac {x^3 \left (b \left (11 b^2+68 a c\right )+10 c \left (11 b^2-28 a c\right ) x\right ) \sqrt {a x^2+b x^3+c x^4}}{4480 c^2}+\frac {x (3 b+14 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{112 c}+\frac {\int \frac {x^2 \left (\frac {3}{4} a b \left (231 b^4-1560 a b^2 c+2416 a^2 c^2\right )+\frac {3}{16} \left (1155 b^6-8988 a b^4 c+18896 a^2 b^2 c^2-6720 a^3 c^3\right ) x\right )}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{26880 c^4} \\ & = \frac {\left (1155 b^6-8988 a b^4 c+18896 a^2 b^2 c^2-6720 a^3 c^3\right ) \sqrt {a x^2+b x^3+c x^4}}{286720 c^5}-\frac {b \left (231 b^4-1560 a b^2 c+2416 a^2 c^2\right ) x \sqrt {a x^2+b x^3+c x^4}}{71680 c^4}+\frac {\left (99 b^4-568 a b^2 c+560 a^2 c^2\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}{35840 c^3}-\frac {x^3 \left (b \left (11 b^2+68 a c\right )+10 c \left (11 b^2-28 a c\right ) x\right ) \sqrt {a x^2+b x^3+c x^4}}{4480 c^2}+\frac {x (3 b+14 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{112 c}-\frac {\int \frac {x \left (\frac {3}{16} a \left (1155 b^6-8988 a b^4 c+18896 a^2 b^2 c^2-6720 a^3 c^3\right )+\frac {3}{32} b \left (3465 b^6-30660 a b^4 c+81648 a^2 b^2 c^2-58816 a^3 c^3\right ) x\right )}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{53760 c^5} \\ & = \frac {\left (1155 b^6-8988 a b^4 c+18896 a^2 b^2 c^2-6720 a^3 c^3\right ) \sqrt {a x^2+b x^3+c x^4}}{286720 c^5}-\frac {b \left (3465 b^6-30660 a b^4 c+81648 a^2 b^2 c^2-58816 a^3 c^3\right ) \sqrt {a x^2+b x^3+c x^4}}{573440 c^6 x}-\frac {b \left (231 b^4-1560 a b^2 c+2416 a^2 c^2\right ) x \sqrt {a x^2+b x^3+c x^4}}{71680 c^4}+\frac {\left (99 b^4-568 a b^2 c+560 a^2 c^2\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}{35840 c^3}-\frac {x^3 \left (b \left (11 b^2+68 a c\right )+10 c \left (11 b^2-28 a c\right ) x\right ) \sqrt {a x^2+b x^3+c x^4}}{4480 c^2}+\frac {x (3 b+14 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{112 c}+\frac {\int \frac {315 \left (b^2-4 a c\right )^2 \left (33 b^4-72 a b^2 c+16 a^2 c^2\right ) x}{64 \sqrt {a x^2+b x^3+c x^4}} \, dx}{53760 c^6} \\ & = \frac {\left (1155 b^6-8988 a b^4 c+18896 a^2 b^2 c^2-6720 a^3 c^3\right ) \sqrt {a x^2+b x^3+c x^4}}{286720 c^5}-\frac {b \left (3465 b^6-30660 a b^4 c+81648 a^2 b^2 c^2-58816 a^3 c^3\right ) \sqrt {a x^2+b x^3+c x^4}}{573440 c^6 x}-\frac {b \left (231 b^4-1560 a b^2 c+2416 a^2 c^2\right ) x \sqrt {a x^2+b x^3+c x^4}}{71680 c^4}+\frac {\left (99 b^4-568 a b^2 c+560 a^2 c^2\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}{35840 c^3}-\frac {x^3 \left (b \left (11 b^2+68 a c\right )+10 c \left (11 b^2-28 a c\right ) x\right ) \sqrt {a x^2+b x^3+c x^4}}{4480 c^2}+\frac {x (3 b+14 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{112 c}+\frac {\left (3 \left (b^2-4 a c\right )^2 \left (33 b^4-72 a b^2 c+16 a^2 c^2\right )\right ) \int \frac {x}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{32768 c^6} \\ & = \frac {\left (1155 b^6-8988 a b^4 c+18896 a^2 b^2 c^2-6720 a^3 c^3\right ) \sqrt {a x^2+b x^3+c x^4}}{286720 c^5}-\frac {b \left (3465 b^6-30660 a b^4 c+81648 a^2 b^2 c^2-58816 a^3 c^3\right ) \sqrt {a