3.936 \(\int x^3 (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=432 \[ -\frac{\left (b^2-4 a c\right )^3 \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{262144 c^{15/2}}+\frac{\left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right )}{131072 c^7}-\frac{\left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right )}{49152 c^6}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right )}{15360 c^5}-\frac{\left (a+b x+c x^2\right )^{7/2} \left (-14 c x \left (-108 a B c-220 A b c+143 b^2 B\right )+1280 a A c^2-1804 a b B c-1980 A b^2 c+1287 b^3 B\right )}{40320 c^4}-\frac{x^2 \left (a+b x+c x^2\right )^{7/2} (13 b B-20 A c)}{180 c^2}+\frac{B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c} \]

[Out]

((b^2 - 4*a*c)^2*(143*b^4*B - 220*A*b^3*c - 264*a*b^2*B*c + 240*a*A*b*c^2 + 48*a
^2*B*c^2)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(131072*c^7) - ((b^2 - 4*a*c)*(143*
b^4*B - 220*A*b^3*c - 264*a*b^2*B*c + 240*a*A*b*c^2 + 48*a^2*B*c^2)*(b + 2*c*x)*
(a + b*x + c*x^2)^(3/2))/(49152*c^6) + ((143*b^4*B - 220*A*b^3*c - 264*a*b^2*B*c
+ 240*a*A*b*c^2 + 48*a^2*B*c^2)*(b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(15360*c^5
) - ((13*b*B - 20*A*c)*x^2*(a + b*x + c*x^2)^(7/2))/(180*c^2) + (B*x^3*(a + b*x
+ c*x^2)^(7/2))/(10*c) - ((1287*b^3*B - 1980*A*b^2*c - 1804*a*b*B*c + 1280*a*A*c
^2 - 14*c*(143*b^2*B - 220*A*b*c - 108*a*B*c)*x)*(a + b*x + c*x^2)^(7/2))/(40320
*c^4) - ((b^2 - 4*a*c)^3*(143*b^4*B - 220*A*b^3*c - 264*a*b^2*B*c + 240*a*A*b*c^
2 + 48*a^2*B*c^2)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(26214
4*c^(15/2))

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Rubi [A]  time = 1.04274, antiderivative size = 432, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ -\frac{\left (b^2-4 a c\right )^3 \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{262144 c^{15/2}}+\frac{\left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right )}{131072 c^7}-\frac{\left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right )}{49152 c^6}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right )}{15360 c^5}-\frac{\left (a+b x+c x^2\right )^{7/2} \left (-14 c x \left (-108 a B c-220 A b c+143 b^2 B\right )+1280 a A c^2-1804 a b B c-1980 A b^2 c+1287 b^3 B\right )}{40320 c^4}-\frac{x^2 \left (a+b x+c x^2\right )^{7/2} (13 b B-20 A c)}{180 c^2}+\frac{B x^3 \left (a+b x+c x^2\right )^{7/2}}{10 c} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(A + B*x)*(a + b*x + c*x^2)^(5/2),x]

[Out]

((b^2 - 4*a*c)^2*(143*b^4*B - 220*A*b^3*c - 264*a*b^2*B*c + 240*a*A*b*c^2 + 48*a
^2*B*c^2)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(131072*c^7) - ((b^2 - 4*a*c)*(143*
b^4*B - 220*A*b^3*c - 264*a*b^2*B*c + 240*a*A*b*c^2 + 48*a^2*B*c^2)*(b + 2*c*x)*
(a + b*x + c*x^2)^(3/2))/(49152*c^6) + ((143*b^4*B - 220*A*b^3*c - 264*a*b^2*B*c
+ 240*a*A*b*c^2 + 48*a^2*B*c^2)*(b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(15360*c^5
) - ((13*b*B - 20*A*c)*x^2*(a + b*x + c*x^2)^(7/2))/(180*c^2) + (B*x^3*(a + b*x
+ c*x^2)^(7/2))/(10*c) - ((1287*b^3*B - 1980*A*b^2*c - 1804*a*b*B*c + 1280*a*A*c
^2 - 14*c*(143*b^2*B - 220*A*b*c - 108*a*B*c)*x)*(a + b*x + c*x^2)^(7/2))/(40320
*c^4) - ((b^2 - 4*a*c)^3*(143*b^4*B - 220*A*b^3*c - 264*a*b^2*B*c + 240*a*A*b*c^
2 + 48*a^2*B*c^2)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(26214
4*c^(15/2))

