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

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

[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 - 6
4*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))

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Rubi [A]  time = 0.662723, 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 \[ \frac{\left (a+b x+c x^2\right )^{7/2} \left (-64 a B c-14 c x (11 b B-18 A c)-162 A b c+99 b^2 B\right )}{2016 c^3}+\frac{5 \left (b^2-4 a c\right )^3 \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{65536 c^{13/2}}-\frac{5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right )}{32768 c^6}+\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right )}{12288 c^5}-\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right )}{768 c^4}+\frac{B x^2 \left (a+b x+c x^2\right )^{7/2}}{9 c} \]

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 - 6
4*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))

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Rubi in Sympy [A]  time = 67.1689, size = 354, normalized size = 1.06 \[ \frac{B x^{2} \left (a + b x + c x^{2}\right )^{\frac{7}{2}}}{9 c} - \frac{\left (a + b x + c x^{2}\right )^{\frac{7}{2}} \left (16 B a c + \frac{9 b \left (18 A c - 11 B b\right )}{4} - \frac{7 c x \left (18 A c - 11 B b\right )}{2}\right )}{504 c^{3}} - \frac{\left (b + 2 c x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}} \left (8 A a c^{2} - 18 A b^{2} c - 12 B a b c + 11 B b^{3}\right )}{768 c^{4}} + \frac{5 \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (8 A a c^{2} - 18 A b^{2} c - 12 B a b c + 11 B b^{3}\right )}{12288 c^{5}} - \frac{5 \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}} \left (8 A a c^{2} - 18 A b^{2} c - 12 B a b c + 11 B b^{3}\right )}{32768 c^{6}} + \frac{5 \left (- 4 a c + b^{2}\right )^{3} \left (8 A a c^{2} - 18 A b^{2} c - 12 B a b c + 11 B b^{3}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{65536 c^{\frac{13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x**2*(a + b*x + c*x**2)**(7/2)/(9*c) - (a + b*x + c*x**2)**(7/2)*(16*B*a*c + 9
*b*(18*A*c - 11*B*b)/4 - 7*c*x*(18*A*c - 11*B*b)/2)/(504*c**3) - (b + 2*c*x)*(a
+ b*x + c*x**2)**(5/2)*(8*A*a*c**2 - 18*A*b**2*c - 12*B*a*b*c + 11*B*b**3)/(768*
c**4) + 5*(b + 2*c*x)*(-4*a*c + b**2)*(a + b*x + c*x**2)**(3/2)*(8*A*a*c**2 - 18
*A*b**2*c - 12*B*a*b*c + 11*B*b**3)/(12288*c**5) - 5*(b + 2*c*x)*(-4*a*c + b**2)
**2*sqrt(a + b*x + c*x**2)*(8*A*a*c**2 - 18*A*b**2*c - 12*B*a*b*c + 11*B*b**3)/(
32768*c**6) + 5*(-4*a*c + b**2)**3*(8*A*a*c**2 - 18*A*b**2*c - 12*B*a*b*c + 11*B
*b**3)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(65536*c**(13/2))

