3.923 \(\int x^3 (A+B x) \left (a+b x+c x^2\right )^{3/2} \, dx\)
Optimal. Leaf size=356 \[ \frac{3 \left (b^2-4 a c\right )^2 \left (16 a^2 B c^2+64 a A b c^2-72 a b^2 B c-48 A b^3 c+33 b^4 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{32768 c^{13/2}}-\frac{3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (16 a^2 B c^2+64 a A b c^2-72 a b^2 B c-48 A b^3 c+33 b^4 B\right )}{16384 c^6}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (16 a^2 B c^2+64 a A b c^2-72 a b^2 B c-48 A b^3 c+33 b^4 B\right )}{2048 c^5}-\frac{\left (a+b x+c x^2\right )^{5/2} \left (-10 c x \left (-28 a B c-48 A b c+33 b^2 B\right )+256 a A c^2-372 a b B c-336 A b^2 c+231 b^3 B\right )}{4480 c^4}-\frac{x^2 \left (a+b x+c x^2\right )^{5/2} (11 b B-16 A c)}{112 c^2}+\frac{B x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c} \]
[Out]
(-3*(b^2 - 4*a*c)*(33*b^4*B - 48*A*b^3*c - 72*a*b^2*B*c + 64*a*A*b*c^2 + 16*a^2*
B*c^2)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(16384*c^6) + ((33*b^4*B - 48*A*b^3*c
- 72*a*b^2*B*c + 64*a*A*b*c^2 + 16*a^2*B*c^2)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2
))/(2048*c^5) - ((11*b*B - 16*A*c)*x^2*(a + b*x + c*x^2)^(5/2))/(112*c^2) + (B*x
^3*(a + b*x + c*x^2)^(5/2))/(8*c) - ((231*b^3*B - 336*A*b^2*c - 372*a*b*B*c + 25
6*a*A*c^2 - 10*c*(33*b^2*B - 48*A*b*c - 28*a*B*c)*x)*(a + b*x + c*x^2)^(5/2))/(4
480*c^4) + (3*(b^2 - 4*a*c)^2*(33*b^4*B - 48*A*b^3*c - 72*a*b^2*B*c + 64*a*A*b*c
^2 + 16*a^2*B*c^2)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(3276
8*c^(13/2))
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Rubi [A] time = 0.889551, antiderivative size = 356, 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
\[ \frac{3 \left (b^2-4 a c\right )^2 \left (16 a^2 B c^2+64 a A b c^2-72 a b^2 B c-48 A b^3 c+33 b^4 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{32768 c^{13/2}}-\frac{3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (16 a^2 B c^2+64 a A b c^2-72 a b^2 B c-48 A b^3 c+33 b^4 B\right )}{16384 c^6}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (16 a^2 B c^2+64 a A b c^2-72 a b^2 B c-48 A b^3 c+33 b^4 B\right )}{2048 c^5}-\frac{\left (a+b x+c x^2\right )^{5/2} \left (-10 c x \left (-28 a B c-48 A b c+33 b^2 B\right )+256 a A c^2-372 a b B c-336 A b^2 c+231 b^3 B\right )}{4480 c^4}-\frac{x^2 \left (a+b x+c x^2\right )^{5/2} (11 b B-16 A c)}{112 c^2}+\frac{B x^3 \left (a+b x+c x^2\right )^{5/2}}{8 c} \]
Antiderivative was successfully verified.
