\(\int (a x^2+b x^3+c x^4)^{3/2} \, dx\) [39]

Optimal result

Integrand size = 20, antiderivative size = 364 \[ \int \left (a x^2+b x^3+c x^4\right )^{3/2} \, dx=-\frac {b \left (105 b^4-728 a b^2 c+1168 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{17920 c^4}+\frac {\left (315 b^6-2520 a b^4 c+5488 a^2 b^2 c^2-2048 a^3 c^3\right ) \sqrt {a x^2+b x^3+c x^4}}{35840 c^5 x}+\frac {\left (7 b^2-32 a c\right ) \left (3 b^2-4 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}{4480 c^3}-\frac {b \left (9 b^2-44 a c\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}{2240 c^2}+\frac {x^3 \left (b^2+24 a c+10 b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{280 c}+\frac {1}{7} x \left (a x^2+b x^3+c x^4\right )^{3/2}-\frac {3 b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) x \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{11/2} \sqrt {a x^2+b x^3+c x^4}} \]

[Out]

1/7*x*(c*x^4+b*x^3+a*x^2)^(3/2)-3/2048*b*(-4*a*c+b^2)^2*(-4*a*c+3*b^2)*x*arctanh(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^(11/2)/(c*x^4+b*x^3+a*x^2)^(1/2)-1/17920*b*(1168*a^2*c^2-728*a*b^2*c+105*b
^4)*(c*x^4+b*x^3+a*x^2)^(1/2)/c^4+1/35840*(-2048*a^3*c^3+5488*a^2*b^2*c^2-2520*a*b^4*c+315*b^6)*(c*x^4+b*x^3+a
*x^2)^(1/2)/c^5/x+1/4480*(-32*a*c+7*b^2)*(-4*a*c+3*b^2)*x*(c*x^4+b*x^3+a*x^2)^(1/2)/c^3-1/2240*b*(-44*a*c+9*b^
2)*x^2*(c*x^4+b*x^3+a*x^2)^(1/2)/c^2+1/280*x^3*(10*b*c*x+24*a*c+b^2)*(c*x^4+b*x^3+a*x^2)^(1/2)/c

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1920, 1959, 1963, 12, 1928, 635, 212} \[ \int \left (a x^2+b x^3+c x^4\right )^{3/2} \, dx=-\frac {b \left (1168 a^2 c^2-728 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{17920 c^4}+\frac {\left (-2048 a^3 c^3+5488 a^2 b^2 c^2-2520 a b^4 c+315 b^6\right ) \sqrt {a x^2+b x^3+c x^4}}{35840 c^5 x}-\frac {3 b x \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{11/2} \sqrt {a x^2+b x^3+c x^4}}+\frac {x \left (7 b^2-32 a c\right ) \left (3 b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4480 c^3}-\frac {b x^2 \left (9 b^2-44 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2240 c^2}+\frac {x^3 \left (24 a c+b^2+10 b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{280 c}+\frac {1}{7} x \left (a x^2+b x^3+c x^4\right )^{3/2} \]

[In]

Int[(a*x^2 + b*x^3 + c*x^4)^(3/2),x]

[Out]

-1/17920*(b*(105*b^4 - 728*a*b^2*c + 1168*a^2*c^2)*Sqrt[a*x^2 + b*x^3 + c*x^4])/c^4 + ((315*b^6 - 2520*a*b^4*c
+ 5488*a^2*b^2*c^2 - 2048*a^3*c^3)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(35840*c^5*x) + ((7*b^2 - 32*a*c)*(3*b^2 - 4*
a*c)*x*Sqrt[a*x^2 + b*x^3 + c*x^4])/(4480*c^3) - (b*(9*b^2 - 44*a*c)*x^2*Sqrt[a*x^2 + b*x^3 + c*x^4])/(2240*c^
2) + (x^3*(b^2 + 24*a*c + 10*b*c*x)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(280*c) + (x*(a*x^2 + b*x^3 + c*x^4)^(3/2))/7
- (3*b*(b^2 - 4*a*c)^2*(3*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])])/(2048*c^(11/2)*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 1920