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Mathematica [A]  time = 1.09158, size = 481, normalized size = 1.44 \[ \frac{315 \left (b^2-4 a c\right )^3 \left (8 a A c^2-12 a b B c-18 A b^2 c+11 b^3 B\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )-2 \sqrt{c} \sqrt{a+x (b+c x)} \left (16 b^4 c^2 \left (10143 a^2 B-3 a c x (791 A+372 B x)+2 c^2 x^3 (81 A+44 B x)\right )-32 b^3 c^3 \left (3 a^2 (2359 A+879 B x)-4 a c x^2 (213 A+107 B x)+8 c^2 x^4 (9 A+5 B x)\right )-192 b^2 c^3 \left (1221 a^3 B-a^2 c x (597 A+266 B x)+4 a c^2 x^3 (27 A+14 B x)+8 c^3 x^5 (243 A+206 B x)\right )-128 b c^4 \left (-13 a^3 (153 A+53 B x)+6 a^2 c x^2 (87 A+41 B x)+24 a c^2 x^4 (307 A+251 B x)+16 c^3 x^6 (297 A+259 B x)\right )-256 c^4 \left (-256 a^4 B+a^3 c x (315 A+128 B x)+6 a^2 c^2 x^3 (413 A+320 B x)+8 a c^3 x^5 (357 A+304 B x)+112 c^4 x^7 (9 A+8 B x)\right )+84 b^6 c (c x (45 A+22 B x)-485 a B)+72 b^5 c^2 \left (7 a (125 A+49 B x)-2 c x^2 (21 A+11 B x)\right )-210 b^7 c (27 A+11 B x)+3465 b^8 B\right )}{4128768 c^{13/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(3465*b^8*B - 210*b^7*c*(27*A + 11*B*x) + 84*b
^6*c*(-485*a*B + c*x*(45*A + 22*B*x)) + 72*b^5*c^2*(-2*c*x^2*(21*A + 11*B*x) + 7
*a*(125*A + 49*B*x)) - 128*b*c^4*(6*a^2*c*x^2*(87*A + 41*B*x) - 13*a^3*(153*A +
53*B*x) + 24*a*c^2*x^4*(307*A + 251*B*x) + 16*c^3*x^6*(297*A + 259*B*x)) - 192*b
^2*c^3*(1221*a^3*B + 4*a*c^2*x^3*(27*A + 14*B*x) + 8*c^3*x^5*(243*A + 206*B*x) -
a^2*c*x*(597*A + 266*B*x)) - 256*c^4*(-256*a^4*B + 112*c^4*x^7*(9*A + 8*B*x) +
a^3*c*x*(315*A + 128*B*x) + 8*a*c^3*x^5*(357*A + 304*B*x) + 6*a^2*c^2*x^3*(413*A
+ 320*B*x)) + 16*b^4*c^2*(10143*a^2*B + 2*c^2*x^3*(81*A + 44*B*x) - 3*a*c*x*(79
1*A + 372*B*x)) - 32*b^3*c^3*(8*c^2*x^4*(9*A + 5*B*x) - 4*a*c*x^2*(213*A + 107*B
*x) + 3*a^2*(2359*A + 879*B*x))) + 315*(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)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(41287
68*c^(13/2))

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Maple [B]  time = 0.017, size = 1277, normalized size = 3.8 \[ \text{result too large to display} \]

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]

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)
-11/144*B*b/c^2*x*(c*x^2+b*x+a)^(7/2)-55/16384*B*b^7/c^5*(c*x^2+b*x+a)^(1/2)*x+1
5/256*B*b/c^(5/2)*a^4*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/64*B*b^2/c^3
*a*(c*x^2+b*x+a)^(5/2)-95/2048*A*b^4/c^3*(c*x^2+b*x+a)^(1/2)*x*a-85/1024*B*b^3/c
^3*(c*x^2+b*x+a)^(1/2)*x*a^2+125/4096*B*b^5/c^4*(c*x^2+b*x+a)^(1/2)*x*a+5/128*B*
b/c^2*a^2*(c*x^2+b*x+a)^(3/2)*x+15/256*B*b/c^2*a^3*(c*x^2+b*x+a)^(1/2)*x-35/768*
B*b^3/c^3*(c*x^2+b*x+a)^(3/2)*x*a+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+55/512*A*b^2/c^2*(c*x^2+b*x+a)^(1/2)*x*a^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-9/1
12*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)+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)-85/2048*B*b^4/c^4*(c*x^2+b*x+a)^(1/2)*a^2+125/8192*B*b^6/c^5*(c
*x^2+b*x+a)^(1/2)*a-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-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^3-55/32768*B*b^8/c^6*(c*x^2+b*x+a)
^(1/2)-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))+1/9*B*x^2*(c*x^2+b*x+a)^(7/2)/c-1/96*A*a/c^2*(c*x^2+
b*x+a)^(5/2)*b+3/64*A*b^2/c^2*(c*x^2+b*x+a)^(5/2)*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+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+25/768*A*b^3/c^3*(c*x^2+b*x+a)^(3/2)*a-5/192*A*a^2/c*(c*x^2+
b*x+a)^(3/2)*x+55/1024*A*b^3/c^3*(c*x^2+b*x+a)^(1/2)*a^2-95/4096*A*b^5/c^4*(c*x^
2+b*x+a)^(1/2)*a+45/8192*A*b^6/c^4*(c*x^2+b*x+a)^(1/2)*x-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-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^2-5/384*A*a^2/c^2*(c*x^2+b*x+a)^(3/2)*b-1
5/1024*A*b^4/c^3*(c*x^2+b*x+a)^(3/2)*x+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^3-1/48*A*a/c*(c*x^2+b*x+a)^(5/2)*x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)*(B*x + A)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.516283, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)*(B*x + A)*x^2,x, algorithm="fricas")

