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

Optimal. Leaf size=198 \[ \frac{\left (b^2-4 a c\right )^2 \left (-4 a B c-12 A b c+7 b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{9/2}}-\frac{\left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 a B c-12 A b c+7 b^2 B\right )}{512 c^4}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a B c-12 A b c+7 b^2 B\right )}{192 c^3}-\frac{\left (a+b x+c x^2\right )^{5/2} (-12 A c+7 b B-10 B c x)}{60 c^2} \]

[Out]

-((b^2 - 4*a*c)*(7*b^2*B - 12*A*b*c - 4*a*B*c)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2]
)/(512*c^4) + ((7*b^2*B - 12*A*b*c - 4*a*B*c)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2
))/(192*c^3) - ((7*b*B - 12*A*c - 10*B*c*x)*(a + b*x + c*x^2)^(5/2))/(60*c^2) +
((b^2 - 4*a*c)^2*(7*b^2*B - 12*A*b*c - 4*a*B*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*S
qrt[a + b*x + c*x^2])])/(1024*c^(9/2))

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Rubi [A]  time = 0.22892, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{\left (b^2-4 a c\right )^2 \left (-4 a B c-12 A b c+7 b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{9/2}}-\frac{\left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 a B c-12 A b c+7 b^2 B\right )}{512 c^4}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a B c-12 A b c+7 b^2 B\right )}{192 c^3}-\frac{\left (a+b x+c x^2\right )^{5/2} (-12 A c+7 b B-10 B c x)}{60 c^2} \]

Antiderivative was successfully verified.

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

[Out]

-((b^2 - 4*a*c)*(7*b^2*B - 12*A*b*c - 4*a*B*c)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2]
)/(512*c^4) + ((7*b^2*B - 12*A*b*c - 4*a*B*c)*(b + 2*c*x)*(a + b*x + c*x^2)^(3/2
))/(192*c^3) - ((7*b*B - 12*A*c - 10*B*c*x)*(a + b*x + c*x^2)^(5/2))/(60*c^2) +
((b^2 - 4*a*c)^2*(7*b^2*B - 12*A*b*c - 4*a*B*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*S
qrt[a + b*x + c*x^2])])/(1024*c^(9/2))

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Rubi in Sympy [A]  time = 23.0126, size = 202, normalized size = 1.02 \[ \frac{\left (a + b x + c x^{2}\right )^{\frac{5}{2}} \left (6 A c - \frac{7 B b}{2} + 5 B c x\right )}{30 c^{2}} + \frac{\left (b + 2 c x\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (- 12 A b c - 4 B a c + 7 B b^{2}\right )}{192 c^{3}} - \frac{\left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}} \left (- 12 A b c - 4 B a c + 7 B b^{2}\right )}{512 c^{4}} + \frac{\left (- 4 a c + b^{2}\right )^{2} \left (- 12 A b c - 4 B a c + 7 B b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{1024 c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x + c*x**2)**(5/2)*(6*A*c - 7*B*b/2 + 5*B*c*x)/(30*c**2) + (b + 2*c*x)*(a
+ b*x + c*x**2)**(3/2)*(-12*A*b*c - 4*B*a*c + 7*B*b**2)/(192*c**3) - (b + 2*c*x
)*(-4*a*c + b**2)*sqrt(a + b*x + c*x**2)*(-12*A*b*c - 4*B*a*c + 7*B*b**2)/(512*c
**4) + (-4*a*c + b**2)**2*(-12*A*b*c - 4*B*a*c + 7*B*b**2)*atanh((b + 2*c*x)/(2*
sqrt(c)*sqrt(a + b*x + c*x**2)))/(1024*c**(9/2))

