3.57 \(\int \frac{x^5}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=153 \[ \frac{2 x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}-\frac{2 b \sqrt{a x^2+b x^3+c x^4}}{c x \left (b^2-4 a c\right )}+\frac{x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{3/2} \sqrt{a x^2+b x^3+c x^4}} \]

[Out]

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

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Rubi [A]  time = 0.274087, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{2 x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt{a x^2+b x^3+c x^4}}-\frac{2 b \sqrt{a x^2+b x^3+c x^4}}{c x \left (b^2-4 a c\right )}+\frac{x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{3/2} \sqrt{a x^2+b x^3+c x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a*x^2 + b*x^3 + c*x^4)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 33.3719, size = 139, normalized size = 0.91 \[ - \frac{2 b \sqrt{a x^{2} + b x^{3} + c x^{4}}}{c x \left (- 4 a c + b^{2}\right )} + \frac{2 x^{2} \left (2 a + b x\right )}{\left (- 4 a c + b^{2}\right ) \sqrt{a x^{2} + b x^{3} + c x^{4}}} + \frac{x \sqrt{a + b x + c x^{2}} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{c^{\frac{3}{2}} \sqrt{a x^{2} + b x^{3} + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(c*x**4+b*x**3+a*x**2)**(3/2),x)

[Out]

-2*b*sqrt(a*x**2 + b*x**3 + c*x**4)/(c*x*(-4*a*c + b**2)) + 2*x**2*(2*a + b*x)/(
(-4*a*c + b**2)*sqrt(a*x**2 + b*x**3 + c*x**4)) + x*sqrt(a + b*x + c*x**2)*atanh
((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(c**(3/2)*sqrt(a*x**2 + b*x**3
+ c*x**4))

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Mathematica [A]  time = 0.180332, size = 110, normalized size = 0.72 \[ -\frac{x \left (2 \sqrt{c} \left (-a b+2 a c x+b^2 (-x)\right )+\left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )\right )}{c^{3/2} \left (4 a c-b^2\right ) \sqrt{x^2 (a+x (b+c x))}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a*x^2 + b*x^3 + c*x^4)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.011, size = 166, normalized size = 1.1 \[{\frac{{x}^{3} \left ( c{x}^{2}+bx+a \right ) }{4\,ac-{b}^{2}} \left ( -4\,{c}^{5/2}xa+2\,{c}^{3/2}x{b}^{2}+4\,\sqrt{c{x}^{2}+bx+a}\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ) a{c}^{2}-\sqrt{c{x}^{2}+bx+a}\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b \right ){\frac{1}{\sqrt{c}}}} \right ){b}^{2}c+2\,{c}^{3/2}ab \right ){c}^{-{\frac{5}{2}}} \left ( c{x}^{4}+b{x}^{3}+a{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(c*x^4+b*x^3+a*x^2)^(3/2),x)

[Out]

x^3*(c*x^2+b*x+a)/c^(5/2)*(-4*c^(5/2)*x*a+2*c^(3/2)*x*b^2+4*(c*x^2+b*x+a)^(1/2)*
ln(1/2*(2*(c*x^2+b*x+a)^(1/2)*c^(1/2)+2*c*x+b)/c^(1/2))*a*c^2-(c*x^2+b*x+a)^(1/2
)*ln(1/2*(2*(c*x^2+b*x+a)^(1/2)*c^(1/2)+2*c*x+b)/c^(1/2))*b^2*c+2*c^(3/2)*a*b)/(
c*x^4+b*x^3+a*x^2)^(3/2)/(4*a*c-b^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(x^5/(c*x^4 + b*x^3 + a*x^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.32409, size = 1, normalized size = 0.01 \[ \left [\frac{{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{3} +{\left (b^{3} - 4 \, a b c\right )} x^{2} +{\left (a b^{2} - 4 \, a^{2} c\right )} x\right )} \sqrt{c} \log \left (-\frac{4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c^{2} x + b c\right )} +{\left (8 \, c^{2} x^{3} + 8 \, b c x^{2} +{\left (b^{2} + 4 \, a c\right )} x\right )} \sqrt{c}}{x}\right ) - 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (a b c +{\left (b^{2} c - 2 \, a c^{2}\right )} x\right )}}{2 \,{\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{3} +{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2} +{\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} x\right )}}, -\frac{{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{3} +{\left (b^{3} - 4 \, a b c\right )} x^{2} +{\left (a b^{2} - 4 \, a^{2} c\right )} x\right )} \sqrt{-c} \arctan \left (\frac{{\left (2 \, c x^{2} + b x\right )} \sqrt{-c}}{2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} c}\right ) + 2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (a b c +{\left (b^{2} c - 2 \, a c^{2}\right )} x\right )}}{{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{3} +{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2} +{\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(c*x^4 + b*x^3 + a*x^2)^(3/2),x, algorithm="fricas")

[Out]

[1/2*(((b^2*c - 4*a*c^2)*x^3 + (b^3 - 4*a*b*c)*x^2 + (a*b^2 - 4*a^2*c)*x)*sqrt(c
)*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*sqrt(c*x^4 + b*x^3 + a*x^2)*(a*b*c + (b^2*c -
2*a*c^2)*x))/((b^2*c^3 - 4*a*c^4)*x^3 + (b^3*c^2 - 4*a*b*c^3)*x^2 + (a*b^2*c^2 -
4*a^2*c^3)*x), -(((b^2*c - 4*a*c^2)*x^3 + (b^3 - 4*a*b*c)*x^2 + (a*b^2 - 4*a^2*
c)*x)*sqrt(-c)*arctan(1/2*(2*c*x^2 + b*x)*sqrt(-c)/(sqrt(c*x^4 + b*x^3 + a*x^2)*
c)) + 2*sqrt(c*x^4 + b*x^3 + a*x^2)*(a*b*c + (b^2*c - 2*a*c^2)*x))/((b^2*c^3 - 4
*a*c^4)*x^3 + (b^3*c^2 - 4*a*b*c^3)*x^2 + (a*b^2*c^2 - 4*a^2*c^3)*x)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(c*x**4+b*x**3+a*x**2)**(3/2),x)

[Out]

Integral(x**5/(x**2*(a + b*x + c*x**2))**(3/2), x)

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GIAC/XCAS [A]  time = 0.292568, size = 149, normalized size = 0.97 \[ -\frac{2 \,{\left (\frac{a b c}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x} + \frac{b^{2} c - 2 \, a c^{2}}{b^{2} c^{2} - 4 \, a c^{3}}\right )}}{\sqrt{c + \frac{b}{x} + \frac{a}{x^{2}}}} - \frac{2 \, \arctan \left (\frac{\sqrt{c + \frac{b}{x} + \frac{a}{x^{2}}} - \frac{\sqrt{a}}{x}}{\sqrt{-c}}\right )}{\sqrt{-c} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(c*x^4 + b*x^3 + a*x^2)^(3/2),x, algorithm="giac")

[Out]

-2*(a*b*c/((b^2*c^2 - 4*a*c^3)*x) + (b^2*c - 2*a*c^2)/(b^2*c^2 - 4*a*c^3))/sqrt(
c + b/x + a/x^2) - 2*arctan((sqrt(c + b/x + a/x^2) - sqrt(a)/x)/sqrt(-c))/(sqrt(
-c)*c)