[Out]

[1/8257536*(4*(229376*B*c^8*x^8 - 3465*B*b^8 + 14336*(37*B*b*c^7 + 18*A*c^8)*x^7
+ 1024*(309*B*b^2*c^6 + 2*(304*B*a + 297*A*b)*c^7)*x^6 + 256*(5*B*b^3*c^5 + 285
6*A*a*c^7 + 6*(502*B*a*b + 243*A*b^2)*c^6)*x^5 - 128*(512*B*a^4 + 1989*A*a^3*b)*
c^4 - 128*(11*B*b^4*c^4 - 24*(160*B*a^2 + 307*A*a*b)*c^6 - 6*(14*B*a*b^2 + 3*A*b
^3)*c^5)*x^4 + 96*(2442*B*a^3*b^2 + 2359*A*a^2*b^3)*c^3 + 16*(99*B*b^5*c^3 + 396
48*A*a^2*c^6 + 48*(41*B*a^2*b + 27*A*a*b^2)*c^5 - 2*(428*B*a*b^3 + 81*A*b^4)*c^4
)*x^3 - 504*(322*B*a^2*b^4 + 125*A*a*b^5)*c^2 - 8*(231*B*b^6*c^2 - 32*(128*B*a^3
+ 261*A*a^2*b)*c^5 + 48*(133*B*a^2*b^2 + 71*A*a*b^3)*c^4 - 18*(124*B*a*b^4 + 21
*A*b^5)*c^3)*x^2 + 210*(194*B*a*b^6 + 27*A*b^7)*c + 2*(1155*B*b^7*c + 40320*A*a^
3*c^5 - 32*(1378*B*a^3*b + 1791*A*a^2*b^2)*c^4 + 24*(1758*B*a^2*b^3 + 791*A*a*b^
4)*c^3 - 126*(98*B*a*b^5 + 15*A*b^6)*c^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) + 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)
*log(-4*(2*c^2*x + b*c)*sqrt(c*x^2 + b*x + a) - (8*c^2*x^2 + 8*b*c*x + b^2 + 4*a
*c)*sqrt(c)))/c^(13/2), 1/4128768*(2*(229376*B*c^8*x^8 - 3465*B*b^8 + 14336*(37*
B*b*c^7 + 18*A*c^8)*x^7 + 1024*(309*B*b^2*c^6 + 2*(304*B*a + 297*A*b)*c^7)*x^6 +
256*(5*B*b^3*c^5 + 2856*A*a*c^7 + 6*(502*B*a*b + 243*A*b^2)*c^6)*x^5 - 128*(512
*B*a^4 + 1989*A*a^3*b)*c^4 - 128*(11*B*b^4*c^4 - 24*(160*B*a^2 + 307*A*a*b)*c^6
- 6*(14*B*a*b^2 + 3*A*b^3)*c^5)*x^4 + 96*(2442*B*a^3*b^2 + 2359*A*a^2*b^3)*c^3 +
16*(99*B*b^5*c^3 + 39648*A*a^2*c^6 + 48*(41*B*a^2*b + 27*A*a*b^2)*c^5 - 2*(428*
B*a*b^3 + 81*A*b^4)*c^4)*x^3 - 504*(322*B*a^2*b^4 + 125*A*a*b^5)*c^2 - 8*(231*B*
b^6*c^2 - 32*(128*B*a^3 + 261*A*a^2*b)*c^5 + 48*(133*B*a^2*b^2 + 71*A*a*b^3)*c^4
- 18*(124*B*a*b^4 + 21*A*b^5)*c^3)*x^2 + 210*(194*B*a*b^6 + 27*A*b^7)*c + 2*(11
55*B*b^7*c + 40320*A*a^3*c^5 - 32*(1378*B*a^3*b + 1791*A*a^2*b^2)*c^4 + 24*(1758
*B*a^2*b^3 + 791*A*a*b^4)*c^3 - 126*(98*B*a*b^5 + 15*A*b^6)*c^2)*x)*sqrt(c*x^2 +
b*x + a)*sqrt(-c) + 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 - 1
8*(8*B*a*b^7 + A*b^8)*c)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*
c)))/(sqrt(-c)*c^6)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{2} \left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}\, dx \]

