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

Optimal. Leaf size=422 \[ \frac {3 x \left (b^2-4 a c\right )^2 \left (16 a^2 c^2-72 a b^2 c+33 b^4\right ) \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{13/2} \sqrt {a x^2+b x^3+c x^4}}-\frac {b x \left (2416 a^2 c^2-1560 a b^2 c+231 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{71680 c^4}+\frac {x^2 \left (560 a^2 c^2-568 a b^2 c+99 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{35840 c^3}-\frac {b \left (-58816 a^3 c^3+81648 a^2 b^2 c^2-30660 a b^4 c+3465 b^6\right ) \sqrt {a x^2+b x^3+c x^4}}{573440 c^6 x}+\frac {\left (-6720 a^3 c^3+18896 a^2 b^2 c^2-8988 a b^4 c+1155 b^6\right ) \sqrt {a x^2+b x^3+c x^4}}{286720 c^5}-\frac {x^3 \left (10 c x \left (11 b^2-28 a c\right )+b \left (68 a c+11 b^2\right )\right ) \sqrt {a x^2+b x^3+c x^4}}{4480 c^2}+\frac {x (3 b+14 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{112 c} \]

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Rubi [A]  time = 1.20, antiderivative size = 422, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1919, 1945, 1949, 12, 1914, 621, 206} \begin {gather*} \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 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 {\left (18896 a^2 b^2 c^2-6720 a^3 c^3-8988 a b^4 c+1155 b^6\right ) \sqrt {a x^2+b x^3+c x^4}}{286720 c^5}-\frac {b \left (81648 a^2 b^2 c^2-58816 a^3 c^3-30660 a b^4 c+3465 b^6\right ) \sqrt {a x^2+b x^3+c x^4}}{573440 c^6 x}+\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} \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}}-\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} \end {gather*}

Antiderivative was successfully verified.

[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 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])

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 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 1919

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 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]

Rubi steps

\begin {align*} \int x \left (a x^2+b x^3+c x^4\right )^{3/2} \, dx &=\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 ) \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 )}{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*}

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Mathematica [A]  time = 0.38, size = 236, normalized size = 0.56 \begin {gather*} \frac {\left (x^2 (a+x (b+c x))\right )^{3/2} \left (\frac {\left (16 a^2 c^2-72 a b^2 c+33 b^4\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 )}{4096 c^{11/2} x^3 (a+x (b+c x))^{3/2}}+\frac {\left (372 a b c-280 a c^2 x-231 b^3+330 b^2 c x\right ) (a+x (b+c x))}{560 c^3 x^3}-\frac {11 b (a+x (b+c x))}{14 c x}+a+b x+c x^2\right )}{8 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((x^2*(a + x*(b + c*x)))^(3/2)*(a + b*x + c*x^2 - (11*b*(a + x*(b + c*x)))/(14*c*x) + ((-231*b^3 + 372*a*b*c +
330*b^2*c*x - 280*a*c^2*x)*(a + x*(b + c*x)))/(560*c^3*x^3) + ((33*b^4 - 72*a*b^2*c + 16*a^2*c^2)*(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)])]))/(4096*c^(11/2)*x^3*(a + x*(b + c*x))^(3/2))))/(8*c)

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IntegrateAlgebraic [A]  time = 5.16, size = 323, normalized size = 0.77 \begin {gather*} \frac {\sqrt {a x^2+b x^3+c x^4} \left (58816 a^3 b c^3-13440 a^3 c^4 x-81648 a^2 b^3 c^2+37792 a^2 b^2 c^3 x-19328 a^2 b c^4 x^2+8960 a^2 c^5 x^3+30660 a b^5 c-17976 a b^4 c^2 x+12480 a b^3 c^3 x^2-9088 a b^2 c^4 x^3+6656 a b c^5 x^4+107520 a c^6 x^5-3465 b^7+2310 b^6 c x-1848 b^5 c^2 x^2+1584 b^4 c^3 x^3-1408 b^3 c^4 x^4+1280 b^2 c^5 x^5+87040 b c^6 x^6+71680 c^7 x^7\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 ) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a} x-\sqrt {a x^2+b x^3+c x^4}}\right )}{16384 c^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a*x^2 + b*x^3 + c*x^4]*(-3465*b^7 + 30660*a*b^5*c - 81648*a^2*b^3*c^2 + 58816*a^3*b*c^3 + 2310*b^6*c*x -
17976*a*b^4*c^2*x + 37792*a^2*b^2*c^3*x - 13440*a^3*c^4*x - 1848*b^5*c^2*x^2 + 12480*a*b^3*c^3*x^2 - 19328*a^
2*b*c^4*x^2 + 1584*b^4*c^3*x^3 - 9088*a*b^2*c^4*x^3 + 8960*a^2*c^5*x^3 - 1408*b^3*c^4*x^4 + 6656*a*b*c^5*x^4 +
1280*b^2*c^5*x^5 + 107520*a*c^6*x^5 + 87040*b*c^6*x^6 + 71680*c^7*x^7))/(573440*c^6*x) - (3*(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)*ArcTanh[(Sqrt[c]*x^2)/(Sqrt[a]*x - Sqrt[a*x^2 + b*x^
3 + c*x^4])])/(16384*c^(13/2))