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Mathematica [A]  time = 0.456412, size = 253, normalized size = 1.28 \[ \frac{15 \left (b^2-4 a c\right )^2 \left (-4 a B c-12 A b c+7 b^2 B\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )-2 \sqrt{c} \sqrt{a+x (b+c x)} \left (-16 b c^2 \left (-81 a^2 B+6 a c x (7 A+3 B x)+4 c^2 x^3 (33 A+26 B x)\right )-32 c^3 \left (3 a^2 (16 A+5 B x)+2 a c x^2 (48 A+35 B x)+8 c^2 x^4 (6 A+5 B x)\right )+8 b^3 c (c x (15 A+7 B x)-95 a B)+48 b^2 c^2 \left (a (25 A+9 B x)-c x^2 (2 A+B x)\right )-10 b^4 c (18 A+7 B x)+105 b^5 B\right )}{15360 c^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(105*b^5*B - 10*b^4*c*(18*A + 7*B*x) + 8*b^3*c
*(-95*a*B + c*x*(15*A + 7*B*x)) + 48*b^2*c^2*(-(c*x^2*(2*A + B*x)) + a*(25*A + 9
*B*x)) - 16*b*c^2*(-81*a^2*B + 6*a*c*x*(7*A + 3*B*x) + 4*c^2*x^3*(33*A + 26*B*x)
) - 32*c^3*(8*c^2*x^4*(6*A + 5*B*x) + 3*a^2*(16*A + 5*B*x) + 2*a*c*x^2*(48*A + 3
5*B*x))) + 15*(b^2 - 4*a*c)^2*(7*b^2*B - 12*A*b*c - 4*a*B*c)*Log[b + 2*c*x + 2*S
qrt[c]*Sqrt[a + x*(b + c*x)]])/(15360*c^(9/2))