[In] Int[x^3*(A + B*x)*(a + b*x + c*x^2)^(3/2),x]
[Out]
(-3*(b^2 - 4*a*c)*(33*b^4*B - 48*A*b^3*c - 72*a*b^2*B*c + 64*a*A*b*c^2 + 16*a^2*
B*c^2)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(16384*c^6) + ((33*b^4*B - 48*A*b^3*c
- 72*a*b^2*B*c + 64*a*A*b*c^2 + 16*a^2*B*c^2)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2
))/(2048*c^5) - ((11*b*B - 16*A*c)*x^2*(a + b*x + c*x^2)^(5/2))/(112*c^2) + (B*x
^3*(a + b*x + c*x^2)^(5/2))/(8*c) - ((231*b^3*B - 336*A*b^2*c - 372*a*b*B*c + 25
6*a*A*c^2 - 10*c*(33*b^2*B - 48*A*b*c - 28*a*B*c)*x)*(a + b*x + c*x^2)^(5/2))/(4
480*c^4) + (3*(b^2 - 4*a*c)^2*(33*b^4*B - 48*A*b^3*c - 72*a*b^2*B*c + 64*a*A*b*c
^2 + 16*a^2*B*c^2)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(3276
8*c^(13/2))
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Rubi in Sympy [A] time = 104.01, size = 381, normalized size = 1.07 \[ \frac{B x^{3} \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{8 c} + \frac{x^{2} \left (16 A c - 11 B b\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{112 c^{2}} - \frac{\left (a + b x + c x^{2}\right )^{\frac{5}{2}} \left (96 A a c^{2} - 126 A b^{2} c - \frac{279 B a b c}{2} + \frac{693 B b^{3}}{8} - \frac{15 c x \left (- 48 A b c - 28 B a c + 33 B b^{2}\right )}{4}\right )}{1680 c^{4}} + \frac{\left (b + 2 c x\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (64 A a b c^{2} - 48 A b^{3} c + 16 B a^{2} c^{2} - 72 B a b^{2} c + 33 B b^{4}\right )}{2048 c^{5}} - \frac{3 \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}} \left (64 A a b c^{2} - 48 A b^{3} c + 16 B a^{2} c^{2} - 72 B a b^{2} c + 33 B b^{4}\right )}{16384 c^{6}} + \frac{3 \left (- 4 a c + b^{2}\right )^{2} \left (64 A a b c^{2} - 48 A b^{3} c + 16 B a^{2} c^{2} - 72 B a b^{2} c + 33 B b^{4}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{32768 c^{\frac{13}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(B*x+A)*(c*x**2+b*x+a)**(3/2),x)
[Out]
B*x**3*(a + b*x + c*x**2)**(5/2)/(8*c) + x**2*(16*A*c - 11*B*b)*(a + b*x + c*x**
2)**(5/2)/(112*c**2) - (a + b*x + c*x**2)**(5/2)*(96*A*a*c**2 - 126*A*b**2*c - 2
79*B*a*b*c/2 + 693*B*b**3/8 - 15*c*x*(-48*A*b*c - 28*B*a*c + 33*B*b**2)/4)/(1680
*c**4) + (b + 2*c*x)*(a + b*x + c*x**2)**(3/2)*(64*A*a*b*c**2 - 48*A*b**3*c + 16
*B*a**2*c**2 - 72*B*a*b**2*c + 33*B*b**4)/(2048*c**5) - 3*(b + 2*c*x)*(-4*a*c +
b**2)*sqrt(a + b*x + c*x**2)*(64*A*a*b*c**2 - 48*A*b**3*c + 16*B*a**2*c**2 - 72*
B*a*b**2*c + 33*B*b**4)/(16384*c**6) + 3*(-4*a*c + b**2)**2*(64*A*a*b*c**2 - 48*
A*b**3*c + 16*B*a**2*c**2 - 72*B*a*b**2*c + 33*B*b**4)*atanh((b + 2*c*x)/(2*sqrt
(c)*sqrt(a + b*x + c*x**2)))/(32768*c**(13/2))
