3.910 \(\int x^4 (A+B x) \sqrt{a+b x+c x^2} \, dx\)

Optimal. Leaf size=367 \[ \frac{\left (a+b x+c x^2\right )^{3/2} \left (1024 a^2 B c^2-6 c x \left (280 a A c^2-444 a b B c-294 A b^2 c+231 b^3 B\right )+2744 a A b c^2-3276 a b^2 B c-1470 A b^3 c+1155 b^4 B\right )}{13440 c^5}-\frac{(b+2 c x) \sqrt{a+b x+c x^2} \left (-32 a^2 A c^3+80 a^2 b B c^2+112 a A b^2 c^2-120 a b^3 B c-42 A b^4 c+33 b^5 B\right )}{1024 c^6}+\frac{\left (b^2-4 a c\right ) \left (-32 a^2 A c^3+80 a^2 b B c^2+112 a A b^2 c^2-120 a b^3 B c-42 A b^4 c+33 b^5 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{13/2}}+\frac{x^2 \left (a+b x+c x^2\right )^{3/2} \left (-32 a B c-42 A b c+33 b^2 B\right )}{280 c^3}-\frac{x^3 \left (a+b x+c x^2\right )^{3/2} (11 b B-14 A c)}{84 c^2}+\frac{B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c} \]

[Out]

-((33*b^5*B - 42*A*b^4*c - 120*a*b^3*B*c + 112*a*A*b^2*c^2 + 80*a^2*b*B*c^2 - 32*a^2*A*c^3)*(b + 2*c*x)*Sqrt[a
+ b*x + c*x^2])/(1024*c^6) + ((33*b^2*B - 42*A*b*c - 32*a*B*c)*x^2*(a + b*x + c*x^2)^(3/2))/(280*c^3) - ((11*
b*B - 14*A*c)*x^3*(a + b*x + c*x^2)^(3/2))/(84*c^2) + (B*x^4*(a + b*x + c*x^2)^(3/2))/(7*c) + ((1155*b^4*B - 1
470*A*b^3*c - 3276*a*b^2*B*c + 2744*a*A*b*c^2 + 1024*a^2*B*c^2 - 6*c*(231*b^3*B - 294*A*b^2*c - 444*a*b*B*c +
280*a*A*c^2)*x)*(a + b*x + c*x^2)^(3/2))/(13440*c^5) + ((b^2 - 4*a*c)*(33*b^5*B - 42*A*b^4*c - 120*a*b^3*B*c +
112*a*A*b^2*c^2 + 80*a^2*b*B*c^2 - 32*a^2*A*c^3)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(204
8*c^(13/2))

________________________________________________________________________________________

Rubi [A]  time = 0.508093, antiderivative size = 367, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {832, 779, 612, 621, 206} \[ \frac{\left (a+b x+c x^2\right )^{3/2} \left (1024 a^2 B c^2-6 c x \left (280 a A c^2-444 a b B c-294 A b^2 c+231 b^3 B\right )+2744 a A b c^2-3276 a b^2 B c-1470 A b^3 c+1155 b^4 B\right )}{13440 c^5}-\frac{(b+2 c x) \sqrt{a+b x+c x^2} \left (-32 a^2 A c^3+80 a^2 b B c^2+112 a A b^2 c^2-120 a b^3 B c-42 A b^4 c+33 b^5 B\right )}{1024 c^6}+\frac{\left (b^2-4 a c\right ) \left (-32 a^2 A c^3+80 a^2 b B c^2+112 a A b^2 c^2-120 a b^3 B c-42 A b^4 c+33 b^5 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{13/2}}+\frac{x^2 \left (a+b x+c x^2\right )^{3/2} \left (-32 a B c-42 A b c+33 b^2 B\right )}{280 c^3}-\frac{x^3 \left (a+b x+c x^2\right )^{3/2} (11 b B-14 A c)}{84 c^2}+\frac{B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(A + B*x)*Sqrt[a + b*x + c*x^2],x]

[Out]