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Rubi in Sympy [A]  time = 146.438, size = 461, normalized size = 1.07 \[ \frac{B x^{3} \left (a + b x + c x^{2}\right )^{\frac{7}{2}}}{10 c} + \frac{x^{2} \left (20 A c - 13 B b\right ) \left (a + b x + c x^{2}\right )^{\frac{7}{2}}}{180 c^{2}} - \frac{\left (a + b x + c x^{2}\right )^{\frac{7}{2}} \left (160 A a c^{2} - \frac{495 A b^{2} c}{2} - \frac{451 B a b c}{2} + \frac{1287 B b^{3}}{8} - \frac{7 c x \left (- 220 A b c - 108 B a c + 143 B b^{2}\right )}{4}\right )}{5040 c^{4}} + \frac{\left (b + 2 c x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}} \left (240 A a b c^{2} - 220 A b^{3} c + 48 B a^{2} c^{2} - 264 B a b^{2} c + 143 B b^{4}\right )}{15360 c^{5}} - \frac{\left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (240 A a b c^{2} - 220 A b^{3} c + 48 B a^{2} c^{2} - 264 B a b^{2} c + 143 B b^{4}\right )}{49152 c^{6}} + \frac{\left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}} \left (240 A a b c^{2} - 220 A b^{3} c + 48 B a^{2} c^{2} - 264 B a b^{2} c + 143 B b^{4}\right )}{131072 c^{7}} - \frac{\left (- 4 a c + b^{2}\right )^{3} \left (240 A a b c^{2} - 220 A b^{3} c + 48 B a^{2} c^{2} - 264 B a b^{2} c + 143 B b^{4}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{262144 c^{\frac{15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(B*x+A)*(c*x**2+b*x+a)**(5/2),x)

[Out]

B*x**3*(a + b*x + c*x**2)**(7/2)/(10*c) + x**2*(20*A*c - 13*B*b)*(a + b*x + c*x*
*2)**(7/2)/(180*c**2) - (a + b*x + c*x**2)**(7/2)*(160*A*a*c**2 - 495*A*b**2*c/2
- 451*B*a*b*c/2 + 1287*B*b**3/8 - 7*c*x*(-220*A*b*c - 108*B*a*c + 143*B*b**2)/4
)/(5040*c**4) + (b + 2*c*x)*(a + b*x + c*x**2)**(5/2)*(240*A*a*b*c**2 - 220*A*b*
*3*c + 48*B*a**2*c**2 - 264*B*a*b**2*c + 143*B*b**4)/(15360*c**5) - (b + 2*c*x)*
(-4*a*c + b**2)*(a + b*x + c*x**2)**(3/2)*(240*A*a*b*c**2 - 220*A*b**3*c + 48*B*
a**2*c**2 - 264*B*a*b**2*c + 143*B*b**4)/(49152*c**6) + (b + 2*c*x)*(-4*a*c + b*
*2)**2*sqrt(a + b*x + c*x**2)*(240*A*a*b*c**2 - 220*A*b**3*c + 48*B*a**2*c**2 -
264*B*a*b**2*c + 143*B*b**4)/(131072*c**7) - (-4*a*c + b**2)**3*(240*A*a*b*c**2
- 220*A*b**3*c + 48*B*a**2*c**2 - 264*B*a*b**2*c + 143*B*b**4)*atanh((b + 2*c*x)
/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(262144*c**(15/2))

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Mathematica [A]  time = 1.36836, size = 585, normalized size = 1.35 \[ \frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (32 b^5 c^2 \left (77742 a^2 B-9 a c x (1715 A+869 B x)+22 c^2 x^3 (45 A+26 B x)\right )-320 b^4 c^3 \left (207 a^2 (49 A+20 B x)-a c x^2 (1116 A+605 B x)+4 c^2 x^4 (22 A+13 B x)\right )-640 b^3 c^3 \left (6885 a^3 B-3 a^2 c x (879 A+431 B x)+4 a c^2 x^3 (107 A+60 B x)-8 c^3 x^5 (5 A+3 B x)\right )+256 b^2 c^4 \left (5 a^3 (3663 A+1433 B x)-15 a^2 c x^2 (266 A+139 B x)+120 a c^2 x^4 (7 A+4 B x)+8 c^3 x^6 (3090 A+2681 B x)\right )+256 b c^4 \left (9295 a^4 B-10 a^3 c x (689 A+323 B x)+60 a^2 c^2 x^3 (41 A+22 B x)+16 a c^3 x^5 (3765 A+3181 B x)+224 c^4 x^7 (185 A+164 B x)\right )+512 c^5 \left (-5 a^4 (512 A+189 B x)+10 a^3 c x^2 (128 A+63 B x)+24 a^2 c^2 x^4 (800 A+651 B x)+16 a c^3 x^6 (1520 A+1323 B x)+896 c^4 x^8 (10 A+9 B x)\right )+1848 b^7 c (c x (25 A+13 B x)-305 a B)+48 b^6 c^2 \left (7 a (2425 A+1023 B x)-11 c x^2 (70 A+39 B x)\right )-2310 b^8 c (30 A+13 B x)+45045 b^9 B\right )-315 \left (b^2-4 a c\right )^3 \left (48 a^2 B c^2+240 a A b c^2-264 a b^2 B c-220 A b^3 c+143 b^4 B\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{82575360 c^{15/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(A + B*x)*(a + b*x + c*x^2)^(5/2),x]

