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

Optimal. Leaf size=364 \[ -\frac{b \left (1168 a^2 c^2-728 a b^2 c+105 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{17920 c^4}+\frac{\left (5488 a^2 b^2 c^2-2048 a^3 c^3-2520 a b^4 c+315 b^6\right ) \sqrt{a x^2+b x^3+c x^4}}{35840 c^5 x}-\frac{b x^2 \left (9 b^2-44 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{2240 c^2}+\frac{x \left (7 b^2-32 a c\right ) \left (3 b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{4480 c^3}-\frac{3 b x \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{11/2} \sqrt{a x^2+b x^3+c x^4}}+\frac{x^3 \left (24 a c+b^2+10 b c x\right ) \sqrt{a x^2+b x^3+c x^4}}{280 c}+\frac{1}{7} x \left (a x^2+b x^3+c x^4\right )^{3/2} \]

[Out]

-(b*(105*b^4 - 728*a*b^2*c + 1168*a^2*c^2)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(17920*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])

________________________________________________________________________________________

Rubi [A]  time = 1.03943, antiderivative size = 364, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {1906, 1945, 1949, 12, 1914, 621, 206} \[ -\frac{b \left (1168 a^2 c^2-728 a b^2 c+105 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{17920 c^4}+\frac{\left (5488 a^2 b^2 c^2-2048 a^3 c^3-2520 a b^4 c+315 b^6\right ) \sqrt{a x^2+b x^3+c x^4}}{35840 c^5 x}-\frac{b x^2 \left (9 b^2-44 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{2240 c^2}+\frac{x \left (7 b^2-32 a c\right ) \left (3 b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{4480 c^3}-\frac{3 b x \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{11/2} \sqrt{a x^2+b x^3+c x^4}}+\frac{x^3 \left (24 a c+b^2+10 b c x\right ) \sqrt{a x^2+b x^3+c x^4}}{280 c}+\frac{1}{7} x \left (a x^2+b x^3+c x^4\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[(a*x^2 + b*x^3 + c*x^4)^(3/2),x]

[Out]

-(b*(105*b^4 - 728*a*b^2*c + 1168*a^2*c^2)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(17920*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 1906

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 1945

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 1949

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]

Rule 12

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

Rule 1914

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 621

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \left (a x^2+b x^3+c x^4\right )^{3/2} \, dx &=\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 ) \operatorname{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]  time = 0.267093, size = 197, normalized size = 0.54 \[ \frac{\left (x^2 (a+x (b+c x))\right )^{3/2} \left (\frac{7 \left (4 a b c-3 b^3\right ) \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left (4 c \left (5 a+2 c x^2\right )-3 b^2+8 b c x\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )}{2048 c^{9/2} (a+x (b+c x))^{3/2}}+\frac{\left (-16 a c+21 b^2-30 b c x\right ) (a+x (b+c x))}{40 c^2}+x^2 (a+x (b+c x))\right )}{7 c x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((x^2*(a + x*(b + c*x)))^(3/2)*(x^2*(a + x*(b + c*x)) + ((21*b^2 - 16*a*c - 30*b*c*x)*(a + x*(b + c*x)))/(40*c
^2) + (7*(-3*b^3 + 4*a*b*c)*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(-3*b^2 + 8*b*c*x + 4*c*(5*a + 2*c*x^
2)) + 3*(b^2 - 4*a*c)^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/(2048*c^(9/2)*(a + x*(b + c*x
))^(3/2))))/(7*c*x^3)

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Maple [A]  time = 0.009, size = 479, normalized size = 1.3 \begin{align*}{\frac{1}{71680\,{x}^{3}} \left ( c{x}^{4}+b{x}^{3}+a{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 10240\,{x}^{2} \left ( c{x}^{2}+bx+a \right ) ^{5/2}{c}^{11/2}-7680\, \left ( c{x}^{2}+bx+a \right ) ^{5/2}{c}^{9/2}xb-4096\, \left ( c{x}^{2}+bx+a \right ) ^{5/2}{c}^{9/2}a+5376\, \left ( c{x}^{2}+bx+a \right ) ^{5/2}{c}^{7/2}{b}^{2}+4480\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{9/2}xab-3360\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{7/2}x{b}^{3}+2240\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{7/2}a{b}^{2}-1680\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{5/2}{b}^{4}+6720\,\sqrt{c{x}^{2}+bx+a}{c}^{9/2}x{a}^{2}b-6720\,\sqrt{c{x}^{2}+bx+a}{c}^{7/2}xa{b}^{3}+1260\,\sqrt{c{x}^{2}+bx+a}{c}^{5/2}x{b}^{5}+3360\,\sqrt{c{x}^{2}+bx+a}{c}^{7/2}{a}^{2}{b}^{2}-3360\,\sqrt{c{x}^{2}+bx+a}{c}^{5/2}a{b}^{4}+630\,\sqrt{c{x}^{2}+bx+a}{c}^{3/2}{b}^{6}+6720\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){a}^{3}b{c}^{4}-8400\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){a}^{2}{b}^{3}{c}^{3}+2940\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ) a{b}^{5}{c}^{2}-315\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){b}^{7}c \right ) \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{13}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/71680*(c*x^4+b*x^3+a*x^2)^(3/2)*(10240*x^2*(c*x^2+b*x+a)^(5/2)*c^(11/2)-7680*(c*x^2+b*x+a)^(5/2)*c^(9/2)*x*b
-4096*(c*x^2+b*x+a)^(5/2)*c^(9/2)*a+5376*(c*x^2+b*x+a)^(5/2)*c^(7/2)*b^2+4480*(c*x^2+b*x+a)^(3/2)*c^(9/2)*x*a*
b-3360*(c*x^2+b*x+a)^(3/2)*c^(7/2)*x*b^3+2240*(c*x^2+b*x+a)^(3/2)*c^(7/2)*a*b^2-1680*(c*x^2+b*x+a)^(3/2)*c^(5/
2)*b^4+6720*(c*x^2+b*x+a)^(1/2)*c^(9/2)*x*a^2*b-6720*(c*x^2+b*x+a)^(1/2)*c^(7/2)*x*a*b^3+1260*(c*x^2+b*x+a)^(1
/2)*c^(5/2)*x*b^5+3360*(c*x^2+b*x+a)^(1/2)*c^(7/2)*a^2*b^2-3360*(c*x^2+b*x+a)^(1/2)*c^(5/2)*a*b^4+630*(c*x^2+b
*x+a)^(1/2)*c^(3/2)*b^6+6720*ln(1/2*(2*(c*x^2+b*x+a)^(1/2)*c^(1/2)+2*c*x+b)/c^(1/2))*a^3*b*c^4-8400*ln(1/2*(2*
(c*x^2+b*x+a)^(1/2)*c^(1/2)+2*c*x+b)/c^(1/2))*a^2*b^3*c^3+2940*ln(1/2*(2*(c*x^2+b*x+a)^(1/2)*c^(1/2)+2*c*x+b)/
c^(1/2))*a*b^5*c^2-315*ln(1/2*(2*(c*x^2+b*x+a)^(1/2)*c^(1/2)+2*c*x+b)/c^(1/2))*b^7*c)/x^3/(c*x^2+b*x+a)^(3/2)/
c^(13/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.98469, size = 1299, normalized size = 3.57 \begin{align*} \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 ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a x^{2} + b x^{3} + c x^{4}\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.30208, size = 579, normalized size = 1.59 \begin{align*} \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}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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*sq
rt(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)*sq
rt(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
))*sgn(x)/c^(11/2)