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

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

[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 = 0.561258, 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 \[ \frac{\left (a+b x+c x^2\right )^{5/2} \left (-48 a B c-10 c x (9 b B-14 A c)-98 A b c+63 b^2 B\right )}{840 c^3}-\frac{\left (b^2-4 a c\right )^2 \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2048 c^{11/2}}+\frac{\left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right )}{1024 c^5}-\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right )}{384 c^4}+\frac{B x^2 \left (a+b x+c x^2\right )^{5/2}}{7 c} \]

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
))

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Rubi in Sympy [A]  time = 49.6611, size = 280, normalized size = 1.04 \[ \frac{B x^{2} \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{7 c} - \frac{\left (a + b x + c x^{2}\right )^{\frac{5}{2}} \left (12 B a c + \frac{7 b \left (14 A c - 9 B b\right )}{4} - \frac{5 c x \left (14 A c - 9 B b\right )}{2}\right )}{210 c^{3}} - \frac{\left (b + 2 c x\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (8 A a c^{2} - 14 A b^{2} c - 12 B a b c + 9 B b^{3}\right )}{384 c^{4}} + \frac{\left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}} \left (8 A a c^{2} - 14 A b^{2} c - 12 B a b c + 9 B b^{3}\right )}{1024 c^{5}} - \frac{\left (- 4 a c + b^{2}\right )^{2} \left (8 A a c^{2} - 14 A b^{2} c - 12 B a b c + 9 B b^{3}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{2048 c^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x**2*(a + b*x + c*x**2)**(5/2)/(7*c) - (a + b*x + c*x**2)**(5/2)*(12*B*a*c + 7
*b*(14*A*c - 9*B*b)/4 - 5*c*x*(14*A*c - 9*B*b)/2)/(210*c**3) - (b + 2*c*x)*(a +
b*x + c*x**2)**(3/2)*(8*A*a*c**2 - 14*A*b**2*c - 12*B*a*b*c + 9*B*b**3)/(384*c**
4) + (b + 2*c*x)*(-4*a*c + b**2)*sqrt(a + b*x + c*x**2)*(8*A*a*c**2 - 14*A*b**2*
c - 12*B*a*b*c + 9*B*b**3)/(1024*c**5) - (-4*a*c + b**2)**2*(8*A*a*c**2 - 14*A*b
**2*c - 12*B*a*b*c + 9*B*b**3)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**
2)))/(2048*c**(11/2))

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Mathematica [A]  time = 0.620383, size = 328, normalized size = 1.22 \[ \frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (48 b^2 c^2 \left (343 a^2 B-2 a c x (63 A+31 B x)+2 c^2 x^3 (7 A+4 B x)\right )+32 b c^3 \left (-3 a^2 (189 A+73 B x)+6 a c x^2 (21 A+11 B x)+8 c^2 x^4 (91 A+75 B x)\right )+64 c^3 \left (-96 a^3 B+3 a^2 c x (35 A+16 B x)+2 a c^2 x^3 (245 A+192 B x)+40 c^3 x^5 (7 A+6 B x)\right )+28 b^4 c (c x (35 A+18 B x)-270 a B)+16 b^3 c^2 \left (7 a (95 A+39 B x)-c x^2 (49 A+27 B x)\right )-210 b^5 c (7 A+3 B x)+945 b^6 B\right )-105 \left (b^2-4 a c\right )^2 \left (8 a A c^2-12 a b B c-14 A b^2 c+9 b^3 B\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{215040 c^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(945*b^6*B - 210*b^5*c*(7*A + 3*B*x) + 28*b^4*c
*(-270*a*B + c*x*(35*A + 18*B*x)) + 48*b^2*c^2*(343*a^2*B + 2*c^2*x^3*(7*A + 4*B
*x) - 2*a*c*x*(63*A + 31*B*x)) + 16*b^3*c^2*(-(c*x^2*(49*A + 27*B*x)) + 7*a*(95*
A + 39*B*x)) + 32*b*c^3*(6*a*c*x^2*(21*A + 11*B*x) - 3*a^2*(189*A + 73*B*x) + 8*
c^2*x^4*(91*A + 75*B*x)) + 64*c^3*(-96*a^3*B + 40*c^3*x^5*(7*A + 6*B*x) + 3*a^2*
c*x*(35*A + 16*B*x) + 2*a*c^2*x^3*(245*A + 192*B*x))) - 105*(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)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x
*(b + c*x)]])/(215040*c^(11/2))

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Maple [B]  time = 0.016, size = 838, normalized size = 3.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7/256*A*b^4/c^3*(c*x^2+b*x+a)^(1/2)*x+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)-9/2048*B*b^7/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^
2+b*x+a)^(1/2))+9/1024*B*b^6/c^5*(c*x^2+b*x+a)^(1/2)-2/35*B*a/c^2*(c*x^2+b*x+a)^
(5/2)+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))+7/96*A*b^2/c^2*(c*x^2+b*x+a)^(3/2)*x+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-3/32*B*b^3/c^3*(c*
x^2+b*x+a)^(1/2)*x*a+1/8*A*b^2/c^2*(c*x^2+b*x+a)^(1/2)*x*a+1/7*B*x^2*(c*x^2+b*x+
a)^(5/2)/c-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-1
/16*A*a^2/c*(c*x^2+b*x+a)^(1/2)*x-1/32*A*a^2/c^2*(c*x^2+b*x+a)^(1/2)*b+1/16*A*b^
3/c^3*(c*x^2+b*x+a)^(1/2)*a+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-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+1/32*B*b^2/
c^3*a*(c*x^2+b*x+a)^(3/2)+3/64*B*b^2/c^3*a^2*(c*x^2+b*x+a)^(1/2)+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