-((33*b^5*B - 42*A*b^4*c - 120*a*b^3*B*c + 112*a*A*b^2*c^2 + 80*a^2*b*B*c^2 - 32*a^2*A*c^3)*(b + 2*c*x)*Sqrt[a
+ b*x + c*x^2])/(1024*c^6) + ((33*b^2*B - 42*A*b*c - 32*a*B*c)*x^2*(a + b*x + c*x^2)^(3/2))/(280*c^3) - ((11*
b*B - 14*A*c)*x^3*(a + b*x + c*x^2)^(3/2))/(84*c^2) + (B*x^4*(a + b*x + c*x^2)^(3/2))/(7*c) + ((1155*b^4*B - 1
470*A*b^3*c - 3276*a*b^2*B*c + 2744*a*A*b*c^2 + 1024*a^2*B*c^2 - 6*c*(231*b^3*B - 294*A*b^2*c - 444*a*b*B*c +
280*a*A*c^2)*x)*(a + b*x + c*x^2)^(3/2))/(13440*c^5) + ((b^2 - 4*a*c)*(33*b^5*B - 42*A*b^4*c - 120*a*b^3*B*c +
112*a*A*b^2*c^2 + 80*a^2*b*B*c^2 - 32*a^2*A*c^3)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(204
8*c^(13/2))

Rule 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m
- 1)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p
+ 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 612

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

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 x^4 (A+B x) \sqrt{a+b x+c x^2} \, dx &=\frac{B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac{\int x^3 \left (-4 a B-\frac{1}{2} (11 b B-14 A c) x\right ) \sqrt{a+b x+c x^2} \, dx}{7 c}\\ &=-\frac{(11 b B-14 A c) x^3 \left (a+b x+c x^2\right )^{3/2}}{84 c^2}+\frac{B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac{\int x^2 \left (\frac{3}{2} a (11 b B-14 A c)+\frac{3}{4} \left (33 b^2 B-42 A b c-32 a B c\right ) x\right ) \sqrt{a+b x+c x^2} \, dx}{42 c^2}\\ &=\frac{\left (33 b^2 B-42 A b c-32 a B c\right ) x^2 \left (a+b x+c x^2\right )^{3/2}}{280 c^3}-\frac{(11 b B-14 A c) x^3 \left (a+b x+c x^2\right )^{3/2}}{84 c^2}+\frac{B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac{\int x \left (-\frac{3}{2} a \left (33 b^2 B-42 A b c-32 a B c\right )-\frac{3}{8} \left (231 b^3 B-294 A b^2 c-444 a b B c+280 a A c^2\right ) x\right ) \sqrt{a+b x+c x^2} \, dx}{210 c^3}\\ &=\frac{\left (33 b^2 B-42 A b c-32 a B c\right ) x^2 \left (a+b x+c x^2\right )^{3/2}}{280 c^3}-\frac{(11 b B-14 A c) x^3 \left (a+b x+c x^2\right )^{3/2}}{84 c^2}+\frac{B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac{\left (1155 b^4 B-1470 A b^3 c-3276 a b^2 B c+2744 a A b c^2+1024 a^2 B c^2-6 c \left (231 b^3 B-294 A b^2 c-444 a b B c+280 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{13440 c^5}-\frac{\left (33 b^5 B-42 A b^4 c-120 a b^3 B c+112 a A b^2 c^2+80 a^2 b B c^2-32 a^2 A c^3\right ) \int \sqrt{a+b x+c x^2} \, dx}{256 c^5}\\ &=-\frac{\left (33 b^5 B-42 A b^4 c-120 a b^3 B c+112 a A b^2 c^2+80 a^2 b B c^2-32 a^2 A c^3\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{1024 c^6}+\frac{\left (33 b^2 B-42 A b c-32 a B c\right ) x^2 \left (a+b x+c x^2\right )^{3/2}}{280 c^3}-\frac{(11 b B-14 A c) x^3 \left (a+b x+c x^2\right )^{3/2}}{84 c^2}+\frac{B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac{\left (1155 b^4 B-1470 A b^3 c-3276 a b^2 B c+2744 a A b c^2+1024 a^2 B c^2-6 c \left (231 b^3 B-294 A b^2 c-444 a b B c+280 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{13440 c^5}+\frac{\left (\left (b^2-4 a c\right ) \left (33 b^5 B-42 A b^4 c-120 a b^3 B c+112 a A b^2 c^2+80 a^2 b B c^2-32 a^2 A c^3\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{2048 c^6}\\ &=-\frac{\left (33 b^5 B-42 A b^4 c-120 a b^3 B c+112 a A b^2 c^2+80 a^2 b B c^2-32 a^2 A c^3\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{1024 c^6}+\frac{\left (33 b^2 B-42 A b c-32 a B c\right ) x^2 \left (a+b x+c x^2\right )^{3/2}}{280 c^3}-\frac{(11 b B-14 A c) x^3 \left (a+b x+c x^2\right )^{3/2}}{84 c^2}+\frac{B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac{\left (1155 b^4 B-1470 A b^3 c-3276 a b^2 B c+2744 a A b c^2+1024 a^2 B c^2-6 c \left (231 b^3 B-294 A b^2 c-444 a b B c+280 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{13440 c^5}+\frac{\left (\left (b^2-4 a c\right ) \left (33 b^5 B-42 A b^4 c-120 a b^3 B c+112 a A b^2 c^2+80 a^2 b B c^2-32 a^2 A c^3\right )\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^6}\\ &=-\frac{\left (33 b^5 B-42 A b^4 c-120 a b^3 B c+112 a A b^2 c^2+80 a^2 b B c^2-32 a^2 A c^3\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{1024 c^6}+\frac{\left (33 b^2 B-42 A b c-32 a B c\right ) x^2 \left (a+b x+c x^2\right )^{3/2}}{280 c^3}-\frac{(11 b B-14 A c) x^3 \left (a+b x+c x^2\right )^{3/2}}{84 c^2}+\frac{B x^4 \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac{\left (1155 b^4 B-1470 A b^3 c-3276 a b^2 B c+2744 a A b c^2+1024 a^2 B c^2-6 c \left (231 b^3 B-294 A b^2 c-444 a b B c+280 a A c^2\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{13440 c^5}+\frac{\left (b^2-4 a c\right ) \left (33 b^5 B-42 A b^4 c-120 a b^3 B c+112 a A b^2 c^2+80 a^2 b B c^2-32 a^2 A c^3\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{13/2}}\\ \end{align*}

