3.30 \(\int x \sqrt{a x^2+b x^3+c x^4} \, dx\)

Optimal. Leaf size=205 \[ -\frac{x \left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{7/2} \sqrt{a x^2+b x^3+c x^4}}+\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{192 c^3 x}-\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{96 c^2}+\frac{x (b+6 c x) \sqrt{a x^2+b x^3+c x^4}}{24 c} \]

[Out]

-((5*b^2 - 12*a*c)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(96*c^2) + (b*(15*b^2 - 52*a*c)*
Sqrt[a*x^2 + b*x^3 + c*x^4])/(192*c^3*x) + (x*(b + 6*c*x)*Sqrt[a*x^2 + b*x^3 + c
*x^4])/(24*c) - ((b^2 - 4*a*c)*(5*b^2 - 4*a*c)*x*Sqrt[a + b*x + c*x^2]*ArcTanh[(
b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(128*c^(7/2)*Sqrt[a*x^2 + b*x^3 +
c*x^4])

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Rubi [A]  time = 0.58968, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{x \left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{7/2} \sqrt{a x^2+b x^3+c x^4}}+\frac{b \left (15 b^2-52 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{192 c^3 x}-\frac{\left (5 b^2-12 a c\right ) \sqrt{a x^2+b x^3+c x^4}}{96 c^2}+\frac{x (b+6 c x) \sqrt{a x^2+b x^3+c x^4}}{24 c} \]

Antiderivative was successfully verified.

[In]  Int[x*Sqrt[a*x^2 + b*x^3 + c*x^4],x]

[Out]

-((5*b^2 - 12*a*c)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(96*c^2) + (b*(15*b^2 - 52*a*c)*
Sqrt[a*x^2 + b*x^3 + c*x^4])/(192*c^3*x) + (x*(b + 6*c*x)*Sqrt[a*x^2 + b*x^3 + c
*x^4])/(24*c) - ((b^2 - 4*a*c)*(5*b^2 - 4*a*c)*x*Sqrt[a + b*x + c*x^2]*ArcTanh[(
b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(128*c^(7/2)*Sqrt[a*x^2 + b*x^3 +
c*x^4])

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Rubi in Sympy [A]  time = 59.5143, size = 190, normalized size = 0.93 \[ \frac{b \left (- 52 a c + 15 b^{2}\right ) \sqrt{a x^{2} + b x^{3} + c x^{4}}}{192 c^{3} x} + \frac{x \left (\frac{b}{2} + 3 c x\right ) \sqrt{a x^{2} + b x^{3} + c x^{4}}}{12 c} - \frac{\left (- 12 a c + 5 b^{2}\right ) \sqrt{a x^{2} + b x^{3} + c x^{4}}}{96 c^{2}} - \frac{x \left (- 4 a c + b^{2}\right ) \left (- 4 a c + 5 b^{2}\right ) \sqrt{a + b x + c x^{2}} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{128 c^{\frac{7}{2}} \sqrt{a x^{2} + b x^{3} + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(c*x**4+b*x**3+a*x**2)**(1/2),x)

[Out]

b*(-52*a*c + 15*b**2)*sqrt(a*x**2 + b*x**3 + c*x**4)/(192*c**3*x) + x*(b/2 + 3*c
*x)*sqrt(a*x**2 + b*x**3 + c*x**4)/(12*c) - (-12*a*c + 5*b**2)*sqrt(a*x**2 + b*x
**3 + c*x**4)/(96*c**2) - x*(-4*a*c + b**2)*(-4*a*c + 5*b**2)*sqrt(a + b*x + c*x
**2)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(128*c**(7/2)*sqrt(a*
x**2 + b*x**3 + c*x**4))

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Mathematica [A]  time = 0.332886, size = 150, normalized size = 0.73 \[ \frac{2 \sqrt{c} x (a+x (b+c x)) \left (b \left (8 c^2 x^2-52 a c\right )+24 c^2 x \left (a+2 c x^2\right )+15 b^3-10 b^2 c x\right )-3 x \left (16 a^2 c^2-24 a b^2 c+5 b^4\right ) \sqrt{a+x (b+c x)} \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{384 c^{7/2} \sqrt{x^2 (a+x (b+c x))}} \]

Antiderivative was successfully verified.

[In]  Integrate[x*Sqrt[a*x^2 + b*x^3 + c*x^4],x]

[Out]

(2*Sqrt[c]*x*(a + x*(b + c*x))*(15*b^3 - 10*b^2*c*x + 24*c^2*x*(a + 2*c*x^2) + b
*(-52*a*c + 8*c^2*x^2)) - 3*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*x*Sqrt[a + x*(b +
c*x)]*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(384*c^(7/2)*Sqrt[x^2*(a
+ x*(b + c*x))])

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Maple [A]  time = 0.011, size = 265, normalized size = 1.3 \[{\frac{1}{384\,x}\sqrt{c{x}^{4}+b{x}^{3}+a{x}^{2}} \left ( 96\,x \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{7/2}-80\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{5/2}b-48\,\sqrt{c{x}^{2}+bx+a}{c}^{7/2}xa+60\,\sqrt{c{x}^{2}+bx+a}{c}^{5/2}x{b}^{2}-24\,\sqrt{c{x}^{2}+bx+a}{c}^{5/2}ab+30\,\sqrt{c{x}^{2}+bx+a}{c}^{3/2}{b}^{3}-48\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){a}^{2}{c}^{3}+72\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ) a{b}^{2}{c}^{2}-15\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){b}^{4}c \right ){\frac{1}{\sqrt{c{x}^{2}+bx+a}}}{c}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(c*x^4+b*x^3+a*x^2)^(1/2),x)

[Out]