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Maple [B]  time = 0.012, size = 644, normalized size = 3.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*B*a/c^2*(c*x^2+b*x+a)^(3/2)*b+9/64*B*b^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c
*x^2+b*x+a)^(1/2))*a^2+1/5*A*(c*x^2+b*x+a)^(5/2)/c+1/16*B*b^3/c^3*(c*x^2+b*x+a)^
(1/2)*a-1/16*B*a^2/c*(c*x^2+b*x+a)^(1/2)*x-3/32*A*b^2/c^2*(c*x^2+b*x+a)^(1/2)*a-
3/16*A*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2+3/32*A*b^3/c^(5
/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-7/256*B*b^4/c^3*(c*x^2+b*x+a)^
(1/2)*x+3/64*A*b^3/c^2*(c*x^2+b*x+a)^(1/2)*x-1/32*B*a^2/c^2*(c*x^2+b*x+a)^(1/2)*
b-15/256*B*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-1/24*B*a/c*
(c*x^2+b*x+a)^(3/2)*x-1/16*A*b^2/c^2*(c*x^2+b*x+a)^(3/2)+3/128*A*b^4/c^3*(c*x^2+
b*x+a)^(1/2)-3/256*A*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/6
*B*x*(c*x^2+b*x+a)^(5/2)/c-7/60*B*b/c^2*(c*x^2+b*x+a)^(5/2)+7/192*B*b^3/c^3*(c*x
^2+b*x+a)^(3/2)-7/512*B*b^5/c^4*(c*x^2+b*x+a)^(1/2)+7/1024*B*b^6/c^(9/2)*ln((1/2
*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-1/16*B*a^3/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(
c*x^2+b*x+a)^(1/2))-1/8*A*b/c*(c*x^2+b*x+a)^(3/2)*x+1/8*B*b^2/c^2*(c*x^2+b*x+a)^
(1/2)*x*a-3/16*A*b/c*(c*x^2+b*x+a)^(1/2)*x*a+7/96*B*b^2/c^2*(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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.354096, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/30720*(4*(1280*B*c^5*x^5 - 105*B*b^5 + 1536*A*a^2*c^3 + 128*(13*B*b*c^4 + 12*
A*c^5)*x^4 + 16*(3*B*b^2*c^3 + 4*(35*B*a + 33*A*b)*c^4)*x^3 - 48*(27*B*a^2*b + 2
5*A*a*b^2)*c^2 - 8*(7*B*b^3*c^2 - 384*A*a*c^4 - 12*(3*B*a*b + A*b^2)*c^3)*x^2 +
20*(38*B*a*b^3 + 9*A*b^4)*c + 2*(35*B*b^4*c + 48*(5*B*a^2 + 7*A*a*b)*c^3 - 12*(1
8*B*a*b^2 + 5*A*b^3)*c^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) - 15*(7*B*b^6 - 64*(B
*a^3 + 3*A*a^2*b)*c^3 + 48*(3*B*a^2*b^2 + 2*A*a*b^3)*c^2 - 12*(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^(9/2), 1/15360*(2*(1280*B*c^5*x^5 - 105*B*b^5 + 1536*A*a^2*c^
3 + 128*(13*B*b*c^4 + 12*A*c^5)*x^4 + 16*(3*B*b^2*c^3 + 4*(35*B*a + 33*A*b)*c^4)
*x^3 - 48*(27*B*a^2*b + 25*A*a*b^2)*c^2 - 8*(7*B*b^3*c^2 - 384*A*a*c^4 - 12*(3*B
*a*b + A*b^2)*c^3)*x^2 + 20*(38*B*a*b^3 + 9*A*b^4)*c + 2*(35*B*b^4*c + 48*(5*B*a
^2 + 7*A*a*b)*c^3 - 12*(18*B*a*b^2 + 5*A*b^3)*c^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt
(-c) + 15*(7*B*b^6 - 64*(B*a^3 + 3*A*a^2*b)*c^3 + 48*(3*B*a^2*b^2 + 2*A*a*b^3)*c
^2 - 12*(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^4)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.288064, size = 448, normalized size = 2.26 \[ \frac{1}{7680} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, B c x + \frac{13 \, B b c^{5} + 12 \, A c^{6}}{c^{5}}\right )} x + \frac{3 \, B b^{2} c^{4} + 140 \, B a c^{5} + 132 \, A b c^{5}}{c^{5}}\right )} x - \frac{7 \, B b^{3} c^{3} - 36 \, B a b c^{4} - 12 \, A b^{2} c^{4} - 384 \, A a c^{5}}{c^{5}}\right )} x + \frac{35 \, B b^{4} c^{2} - 216 \, B a b^{2} c^{3} - 60 \, A b^{3} c^{3} + 240 \, B a^{2} c^{4} + 336 \, A a b c^{4}}{c^{5}}\right )} x - \frac{105 \, B b^{5} c - 760 \, B a b^{3} c^{2} - 180 \, A b^{4} c^{2} + 1296 \, B a^{2} b c^{3} + 1200 \, A a b^{2} c^{3} - 1536 \, A a^{2} c^{4}}{c^{5}}\right )} - \frac{{\left (7 \, B b^{6} - 60 \, B a b^{4} c - 12 \, A b^{5} c + 144 \, B a^{2} b^{2} c^{2} + 96 \, A a b^{3} c^{2} - 64 \, B a^{3} c^{3} - 192 \, A a^{2} b c^{3}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{1024 \, c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7680*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*B*c*x + (13*B*b*c^5 + 12*A*c^6)/c^5
)*x + (3*B*b^2*c^4 + 140*B*a*c^5 + 132*A*b*c^5)/c^5)*x - (7*B*b^3*c^3 - 36*B*a*b
*c^4 - 12*A*b^2*c^4 - 384*A*a*c^5)/c^5)*x + (35*B*b^4*c^2 - 216*B*a*b^2*c^3 - 60
*A*b^3*c^3 + 240*B*a^2*c^4 + 336*A*a*b*c^4)/c^5)*x - (105*B*b^5*c - 760*B*a*b^3*
c^2 - 180*A*b^4*c^2 + 1296*B*a^2*b*c^3 + 1200*A*a*b^2*c^3 - 1536*A*a^2*c^4)/c^5)
- 1/1024*(7*B*b^6 - 60*B*a*b^4*c - 12*A*b^5*c + 144*B*a^2*b^2*c^2 + 96*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^(9/2)