Mathematica [A]  time = 0.60224, size = 312, normalized size = 0.85 \[ \frac{\frac{7 \left (32 a^2 A c^3-80 a^2 b B c^2-112 a A b^2 c^2+120 a b^3 B c+42 A b^4 c-33 b^5 B\right ) \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )}{2048 c^{11/2}}+\frac{x^2 (a+x (b+c x))^{3/2} \left (-32 a B c-42 A b c+33 b^2 B\right )}{40 c^2}+\frac{(a+x (b+c x))^{3/2} \left (252 b^2 c (7 A c x-13 a B)+8 a b c^2 (343 A+333 B x)+16 a c^2 (64 a B-105 A c x)-42 b^3 c (35 A+33 B x)+1155 b^4 B\right )}{1920 c^4}+\frac{x^3 (a+x (b+c x))^{3/2} (14 A c-11 b B)}{12 c}+B x^4 (a+x (b+c x))^{3/2}}{7 c} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(A + B*x)*Sqrt[a + b*x + c*x^2],x]

[Out]

(((33*b^2*B - 42*A*b*c - 32*a*B*c)*x^2*(a + x*(b + c*x))^(3/2))/(40*c^2) + ((-11*b*B + 14*A*c)*x^3*(a + x*(b +
c*x))^(3/2))/(12*c) + B*x^4*(a + x*(b + c*x))^(3/2) + ((a + x*(b + c*x))^(3/2)*(1155*b^4*B - 42*b^3*c*(35*A +
33*B*x) + 8*a*b*c^2*(343*A + 333*B*x) + 16*a*c^2*(64*a*B - 105*A*c*x) + 252*b^2*c*(-13*a*B + 7*A*c*x)))/(1920
*c^4) + (7*(-33*b^5*B + 42*A*b^4*c + 120*a*b^3*B*c - 112*a*A*b^2*c^2 - 80*a^2*b*B*c^2 + 32*a^2*A*c^3)*(2*Sqrt[
c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)] - (b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]))/
(2048*c^(11/2)))/(7*c)

________________________________________________________________________________________

Maple [B]  time = 0.016, size = 872, normalized size = 2.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(B*x+A)*(c*x^2+b*x+a)^(1/2),x)

[Out]

