3.42 \(\int \frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^3} \, dx\)

Optimal. Leaf size=165 \[ \frac{3 x \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{5/2} \sqrt{a x^2+b x^3+c x^4}}-\frac{3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{64 c^2 x}+\frac{(b+2 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{8 c x^3} \]

[Out]

(-3*(b^2 - 4*a*c)*(b + 2*c*x)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(64*c^2*x) + ((b + 2*
c*x)*(a*x^2 + b*x^3 + c*x^4)^(3/2))/(8*c*x^3) + (3*(b^2 - 4*a*c)^2*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^(5/2)*
Sqrt[a*x^2 + b*x^3 + c*x^4])

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Rubi [A]  time = 0.199709, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{3 x \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{5/2} \sqrt{a x^2+b x^3+c x^4}}-\frac{3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{64 c^2 x}+\frac{(b+2 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{8 c x^3} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(b^2 - 4*a*c)*(b + 2*c*x)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(64*c^2*x) + ((b + 2*
c*x)*(a*x^2 + b*x^3 + c*x^4)^(3/2))/(8*c*x^3) + (3*(b^2 - 4*a*c)^2*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^(5/2)*
Sqrt[a*x^2 + b*x^3 + c*x^4])

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Rubi in Sympy [A]  time = 30.5433, size = 155, normalized size = 0.94 \[ \frac{\left (b + 2 c x\right ) \left (a x^{2} + b x^{3} + c x^{4}\right )^{\frac{3}{2}}}{8 c x^{3}} - \frac{3 \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \sqrt{a x^{2} + b x^{3} + c x^{4}}}{64 c^{2} x} + \frac{3 x \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{128 c^{\frac{5}{2}} \sqrt{a x^{2} + b x^{3} + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b + 2*c*x)*(a*x**2 + b*x**3 + c*x**4)**(3/2)/(8*c*x**3) - 3*(b + 2*c*x)*(-4*a*c
+ b**2)*sqrt(a*x**2 + b*x**3 + c*x**4)/(64*c**2*x) + 3*x*(-4*a*c + b**2)**2*sqr
t(a + b*x + c*x**2)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(128*c
**(5/2)*sqrt(a*x**2 + b*x**3 + c*x**4))

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Mathematica [A]  time = 0.211458, size = 130, normalized size = 0.79 \[ \frac{x \sqrt{a+x (b+c x)} \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left (4 c \left (5 a+2 c x^2\right )-3 b^2+8 b c x\right )+3 \left (b^2-4 a c\right )^2 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )\right )}{128 c^{5/2} \sqrt{x^2 (a+x (b+c x))}} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[a + x*(b + c*x)]*(2*Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(-3*b^2 +
8*b*c*x + 4*c*(5*a + 2*c*x^2)) + 3*(b^2 - 4*a*c)^2*Log[b + 2*c*x + 2*Sqrt[c]*Sqr
t[a + x*(b + c*x)]]))/(128*c^(5/2)*Sqrt[x^2*(a + x*(b + c*x))])

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Maple [A]  time = 0.007, size = 265, normalized size = 1.6 \[{\frac{1}{128\,{x}^{3}} \left ( c{x}^{4}+b{x}^{3}+a{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 32\,x \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{7/2}+16\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{5/2}b+48\,\sqrt{c{x}^{2}+bx+a}{c}^{7/2}xa-12\,\sqrt{c{x}^{2}+bx+a}{c}^{5/2}x{b}^{2}+24\,\sqrt{c{x}^{2}+bx+a}{c}^{5/2}ab-6\,\sqrt{c{x}^{2}+bx+a}{c}^{3/2}{b}^{3}+48\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){a}^{2}{c}^{3}-24\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ) a{b}^{2}{c}^{2}+3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){b}^{4}c \right ) \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/128*(c*x^4+b*x^3+a*x^2)^(3/2)*(32*x*(c*x^2+b*x+a)^(3/2)*c^(7/2)+16*(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-12*(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-6*(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^3-24*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+3*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^3/(c*x^2+b*x+a)^(3/2)/c^(7/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.298431, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{c} x \log \left (-\frac{4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c^{2} x + b c\right )} +{\left (8 \, c^{2} x^{3} + 8 \, b c x^{2} +{\left (b^{2} + 4 \, a c\right )} x\right )} \sqrt{c}}{x}\right ) + 4 \,{\left (16 \, c^{4} x^{3} + 24 \, b c^{3} x^{2} - 3 \, b^{3} c + 20 \, a b c^{2} + 2 \,{\left (b^{2} c^{2} + 20 \, a c^{3}\right )} x\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{256 \, c^{3} x}, -\frac{3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-c} x \arctan \left (\frac{{\left (2 \, c x^{2} + b x\right )} \sqrt{-c}}{2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} c}\right ) - 2 \,{\left (16 \, c^{4} x^{3} + 24 \, b c^{3} x^{2} - 3 \, b^{3} c + 20 \, a b c^{2} + 2 \,{\left (b^{2} c^{2} + 20 \, a c^{3}\right )} x\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{128 \, c^{3} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/256*(3*(b^4 - 8*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*(16*c^4*x^3 + 24*b*c^3*x^2 - 3*b^3*c + 20*a*b*c^2 + 2*(b^2*c^2 + 20*a*c^3)*x)
*sqrt(c*x^4 + b*x^3 + a*x^2))/(c^3*x), -1/128*(3*(b^4 - 8*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*(16*c^4*x^3 + 24*b*c^3*x^2 - 3*b^3*c + 20*a*b*c^2 + 2*(b^2*c^2 + 20*a*c^3)*x
)*sqrt(c*x^4 + b*x^3 + a*x^2))/(c^3*x)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.318725, size = 313, normalized size = 1.9 \[ \frac{1}{64} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \, c x{\rm sign}\left (x\right ) + 3 \, b{\rm sign}\left (x\right )\right )} x + \frac{b^{2} c^{2}{\rm sign}\left (x\right ) + 20 \, a c^{3}{\rm sign}\left (x\right )}{c^{3}}\right )} x - \frac{3 \, b^{3} c{\rm sign}\left (x\right ) - 20 \, a b c^{2}{\rm sign}\left (x\right )}{c^{3}}\right )} - \frac{3 \,{\left (b^{4}{\rm sign}\left (x\right ) - 8 \, a b^{2} c{\rm sign}\left (x\right ) + 16 \, a^{2} c^{2}{\rm sign}\left (x\right )\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{5}{2}}} + \frac{{\left (3 \, b^{4}{\rm ln}\left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 24 \, a b^{2} c{\rm ln}\left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 48 \, a^{2} c^{2}{\rm ln}\left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 6 \, \sqrt{a} b^{3} \sqrt{c} - 40 \, a^{\frac{3}{2}} b c^{\frac{3}{2}}\right )}{\rm sign}\left (x\right )}{128 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/64*sqrt(c*x^2 + b*x + a)*(2*(4*(2*c*x*sign(x) + 3*b*sign(x))*x + (b^2*c^2*sign
(x) + 20*a*c^3*sign(x))/c^3)*x - (3*b^3*c*sign(x) - 20*a*b*c^2*sign(x))/c^3) - 3
/128*(b^4*sign(x) - 8*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^(5/2) + 1/128*(3*b^4*ln(abs(-b + 2*sqr
t(a)*sqrt(c))) - 24*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))) + 6*sqrt(a)*b^3*sqrt(c) - 40*a^(3/2)*b*c^(3/2))*sign(x)
/c^(5/2)