3.937 \(\int x^2 (A+B x) (a+b x+c x^2)^{5/2} \, dx\)

Optimal. Leaf size=333 \[ \frac{\left (a+b x+c x^2\right )^{7/2} \left (-64 a B c-14 c x (11 b B-18 A c)-162 A b c+99 b^2 B\right )}{2016 c^3}-\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right )}{768 c^4}+\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right )}{12288 c^5}-\frac{5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right )}{32768 c^6}+\frac{5 \left (b^2-4 a c\right )^3 \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{65536 c^{13/2}}+\frac{B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c} \]

[Out]

(-5*(b^2 - 4*a*c)^2*(11*b^3*B - 18*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])/(32768
*c^6) + (5*(b^2 - 4*a*c)*(11*b^3*B - 18*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))
/(12288*c^5) - ((11*b^3*B - 18*A*b^2*c - 12*a*b*B*c + 8*a*A*c^2)*(b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(768*c^4
) + (B*x^2*(a + b*x + c*x^2)^(7/2))/(9*c) + ((99*b^2*B - 162*A*b*c - 64*a*B*c - 14*c*(11*b*B - 18*A*c)*x)*(a +
b*x + c*x^2)^(7/2))/(2016*c^3) + (5*(b^2 - 4*a*c)^3*(11*b^3*B - 18*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])])/(65536*c^(13/2))

________________________________________________________________________________________

Rubi [A]  time = 0.309255, antiderivative size = 333, 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} \[ \frac{\left (a+b x+c x^2\right )^{7/2} \left (-64 a B c-14 c x (11 b B-18 A c)-162 A b c+99 b^2 B\right )}{2016 c^3}-\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right )}{768 c^4}+\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right )}{12288 c^5}-\frac{5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right )}{32768 c^6}+\frac{5 \left (b^2-4 a c\right )^3 \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{65536 c^{13/2}}+\frac{B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(A + B*x)*(a + b*x + c*x^2)^(5/2),x]

[Out]

(-5*(b^2 - 4*a*c)^2*(11*b^3*B - 18*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])/(32768
*c^6) + (5*(b^2 - 4*a*c)*(11*b^3*B - 18*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))
/(12288*c^5) - ((11*b^3*B - 18*A*b^2*c - 12*a*b*B*c + 8*a*A*c^2)*(b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(768*c^4
) + (B*x^2*(a + b*x + c*x^2)^(7/2))/(9*c) + ((99*b^2*B - 162*A*b*c - 64*a*B*c - 14*c*(11*b*B - 18*A*c)*x)*(a +
b*x + c*x^2)^(7/2))/(2016*c^3) + (5*(b^2 - 4*a*c)^3*(11*b^3*B - 18*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])])/(65536*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^2 (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac{B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac{\int x \left (-2 a B-\frac{1}{2} (11 b B-18 A c) x\right ) \left (a+b x+c x^2\right )^{5/2} \, dx}{9 c}\\ &=\frac{B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac{\left (99 b^2 B-162 A b c-64 a B c-14 c (11 b B-18 A c) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}-\frac{\left (11 b^3 B-18 A b^2 c-12 a b B c+8 a A c^2\right ) \int \left (a+b x+c x^2\right )^{5/2} \, dx}{64 c^3}\\ &=-\frac{\left (11 b^3 B-18 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 )^{5/2}}{768 c^4}+\frac{B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac{\left (99 b^2 B-162 A b c-64 a B c-14 c (11 b B-18 A c) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}+\frac{\left (5 \left (b^2-4 a c\right ) \left (11 b^3 B-18 A b^2 c-12 a b B c+8 a A c^2\right )\right ) \int \left (a+b x+c x^2\right )^{3/2} \, dx}{1536 c^4}\\ &=\frac{5 \left (b^2-4 a c\right ) \left (11 b^3 B-18 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}}{12288 c^5}-\frac{\left (11 b^3 B-18 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 )^{5/2}}{768 c^4}+\frac{B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac{\left (99 b^2 B-162 A b c-64 a B c-14 c (11 b B-18 A c) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}-\frac{\left (5 \left (b^2-4 a c\right )^2 \left (11 b^3 B-18 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}{8192 c^5}\\ &=-\frac{5 \left (b^2-4 a c\right )^2 \left (11 b^3 B-18 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}}{32768 c^6}+\frac{5 \left (b^2-4 a c\right ) \left (11 b^3 B-18 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}}{12288 c^5}-\frac{\left (11 b^3 B-18 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 )^{5/2}}{768 c^4}+\frac{B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac{\left (99 b^2 B-162 A b c-64 a B c-14 c (11 b B-18 A c) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}+\frac{\left (5 \left (b^2-4 a c\right )^3 \left (11 b^3 B-18 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}{65536 c^6}\\ &=-\frac{5 \left (b^2-4 a c\right )^2 \left (11 b^3 B-18 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}}{32768 c^6}+\frac{5 \left (b^2-4 a c\right ) \left (11 b^3 B-18 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}}{12288 c^5}-\frac{\left (11 b^3 B-18 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 )^{5/2}}{768 c^4}+\frac{B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac{\left (99 b^2 B-162 A b c-64 a B c-14 c (11 b B-18 A c) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}+\frac{\left (5 \left (b^2-4 a c\right )^3 \left (11 b^3 B-18 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 )}{32768 c^6}\\ &=-\frac{5 \left (b^2-4 a c\right )^2 \left (11 b^3 B-18 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}}{32768 c^6}+\frac{5 \left (b^2-4 a c\right ) \left (11 b^3 B-18 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}}{12288 c^5}-\frac{\left (11 b^3 B-18 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 )^{5/2}}{768 c^4}+\frac{B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c}+\frac{\left (99 b^2 B-162 A b c-64 a B c-14 c (11 b B-18 A c) x\right ) \left (a+b x+c x^2\right )^{7/2}}{2016 c^3}+\frac{5 \left (b^2-4 a c\right )^3 \left (11 b^3 B-18 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 )}{65536 c^{13/2}}\\ \end{align*}