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fricas [A]  time = 1.38, size = 664, normalized size = 1.57 \begin {gather*} \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 ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]

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giac [A]  time = 1.46, size = 521, normalized size = 1.23 \begin {gather*} \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}\relax (x) + 17 \, b \mathrm {sgn}\relax (x)\right )} x + \frac {b^{2} c^{6} \mathrm {sgn}\relax (x) + 84 \, a c^{7} \mathrm {sgn}\relax (x)}{c^{7}}\right )} x - \frac {11 \, b^{3} c^{5} \mathrm {sgn}\relax (x) - 52 \, a b c^{6} \mathrm {sgn}\relax (x)}{c^{7}}\right )} x + \frac {99 \, b^{4} c^{4} \mathrm {sgn}\relax (x) - 568 \, a b^{2} c^{5} \mathrm {sgn}\relax (x) + 560 \, a^{2} c^{6} \mathrm {sgn}\relax (x)}{c^{7}}\right )} x - \frac {231 \, b^{5} c^{3} \mathrm {sgn}\relax (x) - 1560 \, a b^{3} c^{4} \mathrm {sgn}\relax (x) + 2416 \, a^{2} b c^{5} \mathrm {sgn}\relax (x)}{c^{7}}\right )} x + \frac {1155 \, b^{6} c^{2} \mathrm {sgn}\relax (x) - 8988 \, a b^{4} c^{3} \mathrm {sgn}\relax (x) + 18896 \, a^{2} b^{2} c^{4} \mathrm {sgn}\relax (x) - 6720 \, a^{3} c^{5} \mathrm {sgn}\relax (x)}{c^{7}}\right )} x - \frac {3465 \, b^{7} c \mathrm {sgn}\relax (x) - 30660 \, a b^{5} c^{2} \mathrm {sgn}\relax (x) + 81648 \, a^{2} b^{3} c^{3} \mathrm {sgn}\relax (x) - 58816 \, a^{3} b c^{4} \mathrm {sgn}\relax (x)}{c^{7}}\right )} - \frac {3 \, {\left (33 \, b^{8} \mathrm {sgn}\relax (x) - 336 \, a b^{6} c \mathrm {sgn}\relax (x) + 1120 \, a^{2} b^{4} c^{2} \mathrm {sgn}\relax (x) - 1280 \, a^{3} b^{2} c^{3} \mathrm {sgn}\relax (x) + 256 \, a^{4} c^{4} \mathrm {sgn}\relax (x)\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}\relax (x)}{1146880 \, c^{\frac {13}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)*s
qrt(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))) + 693
0*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)