35/128*B*b^3/c^(9/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-5/32*B*b/c^(7/2)*a^3*ln((1/2*b+c*x)/c^(1/
2)+(c*x^2+b*x+a)^(1/2))-63/512*B*b^5/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-3/20*A*b/c^2*x^2*(
c*x^2+b*x+a)^(3/2)-33/320*B*b^3/c^4*x*(c*x^2+b*x+a)^(3/2)-11/84*B*b/c^2*x^3*(c*x^2+b*x+a)^(3/2)+33/280*B*b^2/c
^3*x^2*(c*x^2+b*x+a)^(3/2)-5/64*B*b^2/c^4*a^2*(c*x^2+b*x+a)^(1/2)+15/128*B*b^4/c^5*a*(c*x^2+b*x+a)^(1/2)-39/16
0*B*b^2/c^4*a*(c*x^2+b*x+a)^(3/2)-33/512*B*b^5/c^5*(c*x^2+b*x+a)^(1/2)*x-4/35*B*a/c^2*x^2*(c*x^2+b*x+a)^(3/2)-
1/8*A*a/c^2*x*(c*x^2+b*x+a)^(3/2)+1/16*A*a^2/c^2*(c*x^2+b*x+a)^(1/2)*x+1/32*A*a^2/c^3*(c*x^2+b*x+a)^(1/2)*b+35
/256*A*b^4/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+21/160*A*b^2/c^3*x*(c*x^2+b*x+a)^(3/2)+21/256
*A*b^4/c^4*(c*x^2+b*x+a)^(1/2)*x+49/240*A*b/c^3*a*(c*x^2+b*x+a)^(3/2)-7/64*A*b^3/c^4*a*(c*x^2+b*x+a)^(1/2)-15/
64*A*b^2/c^(7/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/7*B*x^4*(c*x^2+b*x+a)^(3/2)/c+1/6*A*x^3*(c*
x^2+b*x+a)^(3/2)/c-7/64*A*b^3/c^4*(c*x^2+b*x+a)^(3/2)+21/512*A*b^5/c^5*(c*x^2+b*x+a)^(1/2)-21/1024*A*b^6/c^(11
/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/16*A*a^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))
+33/2048*B*b^7/c^(13/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+8/105*B*a^2/c^3*(c*x^2+b*x+a)^(3/2)+11/128
*B*b^4/c^5*(c*x^2+b*x+a)^(3/2)-33/1024*B*b^6/c^6*(c*x^2+b*x+a)^(1/2)+15/64*B*b^3/c^4*a*(c*x^2+b*x+a)^(1/2)*x-7
/32*A*b^2/c^3*a*(c*x^2+b*x+a)^(1/2)*x+111/560*B*b/c^3*a*x*(c*x^2+b*x+a)^(3/2)-5/32*B*b/c^3*a^2*(c*x^2+b*x+a)^(
1/2)*x

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)*(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 2.81441, size = 2040, normalized size = 5.56 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)*(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[1/430080*(105*(33*B*b^7 + 128*A*a^3*c^4 - 160*(2*B*a^3*b + 3*A*a^2*b^2)*c^3 + 280*(2*B*a^2*b^3 + A*a*b^4)*c^2
- 42*(6*B*a*b^5 + A*b^6)*c)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt
(c) - 4*a*c) + 4*(15360*B*c^7*x^6 - 3465*B*b^6*c + 1280*(B*b*c^6 + 14*A*c^7)*x^5 + 32*(256*B*a^3 + 791*A*a^2*b
)*c^4 - 128*(11*B*b^2*c^5 - 2*(12*B*a + 7*A*b)*c^6)*x^4 - 336*(103*B*a^2*b^2 + 70*A*a*b^3)*c^3 + 16*(99*B*b^3*
c^4 + 280*A*a*c^6 - 2*(158*B*a*b + 63*A*b^2)*c^5)*x^3 + 210*(104*B*a*b^4 + 21*A*b^5)*c^2 - 8*(231*B*b^4*c^3 +
8*(64*B*a^2 + 119*A*a*b)*c^5 - 6*(162*B*a*b^2 + 49*A*b^3)*c^4)*x^2 + 2*(1155*B*b^5*c^2 - 3360*A*a^2*c^5 + 16*(
397*B*a^2*b + 392*A*a*b^2)*c^4 - 42*(144*B*a*b^3 + 35*A*b^4)*c^3)*x)*sqrt(c*x^2 + b*x + a))/c^7, -1/215040*(10
5*(33*B*b^7 + 128*A*a^3*c^4 - 160*(2*B*a^3*b + 3*A*a^2*b^2)*c^3 + 280*(2*B*a^2*b^3 + A*a*b^4)*c^2 - 42*(6*B*a*
b^5 + A*b^6)*c)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) - 2*(1
5360*B*c^7*x^6 - 3465*B*b^6*c + 1280*(B*b*c^6 + 14*A*c^7)*x^5 + 32*(256*B*a^3 + 791*A*a^2*b)*c^4 - 128*(11*B*b
^2*c^5 - 2*(12*B*a + 7*A*b)*c^6)*x^4 - 336*(103*B*a^2*b^2 + 70*A*a*b^3)*c^3 + 16*(99*B*b^3*c^4 + 280*A*a*c^6 -
2*(158*B*a*b + 63*A*b^2)*c^5)*x^3 + 210*(104*B*a*b^4 + 21*A*b^5)*c^2 - 8*(231*B*b^4*c^3 + 8*(64*B*a^2 + 119*A
*a*b)*c^5 - 6*(162*B*a*b^2 + 49*A*b^3)*c^4)*x^2 + 2*(1155*B*b^5*c^2 - 3360*A*a^2*c^5 + 16*(397*B*a^2*b + 392*A
*a*b^2)*c^4 - 42*(144*B*a*b^3 + 35*A*b^4)*c^3)*x)*sqrt(c*x^2 + b*x + a))/c^7]