Int[((b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(p_), x_Symbol] :> Simp[x*((a*x^q + b*x^n + c*x^(
2*n - q))^p/(p*(2*n - q) + 1)), x] + Dist[(n - q)*(p/(p*(2*n - q) + 1)), Int[x^q*(2*a + b*x^(n - q))*(a*x^q +
b*x^n + c*x^(2*n - q))^(p - 1), x], x] /; FreeQ[{a, b, c, n, q}, x] && EqQ[r, 2*n - q] && PosQ[n - q] &&  !Int
egerQ[p] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && NeQ[p*(2*n - q) + 1, 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 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 {1}{7} x \left (a x^2+b x^3+c x^4\right )^{3/2}+\frac {3}{14} \int x^2 (2 a+b x) \sqrt {a x^2+b x^3+c x^4} \, dx \\ & = \frac {x^3 \left (b^2+24 a c+10 b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{280 c}+\frac {1}{7} x \left (a x^2+b x^3+c x^4\right )^{3/2}+\frac {\int \frac {x^4 \left (-4 a \left (b^2-6 a c\right )-\frac {1}{2} b \left (9 b^2-44 a c\right ) x\right )}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{280 c} \\ & = -\frac {b \left (9 b^2-44 a c\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}{2240 c^2}+\frac {x^3 \left (b^2+24 a c+10 b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{280 c}+\frac {1}{7} x \left (a x^2+b x^3+c x^4\right )^{3/2}-\frac {\int \frac {x^3 \left (-\frac {3}{2} a b \left (9 b^2-44 a c\right )-\frac {3}{4} \left (7 b^2-32 a c\right ) \left (3 b^2-4 a c\right ) x\right )}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{1120 c^2} \\ & = \frac {\left (7 b^2-32 a c\right ) \left (3 b^2-4 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}{4480 c^3}-\frac {b \left (9 b^2-44 a c\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}{2240 c^2}+\frac {x^3 \left (b^2+24 a c+10 b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{280 c}+\frac {1}{7} x \left (a x^2+b x^3+c x^4\right )^{3/2}+\frac {\int \frac {x^2 \left (-\frac {3}{2} a \left (7 b^2-32 a c\right ) \left (3 b^2-4 a c\right )-\frac {3}{8} b \left (105 b^4-728 a b^2 c+1168 a^2 c^2\right ) x\right )}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{3360 c^3} \\ & = -\frac {b \left (105 b^4-728 a b^2 c+1168 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{17920 c^4}+\frac {\left (7 b^2-32 a c\right ) \left (3 b^2-4 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}{4480 c^3}-\frac {b \left (9 b^2-44 a c\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}{2240 c^2}+\frac {x^3 \left (b^2+24 a c+10 b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{280 c}+\frac {1}{7} x \left (a x^2+b x^3+c x^4\right )^{3/2}-\frac {\int \frac {x \left (-\frac {3}{8} a b \left (105 b^4-728 a b^2 c+1168 a^2 c^2\right )-\frac {3}{16} \left (315 b^6-2520 a b^4 c+5488 a^2 b^2 c^2-2048 a^3 c^3\right ) x\right )}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{6720 c^4} \\ & = -\frac {b \left (105 b^4-728 a b^2 c+1168 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{17920 c^4}+\frac {\left (315 b^6-2520 a b^4 c+5488 a^2 b^2 c^2-2048 a^3 c^3\right ) \sqrt {a x^2+b x^3+c x^4}}{35840 c^5 x}+\frac {\left (7 b^2-32 a c\right ) \left (3 b^2-4 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}{4480 c^3}-\frac {b \left (9 b^2-44 a c\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}{2240 c^2}+\frac {x^3 \left (b^2+24 a c+10 b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{280 c}+\frac {1}{7} x \left (a x^2+b x^3+c x^4\right )^{3/2}+\frac {\int -\frac {315 b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) x}{32 \sqrt {a x^2+b x^3+c x^4}} \, dx}{6720 c^5} \\ & = -\frac {b \left (105 b^4-728 a b^2 c+1168 