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

Optimal. Leaf size=197 \[ \frac{3 \left (-4 a B c-4 A b c+5 b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{7/2}}-\frac{2 x^2 \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{\sqrt{a+b x+c x^2} \left (-2 c x \left (-12 a B c-4 A b c+5 b^2 B\right )+32 a A c^2-52 a b B c-12 A b^2 c+15 b^3 B\right )}{4 c^3 \left (b^2-4 a c\right )} \]

[Out]

(-2*x^2*(a*(b*B - 2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x))/(c*(b^2 - 4*a*c)*Sqrt[a
+ b*x + c*x^2]) - ((15*b^3*B - 12*A*b^2*c - 52*a*b*B*c + 32*a*A*c^2 - 2*c*(5*b^
2*B - 4*A*b*c - 12*a*B*c)*x)*Sqrt[a + b*x + c*x^2])/(4*c^3*(b^2 - 4*a*c)) + (3*(
5*b^2*B - 4*A*b*c - 4*a*B*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2
])])/(8*c^(7/2))

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Rubi [A]  time = 0.376436, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{3 \left (-4 a B c-4 A b c+5 b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{7/2}}-\frac{2 x^2 \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{\sqrt{a+b x+c x^2} \left (-2 c x \left (-12 a B c-4 A b c+5 b^2 B\right )+32 a A c^2-52 a b B c-12 A b^2 c+15 b^3 B\right )}{4 c^3 \left (b^2-4 a c\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-2*x^2*(a*(b*B - 2*A*c) + (b^2*B - A*b*c - 2*a*B*c)*x))/(c*(b^2 - 4*a*c)*Sqrt[a
+ b*x + c*x^2]) - ((15*b^3*B - 12*A*b^2*c - 52*a*b*B*c + 32*a*A*c^2 - 2*c*(5*b^
2*B - 4*A*b*c - 12*a*B*c)*x)*Sqrt[a + b*x + c*x^2])/(4*c^3*(b^2 - 4*a*c)) + (3*(
5*b^2*B - 4*A*b*c - 4*a*B*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2
])])/(8*c^(7/2))

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Rubi in Sympy [A]  time = 39.5663, size = 199, normalized size = 1.01 \[ \frac{2 x^{2} \left (a \left (2 A c - B b\right ) - x \left (- A b c - 2 B a c + B b^{2}\right )\right )}{c \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}} - \frac{\sqrt{a + b x + c x^{2}} \left (8 A a c^{2} - 3 A b^{2} c - 13 B a b c + \frac{15 B b^{3}}{4} - \frac{c x \left (- 4 A b c - 12 B a c + 5 B b^{2}\right )}{2}\right )}{c^{3} \left (- 4 a c + b^{2}\right )} + \frac{3 \left (- 4 A b c - 4 B a c + 5 B b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{8 c^{\frac{7}{2}}} \]

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]

2*x**2*(a*(2*A*c - B*b) - x*(-A*b*c - 2*B*a*c + B*b**2))/(c*(-4*a*c + b**2)*sqrt
(a + b*x + c*x**2)) - sqrt(a + b*x + c*x**2)*(8*A*a*c**2 - 3*A*b**2*c - 13*B*a*b
*c + 15*B*b**3/4 - c*x*(-4*A*b*c - 12*B*a*c + 5*B*b**2)/2)/(c**3*(-4*a*c + b**2)
) + 3*(-4*A*b*c - 4*B*a*c + 5*B*b**2)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x
+ c*x**2)))/(8*c**(7/2))