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maple [A]  time = 0.01, size = 649, normalized size = 1.54 \begin {gather*} \frac {\left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (26880 a^{4} c^{5} \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )-134400 a^{3} b^{2} c^{4} \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )+117600 a^{2} b^{4} c^{3} \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )-35280 a \,b^{6} c^{2} \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )+3465 b^{8} c \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )+26880 \sqrt {c \,x^{2}+b x +a}\, a^{3} c^{\frac {11}{2}} x -127680 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{2} c^{\frac {9}{2}} x +85680 \sqrt {c \,x^{2}+b x +a}\, a \,b^{4} c^{\frac {7}{2}} x -13860 \sqrt {c \,x^{2}+b x +a}\, b^{6} c^{\frac {5}{2}} x +143360 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} c^{\frac {13}{2}} x^{3}+13440 \sqrt {c \,x^{2}+b x +a}\, a^{3} b \,c^{\frac {9}{2}}-63840 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{3} c^{\frac {7}{2}}+17920 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{2} c^{\frac {11}{2}} x +42840 \sqrt {c \,x^{2}+b x +a}\, a \,b^{5} c^{\frac {5}{2}}-80640 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \,b^{2} c^{\frac {9}{2}} x -6930 \sqrt {c \,x^{2}+b x +a}\, b^{7} c^{\frac {3}{2}}+36960 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{4} c^{\frac {7}{2}} x -112640 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b \,c^{\frac {11}{2}} x^{2}+8960 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a^{2} b \,c^{\frac {9}{2}}-40320 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \,b^{3} c^{\frac {7}{2}}-71680 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a \,c^{\frac {11}{2}} x +18480 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{5} c^{\frac {5}{2}}+84480 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b^{2} c^{\frac {9}{2}} x +95232 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} a b \,c^{\frac {9}{2}}-59136 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b^{3} c^{\frac {7}{2}}\right )}{1146880 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{\frac {15}{2}} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1146880*(c*x^4+b*x^3+a*x^2)^(3/2)*(26880*ln(1/2*(2*c*x+b+2*(c*x^2+b*x+a)^(1/2)*c^(1/2))/c^(1/2))*a^4*c^5+346
5*ln(1/2*(2*c*x+b+2*(c*x^2+b*x+a)^(1/2)*c^(1/2))/c^(1/2))*b^8*c+143360*x^3*(c*x^2+b*x+a)^(5/2)*c^(13/2)-59136*
(c*x^2+b*x+a)^(5/2)*c^(7/2)*b^3+18480*(c*x^2+b*x+a)^(3/2)*c^(5/2)*b^5-6930*(c*x^2+b*x+a)^(1/2)*c^(3/2)*b^7+428
40*(c*x^2+b*x+a)^(1/2)*c^(5/2)*a*b^5+8960*(c*x^2+b*x+a)^(3/2)*c^(9/2)*a^2*b-40320*(c*x^2+b*x+a)^(3/2)*c^(7/2)*
a*b^3-134400*ln(1/2*(2*c*x+b+2*(c*x^2+b*x+a)^(1/2)*c^(1/2))/c^(1/2))*a^3*b^2*c^4+117600*ln(1/2*(2*c*x+b+2*(c*x
^2+b*x+a)^(1/2)*c^(1/2))/c^(1/2))*a^2*b^4*c^3-35280*ln(1/2*(2*c*x+b+2*(c*x^2+b*x+a)^(1/2)*c^(1/2))/c^(1/2))*a*
b^6*c^2-13860*(c*x^2+b*x+a)^(1/2)*c^(5/2)*x*b^6+13440*(c*x^2+b*x+a)^(1/2)*c^(9/2)*a^3*b-63840*(c*x^2+b*x+a)^(1
/2)*c^(7/2)*a^2*b^3-112640*(c*x^2+b*x+a)^(5/2)*c^(11/2)*x^2*b-71680*(c*x^2+b*x+a)^(5/2)*c^(11/2)*x*a+84480*(c*
x^2+b*x+a)^(5/2)*c^(9/2)*x*b^2+95232*(c*x^2+b*x+a)^(5/2)*c^(9/2)*a*b+17920*(c*x^2+b*x+a)^(3/2)*c^(11/2)*x*a^2+
36960*(c*x^2+b*x+a)^(3/2)*c^(7/2)*x*b^4+26880*(c*x^2+b*x+a)^(1/2)*c^(11/2)*x*a^3-80640*(c*x^2+b*x+a)^(3/2)*c^(
9/2)*x*a*b^2-127680*(c*x^2+b*x+a)^(1/2)*c^(9/2)*x*a^2*b^2+85680*(c*x^2+b*x+a)^(1/2)*c^(7/2)*x*a*b^4)/x^3/(c*x^
2+b*x+a)^(3/2)/c^(15/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac {3}{2}} x\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x\,{\left (c\,x^4+b\,x^3+a\,x^2\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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