3.911 \(\int x^3 (A+B x) \sqrt{a+b x+c x^2} \, dx\)

Optimal. Leaf size=280 \[ -\frac{\left (b^2-4 a c\right ) \left (16 a^2 B c^2+48 a A b c^2-56 a b^2 B c-28 A b^3 c+21 b^4 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{11/2}}+\frac{(b+2 c x) \sqrt{a+b x+c x^2} \left (16 a^2 B c^2+48 a A b c^2-56 a b^2 B c-28 A b^3 c+21 b^4 B\right )}{512 c^5}-\frac{\left (a+b x+c x^2\right )^{3/2} \left (-6 c x \left (-20 a B c-28 A b c+21 b^2 B\right )+128 a A c^2-196 a b B c-140 A b^2 c+105 b^3 B\right )}{960 c^4}-\frac{x^2 \left (a+b x+c x^2\right )^{3/2} (3 b B-4 A c)}{20 c^2}+\frac{B x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c} \]

[Out]

((21*b^4*B - 28*A*b^3*c - 56*a*b^2*B*c + 48*a*A*b*c^2 + 16*a^2*B*c^2)*(b + 2*c*x
)*Sqrt[a + b*x + c*x^2])/(512*c^5) - ((3*b*B - 4*A*c)*x^2*(a + b*x + c*x^2)^(3/2
))/(20*c^2) + (B*x^3*(a + b*x + c*x^2)^(3/2))/(6*c) - ((105*b^3*B - 140*A*b^2*c
- 196*a*b*B*c + 128*a*A*c^2 - 6*c*(21*b^2*B - 28*A*b*c - 20*a*B*c)*x)*(a + b*x +
c*x^2)^(3/2))/(960*c^4) - ((b^2 - 4*a*c)*(21*b^4*B - 28*A*b^3*c - 56*a*b^2*B*c
+ 48*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])])/(1024*c^(11/2))

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Rubi [A]  time = 0.759846, antiderivative size = 280, 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 \[ -\frac{\left (b^2-4 a c\right ) \left (16 a^2 B c^2+48 a A b c^2-56 a b^2 B c-28 A b^3 c+21 b^4 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{11/2}}+\frac{(b+2 c x) \sqrt{a+b x+c x^2} \left (16 a^2 B c^2+48 a A b c^2-56 a b^2 B c-28 A b^3 c+21 b^4 B\right )}{512 c^5}-\frac{\left (a+b x+c x^2\right )^{3/2} \left (-6 c x \left (-20 a B c-28 A b c+21 b^2 B\right )+128 a A c^2-196 a b B c-140 A b^2 c+105 b^3 B\right )}{960 c^4}-\frac{x^2 \left (a+b x+c x^2\right )^{3/2} (3 b B-4 A c)}{20 c^2}+\frac{B x^3 \left (a+b x+c x^2\right )^{3/2}}{6 c} \]

Antiderivative was successfully verified.

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

[Out]

((21*b^4*B - 28*A*b^3*c - 56*a*b^2*B*c + 48*a*A*b*c^2 + 16*a^2*B*c^2)*(b + 2*c*x
)*Sqrt[a + b*x + c*x^2])/(512*c^5) - ((3*b*B - 4*A*c)*x^2*(a + b*x + c*x^2)^(3/2
))/(20*c^2) + (B*x^3*(a + b*x + c*x^2)^(3/2))/(6*c) - ((105*b^3*B - 140*A*b^2*c
- 196*a*b*B*c + 128*a*A*c^2 - 6*c*(21*b^2*B - 28*A*b*c - 20*a*B*c)*x)*(a + b*x +
c*x^2)^(3/2))/(960*c^4) - ((b^2 - 4*a*c)*(21*b^4*B - 28*A*b^3*c - 56*a*b^2*B*c
+ 48*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])])/(1024*c^(11/2))

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Rubi in Sympy [A]  time = 64.4336, size = 298, normalized size = 1.06 \[ \frac{B x^{3} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{6 c} + \frac{x^{2} \left (4 A c - 3 B b\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{20 c^{2}} - \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (48 A a c^{2} - \frac{105 A b^{2} c}{2} - \frac{147 B a b c}{2} + \frac{315 B b^{3}}{8} - \frac{9 c x \left (- 28 A b c - 20 B a c + 21 B b^{2}\right )}{4}\right )}{360 c^{4}} + \frac{\left (b + 2 c x\right ) \sqrt{a + b x + c x^{2}} \left (48 A a b c^{2} - 28 A b^{3} c + 16 B a^{2} c^{2} - 56 B a b^{2} c + 21 B b^{4}\right )}{512 c^{5}} - \frac{\left (- 4 a c + b^{2}\right ) \left (48 A a b c^{2} - 28 A b^{3} c + 16 B a^{2} c^{2} - 56 B a b^{2} c + 21 B b^{4}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{1024 c^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(B*x+A)*(c*x**2+b*x+a)**(1/2),x)

[Out]