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Mathematica [A]  time = 0.415782, size = 219, normalized size = 1.11 \[ \frac{2 \sqrt{c} \left (4 a^2 c (8 A c-13 b B+6 B c x)+a \left (-2 b^2 c (6 A+31 B x)-20 b c^2 x (B x-2 A)+8 c^3 x^2 (2 A+B x)+15 b^3 B\right )+b^2 x \left (b (5 B c x-12 A c)-2 c^2 x (2 A+B x)+15 b^2 B\right )\right )-3 \left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} \left (-4 a B c-4 A b c+5 b^2 B\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{8 c^{7/2} \left (4 a c-b^2\right ) \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c]*(4*a^2*c*(-13*b*B + 8*A*c + 6*B*c*x) + a*(15*b^3*B - 20*b*c^2*x*(-2*A
+ B*x) + 8*c^3*x^2*(2*A + B*x) - 2*b^2*c*(6*A + 31*B*x)) + b^2*x*(15*b^2*B - 2*
c^2*x*(2*A + B*x) + b*(-12*A*c + 5*B*c*x))) - 3*(b^2 - 4*a*c)*(5*b^2*B - 4*A*b*c
- 4*a*B*c)*Sqrt[a + x*(b + c*x)]*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)
]])/(8*c^(7/2)*(-b^2 + 4*a*c)*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.015, size = 576, normalized size = 2.9 \[{\frac{A{x}^{2}}{c}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{3\,Abx}{2\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{3\,{b}^{2}A}{4\,{c}^{3}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{3\,A{b}^{3}x}{2\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{3\,A{b}^{4}}{4\,{c}^{3} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{3\,Ab}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}+2\,{\frac{aA}{{c}^{2}\sqrt{c{x}^{2}+bx+a}}}+4\,{\frac{aAbx}{c \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}+2\,{\frac{a{b}^{2}A}{{c}^{2} \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}+{\frac{{x}^{3}B}{2\,c}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{5\,Bb{x}^{2}}{4\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{15\,{b}^{2}Bx}{8\,{c}^{3}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{15\,B{b}^{3}}{16\,{c}^{4}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{15\,{b}^{4}Bx}{8\,{c}^{3} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{15\,B{b}^{5}}{16\,{c}^{4} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{15\,{b}^{2}B}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{7}{2}}}}-{\frac{13\,abB}{4\,{c}^{3}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{13\,Bxa{b}^{2}}{2\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{13\,Ba{b}^{3}}{4\,{c}^{3} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{3\,aBx}{2\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-{\frac{3\,Ba}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}} \]

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]