x^2+b x^3+c x^4}}{573440 c^6 x}-\frac {b \left (231 b^4-1560 a b^2 c+2416 a^2 c^2\right ) x \sqrt {a x^2+b x^3+c x^4}}{71680 c^4}+\frac {\left (99 b^4-568 a b^2 c+560 a^2 c^2\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}{35840 c^3}-\frac {x^3 \left (b \left (11 b^2+68 a c\right )+10 c \left (11 b^2-28 a c\right ) x\right ) \sqrt {a x^2+b x^3+c x^4}}{4480 c^2}+\frac {x (3 b+14 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{112 c}+\frac {\left (3 \left (b^2-4 a c\right )^2 \left (33 b^4-72 a b^2 c+16 a^2 c^2\right ) x \sqrt {a+b x+c x^2}\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{32768 c^6 \sqrt {a x^2+b x^3+c x^4}} \\ & = \frac {\left (1155 b^6-8988 a b^4 c+18896 a^2 b^2 c^2-6720 a^3 c^3\right ) \sqrt {a x^2+b x^3+c x^4}}{286720 c^5}-\frac {b \left (3465 b^6-30660 a b^4 c+81648 a^2 b^2 c^2-58816 a^3 c^3\right ) \sqrt {a x^2+b x^3+c x^4}}{573440 c^6 x}-\frac {b \left (231 b^4-1560 a b^2 c+2416 a^2 c^2\right ) x \sqrt {a x^2+b x^3+c x^4}}{71680 c^4}+\frac {\left (99 b^4-568 a b^2 c+560 a^2 c^2\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}{35840 c^3}-\frac {x^3 \left (b \left (11 b^2+68 a c\right )+10 c \left (11 b^2-28 a c\right ) x\right ) \sqrt {a x^2+b x^3+c x^4}}{4480 c^2}+\frac {x (3 b+14 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{112 c}+\frac {\left (3 \left (b^2-4 a c\right )^2 \left (33 b^4-72 a b^2 c+16 a^2 c^2\right ) x \sqrt {a+b x+c x^2}\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{16384 c^6 \sqrt {a x^2+b x^3+c x^4}} \\ & = \frac {\left (1155 b^6-8988 a b^4 c+18896 a^2 b^2 c^2-6720 a^3 c^3\right ) \sqrt {a x^2+b x^3+c x^4}}{286720 c^5}-\frac {b \left (3465 b^6-30660 a b^4 c+81648 a^2 b^2 c^2-58816 a^3 c^3\right ) \sqrt {a x^2+b x^3+c x^4}}{573440 c^6 x}-\frac {b \left (231 b^4-1560 a b^2 c+2416 a^2 c^2\right ) x \sqrt {a x^2+b x^3+c x^4}}{71680 c^4}+\frac {\left (99 b^4-568 a b^2 c+560 a^2 c^2\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}{35840 c^3}-\frac {x^3 \left (b \left (11 b^2+68 a c\right )+10 c \left (11 b^2-28 a c\right ) x\right ) \sqrt {a x^2+b x^3+c x^4}}{4480 c^2}+\frac {x (3 b+14 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{112 c}+\frac {3 \left (b^2-4 a c\right )^2 \left (33 b^4-72 a b^2 c+16 a^2 c^2\right ) x \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{13/2} \sqrt {a x^2+b x^3+c x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.95 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.72 \[ \int x \left (a x^2+b x^3+c x^4\right )^{3/2} \, dx=\frac {x \sqrt {a+x (b+c x)} \left (2 \sqrt {c} \sqrt {a+x (b+c x)} \left (-3465 b^7+2310 b^6 c x+84 b^5 c \left (365 a-22 c x^2\right )+24 b^4 c^2 x \left (-749 a+66 c x^2\right )+32 b^2 c^3 x \left (1181 a^2-284 a c x^2+40 c^2 x^4\right )-16 b^3 c^2 \left (5103 a^2-780 a c x^2+88 c^2 x^4\right )+4480 c^4 x \left (-3 a^3+2 a^2 c x^2+24 a c^2 x^4+16 c^3 x^6\right )+64 b c^3 \left (919 a^3-302 a^2 c x^2+104 a c^2 x^4+1360 c^3 x^6\right )\right )-105 \left (b^2-4 a c\right )^2 \left (33 b^4-72 a b^2 c+16 a^2 c^2\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{1146880 c^{13/2} \sqrt {x^2 (a+x (b+c x))}} \]