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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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.41847, 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^2,x, algorithm="fricas")

[Out]

[1/430080*(4*(15360*B*c^6*x^6 + 945*B*b^6 + 1280*(15*B*b*c^5 + 14*A*c^6)*x^5 + 1
28*(3*B*b^2*c^4 + 2*(96*B*a + 91*A*b)*c^5)*x^4 - 96*(64*B*a^3 + 189*A*a^2*b)*c^3
- 16*(27*B*b^3*c^3 - 1960*A*a*c^5 - 6*(22*B*a*b + 7*A*b^2)*c^4)*x^3 + 112*(147*
B*a^2*b^2 + 95*A*a*b^3)*c^2 + 8*(63*B*b^4*c^2 + 24*(16*B*a^2 + 21*A*a*b)*c^4 - 2
*(186*B*a*b^2 + 49*A*b^3)*c^3)*x^2 - 210*(36*B*a*b^4 + 7*A*b^5)*c - 2*(315*B*b^5
*c - 3360*A*a^2*c^4 + 48*(73*B*a^2*b + 63*A*a*b^2)*c^3 - 14*(156*B*a*b^3 + 35*A*
b^4)*c^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) + 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)*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^(11/2), 1/215040*(2*(15360*B*c^6*x^6 + 945*B*b^6 + 1280
*(15*B*b*c^5 + 14*A*c^6)*x^5 + 128*(3*B*b^2*c^4 + 2*(96*B*a + 91*A*b)*c^5)*x^4 -
96*(64*B*a^3 + 189*A*a^2*b)*c^3 - 16*(27*B*b^3*c^3 - 1960*A*a*c^5 - 6*(22*B*a*b
+ 7*A*b^2)*c^4)*x^3 + 112*(147*B*a^2*b^2 + 95*A*a*b^3)*c^2 + 8*(63*B*b^4*c^2 +
24*(16*B*a^2 + 21*A*a*b)*c^4 - 2*(186*B*a*b^2 + 49*A*b^3)*c^3)*x^2 - 210*(36*B*a
*b^4 + 7*A*b^5)*c - 2*(315*B*b^5*c - 3360*A*a^2*c^4 + 48*(73*B*a^2*b + 63*A*a*b^
2)*c^3 - 14*(156*B*a*b^3 + 35*A*b^4)*c^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) - 10
5*(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)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqr
t(c*x^2 + b*x + a)*c)))/(sqrt(-c)*c^5)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.28466, size = 570, normalized size = 2.12 \[ \frac{1}{107520} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \,{\left (12 \, B c x + \frac{15 \, B b c^{6} + 14 \, A c^{7}}{c^{6}}\right )} x + \frac{3 \, B b^{2} c^{5} + 192 \, B a c^{6} + 182 \, A b c^{6}}{c^{6}}\right )} x - \frac{27 \, B b^{3} c^{4} - 132 \, B a b c^{5} - 42 \, A b^{2} c^{5} - 1960 \, A a c^{6}}{c^{6}}\right )} x + \frac{63 \, B b^{4} c^{3} - 372 \, B a b^{2} c^{4} - 98 \, A b^{3} c^{4} + 384 \, B a^{2} c^{5} + 504 \, A a b c^{5}}{c^{6}}\right )} x - \frac{315 \, B b^{5} c^{2} - 2184 \, B a b^{3} c^{3} - 490 \, A b^{4} c^{3} + 3504 \, B a^{2} b c^{4} + 3024 \, A a b^{2} c^{4} - 3360 \, A a^{2} c^{5}}{c^{6}}\right )} x + \frac{945 \, B b^{6} c - 7560 \, B a b^{4} c^{2} - 1470 \, A b^{5} c^{2} + 16464 \, B a^{2} b^{2} c^{3} + 10640 \, A a b^{3} c^{3} - 6144 \, B a^{3} c^{4} - 18144 \, A a^{2} b c^{4}}{c^{6}}\right )} + \frac{{\left (9 \, B b^{7} - 84 \, B a b^{5} c - 14 \, A b^{6} c + 240 \, B a^{2} b^{3} c^{2} + 120 \, A a b^{4} c^{2} - 192 \, B a^{3} b c^{3} - 288 \, A a^{2} b^{2} c^{3} + 128 \, A a^{3} c^{4}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{2048 \, c^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(3/2)*(B*x + A)*x^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 - 1
32*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 - 2
184*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)*ln(abs(-2*(sqrt(c)*x - sqrt(c*
x^2 + b*x + a))*sqrt(c) - b))/c^(11/2)