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Mathematica [A] time = 0.820573, size = 412, normalized size = 1.16 \[ \frac{105 \left (b^2-4 a c\right )^2 \left (16 a^2 B c^2+64 a A b c^2-72 a b^2 B c-48 A b^3 c+33 b^4 B\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )-2 \sqrt{c} \sqrt{a+x (b+c x)} \left (16 b^3 c^2 \left (5103 a^2 B-52 a c x (28 A+15 B x)+8 c^2 x^3 (18 A+11 B x)\right )-32 b^2 c^3 \left (a^2 (2744 A+1181 B x)-4 a c x^2 (124 A+71 B x)+8 c^2 x^4 (8 A+5 B x)\right )-64 b c^3 \left (919 a^3 B-2 a^2 c x (292 A+151 B x)+8 a c^2 x^3 (22 A+13 B x)+80 c^3 x^5 (20 A+17 B x)\right )-128 c^4 \left (-a^3 (256 A+105 B x)+2 a^2 c x^2 (64 A+35 B x)+8 a c^2 x^4 (128 A+105 B x)+80 c^3 x^6 (8 A+7 B x)\right )+84 b^5 c (2 c x (20 A+11 B x)-365 a B)+24 b^4 c^2 \left (7 a (240 A+107 B x)-2 c x^2 (56 A+33 B x)\right )-210 b^6 c (24 A+11 B x)+3465 b^7 B\right )}{1146880 c^{13/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^3*(A + B*x)*(a + b*x + c*x^2)^(3/2),x]
[Out]
(-2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(3465*b^7*B - 210*b^6*c*(24*A + 11*B*x) + 84*b
^5*c*(-365*a*B + 2*c*x*(20*A + 11*B*x)) + 16*b^3*c^2*(5103*a^2*B + 8*c^2*x^3*(18
*A + 11*B*x) - 52*a*c*x*(28*A + 15*B*x)) - 128*c^4*(80*c^3*x^6*(8*A + 7*B*x) + 2
*a^2*c*x^2*(64*A + 35*B*x) + 8*a*c^2*x^4*(128*A + 105*B*x) - a^3*(256*A + 105*B*
x)) + 24*b^4*c^2*(-2*c*x^2*(56*A + 33*B*x) + 7*a*(240*A + 107*B*x)) - 64*b*c^3*(
919*a^3*B + 8*a*c^2*x^3*(22*A + 13*B*x) + 80*c^3*x^5*(20*A + 17*B*x) - 2*a^2*c*x
*(292*A + 151*B*x)) - 32*b^2*c^3*(8*c^2*x^4*(8*A + 5*B*x) - 4*a*c*x^2*(124*A + 7
1*B*x) + a^2*(2744*A + 1181*B*x))) + 105*(b^2 - 4*a*c)^2*(33*b^4*B - 48*A*b^3*c
- 72*a*b^2*B*c + 64*a*A*b*c^2 + 16*a^2*B*c^2)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a +
x*(b + c*x)]])/(1146880*c^(13/2))
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Maple [B] time = 0.019, size = 1061, normalized size = 3. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(B*x+A)*(c*x^2+b*x+a)^(3/2),x)
[Out]
21/512*A*b^5/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^3/
c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2-3/64*A*b^3/c^3*(c*x^2+b*
x+a)^(3/2)*x+9/512*A*b^5/c^4*(c*x^2+b*x+a)^(1/2)*x-3/64*A*b^4/c^4*(c*x^2+b*x+a)^
(1/2)*a+1/32*A*b^2/c^3*a*(c*x^2+b*x+a)^(3/2)-3/28*A*b/c^2*x*(c*x^2+b*x+a)^(5/2)-
1/16*B*a/c^2*x*(c*x^2+b*x+a)^(5/2)+1/64*B*a^2/c^2*(c*x^2+b*x+a)^(3/2)*x+1/128*B*
a^2/c^3*(c*x^2+b*x+a)^(3/2)*b+3/128*B*a^3/c^2*(c*x^2+b*x+a)^(1/2)*x+3/256*B*a^3/
c^3*(c*x^2+b*x+a)^(1/2)*b+33/448*B*b^2/c^3*x*(c*x^2+b*x+a)^(5/2)+33/1024*B*b^4/c
^4*(c*x^2+b*x+a)^(3/2)*x-99/8192*B*b^6/c^5*(c*x^2+b*x+a)^(1/2)*x+153/4096*B*b^5/
c^5*(c*x^2+b*x+a)^(1/2)*a-9/256*B*b^3/c^4*a*(c*x^2+b*x+a)^(3/2)-57/1024*B*b^3/c^
4*a^2*(c*x^2+b*x+a)^(1/2)-99/16384*B*b^7/c^6*(c*x^2+b*x+a)^(1/2)+3/128*B*a^4/c^(
5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+99/32768*B*b^8/c^(13/2)*ln((1/2
*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+3/40*A*b^2/c^3*(c*x^2+b*x+a)^(5/2)-3/32*A*b
^3/c^3*(c*x^2+b*x+a)^(1/2)*x*a-57/512*B*b^2/c^3*a^2*(c*x^2+b*x+a)^(1/2)*x+153/20