[In]

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

[Out]

(x*Sqrt[a + x*(b + c*x)]*(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-3465*b^7 + 2310*b^6*c*x + 84*b^5*c*(365*a - 22*c*x
^2) + 24*b^4*c^2*x*(-749*a + 66*c*x^2) + 32*b^2*c^3*x*(1181*a^2 - 284*a*c*x^2 + 40*c^2*x^4) - 16*b^3*c^2*(5103
*a^2 - 780*a*c*x^2 + 88*c^2*x^4) + 4480*c^4*x*(-3*a^3 + 2*a^2*c*x^2 + 24*a*c^2*x^4 + 16*c^3*x^6) + 64*b*c^3*(9
19*a^3 - 302*a^2*c*x^2 + 104*a*c^2*x^4 + 1360*c^3*x^6)) - 105*(b^2 - 4*a*c)^2*(33*b^4 - 72*a*b^2*c + 16*a^2*c^
2)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]]))/(1146880*c^(13/2)*Sqrt[x^2*(a + x*(b + c*x))])

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.78

method	result	size
risch	\(\frac {\left (71680 c^{7} x^{7}+87040 b \,c^{6} x^{6}+107520 a \,c^{6} x^{5}+1280 b^{2} c^{5} x^{5}+6656 a b \,c^{5} x^{4}-1408 b^{3} c^{4} x^{4}+8960 a^{2} c^{5} x^{3}-9088 a \,b^{2} c^{4} x^{3}+1584 b^{4} c^{3} x^{3}-19328 a^{2} b \,c^{4} x^{2}+12480 a \,b^{3} c^{3} x^{2}-1848 b^{5} c^{2} x^{2}-13440 a^{3} c^{4} x +37792 a^{2} b^{2} c^{3} x -17976 a \,b^{4} c^{2} x +2310 b^{6} c x +58816 b \,c^{3} a^{3}-81648 b^{3} c^{2} a^{2}+30660 b^{5} c a -3465 b^{7}\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{573440 c^{6} x}+\frac {3 \left (256 a^{4} c^{4}-1280 a^{3} b^{2} c^{3}+1120 a^{2} b^{4} c^{2}-336 a \,b^{6} c +33 b^{8}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{32768 c^{\frac {13}{2}} x \sqrt {c \,x^{2}+b x +a}}\)	\(328\)
default	\(\frac {\left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (-80640 c^{\frac {9}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \,b^{2} x -127680 c^{\frac {9}{2}} \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{2} x +85680 c^{\frac {7}{2}} \sqrt {c \,x^{2}+b x +a}\, a \,b^{4} x +143360 x^{3} \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} c^{\frac {13}{2}}-59136 c^{\frac {7}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b^{3}+18480 c^{\frac {5}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{5}-134400 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) a^{3} b^{2} c^{4}+117600 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) a^{2} b^{4} c^{3}-35280 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) a \,b^{6} c^{2}-40320 c^{\frac {7}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \,b^{3}+26880 c^{\frac {11}{2}} \sqrt {c \,x^{2}+b x +a}\, a^{3} x -13860 c^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}\, b^{6} x +13440 c^{\frac {9}{2}} \sqrt {c \,x^{2}+b x +a}\, a^{3} b -63840 c^{\frac {7}{2}} \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{3}+42840 c^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}\, a \,b^{5}+36960 c^{\frac {7}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{4} x +8960 c^{\frac {9}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{2} b -112640 c^{\frac {11}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b \,x^{2}-71680 c^{\frac {11}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a x +84480 c^{\frac {9}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b^{2} x +95232 c^{\frac {9}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a b +17920 c^{\frac {11}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{2} x +3465 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) b^{8} c -6930 c^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}\, b^{7}+26880 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) a^{4} c^{5}\right )}{1146880 x^{3} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{\frac {15}{2}}}\)	\(649\)