A*x^2/c/(c*x^2+b*x+a)^(1/2)+3/2*A*b/c^2*x/(c*x^2+b*x+a)^(1/2)-3/4*A*b^2/c^3/(c*x
^2+b*x+a)^(1/2)-3/2*A*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-3/4*A*b^4/c^3/(4
*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-3/2*A*b/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+
a)^(1/2))+2*A*a/c^2/(c*x^2+b*x+a)^(1/2)+4*A*a/c*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2
)*x+2*A*a/c^2*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+1/2*B*x^3/c/(c*x^2+b*x+a)^(1/2
)-5/4*B*b/c^2*x^2/(c*x^2+b*x+a)^(1/2)-15/8*B*b^2/c^3*x/(c*x^2+b*x+a)^(1/2)+15/16
*B*b^3/c^4/(c*x^2+b*x+a)^(1/2)+15/8*B*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+
15/16*B*b^5/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+15/8*B*b^2/c^(7/2)*ln((1/2*b+c*x
)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-13/4*B*b/c^3*a/(c*x^2+b*x+a)^(1/2)-13/2*B*b^2/c^2
*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x-13/4*B*b^3/c^3*a/(4*a*c-b^2)/(c*x^2+b*x+a)^
(1/2)+3/2*B*a/c^2*x/(c*x^2+b*x+a)^(1/2)-3/2*B*a/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(
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((B*x + A)*x^3/(c*x^2 + b*x + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.495165, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/16*(4*(15*B*a*b^3 + 32*A*a^2*c^2 - 2*(B*b^2*c^2 - 4*B*a*c^3)*x^3 + (5*B*b^3*
c + 16*A*a*c^3 - 4*(5*B*a*b + A*b^2)*c^2)*x^2 - 4*(13*B*a^2*b + 3*A*a*b^2)*c + (
15*B*b^4 + 8*(3*B*a^2 + 5*A*a*b)*c^2 - 2*(31*B*a*b^2 + 6*A*b^3)*c)*x)*sqrt(c*x^2
+ b*x + a)*sqrt(c) + 3*(5*B*a*b^4 + 16*(B*a^3 + A*a^2*b)*c^2 + (5*B*b^4*c + 16*
(B*a^2 + A*a*b)*c^3 - 4*(6*B*a*b^2 + A*b^3)*c^2)*x^2 - 4*(6*B*a^2*b^2 + A*a*b^3)
*c + (5*B*b^5 + 16*(B*a^2*b + A*a*b^2)*c^2 - 4*(6*B*a*b^3 + A*b^4)*c)*x)*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)))/((a*b^2*c^3 - 4*a^2*c^4 + (b^2*c^4 - 4*a*c^5)*x^2 + (b^3*c^3 - 4*a*b*c^4)*x
)*sqrt(c)), -1/8*(2*(15*B*a*b^3 + 32*A*a^2*c^2 - 2*(B*b^2*c^2 - 4*B*a*c^3)*x^3 +
(5*B*b^3*c + 16*A*a*c^3 - 4*(5*B*a*b + A*b^2)*c^2)*x^2 - 4*(13*B*a^2*b + 3*A*a*
b^2)*c + (15*B*b^4 + 8*(3*B*a^2 + 5*A*a*b)*c^2 - 2*(31*B*a*b^2 + 6*A*b^3)*c)*x)*
sqrt(c*x^2 + b*x + a)*sqrt(-c) - 3*(5*B*a*b^4 + 16*(B*a^3 + A*a^2*b)*c^2 + (5*B*
b^4*c + 16*(B*a^2 + A*a*b)*c^3 - 4*(6*B*a*b^2 + A*b^3)*c^2)*x^2 - 4*(6*B*a^2*b^2
+ A*a*b^3)*c + (5*B*b^5 + 16*(B*a^2*b + A*a*b^2)*c^2 - 4*(6*B*a*b^3 + A*b^4)*c)
*x)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/((a*b^2*c^3 - 4*
a^2*c^4 + (b^2*c^4 - 4*a*c^5)*x^2 + (b^3*c^3 - 4*a*b*c^4)*x)*sqrt(-c))]

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

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 = 0.290038, size = 362, normalized size = 1.84 \[ \frac{{\left ({\left (\frac{2 \,{\left (B b^{2} c^{2} - 4 \, B a c^{3}\right )} x}{b^{2} c^{3} - 4 \, a c^{4}} - \frac{5 \, B b^{3} c - 20 \, B a b c^{2} - 4 \, A b^{2} c^{2} + 16 \, A a c^{3}}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x - \frac{15 \, B b^{4} - 62 \, B a b^{2} c - 12 \, A b^{3} c + 24 \, B a^{2} c^{2} + 40 \, A a b c^{2}}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x - \frac{15 \, B a b^{3} - 52 \, B a^{2} b c - 12 \, A a b^{2} c + 32 \, A a^{2} c^{2}}{b^{2} c^{3} - 4 \, a c^{4}}}{4 \, \sqrt{c x^{2} + b x + a}} - \frac{3 \,{\left (5 \, B b^{2} - 4 \, B a c - 4 \, A b c\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{8 \, c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(((2*(B*b^2*c^2 - 4*B*a*c^3)*x/(b^2*c^3 - 4*a*c^4) - (5*B*b^3*c - 20*B*a*b*c
^2 - 4*A*b^2*c^2 + 16*A*a*c^3)/(b^2*c^3 - 4*a*c^4))*x - (15*B*b^4 - 62*B*a*b^2*c
- 12*A*b^3*c + 24*B*a^2*c^2 + 40*A*a*b*c^2)/(b^2*c^3 - 4*a*c^4))*x - (15*B*a*b^
3 - 52*B*a^2*b*c - 12*A*a*b^2*c + 32*A*a^2*c^2)/(b^2*c^3 - 4*a*c^4))/sqrt(c*x^2
+ b*x + a) - 3/8*(5*B*b^2 - 4*B*a*c - 4*A*b*c)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))*sqrt(c) - b))/c^(7/2)