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(45045*b^9*B - 2310*b^8*c*(30*A + 13*B*x) + 184
8*b^7*c*(-305*a*B + c*x*(25*A + 13*B*x)) - 640*b^3*c^3*(6885*a^3*B - 8*c^3*x^5*(
5*A + 3*B*x) + 4*a*c^2*x^3*(107*A + 60*B*x) - 3*a^2*c*x*(879*A + 431*B*x)) - 320
*b^4*c^3*(4*c^2*x^4*(22*A + 13*B*x) + 207*a^2*(49*A + 20*B*x) - a*c*x^2*(1116*A
+ 605*B*x)) + 32*b^5*c^2*(77742*a^2*B + 22*c^2*x^3*(45*A + 26*B*x) - 9*a*c*x*(17
15*A + 869*B*x)) + 48*b^6*c^2*(-11*c*x^2*(70*A + 39*B*x) + 7*a*(2425*A + 1023*B*
x)) + 512*c^5*(896*c^4*x^8*(10*A + 9*B*x) + 10*a^3*c*x^2*(128*A + 63*B*x) - 5*a^
4*(512*A + 189*B*x) + 24*a^2*c^2*x^4*(800*A + 651*B*x) + 16*a*c^3*x^6*(1520*A +
1323*B*x)) + 256*b^2*c^4*(120*a*c^2*x^4*(7*A + 4*B*x) - 15*a^2*c*x^2*(266*A + 13
9*B*x) + 5*a^3*(3663*A + 1433*B*x) + 8*c^3*x^6*(3090*A + 2681*B*x)) + 256*b*c^4*
(9295*a^4*B + 60*a^2*c^2*x^3*(41*A + 22*B*x) + 224*c^4*x^7*(185*A + 164*B*x) - 1
0*a^3*c*x*(689*A + 323*B*x) + 16*a*c^3*x^5*(3765*A + 3181*B*x))) - 315*(b^2 - 4*
a*c)^3*(143*b^4*B - 220*A*b^3*c - 264*a*b^2*B*c + 240*a*A*b*c^2 + 48*a^2*B*c^2)*
Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(82575360*c^(15/2))

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Maple [B]  time = 0.02, size = 1549, normalized size = 3.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(B*x+A)*(c*x^2+b*x+a)^(5/2),x)

[Out]

