3.924 \(\int x^2 (A+B x) (a+b x+c x^2)^{3/2} \, dx\)

Optimal. Leaf size=269 \[ \frac{\left (a+b x+c x^2\right )^{5/2} \left (-48 a B c-10 c x (9 b B-14 A c)-98 A b c+63 b^2 B\right )}{840 c^3}-\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right )}{384 c^4}+\frac{\left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right )}{1024 c^5}-\frac{\left (b^2-4 a c\right )^2 \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{11/2}}+\frac{B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c} \]

[Out]

((b^2 - 4*a*c)*(9*b^3*B - 14*A*b^2*c - 12*a*b*B*c + 8*a*A*c^2)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(1024*c^5) -
((9*b^3*B - 14*A*b^2*c - 12*a*b*B*c + 8*a*A*c^2)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(384*c^4) + (B*x^2*(a +
b*x + c*x^2)^(5/2))/(7*c) + ((63*b^2*B - 98*A*b*c - 48*a*B*c - 10*c*(9*b*B - 14*A*c)*x)*(a + b*x + c*x^2)^(5/
2))/(840*c^3) - ((b^2 - 4*a*c)^2*(9*b^3*B - 14*A*b^2*c - 12*a*b*B*c + 8*a*A*c^2)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c
]*Sqrt[a + b*x + c*x^2])])/(2048*c^(11/2))

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Rubi [A]  time = 0.248315, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{\left (a+b x+c x^2\right )^{5/2} \left (-48 a B c-10 c x (9 b B-14 A c)-98 A b c+63 b^2 B\right )}{840 c^3}-\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right )}{384 c^4}+\frac{\left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right )}{1024 c^5}-\frac{\left (b^2-4 a c\right )^2 \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{11/2}}+\frac{B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b^2 - 4*a*c)*(9*b^3*B - 14*A*b^2*c - 12*a*b*B*c + 8*a*A*c^2)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(1024*c^5) -
((9*b^3*B - 14*A*b^2*c - 12*a*b*B*c + 8*a*A*c^2)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(384*c^4) + (B*x^2*(a +
b*x + c*x^2)^(5/2))/(7*c) + ((63*b^2*B - 98*A*b*c - 48*a*B*c - 10*c*(9*b*B - 14*A*c)*x)*(a + b*x + c*x^2)^(5/
2))/(840*c^3) - ((b^2 - 4*a*c)^2*(9*b^3*B - 14*A*b^2*c - 12*a*b*B*c + 8*a*A*c^2)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c
]*Sqrt[a + b*x + c*x^2])])/(2048*c^(11/2))

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^2 (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac{B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac{\int x \left (-2 a B-\frac{1}{2} (9 b B-14 A c) x\right ) \left (a+b x+c x^2\right )^{3/2} \, dx}{7 c}\\ &=\frac{B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac{\left (63 b^2 B-98 A b c-48 a B c-10 c (9 b B-14 A c) x\right ) \left (a+b x+c x^2\right )^{5/2}}{840 c^3}-\frac{\left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{48 c^3}\\ &=-\frac{\left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^4}+\frac{B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac{\left (63 b^2 B-98 A b c-48 a B c-10 c (9 b B-14 A c) x\right ) \left (a+b x+c x^2\right )^{5/2}}{840 c^3}+\frac{\left (\left (b^2-4 a c\right ) \left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right )\right ) \int \sqrt{a+b x+c x^2} \, dx}{256 c^4}\\ &=\frac{\left (b^2-4 a c\right ) \left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{1024 c^5}-\frac{\left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^4}+\frac{B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac{\left (63 b^2 B-98 A b c-48 a B c-10 c (9 b B-14 A c) x\right ) \left (a+b x+c x^2\right )^{5/2}}{840 c^3}-\frac{\left (\left (b^2-4 a c\right )^2 \left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{2048 c^5}\\ &=\frac{\left (b^2-4 a c\right ) \left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{1024 c^5}-\frac{\left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^4}+\frac{B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac{\left (63 b^2 B-98 A b c-48 a B c-10 c (9 b B-14 A c) x\right ) \left (a+b x+c x^2\right )^{5/2}}{840 c^3}-\frac{\left (\left (b^2-4 a c\right )^2 \left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\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^5}\\ &=\frac{\left (b^2-4 a c\right ) \left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{1024 c^5}-\frac{\left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^4}+\frac{B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac{\left (63 b^2 B-98 A b c-48 a B c-10 c (9 b B-14 A c) x\right ) \left (a+b x+c x^2\right )^{5/2}}{840 c^3}-\frac{\left (b^2-4 a c\right )^2 \left (9 b^3 B-14 A b^2 c-12 a b B c+8 a A c^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.413515, size = 206, normalized size = 0.77 \[ \frac{-\frac{7 \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right ) \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left (4 c \left (5 a+2 c x^2\right )-3 b^2+8 b c x\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )}{6144 c^{9/2}}+\frac{(a+x (b+c x))^{5/2} \left (4 c (35 A c x-12 a B)-2 b c (49 A+45 B x)+63 b^2 B\right )}{120 c^2}+B x^2 (a+x (b+c x))^{5/2}}{7 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(B*x^2*(a + x*(b + c*x))^(5/2) + ((a + x*(b + c*x))^(5/2)*(63*b^2*B - 2*b*c*(49*A + 45*B*x) + 4*c*(-12*a*B + 3
5*A*c*x)))/(120*c^2) - (7*(9*b^3*B - 14*A*b^2*c - 12*a*b*B*c + 8*a*A*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)])]))/(6144*c^(9/2)))/(7*c)