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{17920 c^4}+\frac {\left (315 b^6-2520 a b^4 c+5488 a^2 b^2 c^2-2048 a^3 c^3\right ) \sqrt {a x^2+b x^3+c x^4}}{35840 c^5 x}+\frac {\left (7 b^2-32 a c\right ) \left (3 b^2-4 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}{4480 c^3}-\frac {b \left (9 b^2-44 a c\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}{2240 c^2}+\frac {x^3 \left (b^2+24 a c+10 b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{280 c}+\frac {1}{7} x \left (a x^2+b x^3+c x^4\right )^{3/2}-\frac {\left (3 b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right )\right ) \int \frac {x}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{2048 c^5} \\ & = -\frac {b \left (105 b^4-728 a b^2 c+1168 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{17920 c^4}+\frac {\left (315 b^6-2520 a b^4 c+5488 a^2 b^2 c^2-2048 a^3 c^3\right ) \sqrt {a x^2+b x^3+c x^4}}{35840 c^5 x}+\frac {\left (7 b^2-32 a c\right ) \left (3 b^2-4 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}{4480 c^3}-\frac {b \left (9 b^2-44 a c\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}{2240 c^2}+\frac {x^3 \left (b^2+24 a c+10 b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{280 c}+\frac {1}{7} x \left (a x^2+b x^3+c x^4\right )^{3/2}-\frac {\left (3 b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) x \sqrt {a+b x+c x^2}\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2048 c^5 \sqrt {a x^2+b x^3+c x^4}} \\ & = -\frac {b \left (105 b^4-728 a b^2 c+1168 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{17920 c^4}+\frac {\left (315 b^6-2520 a b^4 c+5488 a^2 b^2 c^2-2048 a^3 c^3\right ) \sqrt {a x^2+b x^3+c x^4}}{35840 c^5 x}+\frac {\left (7 b^2-32 a c\right ) \left (3 b^2-4 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}{4480 c^3}-\frac {b \left (9 b^2-44 a c\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}{2240 c^2}+\frac {x^3 \left (b^2+24 a c+10 b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{280 c}+\frac {1}{7} x \left (a x^2+b x^3+c x^4\right )^{3/2}-\frac {\left (3 b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\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 )}{1024 c^5 \sqrt {a x^2+b x^3+c x^4}} \\ & = -\frac {b \left (105 b^4-728 a b^2 c+1168 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{17920 c^4}+\frac {\left (315 b^6-2520 a b^4 c+5488 a^2 b^2 c^2-2048 a^3 c^3\right ) \sqrt {a x^2+b x^3+c x^4}}{35840 c^5 x}+\frac {\left (7 b^2-32 a c\right ) \left (3 b^2-4 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}{4480 c^3}-\frac {b \left (9 b^2-44 a c\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}{2240 c^2}+\frac {x^3 \left (b^2+24 a c+10 b c x\right ) \sqrt {a x^2+b x^3+c x^4}}{280 c}+\frac {1}{7} x \left (a x^2+b x^3+c x^4\right )^{3/2}-\frac {3 b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\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 )}{2048 c^{11/2} \sqrt {a x^2+b x^3+c x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.70 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.66 \[ \int \left (a x^2+b x^3+c x^4\right )^{3/2} \, dx=\frac {x \sqrt {a+x (b+c x)} \left (2 \sqrt {c} \sqrt {a+x (b+c x)} \left (315 b^6-210 b^5 c x+16 b^3 c^2 x \left (91 a-9 c x^2\right )+168 b^4 c \left (-15 a+c x^2\right )+1024 c^3 \left (a+c x^2\right )^2 \left (-2 a+5 c x^2\right )+16 b^2 c^2 \left (343 a^2-62 a c x^2+8 c^2 x^4\right )+32 b c^3 x \left (-73 a^2+22 a c x^2+200 c^2 x^4\right )\right )+105 b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{71680 c^{11/2} \sqrt {x^2 (a+x (b+c x))}} \]