1/256*B*a^3/c^3*(c*x^2+b*x+a)^(3/2)*b+3/256*B*a^4/c^2*(c*x^2+b*x+a)^(1/2)*x+3/51
2*B*a^4/c^3*(c*x^2+b*x+a)^(1/2)*b+143/65536*B*b^8/c^6*(c*x^2+b*x+a)^(1/2)*x+1/16
0*B*a^2/c^2*(c*x^2+b*x+a)^(5/2)*x-3/80*B*a/c^2*x*(c*x^2+b*x+a)^(7/2)+495/65536*B
*b^8/c^(13/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+175/2048*B*b^4/c^(9/
2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3-315/8192*B*b^6/c^(11/2)*ln((1
/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2-75/1024*B*b^2/c^(7/2)*a^4*ln((1/2*b+c
*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+451/10080*B*b/c^3*a*(c*x^2+b*x+a)^(7/2)-13/180*
B*b/c^2*x^2*(c*x^2+b*x+a)^(7/2)+143/2880*B*b^2/c^3*x*(c*x^2+b*x+a)^(7/2)+143/768
0*B*b^4/c^4*(c*x^2+b*x+a)^(5/2)*x-143/24576*B*b^6/c^5*(c*x^2+b*x+a)^(3/2)*x+209/
12288*B*b^5/c^5*(c*x^2+b*x+a)^(3/2)*a+5/256*A*b^2/c^3*a^2*(c*x^2+b*x+a)^(3/2)+13
9/4096*B*b^5/c^5*(c*x^2+b*x+a)^(1/2)*a^2-11/1024*B*b^7/c^6*(c*x^2+b*x+a)^(1/2)*a
-55/32768*A*b^8/c^6*(c*x^2+b*x+a)^(1/2)-143/4480*B*b^3/c^4*(c*x^2+b*x+a)^(7/2)+1
43/15360*B*b^5/c^5*(c*x^2+b*x+a)^(5/2)-143/49152*B*b^7/c^6*(c*x^2+b*x+a)^(3/2)+1
43/131072*B*b^9/c^7*(c*x^2+b*x+a)^(1/2)-143/262144*B*b^10/c^(15/2)*ln((1/2*b+c*x
)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+3/256*B*a^5/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2
+b*x+a)^(1/2))-2/63*A*a/c^2*(c*x^2+b*x+a)^(7/2)+55/65536*A*b^9/c^(13/2)*ln((1/2*
b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/9*A*x^2*(c*x^2+b*x+a)^(7/2)/c+11/224*A*b^2
/c^3*(c*x^2+b*x+a)^(7/2)-11/768*A*b^4/c^4*(c*x^2+b*x+a)^(5/2)+55/12288*A*b^6/c^5
*(c*x^2+b*x+a)^(3/2)+1/128*B*a^3/c^2*(c*x^2+b*x+a)^(3/2)*x+15/512*A*b^2/c^3*a^3*
(c*x^2+b*x+a)^(1/2)+15/256*A*b/c^(5/2)*a^4*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^
(1/2))-45/4096*A*b^7/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-25/2
56*A*b^3/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3+105/2048*A*b^5/
c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2-11/144*A*b/c^2*x*(c*x^2+
b*x+a)^(7/2)-11/384*A*b^3/c^3*(c*x^2+b*x+a)^(5/2)*x+55/6144*A*b^5/c^4*(c*x^2+b*x
+a)^(3/2)*x-35/1536*A*b^4/c^4*(c*x^2+b*x+a)^(3/2)*a-55/16384*A*b^7/c^5*(c*x^2+b*
x+a)^(1/2)*x-85/2048*A*b^4/c^4*(c*x^2+b*x+a)^(1/2)*a^2+125/8192*A*b^6/c^5*(c*x^2
+b*x+a)^(1/2)*a+1/64*A*b^2/c^3*a*(c*x^2+b*x+a)^(5/2)+1/10*B*x^3*(c*x^2+b*x+a)^(7
/2)/c-35/768*A*b^3/c^3*(c*x^2+b*x+a)^(3/2)*x*a-85/1024*A*b^3/c^3*(c*x^2+b*x+a)^(
1/2)*x*a^2+15/256*A*b/c^2*a^3*(c*x^2+b*x+a)^(1/2)*x+125/4096*A*b^5/c^4*(c*x^2+b*
x+a)^(1/2)*x*a+139/2048*B*b^4/c^4*(c*x^2+b*x+a)^(1/2)*x*a^2-11/512*B*b^6/c^5*(c*
x^2+b*x+a)^(1/2)*x*a-23/512*B*b^2/c^3*a^2*(c*x^2+b*x+a)^(3/2)*x-9/128*B*b^2/c^3*
a^3*(c*x^2+b*x+a)^(1/2)*x+209/6144*B*b^4/c^4*(c*x^2+b*x+a)^(3/2)*x*a+1/32*A*b/c^
2*a*(c*x^2+b*x+a)^(5/2)*x+5/128*A*b/c^2*a^2*(c*x^2+b*x+a)^(3/2)*x-11/320*B*b^2/c
^3*a*(c*x^2+b*x+a)^(5/2)*x+1/320*B*a^2/c^3*(c*x^2+b*x+a)^(5/2)*b-11/640*B*b^3/c^
4*a*(c*x^2+b*x+a)^(5/2)-23/1024*B*b^3/c^4*a^2*(c*x^2+b*x+a)^(3/2)-9/256*B*b^3/c^
4*a^3*(c*x^2+b*x+a)^(1/2)

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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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.639032, 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^3,x, algorithm="fricas")

[Out]