48*B*b^4/c^4*(c*x^2+b*x+a)^(1/2)*x*a-11/112*B*b/c^2*x^2*(c*x^2+b*x+a)^(5/2)+93/1
120*B*b/c^3*a*(c*x^2+b*x+a)^(5/2)-15/128*B*b^2/c^(7/2)*a^3*ln((1/2*b+c*x)/c^(1/2
)+(c*x^2+b*x+a)^(1/2))+105/1024*B*b^4/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+
a)^(1/2))*a^2-63/2048*B*b^6/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))
*a+3/32*A*b/c^(5/2)*a^3*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-9/128*B*b^2/
c^3*a*(c*x^2+b*x+a)^(3/2)*x+3/32*A*b/c^2*a^2*(c*x^2+b*x+a)^(1/2)*x-3/128*A*b^4/c
^4*(c*x^2+b*x+a)^(3/2)+9/1024*A*b^6/c^5*(c*x^2+b*x+a)^(1/2)-9/2048*A*b^7/c^(11/2
)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/7*A*x^2*(c*x^2+b*x+a)^(5/2)/c-2/
35*A*a/c^2*(c*x^2+b*x+a)^(5/2)-33/640*B*b^3/c^4*(c*x^2+b*x+a)^(5/2)+33/2048*B*b^
5/c^5*(c*x^2+b*x+a)^(3/2)+1/8*B*x^3*(c*x^2+b*x+a)^(5/2)/c+3/64*A*b^2/c^3*a^2*(c*
x^2+b*x+a)^(1/2)+1/16*A*b/c^2*a*(c*x^2+b*x+a)^(3/2)*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)^(3/2)*(B*x + A)*x^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.463332, 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)^(3/2)*(B*x + A)*x^3,x, algorithm="fricas")
[Out]
[1/2293760*(4*(71680*B*c^7*x^7 - 3465*B*b^7 - 32768*A*a^3*c^4 + 5120*(17*B*b*c^6
+ 16*A*c^7)*x^6 + 1280*(B*b^2*c^5 + 4*(21*B*a + 20*A*b)*c^6)*x^5 - 128*(11*B*b^
3*c^4 - 1024*A*a*c^6 - 4*(13*B*a*b + 4*A*b^2)*c^5)*x^4 + 64*(919*B*a^3*b + 1372*
A*a^2*b^2)*c^3 + 16*(99*B*b^4*c^3 + 16*(35*B*a^2 + 44*A*a*b)*c^5 - 8*(71*B*a*b^2
+ 18*A*b^3)*c^4)*x^3 - 1008*(81*B*a^2*b^3 + 40*A*a*b^4)*c^2 - 8*(231*B*b^5*c^2
- 2048*A*a^2*c^5 + 16*(151*B*a^2*b + 124*A*a*b^2)*c^4 - 24*(65*B*a*b^3 + 14*A*b^
4)*c^3)*x^2 + 420*(73*B*a*b^5 + 12*A*b^6)*c + 2*(1155*B*b^6*c - 64*(105*B*a^3 +
292*A*a^2*b)*c^4 + 16*(1181*B*a^2*b^2 + 728*A*a*b^3)*c^3 - 84*(107*B*a*b^4 + 20*
A*b^5)*c^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) + 105*(33*B*b^8 + 256*(B*a^4 + 4*A*
a^3*b)*c^4 - 1280*(B*a^3*b^2 + A*a^2*b^3)*c^3 + 224*(5*B*a^2*b^4 + 2*A*a*b^5)*c^
2 - 48*(7*B*a*b^6 + A*b^7)*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), 1/1146880*(2*(71680*B*c^7*x
^7 - 3465*B*b^7 - 32768*A*a^3*c^4 + 5120*(17*B*b*c^6 + 16*A*c^7)*x^6 + 1280*(B*b
^2*c^5 + 4*(21*B*a + 20*A*b)*c^6)*x^5 - 128*(11*B*b^3*c^4 - 1024*A*a*c^6 - 4*(13
*B*a*b + 4*A*b^2)*c^5)*x^4 + 64*(919*B*a^3*b + 1372*A*a^2*b^2)*c^3 + 16*(99*B*b^
4*c^3 + 16*(35*B*a^2 + 44*A*a*b)*c^5 - 8*(71*B*a*b^2 + 18*A*b^3)*c^4)*x^3 - 1008
*(81*B*a^2*b^3 + 40*A*a*b^4)*c^2 - 8*(231*B*b^5*c^2 - 2048*A*a^2*c^5 + 16*(151*B
*a^2*b + 124*A*a*b^2)*c^4 - 24*(65*B*a*b^3 + 14*A*b^4)*c^3)*x^2 + 420*(73*B*a*b^
5 + 12*A*b^6)*c + 2*(1155*B*b^6*c - 64*(105*B*a^3 + 292*A*a^2*b)*c^4 + 16*(1181*
B*a^2*b^2 + 728*A*a*b^3)*c^3 - 84*(107*B*a*b^4 + 20*A*b^5)*c^2)*x)*sqrt(c*x^2 +
b*x + a)*sqrt(-c) + 105*(33*B*b^8 + 256*(B*a^4 + 4*A*a^3*b)*c^4 - 1280*(B*a^3*b^
2 + A*a^2*b^3)*c^3 + 224*(5*B*a^2*b^4 + 2*A*a*b^5)*c^2 - 48*(7*B*a*b^6 + A*b^7)*