Mathematica [A]  time = 0.488308, size = 254, normalized size = 0.76 \[ \frac{\frac{(a+x (b+c x))^{7/2} \left (4 c (63 A c x-16 a B)-2 b c (81 A+77 B x)+99 b^2 B\right )}{224 c^2}+\frac{3 \left (-8 a A c^2+12 a b B c+18 A b^2 c-11 b^3 B\right ) \left (256 c^{5/2} (b+2 c x) (a+x (b+c x))^{5/2}-5 \left (b^2-4 a c\right ) \left (16 c^{3/2} (b+2 c x) (a+x (b+c x))^{3/2}-3 \left (b^2-4 a c\right ) \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}-\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )\right )\right )}{65536 c^{11/2}}+B x^2 (a+x (b+c x))^{7/2}}{9 c} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(A + B*x)*(a + b*x + c*x^2)^(5/2),x]

[Out]

(B*x^2*(a + x*(b + c*x))^(7/2) + ((a + x*(b + c*x))^(7/2)*(99*b^2*B - 2*b*c*(81*A + 77*B*x) + 4*c*(-16*a*B + 6
3*A*c*x)))/(224*c^2) + (3*(-11*b^3*B + 18*A*b^2*c + 12*a*b*B*c - 8*a*A*c^2)*(256*c^(5/2)*(b + 2*c*x)*(a + x*(b
+ c*x))^(5/2) - 5*(b^2 - 4*a*c)*(16*c^(3/2)*(b + 2*c*x)*(a + x*(b + c*x))^(3/2) - 3*(b^2 - 4*a*c)*(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)])]))))/(
65536*c^(11/2)))/(9*c)

________________________________________________________________________________________

Maple [B]  time = 0.01, size = 1277, normalized size = 3.8 \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)^(5/2),x)

[Out]