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Maple [B]  time = 0.01, size = 838, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/28*B*b/c^2*x*(c*x^2+b*x+a)^(5/2)-3/64*B*b^3/c^3*(c*x^2+b*x+a)^(3/2)*x+9/512*B*b^5/c^4*(c*x^2+b*x+a)^(1/2)*x
-3/64*B*b^4/c^4*(c*x^2+b*x+a)^(1/2)*a+3/32*B*b/c^(5/2)*a^3*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-15/128*
B*b^3/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2+21/512*B*b^5/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x
^2+b*x+a)^(1/2))*a+3/64*B*b^2/c^3*a^2*(c*x^2+b*x+a)^(1/2)+7/96*A*b^2/c^2*(c*x^2+b*x+a)^(3/2)*x-7/256*A*b^4/c^3
*(c*x^2+b*x+a)^(1/2)*x+1/16*A*b^3/c^3*(c*x^2+b*x+a)^(1/2)*a+1/32*B*b^2/c^3*a*(c*x^2+b*x+a)^(3/2)-1/24*A*a/c*(c
*x^2+b*x+a)^(3/2)*x-1/48*A*a/c^2*(c*x^2+b*x+a)^(3/2)*b+9/64*A*b^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)
^(1/2))*a^2-15/256*A*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-1/32*A*a^2/c^2*(c*x^2+b*x+a)^(1
/2)*b-1/16*A*a^2/c*(c*x^2+b*x+a)^(1/2)*x+1/7*B*x^2*(c*x^2+b*x+a)^(5/2)/c+1/16*B*b/c^2*a*(c*x^2+b*x+a)^(3/2)*x+
3/32*B*b/c^2*a^2*(c*x^2+b*x+a)^(1/2)*x+1/8*A*b^2/c^2*(c*x^2+b*x+a)^(1/2)*x*a-3/32*B*b^3/c^3*(c*x^2+b*x+a)^(1/2
)*x*a+1/6*A*x*(c*x^2+b*x+a)^(5/2)/c-7/60*A*b/c^2*(c*x^2+b*x+a)^(5/2)+7/192*A*b^3/c^3*(c*x^2+b*x+a)^(3/2)-7/512
*A*b^5/c^4*(c*x^2+b*x+a)^(1/2)+7/1024*A*b^6/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-1/16*A*a^3/c^(
3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-9/2048*B*b^7/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1
/2))-2/35*B*a/c^2*(c*x^2+b*x+a)^(5/2)+3/40*B*b^2/c^3*(c*x^2+b*x+a)^(5/2)-3/128*B*b^4/c^4*(c*x^2+b*x+a)^(3/2)+9
/1024*B*b^6/c^5*(c*x^2+b*x+a)^(1/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.64605, size = 2017, normalized size = 7.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/430080*(105*(9*B*b^7 + 128*A*a^3*c^4 - 96*(2*B*a^3*b + 3*A*a^2*b^2)*c^3 + 120*(2*B*a^2*b^3 + A*a*b^4)*c^2 -
14*(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 + 945*B*b^6*c + 1280*(15*B*b*c^6 + 14*A*c^7)*x^5 - 96*(64*B*a^3 + 189*A*a^2*b)
*c^4 + 128*(3*B*b^2*c^5 + 2*(96*B*a + 91*A*b)*c^6)*x^4 + 112*(147*B*a^2*b^2 + 95*A*a*b^3)*c^3 - 16*(27*B*b^3*c
^4 - 1960*A*a*c^6 - 6*(22*B*a*b + 7*A*b^2)*c^5)*x^3 - 210*(36*B*a*b^4 + 7*A*b^5)*c^2 + 8*(63*B*b^4*c^3 + 24*(1
6*B*a^2 + 21*A*a*b)*c^5 - 2*(186*B*a*b^2 + 49*A*b^3)*c^4)*x^2 - 2*(315*B*b^5*c^2 - 3360*A*a^2*c^5 + 48*(73*B*a
^2*b + 63*A*a*b^2)*c^4 - 14*(156*B*a*b^3 + 35*A*b^4)*c^3)*x)*sqrt(c*x^2 + b*x + a))/c^6, 1/215040*(105*(9*B*b^