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. \[ \int x^{3} \left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(B*x+A)*(c*x**2+b*x+a)**(3/2),x)
[Out]
Integral(x**3*(A + B*x)*(a + b*x + c*x**2)**(3/2), x)
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GIAC/XCAS [A] time = 0.285392, size = 707, normalized size = 1.99 \[ \frac{1}{573440} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \,{\left (4 \,{\left (14 \, B c x + \frac{17 \, B b c^{7} + 16 \, A c^{8}}{c^{7}}\right )} x + \frac{B b^{2} c^{6} + 84 \, B a c^{7} + 80 \, A b c^{7}}{c^{7}}\right )} x - \frac{11 \, B b^{3} c^{5} - 52 \, B a b c^{6} - 16 \, A b^{2} c^{6} - 1024 \, A a c^{7}}{c^{7}}\right )} x + \frac{99 \, B b^{4} c^{4} - 568 \, B a b^{2} c^{5} - 144 \, A b^{3} c^{5} + 560 \, B a^{2} c^{6} + 704 \, A a b c^{6}}{c^{7}}\right )} x - \frac{231 \, B b^{5} c^{3} - 1560 \, B a b^{3} c^{4} - 336 \, A b^{4} c^{4} + 2416 \, B a^{2} b c^{5} + 1984 \, A a b^{2} c^{5} - 2048 \, A a^{2} c^{6}}{c^{7}}\right )} x + \frac{1155 \, B b^{6} c^{2} - 8988 \, B a b^{4} c^{3} - 1680 \, A b^{5} c^{3} + 18896 \, B a^{2} b^{2} c^{4} + 11648 \, A a b^{3} c^{4} - 6720 \, B a^{3} c^{5} - 18688 \, A a^{2} b c^{5}}{c^{7}}\right )} x - \frac{3465 \, B b^{7} c - 30660 \, B a b^{5} c^{2} - 5040 \, A b^{6} c^{2} + 81648 \, B a^{2} b^{3} c^{3} + 40320 \, A a b^{4} c^{3} - 58816 \, B a^{3} b c^{4} - 87808 \, A a^{2} b^{2} c^{4} + 32768 \, A a^{3} c^{5}}{c^{7}}\right )} - \frac{3 \,{\left (33 \, B b^{8} - 336 \, B a b^{6} c - 48 \, A b^{7} c + 1120 \, B a^{2} b^{4} c^{2} + 448 \, A a b^{5} c^{2} - 1280 \, B a^{3} b^{2} c^{3} - 1280 \, A a^{2} b^{3} c^{3} + 256 \, B a^{4} c^{4} + 1024 \, A a^{3} b c^{4}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{32768 \, c^{\frac{13}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^(3/2)*(B*x + A)*x^3,x, algorithm="giac")
[Out]
1/573440*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(4*(14*B*c*x + (17*B*b*c^7 + 16*A
*c^8)/c^7)*x + (B*b^2*c^6 + 84*B*a*c^7 + 80*A*b*c^7)/c^7)*x - (11*B*b^3*c^5 - 52
*B*a*b*c^6 - 16*A*b^2*c^6 - 1024*A*a*c^7)/c^7)*x + (99*B*b^4*c^4 - 568*B*a*b^2*c
^5 - 144*A*b^3*c^5 + 560*B*a^2*c^6 + 704*A*a*b*c^6)/c^7)*x - (231*B*b^5*c^3 - 15
60*B*a*b^3*c^4 - 336*A*b^4*c^4 + 2416*B*a^2*b*c^5 + 1984*A*a*b^2*c^5 - 2048*A*a^
2*c^6)/c^7)*x + (1155*B*b^6*c^2 - 8988*B*a*b^4*c^3 - 1680*A*b^5*c^3 + 18896*B*a^
2*b^2*c^4 + 11648*A*a*b^3*c^4 - 6720*B*a^3*c^5 - 18688*A*a^2*b*c^5)/c^7)*x - (34
65*B*b^7*c - 30660*B*a*b^5*c^2 - 5040*A*b^6*c^2 + 81648*B*a^2*b^3*c^3 + 40320*A*
a*b^4*c^3 - 58816*B*a^3*b*c^4 - 87808*A*a^2*b^2*c^4 + 32768*A*a^3*c^5)/c^7) - 3/
32768*(33*B*b^8 - 336*B*a*b^6*c - 48*A*b^7*c + 1120*B*a^2*b^4*c^2 + 448*A*a*b^5*
c^2 - 1280*B*a^3*b^2*c^3 - 1280*A*a^2*b^3*c^3 + 256*B*a^4*c^4 + 1024*A*a^3*b*c^4
)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(13/2)