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(B*x+A)*(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral(x**4*(A + B*x)*sqrt(a + b*x + c*x**2), x)

________________________________________________________________________________________

Giac [A]  time = 1.37501, size = 559, normalized size = 1.52 \begin{align*} \frac{1}{107520} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \,{\left (12 \, B x + \frac{B b c^{5} + 14 \, A c^{6}}{c^{6}}\right )} x - \frac{11 \, B b^{2} c^{4} - 24 \, B a c^{5} - 14 \, A b c^{5}}{c^{6}}\right )} x + \frac{99 \, B b^{3} c^{3} - 316 \, B a b c^{4} - 126 \, A b^{2} c^{4} + 280 \, A a c^{5}}{c^{6}}\right )} x - \frac{231 \, B b^{4} c^{2} - 972 \, B a b^{2} c^{3} - 294 \, A b^{3} c^{3} + 512 \, B a^{2} c^{4} + 952 \, A a b c^{4}}{c^{6}}\right )} x + \frac{1155 \, B b^{5} c - 6048 \, B a b^{3} c^{2} - 1470 \, A b^{4} c^{2} + 6352 \, B a^{2} b c^{3} + 6272 \, A a b^{2} c^{3} - 3360 \, A a^{2} c^{4}}{c^{6}}\right )} x - \frac{3465 \, B b^{6} - 21840 \, B a b^{4} c - 4410 \, A b^{5} c + 34608 \, B a^{2} b^{2} c^{2} + 23520 \, A a b^{3} c^{2} - 8192 \, B a^{3} c^{3} - 25312 \, A a^{2} b c^{3}}{c^{6}}\right )} - \frac{{\left (33 \, B b^{7} - 252 \, B a b^{5} c - 42 \, A b^{6} c + 560 \, B a^{2} b^{3} c^{2} + 280 \, A a b^{4} c^{2} - 320 \, B a^{3} b c^{3} - 480 \, A a^{2} b^{2} c^{3} + 128 \, A a^{3} c^{4}\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{13}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(B*x+A)*(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

1/107520*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(12*B*x + (B*b*c^5 + 14*A*c^6)/c^6)*x - (11*B*b^2*c^4 - 24*B*a*
c^5 - 14*A*b*c^5)/c^6)*x + (99*B*b^3*c^3 - 316*B*a*b*c^4 - 126*A*b^2*c^4 + 280*A*a*c^5)/c^6)*x - (231*B*b^4*c^
2 - 972*B*a*b^2*c^3 - 294*A*b^3*c^3 + 512*B*a^2*c^4 + 952*A*a*b*c^4)/c^6)*x + (1155*B*b^5*c - 6048*B*a*b^3*c^2
- 1470*A*b^4*c^2 + 6352*B*a^2*b*c^3 + 6272*A*a*b^2*c^3 - 3360*A*a^2*c^4)/c^6)*x - (3465*B*b^6 - 21840*B*a*b^4
*c - 4410*A*b^5*c + 34608*B*a^2*b^2*c^2 + 23520*A*a*b^3*c^2 - 8192*B*a^3*c^3 - 25312*A*a^2*b*c^3)/c^6) - 1/204
8*(33*B*b^7 - 252*B*a*b^5*c - 42*A*b^6*c + 560*B*a^2*b^3*c^2 + 280*A*a*b^4*c^2 - 320*B*a^3*b*c^3 - 480*A*a^2*b
^2*c^3 + 128*A*a^3*c^4)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(13/2)