55/1024*A*b^3/c^3*(c*x^2+b*x+a)^(1/2)*a^2+45/8192*A*b^6/c^4*(c*x^2+b*x+a)^(1/2)*x+1/9*B*x^2*(c*x^2+b*x+a)^(7/2
)/c+125/4096*B*b^5/c^4*(c*x^2+b*x+a)^(1/2)*x*a+15/256*B*b/c^2*a^3*(c*x^2+b*x+a)^(1/2)*x+5/128*B*b/c^2*a^2*(c*x
^2+b*x+a)^(3/2)*x+1/32*B*b/c^2*a*(c*x^2+b*x+a)^(5/2)*x+25/384*A*b^2/c^2*(c*x^2+b*x+a)^(3/2)*x*a-95/2048*A*b^4/
c^3*(c*x^2+b*x+a)^(1/2)*x*a+55/512*A*b^2/c^2*(c*x^2+b*x+a)^(1/2)*x*a^2-85/1024*B*b^3/c^3*(c*x^2+b*x+a)^(1/2)*x
*a^2-35/768*B*b^3/c^3*(c*x^2+b*x+a)^(3/2)*x*a-9/112*A*b/c^2*(c*x^2+b*x+a)^(7/2)+3/128*A*b^3/c^3*(c*x^2+b*x+a)^
(5/2)-15/2048*A*b^5/c^4*(c*x^2+b*x+a)^(3/2)+45/16384*A*b^7/c^5*(c*x^2+b*x+a)^(1/2)-5/128*A*a^4/c^(3/2)*ln((1/2
*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-45/32768*A*b^8/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/8*A
*x*(c*x^2+b*x+a)^(7/2)/c-2/63*B*a/c^2*(c*x^2+b*x+a)^(7/2)+55/65536*B*b^9/c^(13/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^
2+b*x+a)^(1/2))+11/224*B*b^2/c^3*(c*x^2+b*x+a)^(7/2)-11/768*B*b^4/c^4*(c*x^2+b*x+a)^(5/2)+55/12288*B*b^6/c^5*(
c*x^2+b*x+a)^(3/2)-55/32768*B*b^8/c^6*(c*x^2+b*x+a)^(1/2)-5/192*A*a^2/c*(c*x^2+b*x+a)^(3/2)*x-5/384*A*a^2/c^2*
(c*x^2+b*x+a)^(3/2)*b-5/128*A*a^3/c*(c*x^2+b*x+a)^(1/2)*x-5/256*A*a^3/c^2*(c*x^2+b*x+a)^(1/2)*b+3/64*A*b^2/c^2
*(c*x^2+b*x+a)^(5/2)*x-15/1024*A*b^4/c^3*(c*x^2+b*x+a)^(3/2)*x+25/768*A*b^3/c^3*(c*x^2+b*x+a)^(3/2)*a-75/1024*
A*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2+35/2048*A*b^6/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*
x^2+b*x+a)^(1/2))*a+15/128*A*b^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3-85/2048*B*b^4/c^4*(c*
x^2+b*x+a)^(1/2)*a^2-25/256*B*b^3/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3+15/256*B*b/c^(5/2)*a
^4*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+125/8192*B*b^6/c^5*(c*x^2+b*x+a)^(1/2)*a-11/144*B*b/c^2*x*(c*x^
2+b*x+a)^(7/2)+1/64*B*b^2/c^3*a*(c*x^2+b*x+a)^(5/2)+5/256*B*b^2/c^3*a^2*(c*x^2+b*x+a)^(3/2)+15/512*B*b^2/c^3*a
^3*(c*x^2+b*x+a)^(1/2)-11/384*B*b^3/c^3*(c*x^2+b*x+a)^(5/2)*x+55/6144*B*b^5/c^4*(c*x^2+b*x+a)^(3/2)*x-35/1536*
B*b^4/c^4*(c*x^2+b*x+a)^(3/2)*a-55/16384*B*b^7/c^5*(c*x^2+b*x+a)^(1/2)*x+105/2048*B*b^5/c^(9/2)*ln((1/2*b+c*x)
/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2-45/4096*B*b^7/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-95/4096
*A*b^5/c^4*(c*x^2+b*x+a)^(1/2)*a-1/48*A*a/c*(c*x^2+b*x+a)^(5/2)*x-1/96*A*a/c^2*(c*x^2+b*x+a)^(5/2)*b

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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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.73838, size = 3071, normalized size = 9.22 \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)^(5/2),x, algorithm="fricas")

[Out]

