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

Optimal. Leaf size=206 \[ \frac{\left (48 a^2 B c^2+96 a A b c^2-120 a b^2 B c-40 A b^3 c+35 b^4 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{9/2}}-\frac{\sqrt{a+b x+c x^2} \left (-2 c x \left (-36 a B c-40 A b c+35 b^2 B\right )+128 a A c^2-220 a b B c-120 A b^2 c+105 b^3 B\right )}{192 c^4}-\frac{x^2 \sqrt{a+b x+c x^2} (7 b B-8 A c)}{24 c^2}+\frac{B x^3 \sqrt{a+b x+c x^2}}{4 c} \]

[Out]

-((7*b*B - 8*A*c)*x^2*Sqrt[a + b*x + c*x^2])/(24*c^2) + (B*x^3*Sqrt[a + b*x + c*
x^2])/(4*c) - ((105*b^3*B - 120*A*b^2*c - 220*a*b*B*c + 128*a*A*c^2 - 2*c*(35*b^
2*B - 40*A*b*c - 36*a*B*c)*x)*Sqrt[a + b*x + c*x^2])/(192*c^4) + ((35*b^4*B - 40
*A*b^3*c - 120*a*b^2*B*c + 96*a*A*b*c^2 + 48*a^2*B*c^2)*ArcTanh[(b + 2*c*x)/(2*S
qrt[c]*Sqrt[a + b*x + c*x^2])])/(128*c^(9/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.599038, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{\left (48 a^2 B c^2+96 a A b c^2-120 a b^2 B c-40 A b^3 c+35 b^4 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{9/2}}-\frac{\sqrt{a+b x+c x^2} \left (-2 c x \left (-36 a B c-40 A b c+35 b^2 B\right )+128 a A c^2-220 a b B c-120 A b^2 c+105 b^3 B\right )}{192 c^4}-\frac{x^2 \sqrt{a+b x+c x^2} (7 b B-8 A c)}{24 c^2}+\frac{B x^3 \sqrt{a+b x+c x^2}}{4 c} \]

Antiderivative was successfully verified.

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

[Out]

-((7*b*B - 8*A*c)*x^2*Sqrt[a + b*x + c*x^2])/(24*c^2) + (B*x^3*Sqrt[a + b*x + c*
x^2])/(4*c) - ((105*b^3*B - 120*A*b^2*c - 220*a*b*B*c + 128*a*A*c^2 - 2*c*(35*b^
2*B - 40*A*b*c - 36*a*B*c)*x)*Sqrt[a + b*x + c*x^2])/(192*c^4) + ((35*b^4*B - 40
*A*b^3*c - 120*a*b^2*B*c + 96*a*A*b*c^2 + 48*a^2*B*c^2)*ArcTanh[(b + 2*c*x)/(2*S
qrt[c]*Sqrt[a + b*x + c*x^2])])/(128*c^(9/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 52.6582, size = 214, normalized size = 1.04 \[ \frac{B x^{3} \sqrt{a + b x + c x^{2}}}{4 c} + \frac{x^{2} \left (8 A c - 7 B b\right ) \sqrt{a + b x + c x^{2}}}{24 c^{2}} - \frac{\sqrt{a + b x + c x^{2}} \left (16 A a c^{2} - 15 A b^{2} c - \frac{55 B a b c}{2} + \frac{105 B b^{3}}{8} - \frac{c x \left (- 40 A b c - 36 B a c + 35 B b^{2}\right )}{4}\right )}{24 c^{4}} + \frac{\left (96 A a b c^{2} - 40 A b^{3} c + 48 B a^{2} c^{2} - 120 B a b^{2} c + 35 B b^{4}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{128 c^{\frac{9}{2}}} \]

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*sqrt(a + b*x + c*x**2)/(4*c) + x**2*(8*A*c - 7*B*b)*sqrt(a + b*x + c*x**2
)/(24*c**2) - sqrt(a + b*x + c*x**2)*(16*A*a*c**2 - 15*A*b**2*c - 55*B*a*b*c/2 +
105*B*b**3/8 - c*x*(-40*A*b*c - 36*B*a*c + 35*B*b**2)/4)/(24*c**4) + (96*A*a*b*
c**2 - 40*A*b**3*c + 48*B*a**2*c**2 - 120*B*a*b**2*c + 35*B*b**4)*atanh((b + 2*c
*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(128*c**(9/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.253161, size = 170, normalized size = 0.83 \[ \frac{3 \left (48 a^2 B c^2+96 a A b c^2-120 a b^2 B c-40 A b^3 c+35 b^4 B\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )-2 \sqrt{c} \sqrt{a+x (b+c x)} \left (4 b c (2 c x (10 A+7 B x)-55 a B)+8 c^2 \left (a (16 A+9 B x)-2 c x^2 (4 A+3 B x)\right )-10 b^2 c (12 A+7 B x)+105 b^3 B\right )}{384 c^{9/2}} \]

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)]*(105*b^3*B - 10*b^2*c*(12*A + 7*B*x) + 4*b*c*(
-55*a*B + 2*c*x*(10*A + 7*B*x)) + 8*c^2*(-2*c*x^2*(4*A + 3*B*x) + a*(16*A + 9*B*
x))) + 3*(35*b^4*B - 40*A*b^3*c - 120*a*b^2*B*c + 96*a*A*b*c^2 + 48*a^2*B*c^2)*L
og[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(384*c^(9/2))