[In]

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

[Out]

1/573440*(71680*c^7*x^7+87040*b*c^6*x^6+107520*a*c^6*x^5+1280*b^2*c^5*x^5+6656*a*b*c^5*x^4-1408*b^3*c^4*x^4+89
60*a^2*c^5*x^3-9088*a*b^2*c^4*x^3+1584*b^4*c^3*x^3-19328*a^2*b*c^4*x^2+12480*a*b^3*c^3*x^2-1848*b^5*c^2*x^2-13
440*a^3*c^4*x+37792*a^2*b^2*c^3*x-17976*a*b^4*c^2*x+2310*b^6*c*x+58816*a^3*b*c^3-81648*a^2*b^3*c^2+30660*a*b^5
*c-3465*b^7)/c^6*(x^2*(c*x^2+b*x+a))^(1/2)/x+3/32768*(256*a^4*c^4-1280*a^3*b^2*c^3+1120*a^2*b^4*c^2-336*a*b^6*
c+33*b^8)/c^(13/2)*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)

none

Time = 0.31 (sec) , antiderivative size = 664, normalized size of antiderivative = 1.57 \[ \int x \left (a x^2+b x^3+c x^4\right )^{3/2} \, dx=\left [\frac {105 \, {\left (33 \, b^{8} - 336 \, a b^{6} c + 1120 \, a^{2} b^{4} c^{2} - 1280 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\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 (71680 \, c^{8} x^{7} + 87040 \, b c^{7} x^{6} - 3465 \, b^{7} c + 30660 \, a b^{5} c^{2} - 81648 \, a^{2} b^{3} c^{3} + 58816 \, a^{3} b c^{4} + 1280 \, {\left (b^{2} c^{6} + 84 \, a c^{7}\right )} x^{5} - 128 \, {\left (11 \, b^{3} c^{5} - 52 \, a b c^{6}\right )} x^{4} + 16 \, {\left (99 \, b^{4} c^{4} - 568 \, a b^{2} c^{5} + 560 \, a^{2} c^{6}\right )} x^{3} - 8 \, {\left (231 \, b^{5} c^{3} - 1560 \, a b^{3} c^{4} + 2416 \, a^{2} b c^{5}\right )} x^{2} + 2 \, {\left (1155 \, b^{6} c^{2} - 8988 \, a b^{4} c^{3} + 18896 \, a^{2} b^{2} c^{4} - 6720 \, a^{3} c^{5}\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{2293760 \, c^{7} x}, -\frac {105 \, {\left (33 \, b^{8} - 336 \, a b^{6} c + 1120 \, a^{2} b^{4} c^{2} - 1280 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\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 (71680 \, c^{8} x^{7} + 87040 \, b c^{7} x^{6} - 3465 \, b^{7} c + 30660 \, a b^{5} c^{2} - 81648 \, a^{2} b^{3} c^{3} + 58816 \, a^{3} b c^{4} + 1280 \, {\left (b^{2} c^{6} + 84 \, a c^{7}\right )} x^{5} - 128 \, {\left (11 \, b^{3} c^{5} - 52 \, a b c^{6}\right )} x^{4} + 16 \, {\left (99 \, b^{4} c^{4} - 568 \, a b^{2} c^{5} + 560 \, a^{2} c^{6}\right )} x^{3} - 8 \, {\left (231 \, b^{5} c^{3} - 1560 \, a b^{3} c^{4} + 2416 \, a^{2} b c^{5}\right )} x^{2} + 2 \, {\left (1155 \, b^{6} c^{2} - 8988 \, a b^{4} c^{3} + 18896 \, a^{2} b^{2} c^{4} - 6720 \, a^{3} c^{5}\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{1146880 \, c^{7} x}\right ] \]