[In]

Integrate[(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)]*(315*b^6 - 210*b^5*c*x + 16*b^3*c^2*x*(91*a - 9*c*x^
2) + 168*b^4*c*(-15*a + c*x^2) + 1024*c^3*(a + c*x^2)^2*(-2*a + 5*c*x^2) + 16*b^2*c^2*(343*a^2 - 62*a*c*x^2 +
8*c^2*x^4) + 32*b*c^3*x*(-73*a^2 + 22*a*c*x^2 + 200*c^2*x^4)) + 105*b*(b^2 - 4*a*c)^2*(3*b^2 - 4*a*c)*Log[b +
2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]]))/(71680*c^(11/2)*Sqrt[x^2*(a + x*(b + c*x))])

Maple [A] (verified)

Time = 0.66 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.74

method	result	size
risch	\(-\frac {\left (-5120 c^{6} x^{6}-6400 b \,c^{5} x^{5}-8192 a \,c^{5} x^{4}-128 b^{2} c^{4} x^{4}-704 a b \,c^{4} x^{3}+144 b^{3} c^{3} x^{3}-1024 a^{2} c^{4} x^{2}+992 a \,b^{2} c^{3} x^{2}-168 b^{4} c^{2} x^{2}+2336 a^{2} b \,c^{3} x -1456 a \,b^{3} c^{2} x +210 b^{5} c x +2048 c^{3} a^{3}-5488 a^{2} b^{2} c^{2}+2520 a \,b^{4} c -315 b^{6}\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{35840 c^{5} x}+\frac {3 b \left (64 c^{3} a^{3}-80 a^{2} b^{2} c^{2}+28 a \,b^{4} c -3 b^{6}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{2048 c^{\frac {11}{2}} x \sqrt {c \,x^{2}+b x +a}}\)	\(271\)
default	\(\frac {\left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (10240 x^{2} \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} c^{\frac {11}{2}}-7680 c^{\frac {9}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b x -4096 c^{\frac {9}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a +4480 c^{\frac {9}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a b x +6720 c^{\frac {9}{2}} \sqrt {c \,x^{2}+b x +a}\, a^{2} b x +5376 c^{\frac {7}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b^{2}-3360 c^{\frac {7}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{3} x +2240 c^{\frac {7}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \,b^{2}-6720 c^{\frac {7}{2}} \sqrt {c \,x^{2}+b x +a}\, a \,b^{3} x +3360 c^{\frac {7}{2}} \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{2}-1680 c^{\frac {5}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{4}+1260 c^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}\, b^{5} x -3360 c^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}\, a \,b^{4}+630 c^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}\, b^{6}+6720 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) a^{3} b \,c^{4}-8400 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) a^{2} b^{3} c^{3}+2940 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) a \,b^{5} c^{2}-315 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) b^{7} c \right )}{71680 x^{3} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{\frac {13}{2}}}\)	\(479\)

[In]

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

[Out]