B*x**3*(a + b*x + c*x**2)**(3/2)/(6*c) + x**2*(4*A*c - 3*B*b)*(a + b*x + c*x**2)
**(3/2)/(20*c**2) - (a + b*x + c*x**2)**(3/2)*(48*A*a*c**2 - 105*A*b**2*c/2 - 14
7*B*a*b*c/2 + 315*B*b**3/8 - 9*c*x*(-28*A*b*c - 20*B*a*c + 21*B*b**2)/4)/(360*c*
*4) + (b + 2*c*x)*sqrt(a + b*x + c*x**2)*(48*A*a*b*c**2 - 28*A*b**3*c + 16*B*a**
2*c**2 - 56*B*a*b**2*c + 21*B*b**4)/(512*c**5) - (-4*a*c + b**2)*(48*A*a*b*c**2
- 28*A*b**3*c + 16*B*a**2*c**2 - 56*B*a*b**2*c + 21*B*b**4)*atanh((b + 2*c*x)/(2
*sqrt(c)*sqrt(a + b*x + c*x**2)))/(1024*c**(11/2))

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Mathematica [A]  time = 0.403528, size = 272, normalized size = 0.97 \[ \frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (16 b c^2 \left (113 a^2 B-2 a c x (29 A+17 B x)+4 c^2 x^3 (3 A+2 B x)\right )-32 c^3 \left (a^2 (32 A+15 B x)-2 a c x^2 (8 A+5 B x)-8 c^2 x^4 (6 A+5 B x)\right )+56 b^3 c (c x (5 A+3 B x)-30 a B)+16 b^2 c^2 \left (a (115 A+56 B x)-c x^2 (14 A+9 B x)\right )-210 b^4 c (2 A+B x)+315 b^5 B\right )-15 \left (b^2-4 a c\right ) \left (16 a^2 B c^2+48 a A b c^2-56 a b^2 B c-28 A b^3 c+21 b^4 B\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{15360 c^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(315*b^5*B - 210*b^4*c*(2*A + B*x) + 56*b^3*c*(
-30*a*B + c*x*(5*A + 3*B*x)) - 32*c^3*(-8*c^2*x^4*(6*A + 5*B*x) - 2*a*c*x^2*(8*A
+ 5*B*x) + a^2*(32*A + 15*B*x)) + 16*b*c^2*(113*a^2*B + 4*c^2*x^3*(3*A + 2*B*x)
- 2*a*c*x*(29*A + 17*B*x)) + 16*b^2*c^2*(-(c*x^2*(14*A + 9*B*x)) + a*(115*A + 5
6*B*x))) - 15*(b^2 - 4*a*c)*(21*b^4*B - 28*A*b^3*c - 56*a*b^2*B*c + 48*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)]])/(15360*c^(11/
2))

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Maple [B]  time = 0.016, size = 671, normalized size = 2.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(B*x+A)*(c*x^2+b*x+a)^(1/2),x)

[Out]