[In]

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

[Out]

[1/2293760*(105*(33*b^8 - 336*a*b^6*c + 1120*a^2*b^4*c^2 - 1280*a^3*b^2*c^3 + 256*a^4*c^4)*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*(71680*c^8*x^
7 + 87040*b*c^7*x^6 - 3465*b^7*c + 30660*a*b^5*c^2 - 81648*a^2*b^3*c^3 + 58816*a^3*b*c^4 + 1280*(b^2*c^6 + 84*
a*c^7)*x^5 - 128*(11*b^3*c^5 - 52*a*b*c^6)*x^4 + 16*(99*b^4*c^4 - 568*a*b^2*c^5 + 560*a^2*c^6)*x^3 - 8*(231*b^
5*c^3 - 1560*a*b^3*c^4 + 2416*a^2*b*c^5)*x^2 + 2*(1155*b^6*c^2 - 8988*a*b^4*c^3 + 18896*a^2*b^2*c^4 - 6720*a^3
*c^5)*x)*sqrt(c*x^4 + b*x^3 + a*x^2))/(c^7*x), -1/1146880*(105*(33*b^8 - 336*a*b^6*c + 1120*a^2*b^4*c^2 - 1280
*a^3*b^2*c^3 + 256*a^4*c^4)*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*(71680*c^8*x^7 + 87040*b*c^7*x^6 - 3465*b^7*c + 30660*a*b^5*c^2 - 81648*a^2*b^3*c^3 + 58
816*a^3*b*c^4 + 1280*(b^2*c^6 + 84*a*c^7)*x^5 - 128*(11*b^3*c^5 - 52*a*b*c^6)*x^4 + 16*(99*b^4*c^4 - 568*a*b^2
*c^5 + 560*a^2*c^6)*x^3 - 8*(231*b^5*c^3 - 1560*a*b^3*c^4 + 2416*a^2*b*c^5)*x^2 + 2*(1155*b^6*c^2 - 8988*a*b^4
*c^3 + 18896*a^2*b^2*c^4 - 6720*a^3*c^5)*x)*sqrt(c*x^4 + b*x^3 + a*x^2))/(c^7*x)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.59 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.21 \[ \int x \left (a x^2+b x^3+c x^4\right )^{3/2} \, dx=\frac {1}{573440} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (4 \, {\left (14 \, c x \mathrm {sgn}\left (x\right ) + 17 \, b \mathrm {sgn}\left (x\right )\right )} x + \frac {b^{2} c^{6} \mathrm {sgn}\left (x\right ) + 84 \, a c^{7} \mathrm {sgn}\left (x\right )}{c^{7}}\right )} x - \frac {11 \, b^{3} c^{5} \mathrm {sgn}\left (x\right ) - 52 \, a b c^{6} \mathrm {sgn}\left (x\right )}{c^{7}}\right )} x + \frac {99 \, b^{4} c^{4} \mathrm {sgn}\left (x\right ) - 568 \, a b^{2} c^{5} \mathrm {sgn}\left (x\right ) + 560 \, a^{2} c^{6} \mathrm {sgn}\left (x\right )}{c^{7}}\right )} x - \frac {231 \, b^{5} c^{3} \mathrm {sgn}\left (x\right ) - 1560 \, a b^{3} c^{4} \mathrm {sgn}\left (x\right ) + 2416 \, a^{2} b c^{5} \mathrm {sgn}\left (x\right )}{c^{7}}\right )} x + \frac {1155 \, b^{6} c^{2} \mathrm {sgn}\left (x\right ) - 8988 \, a b^{4} c^{3} \mathrm {sgn}\left (x\right ) + 18896 \, a^{2} b^{2} c^{4} \mathrm {sgn}\left (x\right ) - 6720 \, a^{3} c^{5} \mathrm {sgn}\left (x\right )}{c^{7}}\right )} x - \frac {3465 \, b^{7} c \mathrm {sgn}\left (x\right ) - 30660 \, a b^{5} c^{2} \mathrm {sgn}\left (x\right ) + 81648 \, a^{2} b^{3} c^{3} \mathrm {sgn}\left (x\right ) - 58816 \, a^{3} b c^{4} \mathrm {sgn}\left (x\right )}{c^{7}}\right )} - \frac {3 \, {\left (33 \, b^{8} \mathrm {sgn}\left (x\right ) - 336 \, a b^{6} c \mathrm {sgn}\left (x\right ) + 1120 \, a^{2} b^{4} c^{2} \mathrm {sgn}\left (x\right ) - 1280 \, a^{3} b^{2} c^{3} \mathrm {sgn}\left (x\right ) + 256 \, a^{4} c^{4} \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 )}{32768 \, c^{\frac {13}{2}}} + \frac {{\left (3465 \, b^{8} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 35280 \, a b^{6} c \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 117600 \, a^{2} b^{4} c^{2} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 134400 \, a^{3} b^{2} c^{3} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 26880 \, a^{4} c^{4} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 6930 \, \sqrt {a} b^{7} \sqrt {c} - 61320 \, a^{\frac {3}{2}} b^{5} c^{\frac {3}{2}} + 163296 \, a^{\frac {5}{2}} b^{3} c^{\frac {5}{2}} - 117632 \, a^{\frac {7}{2}} b c^{\frac {7}{2}}\right )} \mathrm {sgn}\left (x\right )}{1146880 \, c^{\frac {13}{2}}} \]