[1/8257536*(315*(11*B*b^9 - 512*A*a^4*c^5 + 768*(B*a^4*b + 2*A*a^3*b^2)*c^4 - 320*(4*B*a^3*b^3 + 3*A*a^2*b^4)*
c^3 + 224*(3*B*a^2*b^5 + A*a*b^6)*c^2 - 18*(8*B*a*b^7 + A*b^8)*c)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*s
qrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) + 4*(229376*B*c^9*x^8 - 3465*B*b^8*c + 14336*(37*B*b*c^8 + 1
8*A*c^9)*x^7 + 1024*(309*B*b^2*c^7 + 2*(304*B*a + 297*A*b)*c^8)*x^6 - 128*(512*B*a^4 + 1989*A*a^3*b)*c^5 + 256
*(5*B*b^3*c^6 + 2856*A*a*c^8 + 6*(502*B*a*b + 243*A*b^2)*c^7)*x^5 + 96*(2442*B*a^3*b^2 + 2359*A*a^2*b^3)*c^4 -
128*(11*B*b^4*c^5 - 24*(160*B*a^2 + 307*A*a*b)*c^7 - 6*(14*B*a*b^2 + 3*A*b^3)*c^6)*x^4 - 504*(322*B*a^2*b^4 +
125*A*a*b^5)*c^3 + 16*(99*B*b^5*c^4 + 39648*A*a^2*c^7 + 48*(41*B*a^2*b + 27*A*a*b^2)*c^6 - 2*(428*B*a*b^3 + 8
1*A*b^4)*c^5)*x^3 + 210*(194*B*a*b^6 + 27*A*b^7)*c^2 - 8*(231*B*b^6*c^3 - 32*(128*B*a^3 + 261*A*a^2*b)*c^6 + 4
8*(133*B*a^2*b^2 + 71*A*a*b^3)*c^5 - 18*(124*B*a*b^4 + 21*A*b^5)*c^4)*x^2 + 2*(1155*B*b^7*c^2 + 40320*A*a^3*c^
6 - 32*(1378*B*a^3*b + 1791*A*a^2*b^2)*c^5 + 24*(1758*B*a^2*b^3 + 791*A*a*b^4)*c^4 - 126*(98*B*a*b^5 + 15*A*b^
6)*c^3)*x)*sqrt(c*x^2 + b*x + a))/c^7, -1/4128768*(315*(11*B*b^9 - 512*A*a^4*c^5 + 768*(B*a^4*b + 2*A*a^3*b^2)
*c^4 - 320*(4*B*a^3*b^3 + 3*A*a^2*b^4)*c^3 + 224*(3*B*a^2*b^5 + A*a*b^6)*c^2 - 18*(8*B*a*b^7 + A*b^8)*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*(229376*B*c^9*x^8 - 346
5*B*b^8*c + 14336*(37*B*b*c^8 + 18*A*c^9)*x^7 + 1024*(309*B*b^2*c^7 + 2*(304*B*a + 297*A*b)*c^8)*x^6 - 128*(51
2*B*a^4 + 1989*A*a^3*b)*c^5 + 256*(5*B*b^3*c^6 + 2856*A*a*c^8 + 6*(502*B*a*b + 243*A*b^2)*c^7)*x^5 + 96*(2442*
B*a^3*b^2 + 2359*A*a^2*b^3)*c^4 - 128*(11*B*b^4*c^5 - 24*(160*B*a^2 + 307*A*a*b)*c^7 - 6*(14*B*a*b^2 + 3*A*b^3
)*c^6)*x^4 - 504*(322*B*a^2*b^4 + 125*A*a*b^5)*c^3 + 16*(99*B*b^5*c^4 + 39648*A*a^2*c^7 + 48*(41*B*a^2*b + 27*
A*a*b^2)*c^6 - 2*(428*B*a*b^3 + 81*A*b^4)*c^5)*x^3 + 210*(194*B*a*b^6 + 27*A*b^7)*c^2 - 8*(231*B*b^6*c^3 - 32*
(128*B*a^3 + 261*A*a^2*b)*c^6 + 48*(133*B*a^2*b^2 + 71*A*a*b^3)*c^5 - 18*(124*B*a*b^4 + 21*A*b^5)*c^4)*x^2 + 2
*(1155*B*b^7*c^2 + 40320*A*a^3*c^6 - 32*(1378*B*a^3*b + 1791*A*a^2*b^2)*c^5 + 24*(1758*B*a^2*b^3 + 791*A*a*b^4
)*c^4 - 126*(98*B*a*b^5 + 15*A*b^6)*c^3)*x)*sqrt(c*x^2 + b*x + a))/c^7]

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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{5}{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)**(5/2),x)

[Out]

Integral(x**2*(A + B*x)*(a + b*x + c*x**2)**(5/2), x)