[1/165150720*(4*(4128768*B*c^9*x^9 + 45045*B*b^9 - 1310720*A*a^4*c^5 + 229376*(4
1*B*b*c^8 + 20*A*c^9)*x^8 + 14336*(383*B*b^2*c^7 + 4*(189*B*a + 185*A*b)*c^8)*x^
7 + 1024*(15*B*b^3*c^6 + 12160*A*a*c^8 + 4*(3181*B*a*b + 1545*A*b^2)*c^7)*x^6 -
256*(65*B*b^4*c^5 - 48*(651*B*a^2 + 1255*A*a*b)*c^7 - 20*(24*B*a*b^2 + 5*A*b^3)*
c^6)*x^5 + 14080*(169*B*a^4*b + 333*A*a^3*b^2)*c^4 + 128*(143*B*b^5*c^4 + 76800*
A*a^2*c^7 + 240*(11*B*a^2*b + 7*A*a*b^2)*c^6 - 20*(60*B*a*b^3 + 11*A*b^4)*c^5)*x
^4 - 2880*(1530*B*a^3*b^3 + 1127*A*a^2*b^4)*c^3 - 16*(1287*B*b^6*c^3 - 960*(21*B
*a^3 + 41*A*a^2*b)*c^6 + 80*(417*B*a^2*b^2 + 214*A*a*b^3)*c^5 - 220*(55*B*a*b^4
+ 9*A*b^5)*c^4)*x^3 + 336*(7404*B*a^2*b^5 + 2425*A*a*b^6)*c^2 + 8*(3003*B*b^7*c^
2 + 81920*A*a^3*c^6 - 6080*(17*B*a^3*b + 21*A*a^2*b^2)*c^5 + 240*(431*B*a^2*b^3
+ 186*A*a*b^4)*c^4 - 132*(237*B*a*b^5 + 35*A*b^6)*c^3)*x^2 - 4620*(122*B*a*b^7 +
15*A*b^8)*c - 2*(15015*B*b^8*c + 1280*(189*B*a^4 + 689*A*a^3*b)*c^5 - 320*(2866
*B*a^3*b^2 + 2637*A*a^2*b^3)*c^4 + 720*(920*B*a^2*b^4 + 343*A*a*b^5)*c^3 - 924*(
186*B*a*b^6 + 25*A*b^7)*c^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) - 315*(143*B*b^10
- 3072*(B*a^5 + 5*A*a^4*b)*c^5 + 6400*(3*B*a^4*b^2 + 4*A*a^3*b^3)*c^4 - 4480*(5*
B*a^3*b^4 + 3*A*a^2*b^5)*c^3 + 1440*(7*B*a^2*b^6 + 2*A*a*b^7)*c^2 - 220*(9*B*a*b
^8 + A*b^9)*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^(15/2), 1/82575360*(2*(4128768*B*c^9*x^9 + 45045*B
*b^9 - 1310720*A*a^4*c^5 + 229376*(41*B*b*c^8 + 20*A*c^9)*x^8 + 14336*(383*B*b^2
*c^7 + 4*(189*B*a + 185*A*b)*c^8)*x^7 + 1024*(15*B*b^3*c^6 + 12160*A*a*c^8 + 4*(
3181*B*a*b + 1545*A*b^2)*c^7)*x^6 - 256*(65*B*b^4*c^5 - 48*(651*B*a^2 + 1255*A*a
*b)*c^7 - 20*(24*B*a*b^2 + 5*A*b^3)*c^6)*x^5 + 14080*(169*B*a^4*b + 333*A*a^3*b^
2)*c^4 + 128*(143*B*b^5*c^4 + 76800*A*a^2*c^7 + 240*(11*B*a^2*b + 7*A*a*b^2)*c^6
- 20*(60*B*a*b^3 + 11*A*b^4)*c^5)*x^4 - 2880*(1530*B*a^3*b^3 + 1127*A*a^2*b^4)*
c^3 - 16*(1287*B*b^6*c^3 - 960*(21*B*a^3 + 41*A*a^2*b)*c^6 + 80*(417*B*a^2*b^2 +
214*A*a*b^3)*c^5 - 220*(55*B*a*b^4 + 9*A*b^5)*c^4)*x^3 + 336*(7404*B*a^2*b^5 +
2425*A*a*b^6)*c^2 + 8*(3003*B*b^7*c^2 + 81920*A*a^3*c^6 - 6080*(17*B*a^3*b + 21*
A*a^2*b^2)*c^5 + 240*(431*B*a^2*b^3 + 186*A*a*b^4)*c^4 - 132*(237*B*a*b^5 + 35*A
*b^6)*c^3)*x^2 - 4620*(122*B*a*b^7 + 15*A*b^8)*c - 2*(15015*B*b^8*c + 1280*(189*
B*a^4 + 689*A*a^3*b)*c^5 - 320*(2866*B*a^3*b^2 + 2637*A*a^2*b^3)*c^4 + 720*(920*
B*a^2*b^4 + 343*A*a*b^5)*c^3 - 924*(186*B*a*b^6 + 25*A*b^7)*c^2)*x)*sqrt(c*x^2 +
b*x + a)*sqrt(-c) - 315*(143*B*b^10 - 3072*(B*a^5 + 5*A*a^4*b)*c^5 + 6400*(3*B*
a^4*b^2 + 4*A*a^3*b^3)*c^4 - 4480*(5*B*a^3*b^4 + 3*A*a^2*b^5)*c^3 + 1440*(7*B*a^
2*b^6 + 2*A*a*b^7)*c^2 - 220*(9*B*a*b^8 + A*b^9)*c)*arctan(1/2*(2*c*x + b)*sqrt(
-c)/(sqrt(c*x^2 + b*x + a)*c)))/(sqrt(-c)*c^7)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{3} \left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(B*x+A)*(c*x**2+b*x+a)**(5/2),x)