[In]

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

[Out]

1/573440*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(4*(14*c*x*sgn(x) + 17*b*sgn(x))*x + (b^2*c^6*sgn(x) + 84*a*c^7
*sgn(x))/c^7)*x - (11*b^3*c^5*sgn(x) - 52*a*b*c^6*sgn(x))/c^7)*x + (99*b^4*c^4*sgn(x) - 568*a*b^2*c^5*sgn(x) +
560*a^2*c^6*sgn(x))/c^7)*x - (231*b^5*c^3*sgn(x) - 1560*a*b^3*c^4*sgn(x) + 2416*a^2*b*c^5*sgn(x))/c^7)*x + (1
155*b^6*c^2*sgn(x) - 8988*a*b^4*c^3*sgn(x) + 18896*a^2*b^2*c^4*sgn(x) - 6720*a^3*c^5*sgn(x))/c^7)*x - (3465*b^
7*c*sgn(x) - 30660*a*b^5*c^2*sgn(x) + 81648*a^2*b^3*c^3*sgn(x) - 58816*a^3*b*c^4*sgn(x))/c^7) - 3/32768*(33*b^
8*sgn(x) - 336*a*b^6*c*sgn(x) + 1120*a^2*b^4*c^2*sgn(x) - 1280*a^3*b^2*c^3*sgn(x) + 256*a^4*c^4*sgn(x))*log(ab
s(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(13/2) + 1/1146880*(3465*b^8*log(abs(b - 2*sqrt(a)*sqr
t(c))) - 35280*a*b^6*c*log(abs(b - 2*sqrt(a)*sqrt(c))) + 117600*a^2*b^4*c^2*log(abs(b - 2*sqrt(a)*sqrt(c))) -
134400*a^3*b^2*c^3*log(abs(b - 2*sqrt(a)*sqrt(c))) + 26880*a^4*c^4*log(abs(b - 2*sqrt(a)*sqrt(c))) + 6930*sqrt
(a)*b^7*sqrt(c) - 61320*a^(3/2)*b^5*c^(3/2) + 163296*a^(5/2)*b^3*c^(5/2) - 117632*a^(7/2)*b*c^(7/2))*sgn(x)/c^
(13/2)

Mupad [F(-1)]

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

[In]

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

[Out]

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