_______________________________________________________________________________________

Maple [B]  time = 0.014, size = 379, normalized size = 1.8 \[{\frac{A{x}^{2}}{3\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{5\,Abx}{12\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,{b}^{2}A}{8\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{5\,A{b}^{3}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{7}{2}}}}+{\frac{3\,abA}{4}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{2\,aA}{3\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{{x}^{3}B}{4\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{7\,Bb{x}^{2}}{24\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{35\,{b}^{2}Bx}{96\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{35\,B{b}^{3}}{64\,{c}^{4}}\sqrt{c{x}^{2}+bx+a}}+{\frac{35\,{b}^{4}B}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{9}{2}}}}-{\frac{15\,a{b}^{2}B}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{7}{2}}}}+{\frac{55\,abB}{48\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,aBx}{8\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,{a}^{2}B}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*A*x^2/c*(c*x^2+b*x+a)^(1/2)-5/12*A*b/c^2*x*(c*x^2+b*x+a)^(1/2)+5/8*A*b^2/c^3
*(c*x^2+b*x+a)^(1/2)-5/16*A*b^3/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/
2))+3/4*A*b/c^(5/2)*a*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-2/3*A*a/c^2*(c
*x^2+b*x+a)^(1/2)+1/4*B*x^3*(c*x^2+b*x+a)^(1/2)/c-7/24*B*b/c^2*x^2*(c*x^2+b*x+a)
^(1/2)+35/96*B*b^2/c^3*x*(c*x^2+b*x+a)^(1/2)-35/64*B*b^3/c^4*(c*x^2+b*x+a)^(1/2)
+35/128*B*b^4/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-15/16*B*b^2/c^
(7/2)*a*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+55/48*B*b/c^3*a*(c*x^2+b*x+a
)^(1/2)-3/8*B*a/c^2*x*(c*x^2+b*x+a)^(1/2)+3/8*B*a^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/
2)+(c*x^2+b*x+a)^(1/2))

_______________________________________________________________________________________

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/sqrt(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.406105, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (48 \, B c^{3} x^{3} - 105 \, B b^{3} - 128 \, A a c^{2} - 8 \,{\left (7 \, B b c^{2} - 8 \, A c^{3}\right )} x^{2} + 20 \,{\left (11 \, B a b + 6 \, A b^{2}\right )} c + 2 \,{\left (35 \, B b^{2} c - 4 \,{\left (9 \, B a + 10 \, A b\right )} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} + 3 \,{\left (35 \, B b^{4} + 48 \,{\left (B a^{2} + 2 \, A a b\right )} c^{2} - 40 \,{\left (3 \, B a b^{2} + A b^{3}\right )} c\right )} \log \left (-4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{768 \, c^{\frac{9}{2}}}, \frac{2 \,{\left (48 \, B c^{3} x^{3} - 105 \, B b^{3} - 128 \, A a c^{2} - 8 \,{\left (7 \, B b c^{2} - 8 \, A c^{3}\right )} x^{2} + 20 \,{\left (11 \, B a b + 6 \, A b^{2}\right )} c + 2 \,{\left (35 \, B b^{2} c - 4 \,{\left (9 \, B a + 10 \, A b\right )} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} + 3 \,{\left (35 \, B b^{4} + 48 \,{\left (B a^{2} + 2 \, A a b\right )} c^{2} - 40 \,{\left (3 \, B a b^{2} + A b^{3}\right )} c\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{384 \, \sqrt{-c} c^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/768*(4*(48*B*c^3*x^3 - 105*B*b^3 - 128*A*a*c^2 - 8*(7*B*b*c^2 - 8*A*c^3)*x^2
+ 20*(11*B*a*b + 6*A*b^2)*c + 2*(35*B*b^2*c - 4*(9*B*a + 10*A*b)*c^2)*x)*sqrt(c*
x^2 + b*x + a)*sqrt(c) + 3*(35*B*b^4 + 48*(B*a^2 + 2*A*a*b)*c^2 - 40*(3*B*a*b^2
+ A*b^3)*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), 1/384*(2*(48*B*c^3*x^3 - 105*B*b^3 - 128*A*a*c
^2 - 8*(7*B*b*c^2 - 8*A*c^3)*x^2 + 20*(11*B*a*b + 6*A*b^2)*c + 2*(35*B*b^2*c - 4
*(9*B*a + 10*A*b)*c^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) + 3*(35*B*b^4 + 48*(B*a
^2 + 2*A*a*b)*c^2 - 40*(3*B*a*b^2 + A*b^3)*c)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(s
qrt(c*x^2 + b*x + a)*c)))/(sqrt(-c)*c^4)]

_______________________________________________________________________________________

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

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)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.297439, size = 247, normalized size = 1.2 \[ \frac{1}{192} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (\frac{6 \, B x}{c} - \frac{7 \, B b c^{2} - 8 \, A c^{3}}{c^{4}}\right )} x + \frac{35 \, B b^{2} c - 36 \, B a c^{2} - 40 \, A b c^{2}}{c^{4}}\right )} x - \frac{105 \, B b^{3} - 220 \, B a b c - 120 \, A b^{2} c + 128 \, A a c^{2}}{c^{4}}\right )} - \frac{{\left (35 \, B b^{4} - 120 \, B a b^{2} c - 40 \, A b^{3} c + 48 \, B a^{2} c^{2} + 96 \, A a b c^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/192*sqrt(c*x^2 + b*x + a)*(2*(4*(6*B*x/c - (7*B*b*c^2 - 8*A*c^3)/c^4)*x + (35*
B*b^2*c - 36*B*a*c^2 - 40*A*b*c^2)/c^4)*x - (105*B*b^3 - 220*B*a*b*c - 120*A*b^2
*c + 128*A*a*c^2)/c^4) - 1/128*(35*B*b^4 - 120*B*a*b^2*c - 40*A*b^3*c + 48*B*a^2
*c^2 + 96*A*a*b*c^2)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))
/c^(9/2)