[Out]

Integral(x**3*(A + B*x)*(a + b*x + c*x**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.288399, size = 1038, normalized size = 2.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)*(B*x + A)*x^3,x, algorithm="giac")

[Out]

1/41287680*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(4*(14*(16*(18*B*c^2*x + (41*B*b
*c^10 + 20*A*c^11)/c^9)*x + (383*B*b^2*c^9 + 756*B*a*c^10 + 740*A*b*c^10)/c^9)*x
+ (15*B*b^3*c^8 + 12724*B*a*b*c^9 + 6180*A*b^2*c^9 + 12160*A*a*c^10)/c^9)*x - (
65*B*b^4*c^7 - 480*B*a*b^2*c^8 - 100*A*b^3*c^8 - 31248*B*a^2*c^9 - 60240*A*a*b*c
^9)/c^9)*x + (143*B*b^5*c^6 - 1200*B*a*b^3*c^7 - 220*A*b^4*c^7 + 2640*B*a^2*b*c^
8 + 1680*A*a*b^2*c^8 + 76800*A*a^2*c^9)/c^9)*x - (1287*B*b^6*c^5 - 12100*B*a*b^4
*c^6 - 1980*A*b^5*c^6 + 33360*B*a^2*b^2*c^7 + 17120*A*a*b^3*c^7 - 20160*B*a^3*c^
8 - 39360*A*a^2*b*c^8)/c^9)*x + (3003*B*b^7*c^4 - 31284*B*a*b^5*c^5 - 4620*A*b^6
*c^5 + 103440*B*a^2*b^3*c^6 + 44640*A*a*b^4*c^6 - 103360*B*a^3*b*c^7 - 127680*A*
a^2*b^2*c^7 + 81920*A*a^3*c^8)/c^9)*x - (15015*B*b^8*c^3 - 171864*B*a*b^6*c^4 -
23100*A*b^7*c^4 + 662400*B*a^2*b^4*c^5 + 246960*A*a*b^5*c^5 - 917120*B*a^3*b^2*c
^6 - 843840*A*a^2*b^3*c^6 + 241920*B*a^4*c^7 + 881920*A*a^3*b*c^7)/c^9)*x + (450
45*B*b^9*c^2 - 563640*B*a*b^7*c^3 - 69300*A*b^8*c^3 + 2487744*B*a^2*b^5*c^4 + 81
4800*A*a*b^6*c^4 - 4406400*B*a^3*b^3*c^5 - 3245760*A*a^2*b^4*c^5 + 2379520*B*a^4
*b*c^6 + 4688640*A*a^3*b^2*c^6 - 1310720*A*a^4*c^7)/c^9) + 1/262144*(143*B*b^10
- 1980*B*a*b^8*c - 220*A*b^9*c + 10080*B*a^2*b^6*c^2 + 2880*A*a*b^7*c^2 - 22400*
B*a^3*b^4*c^3 - 13440*A*a^2*b^5*c^3 + 19200*B*a^4*b^2*c^4 + 25600*A*a^3*b^3*c^4
- 3072*B*a^5*c^5 - 15360*A*a^4*b*c^5)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*sqrt(c) - b))/c^(15/2)