-1/35840*(-5120*c^6*x^6-6400*b*c^5*x^5-8192*a*c^5*x^4-128*b^2*c^4*x^4-704*a*b*c^4*x^3+144*b^3*c^3*x^3-1024*a^2
*c^4*x^2+992*a*b^2*c^3*x^2-168*b^4*c^2*x^2+2336*a^2*b*c^3*x-1456*a*b^3*c^2*x+210*b^5*c*x+2048*a^3*c^3-5488*a^2
*b^2*c^2+2520*a*b^4*c-315*b^6)/c^5*(x^2*(c*x^2+b*x+a))^(1/2)/x+3/2048*b*(64*a^3*c^3-80*a^2*b^2*c^2+28*a*b^4*c-
3*b^6)/c^(11/2)*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 = 0.32 (sec) , antiderivative size = 558, normalized size of antiderivative = 1.53 \[ \int \left (a x^2+b x^3+c x^4\right )^{3/2} \, dx=\left [-\frac {105 \, {\left (3 \, b^{7} - 28 \, a b^{5} c + 80 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} \sqrt {c} x \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) - 4 \, {\left (5120 \, c^{7} x^{6} + 6400 \, b c^{6} x^{5} + 315 \, b^{6} c - 2520 \, a b^{4} c^{2} + 5488 \, a^{2} b^{2} c^{3} - 2048 \, a^{3} c^{4} + 128 \, {\left (b^{2} c^{5} + 64 \, a c^{6}\right )} x^{4} - 16 \, {\left (9 \, b^{3} c^{4} - 44 \, a b c^{5}\right )} x^{3} + 8 \, {\left (21 \, b^{4} c^{3} - 124 \, a b^{2} c^{4} + 128 \, a^{2} c^{5}\right )} x^{2} - 2 \, {\left (105 \, b^{5} c^{2} - 728 \, a b^{3} c^{3} + 1168 \, a^{2} b c^{4}\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{143360 \, c^{6} x}, \frac {105 \, {\left (3 \, b^{7} - 28 \, a b^{5} c + 80 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} \sqrt {-c} x \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) + 2 \, {\left (5120 \, c^{7} x^{6} + 6400 \, b c^{6} x^{5} + 315 \, b^{6} c - 2520 \, a b^{4} c^{2} + 5488 \, a^{2} b^{2} c^{3} - 2048 \, a^{3} c^{4} + 128 \, {\left (b^{2} c^{5} + 64 \, a c^{6}\right )} x^{4} - 16 \, {\left (9 \, b^{3} c^{4} - 44 \, a b c^{5}\right )} x^{3} + 8 \, {\left (21 \, b^{4} c^{3} - 124 \, a b^{2} c^{4} + 128 \, a^{2} c^{5}\right )} x^{2} - 2 \, {\left (105 \, b^{5} c^{2} - 728 \, a b^{3} c^{3} + 1168 \, a^{2} b c^{4}\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{71680 \, c^{6} x}\right ] \]

[In]

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

[Out]

[-1/143360*(105*(3*b^7 - 28*a*b^5*c + 80*a^2*b^3*c^2 - 64*a^3*b*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*(5120*c^7*x^6 + 6400*b*c^6*x^5 + 31
5*b^6*c - 2520*a*b^4*c^2 + 5488*a^2*b^2*c^3 - 2048*a^3*c^4 + 128*(b^2*c^5 + 64*a*c^6)*x^4 - 16*(9*b^3*c^4 - 44
*a*b*c^5)*x^3 + 8*(21*b^4*c^3 - 124*a*b^2*c^4 + 128*a^2*c^5)*x^2 - 2*(105*b^5*c^2 - 728*a*b^3*c^3 + 1168*a^2*b
*c^4)*x)*sqrt(c*x^4 + b*x^3 + a*x^2))/(c^6*x), 1/71680*(105*(3*b^7 - 28*a*b^5*c + 80*a^2*b^3*c^2 - 64*a^3*b*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*(51
20*c^7*x^6 + 6400*b*c^6*x^5 + 315*b^6*c - 2520*a*b^4*c^2 + 5488*a^2*b^2*c^3 - 2048*a^3*c^4 + 128*(b^2*c^5 + 64
*a*c^6)*x^4 - 16*(9*b^3*c^4 - 44*a*b*c^5)*x^3 + 8*(21*b^4*c^3 - 124*a*b^2*c^4 + 128*a^2*c^5)*x^2 - 2*(105*b^5*
c^2 - 728*a*b^3*c^3 + 1168*a^2*b*c^4)*x)*sqrt(c*x^4 + b*x^3 + a*x^2))/(c^6*x)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate((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)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 419, normalized size of antiderivative = 1.15 \[ \int \left (a x^2+b x^3+c x^4\right )^{3/2} \, dx=\frac {1}{35840} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (4 \, c x \mathrm {sgn}\left (x\right ) + 5 \, b \mathrm {sgn}\left (x\right )\right )} x + \frac {b^{2} c^{5} \mathrm {sgn}\left (x\right ) + 64 \, a c^{6} \mathrm {sgn}\left (x\right )}{c^{6}}\right )} x - \frac {9 \, b^{3} c^{4} \mathrm {sgn}\left (x\right ) - 44 \, a b c^{5} \mathrm {sgn}\left (x\right )}{c^{6}}\right )} x + \frac {21 \, b^{4} c^{3} \mathrm {sgn}\left (x\right ) - 124 \, a b^{2} c^{4} \mathrm {sgn}\left (x\right ) + 128 \, a^{2} c^{5} \mathrm {sgn}\left (x\right )}{c^{6}}\right )} x - \frac {105 \, b^{5} c^{2} \mathrm {sgn}\left (x\right ) - 728 \, a b^{3} c^{3} \mathrm {sgn}\left (x\right ) + 1168 \, a^{2} b c^{4} \mathrm {sgn}\left (x\right )}{c^{6}}\right )} x + \frac {315 \, b^{6} c \mathrm {sgn}\left (x\right ) - 2520 \, a b^{4} c^{2} \mathrm {sgn}\left (x\right ) + 5488 \, a^{2} b^{2} c^{3} \mathrm {sgn}\left (x\right ) - 2048 \, a^{3} c^{4} \mathrm {sgn}\left (x\right )}{c^{6}}\right )} + \frac {3 \, {\left (3 \, b^{7} \mathrm {sgn}\left (x\right ) - 28 \, a b^{5} c \mathrm {sgn}\left (x\right ) + 80 \, a^{2} b^{3} c^{2} \mathrm {sgn}\left (x\right ) - 64 \, a^{3} b 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 )}{2048 \, c^{\frac {11}{2}}} - \frac {{\left (315 \, b^{7} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 2940 \, a b^{5} c \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 8400 \, a^{2} b^{3} c^{2} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 6720 \, a^{3} b c^{3} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 630 \, \sqrt {a} b^{6} \sqrt {c} - 5040 \, a^{\frac {3}{2}} b^{4} c^{\frac {3}{2}} + 10976 \, a^{\frac {5}{2}} b^{2} c^{\frac {5}{2}} - 4096 \, a^{\frac {7}{2}} c^{\frac {7}{2}}\right )} \mathrm {sgn}\left (x\right )}{71680 \, c^{\frac {11}{2}}} \]