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Giac [B]  time = 1.35669, size = 868, normalized size = 2.61 \begin{align*} \frac{1}{2064384} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (4 \,{\left (14 \,{\left (16 \, B c^{2} x + \frac{37 \, B b c^{9} + 18 \, A c^{10}}{c^{8}}\right )} x + \frac{309 \, B b^{2} c^{8} + 608 \, B a c^{9} + 594 \, A b c^{9}}{c^{8}}\right )} x + \frac{5 \, B b^{3} c^{7} + 3012 \, B a b c^{8} + 1458 \, A b^{2} c^{8} + 2856 \, A a c^{9}}{c^{8}}\right )} x - \frac{11 \, B b^{4} c^{6} - 84 \, B a b^{2} c^{7} - 18 \, A b^{3} c^{7} - 3840 \, B a^{2} c^{8} - 7368 \, A a b c^{8}}{c^{8}}\right )} x + \frac{99 \, B b^{5} c^{5} - 856 \, B a b^{3} c^{6} - 162 \, A b^{4} c^{6} + 1968 \, B a^{2} b c^{7} + 1296 \, A a b^{2} c^{7} + 39648 \, A a^{2} c^{8}}{c^{8}}\right )} x - \frac{231 \, B b^{6} c^{4} - 2232 \, B a b^{4} c^{5} - 378 \, A b^{5} c^{5} + 6384 \, B a^{2} b^{2} c^{6} + 3408 \, A a b^{3} c^{6} - 4096 \, B a^{3} c^{7} - 8352 \, A a^{2} b c^{7}}{c^{8}}\right )} x + \frac{1155 \, B b^{7} c^{3} - 12348 \, B a b^{5} c^{4} - 1890 \, A b^{6} c^{4} + 42192 \, B a^{2} b^{3} c^{5} + 18984 \, A a b^{4} c^{5} - 44096 \, B a^{3} b c^{6} - 57312 \, A a^{2} b^{2} c^{6} + 40320 \, A a^{3} c^{7}}{c^{8}}\right )} x - \frac{3465 \, B b^{8} c^{2} - 40740 \, B a b^{6} c^{3} - 5670 \, A b^{7} c^{3} + 162288 \, B a^{2} b^{4} c^{4} + 63000 \, A a b^{5} c^{4} - 234432 \, B a^{3} b^{2} c^{5} - 226464 \, A a^{2} b^{3} c^{5} + 65536 \, B a^{4} c^{6} + 254592 \, A a^{3} b c^{6}}{c^{8}}\right )} - \frac{5 \,{\left (11 \, B b^{9} - 144 \, B a b^{7} c - 18 \, A b^{8} c + 672 \, B a^{2} b^{5} c^{2} + 224 \, A a b^{6} c^{2} - 1280 \, B a^{3} b^{3} c^{3} - 960 \, A a^{2} b^{4} c^{3} + 768 \, B a^{4} b c^{4} + 1536 \, A a^{3} b^{2} c^{4} - 512 \, A a^{4} c^{5}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{65536 \, c^{\frac{13}{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)^(5/2),x, algorithm="giac")

[Out]

1/2064384*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(4*(14*(16*B*c^2*x + (37*B*b*c^9 + 18*A*c^10)/c^8)*x + (309*B*b
^2*c^8 + 608*B*a*c^9 + 594*A*b*c^9)/c^8)*x + (5*B*b^3*c^7 + 3012*B*a*b*c^8 + 1458*A*b^2*c^8 + 2856*A*a*c^9)/c^
8)*x - (11*B*b^4*c^6 - 84*B*a*b^2*c^7 - 18*A*b^3*c^7 - 3840*B*a^2*c^8 - 7368*A*a*b*c^8)/c^8)*x + (99*B*b^5*c^5
- 856*B*a*b^3*c^6 - 162*A*b^4*c^6 + 1968*B*a^2*b*c^7 + 1296*A*a*b^2*c^7 + 39648*A*a^2*c^8)/c^8)*x - (231*B*b^
6*c^4 - 2232*B*a*b^4*c^5 - 378*A*b^5*c^5 + 6384*B*a^2*b^2*c^6 + 3408*A*a*b^3*c^6 - 4096*B*a^3*c^7 - 8352*A*a^2
*b*c^7)/c^8)*x + (1155*B*b^7*c^3 - 12348*B*a*b^5*c^4 - 1890*A*b^6*c^4 + 42192*B*a^2*b^3*c^5 + 18984*A*a*b^4*c^
5 - 44096*B*a^3*b*c^6 - 57312*A*a^2*b^2*c^6 + 40320*A*a^3*c^7)/c^8)*x - (3465*B*b^8*c^2 - 40740*B*a*b^6*c^3 -
5670*A*b^7*c^3 + 162288*B*a^2*b^4*c^4 + 63000*A*a*b^5*c^4 - 234432*B*a^3*b^2*c^5 - 226464*A*a^2*b^3*c^5 + 6553
6*B*a^4*c^6 + 254592*A*a^3*b*c^6)/c^8) - 5/65536*(11*B*b^9 - 144*B*a*b^7*c - 18*A*b^8*c + 672*B*a^2*b^5*c^2 +
224*A*a*b^6*c^2 - 1280*B*a^3*b^3*c^3 - 960*A*a^2*b^4*c^3 + 768*B*a^4*b*c^4 + 1536*A*a^3*b^2*c^4 - 512*A*a^4*c^
5)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(13/2)