7 + 128*A*a^3*c^4 - 96*(2*B*a^3*b + 3*A*a^2*b^2)*c^3 + 120*(2*B*a^2*b^3 + A*a*b^4)*c^2 - 14*(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*(15360*B*c^7*
x^6 + 945*B*b^6*c + 1280*(15*B*b*c^6 + 14*A*c^7)*x^5 - 96*(64*B*a^3 + 189*A*a^2*b)*c^4 + 128*(3*B*b^2*c^5 + 2*
(96*B*a + 91*A*b)*c^6)*x^4 + 112*(147*B*a^2*b^2 + 95*A*a*b^3)*c^3 - 16*(27*B*b^3*c^4 - 1960*A*a*c^6 - 6*(22*B*
a*b + 7*A*b^2)*c^5)*x^3 - 210*(36*B*a*b^4 + 7*A*b^5)*c^2 + 8*(63*B*b^4*c^3 + 24*(16*B*a^2 + 21*A*a*b)*c^5 - 2*
(186*B*a*b^2 + 49*A*b^3)*c^4)*x^2 - 2*(315*B*b^5*c^2 - 3360*A*a^2*c^5 + 48*(73*B*a^2*b + 63*A*a*b^2)*c^4 - 14*
(156*B*a*b^3 + 35*A*b^4)*c^3)*x)*sqrt(c*x^2 + b*x + a))/c^6]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.34854, size = 570, normalized size = 2.12 \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 c x + \frac{15 \, B b c^{6} + 14 \, A c^{7}}{c^{6}}\right )} x + \frac{3 \, B b^{2} c^{5} + 192 \, B a c^{6} + 182 \, A b c^{6}}{c^{6}}\right )} x - \frac{27 \, B b^{3} c^{4} - 132 \, B a b c^{5} - 42 \, A b^{2} c^{5} - 1960 \, A a c^{6}}{c^{6}}\right )} x + \frac{63 \, B b^{4} c^{3} - 372 \, B a b^{2} c^{4} - 98 \, A b^{3} c^{4} + 384 \, B a^{2} c^{5} + 504 \, A a b c^{5}}{c^{6}}\right )} x - \frac{315 \, B b^{5} c^{2} - 2184 \, B a b^{3} c^{3} - 490 \, A b^{4} c^{3} + 3504 \, B a^{2} b c^{4} + 3024 \, A a b^{2} c^{4} - 3360 \, A a^{2} c^{5}}{c^{6}}\right )} x + \frac{945 \, B b^{6} c - 7560 \, B a b^{4} c^{2} - 1470 \, A b^{5} c^{2} + 16464 \, B a^{2} b^{2} c^{3} + 10640 \, A a b^{3} c^{3} - 6144 \, B a^{3} c^{4} - 18144 \, A a^{2} b c^{4}}{c^{6}}\right )} + \frac{{\left (9 \, B b^{7} - 84 \, B a b^{5} c - 14 \, A b^{6} c + 240 \, B a^{2} b^{3} c^{2} + 120 \, A a b^{4} c^{2} - 192 \, B a^{3} b c^{3} - 288 \, 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{11}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/107520*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(12*B*c*x + (15*B*b*c^6 + 14*A*c^7)/c^6)*x + (3*B*b^2*c^5 + 192
*B*a*c^6 + 182*A*b*c^6)/c^6)*x - (27*B*b^3*c^4 - 132*B*a*b*c^5 - 42*A*b^2*c^5 - 1960*A*a*c^6)/c^6)*x + (63*B*b
^4*c^3 - 372*B*a*b^2*c^4 - 98*A*b^3*c^4 + 384*B*a^2*c^5 + 504*A*a*b*c^5)/c^6)*x - (315*B*b^5*c^2 - 2184*B*a*b^
3*c^3 - 490*A*b^4*c^3 + 3504*B*a^2*b*c^4 + 3024*A*a*b^2*c^4 - 3360*A*a^2*c^5)/c^6)*x + (945*B*b^6*c - 7560*B*a
*b^4*c^2 - 1470*A*b^5*c^2 + 16464*B*a^2*b^2*c^3 + 10640*A*a*b^3*c^3 - 6144*B*a^3*c^4 - 18144*A*a^2*b*c^4)/c^6)
+ 1/2048*(9*B*b^7 - 84*B*a*b^5*c - 14*A*b^6*c + 240*B*a^2*b^3*c^2 + 120*A*a*b^4*c^2 - 192*B*a^3*b*c^3 - 288*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^(11/2)