[In]

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

[Out]

1/35840*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(4*c*x*sgn(x) + 5*b*sgn(x))*x + (b^2*c^5*sgn(x) + 64*a*c^6*sgn(x
))/c^6)*x - (9*b^3*c^4*sgn(x) - 44*a*b*c^5*sgn(x))/c^6)*x + (21*b^4*c^3*sgn(x) - 124*a*b^2*c^4*sgn(x) + 128*a^
2*c^5*sgn(x))/c^6)*x - (105*b^5*c^2*sgn(x) - 728*a*b^3*c^3*sgn(x) + 1168*a^2*b*c^4*sgn(x))/c^6)*x + (315*b^6*c
*sgn(x) - 2520*a*b^4*c^2*sgn(x) + 5488*a^2*b^2*c^3*sgn(x) - 2048*a^3*c^4*sgn(x))/c^6) + 3/2048*(3*b^7*sgn(x) -
28*a*b^5*c*sgn(x) + 80*a^2*b^3*c^2*sgn(x) - 64*a^3*b*c^3*sgn(x))*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)*sqrt(c) + b))/c^(11/2) - 1/71680*(315*b^7*log(abs(b - 2*sqrt(a)*sqrt(c))) - 2940*a*b^5*c*log(abs(b - 2*sqrt(
a)*sqrt(c))) + 8400*a^2*b^3*c^2*log(abs(b - 2*sqrt(a)*sqrt(c))) - 6720*a^3*b*c^3*log(abs(b - 2*sqrt(a)*sqrt(c)
)) + 630*sqrt(a)*b^6*sqrt(c) - 5040*a^(3/2)*b^4*c^(3/2) + 10976*a^(5/2)*b^2*c^(5/2) - 4096*a^(7/2)*c^(7/2))*sg
n(x)/c^(11/2)

Mupad [F(-1)]

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

[In]

int((a*x^2 + b*x^3 + c*x^4)^(3/2),x)

[Out]

int((a*x^2 + b*x^3 + c*x^4)^(3/2), x)