1/6*B*x^3*(c*x^2+b*x+a)^(3/2)/c-15/64*B*b^2/c^(7/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(
c*x^2+b*x+a)^(1/2))+49/240*B*b/c^3*a*(c*x^2+b*x+a)^(3/2)-1/8*B*a/c^2*x*(c*x^2+b*
x+a)^(3/2)+1/32*B*a^2/c^3*(c*x^2+b*x+a)^(1/2)*b+3/32*A*b^2/c^3*a*(c*x^2+b*x+a)^(
1/2)+3/16*A*b/c^(5/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/16*B*a^2
/c^2*(c*x^2+b*x+a)^(1/2)*x-3/20*B*b/c^2*x^2*(c*x^2+b*x+a)^(3/2)+3/16*A*b/c^2*a*(
c*x^2+b*x+a)^(1/2)*x-7/32*B*b^2/c^3*a*(c*x^2+b*x+a)^(1/2)*x+1/5*A*x^2*(c*x^2+b*x
+a)^(3/2)/c+7/48*A*b^2/c^3*(c*x^2+b*x+a)^(3/2)-7/128*A*b^4/c^4*(c*x^2+b*x+a)^(1/
2)+7/256*A*b^5/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-2/15*A*a/c^2*
(c*x^2+b*x+a)^(3/2)+21/160*B*b^2/c^3*x*(c*x^2+b*x+a)^(3/2)-7/64*B*b^3/c^4*(c*x^2
+b*x+a)^(3/2)+21/512*B*b^5/c^5*(c*x^2+b*x+a)^(1/2)-21/1024*B*b^6/c^(11/2)*ln((1/
2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/16*B*a^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+
(c*x^2+b*x+a)^(1/2))+21/256*B*b^4/c^4*(c*x^2+b*x+a)^(1/2)*x+35/256*B*b^4/c^(9/2)
*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-7/64*B*b^3/c^4*a*(c*x^2+b*x+a)^(1
/2)-5/32*A*b^3/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-7/40*A*b/c^
2*x*(c*x^2+b*x+a)^(3/2)-7/64*A*b^3/c^3*(c*x^2+b*x+a)^(1/2)*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(sqrt(c*x^2 + b*x + a)*(B*x + A)*x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.389371, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/30720*(4*(1280*B*c^5*x^5 + 315*B*b^5 - 1024*A*a^2*c^3 + 128*(B*b*c^4 + 12*A*c
^5)*x^4 - 16*(9*B*b^2*c^3 - 4*(5*B*a + 3*A*b)*c^4)*x^3 + 16*(113*B*a^2*b + 115*A
*a*b^2)*c^2 + 8*(21*B*b^3*c^2 + 64*A*a*c^4 - 4*(17*B*a*b + 7*A*b^2)*c^3)*x^2 - 4
20*(4*B*a*b^3 + A*b^4)*c - 2*(105*B*b^4*c + 16*(15*B*a^2 + 29*A*a*b)*c^3 - 28*(1
6*B*a*b^2 + 5*A*b^3)*c^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) - 15*(21*B*b^6 - 64*(
B*a^3 + 3*A*a^2*b)*c^3 + 80*(3*B*a^2*b^2 + 2*A*a*b^3)*c^2 - 28*(5*B*a*b^4 + A*b^
5)*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), 1/15360*(2*(1280*B*c^5*x^5 + 315*B*b^5 - 1024*A*a^2
*c^3 + 128*(B*b*c^4 + 12*A*c^5)*x^4 - 16*(9*B*b^2*c^3 - 4*(5*B*a + 3*A*b)*c^4)*x
^3 + 16*(113*B*a^2*b + 115*A*a*b^2)*c^2 + 8*(21*B*b^3*c^2 + 64*A*a*c^4 - 4*(17*B
*a*b + 7*A*b^2)*c^3)*x^2 - 420*(4*B*a*b^3 + A*b^4)*c - 2*(105*B*b^4*c + 16*(15*B
*a^2 + 29*A*a*b)*c^3 - 28*(16*B*a*b^2 + 5*A*b^3)*c^2)*x)*sqrt(c*x^2 + b*x + a)*s
qrt(-c) - 15*(21*B*b^6 - 64*(B*a^3 + 3*A*a^2*b)*c^3 + 80*(3*B*a^2*b^2 + 2*A*a*b^
3)*c^2 - 28*(5*B*a*b^4 + A*b^5)*c)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 +
b*x + a)*c)))/(sqrt(-c)*c^5)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(B*x+A)*(c*x**2+b*x+a)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.283315, size = 436, normalized size = 1.56 \[ \frac{1}{7680} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, B x + \frac{B b c^{4} + 12 \, A c^{5}}{c^{5}}\right )} x - \frac{9 \, B b^{2} c^{3} - 20 \, B a c^{4} - 12 \, A b c^{4}}{c^{5}}\right )} x + \frac{21 \, B b^{3} c^{2} - 68 \, B a b c^{3} - 28 \, A b^{2} c^{3} + 64 \, A a c^{4}}{c^{5}}\right )} x - \frac{105 \, B b^{4} c - 448 \, B a b^{2} c^{2} - 140 \, A b^{3} c^{2} + 240 \, B a^{2} c^{3} + 464 \, A a b c^{3}}{c^{5}}\right )} x + \frac{315 \, B b^{5} - 1680 \, B a b^{3} c - 420 \, A b^{4} c + 1808 \, B a^{2} b c^{2} + 1840 \, A a b^{2} c^{2} - 1024 \, A a^{2} c^{3}}{c^{5}}\right )} + \frac{{\left (21 \, B b^{6} - 140 \, B a b^{4} c - 28 \, A b^{5} c + 240 \, B a^{2} b^{2} c^{2} + 160 \, A a b^{3} c^{2} - 64 \, B a^{3} c^{3} - 192 \, A a^{2} b c^{3}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{1024 \, c^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7680*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*B*x + (B*b*c^4 + 12*A*c^5)/c^5)*x -
(9*B*b^2*c^3 - 20*B*a*c^4 - 12*A*b*c^4)/c^5)*x + (21*B*b^3*c^2 - 68*B*a*b*c^3 -
28*A*b^2*c^3 + 64*A*a*c^4)/c^5)*x - (105*B*b^4*c - 448*B*a*b^2*c^2 - 140*A*b^3*
c^2 + 240*B*a^2*c^3 + 464*A*a*b*c^3)/c^5)*x + (315*B*b^5 - 1680*B*a*b^3*c - 420*
A*b^4*c + 1808*B*a^2*b*c^2 + 1840*A*a*b^2*c^2 - 1024*A*a^2*c^3)/c^5) + 1/1024*(2
1*B*b^6 - 140*B*a*b^4*c - 28*A*b^5*c + 240*B*a^2*b^2*c^2 + 160*A*a*b^3*c^2 - 64*
B*a^3*c^3 - 192*A*a^2*b*c^3)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(
c) - b))/c^(11/2)