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/XCAS [A]  time = 0.291733, size = 868, normalized size = 2.61 \[ \frac{1}{2064384} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (4 \,{\left (14 \,{\left (16 \, B c^{2} x + \frac{37 \, B b c^{9} + 18 \, A c^{10}}{c^{8}}\right )} x + \frac{309 \, B b^{2} c^{8} + 608 \, B a c^{9} + 594 \, A b c^{9}}{c^{8}}\right )} x + \frac{5 \, B b^{3} c^{7} + 3012 \, B a b c^{8} + 1458 \, A b^{2} c^{8} + 2856 \, A a c^{9}}{c^{8}}\right )} x - \frac{11 \, B b^{4} c^{6} - 84 \, B a b^{2} c^{7} - 18 \, A b^{3} c^{7} - 3840 \, B a^{2} c^{8} - 7368 \, A a b c^{8}}{c^{8}}\right )} x + \frac{99 \, B b^{5} c^{5} - 856 \, B a b^{3} c^{6} - 162 \, A b^{4} c^{6} + 1968 \, B a^{2} b c^{7} + 1296 \, A a b^{2} c^{7} + 39648 \, A a^{2} c^{8}}{c^{8}}\right )} x - \frac{231 \, B b^{6} c^{4} - 2232 \, B a b^{4} c^{5} - 378 \, A b^{5} c^{5} + 6384 \, B a^{2} b^{2} c^{6} + 3408 \, A a b^{3} c^{6} - 4096 \, B a^{3} c^{7} - 8352 \, A a^{2} b c^{7}}{c^{8}}\right )} x + \frac{1155 \, B b^{7} c^{3} - 12348 \, B a b^{5} c^{4} - 1890 \, A b^{6} c^{4} + 42192 \, B a^{2} b^{3} c^{5} + 18984 \, A a b^{4} c^{5} - 44096 \, B a^{3} b c^{6} - 57312 \, A a^{2} b^{2} c^{6} + 40320 \, A a^{3} c^{7}}{c^{8}}\right )} x - \frac{3465 \, B b^{8} c^{2} - 40740 \, B a b^{6} c^{3} - 5670 \, A b^{7} c^{3} + 162288 \, B a^{2} b^{4} c^{4} + 63000 \, A a b^{5} c^{4} - 234432 \, B a^{3} b^{2} c^{5} - 226464 \, A a^{2} b^{3} c^{5} + 65536 \, B a^{4} c^{6} + 254592 \, A a^{3} b c^{6}}{c^{8}}\right )} - \frac{5 \,{\left (11 \, B b^{9} - 144 \, B a b^{7} c - 18 \, A b^{8} c + 672 \, B a^{2} b^{5} c^{2} + 224 \, A a b^{6} c^{2} - 1280 \, B a^{3} b^{3} c^{3} - 960 \, A a^{2} b^{4} c^{3} + 768 \, B a^{4} b c^{4} + 1536 \, A a^{3} b^{2} c^{4} - 512 \, A a^{4} c^{5}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{65536 \, c^{\frac{13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)*(B*x + A)*x^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)*ln(abs(-2*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(13/2)