1/384*(c*x^4+b*x^3+a*x^2)^(1/2)*(96*x*(c*x^2+b*x+a)^(3/2)*c^(7/2)-80*(c*x^2+b*x+
a)^(3/2)*c^(5/2)*b-48*(c*x^2+b*x+a)^(1/2)*c^(7/2)*x*a+60*(c*x^2+b*x+a)^(1/2)*c^(
5/2)*x*b^2-24*(c*x^2+b*x+a)^(1/2)*c^(5/2)*a*b+30*(c*x^2+b*x+a)^(1/2)*c^(3/2)*b^3
-48*ln(1/2*(2*(c*x^2+b*x+a)^(1/2)*c^(1/2)+2*c*x+b)/c^(1/2))*a^2*c^3+72*ln(1/2*(2
*(c*x^2+b*x+a)^(1/2)*c^(1/2)+2*c*x+b)/c^(1/2))*a*b^2*c^2-15*ln(1/2*(2*(c*x^2+b*x
+a)^(1/2)*c^(1/2)+2*c*x+b)/c^(1/2))*b^4*c)/x/(c*x^2+b*x+a)^(1/2)/c^(9/2)

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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(sqrt(c*x^4 + b*x^3 + a*x^2)*x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.292408, size = 1, normalized size = 0. \[ \left [\frac{3 \,{\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{c} x \log \left (\frac{4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c^{2} x + b c\right )} -{\left (8 \, c^{2} x^{3} + 8 \, b c x^{2} +{\left (b^{2} + 4 \, a c\right )} x\right )} \sqrt{c}}{x}\right ) + 4 \,{\left (48 \, c^{4} x^{3} + 8 \, b c^{3} x^{2} + 15 \, b^{3} c - 52 \, a b c^{2} - 2 \,{\left (5 \, b^{2} c^{2} - 12 \, a c^{3}\right )} x\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{768 \, c^{4} x}, \frac{3 \,{\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-c} x \arctan \left (\frac{{\left (2 \, c x^{2} + b x\right )} \sqrt{-c}}{2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} c}\right ) + 2 \,{\left (48 \, c^{4} x^{3} + 8 \, b c^{3} x^{2} + 15 \, b^{3} c - 52 \, a b c^{2} - 2 \,{\left (5 \, b^{2} c^{2} - 12 \, a c^{3}\right )} x\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{384 \, c^{4} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^3 + a*x^2)*x,x, algorithm="fricas")

[Out]

[1/768*(3*(5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*sqrt(c)*x*log((4*sqrt(c*x^4 + b*x^3
+ a*x^2)*(2*c^2*x + b*c) - (8*c^2*x^3 + 8*b*c*x^2 + (b^2 + 4*a*c)*x)*sqrt(c))/x)
+ 4*(48*c^4*x^3 + 8*b*c^3*x^2 + 15*b^3*c - 52*a*b*c^2 - 2*(5*b^2*c^2 - 12*a*c^3
)*x)*sqrt(c*x^4 + b*x^3 + a*x^2))/(c^4*x), 1/384*(3*(5*b^4 - 24*a*b^2*c + 16*a^2
*c^2)*sqrt(-c)*x*arctan(1/2*(2*c*x^2 + b*x)*sqrt(-c)/(sqrt(c*x^4 + b*x^3 + a*x^2
)*c)) + 2*(48*c^4*x^3 + 8*b*c^3*x^2 + 15*b^3*c - 52*a*b*c^2 - 2*(5*b^2*c^2 - 12*
a*c^3)*x)*sqrt(c*x^4 + b*x^3 + a*x^2))/(c^4*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(c*x**4+b*x**3+a*x**2)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.303799, size = 311, normalized size = 1.52 \[ \frac{1}{192} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (6 \, x{\rm sign}\left (x\right ) + \frac{b{\rm sign}\left (x\right )}{c}\right )} x - \frac{5 \, b^{2} c{\rm sign}\left (x\right ) - 12 \, a c^{2}{\rm sign}\left (x\right )}{c^{3}}\right )} x + \frac{15 \, b^{3}{\rm sign}\left (x\right ) - 52 \, a b c{\rm sign}\left (x\right )}{c^{3}}\right )} + \frac{{\left (5 \, b^{4}{\rm sign}\left (x\right ) - 24 \, a b^{2} c{\rm sign}\left (x\right ) + 16 \, a^{2} c^{2}{\rm sign}\left (x\right )\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{7}{2}}} - \frac{{\left (15 \, b^{4}{\rm ln}\left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 72 \, a b^{2} c{\rm ln}\left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 48 \, a^{2} c^{2}{\rm ln}\left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 30 \, \sqrt{a} b^{3} \sqrt{c} - 104 \, a^{\frac{3}{2}} b c^{\frac{3}{2}}\right )}{\rm sign}\left (x\right )}{384 \, c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + b*x^3 + a*x^2)*x,x, algorithm="giac")

[Out]

1/192*sqrt(c*x^2 + b*x + a)*(2*(4*(6*x*sign(x) + b*sign(x)/c)*x - (5*b^2*c*sign(
x) - 12*a*c^2*sign(x))/c^3)*x + (15*b^3*sign(x) - 52*a*b*c*sign(x))/c^3) + 1/128
*(5*b^4*sign(x) - 24*a*b^2*c*sign(x) + 16*a^2*c^2*sign(x))*ln(abs(-2*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(7/2) - 1/384*(15*b^4*ln(abs(-b + 2*sqr
t(a)*sqrt(c))) - 72*a*b^2*c*ln(abs(-b + 2*sqrt(a)*sqrt(c))) + 48*a^2*c^2*ln(abs(
-b + 2*sqrt(a)*sqrt(c))) + 30*sqrt(a)*b^3*sqrt(c) - 104*a^(3/2)*b*c^(3/2))*sign(
x)/c^(7/2)