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

Optimal. Leaf size=219 \[ \frac{3 x \left (4 a c+b^2\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 \sqrt{c} \sqrt{a x^2+b x^3+c x^4}}-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^4}+\frac{3 (3 b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{4 x}-\frac{3 \sqrt{a} b x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{a x^2+b x^3+c x^4}} \]

[Out]

(3*(3*b + 2*c*x)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(4*x) - (a*x^2 + b*x^3 + c*x^4)^(3
/2)/x^4 - (3*Sqrt[a]*b*x*Sqrt[a + b*x + c*x^2]*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sq
rt[a + b*x + c*x^2])])/(2*Sqrt[a*x^2 + b*x^3 + c*x^4]) + (3*(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])])/(8*Sqr
t[c]*Sqrt[a*x^2 + b*x^3 + c*x^4])

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Rubi [A]  time = 0.460006, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{3 x \left (4 a c+b^2\right ) \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 \sqrt{c} \sqrt{a x^2+b x^3+c x^4}}-\frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^4}+\frac{3 (3 b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{4 x}-\frac{3 \sqrt{a} b x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{a x^2+b x^3+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(3*b + 2*c*x)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(4*x) - (a*x^2 + b*x^3 + c*x^4)^(3
/2)/x^4 - (3*Sqrt[a]*b*x*Sqrt[a + b*x + c*x^2]*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sq
rt[a + b*x + c*x^2])])/(2*Sqrt[a*x^2 + b*x^3 + c*x^4]) + (3*(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])])/(8*Sqr
t[c]*Sqrt[a*x^2 + b*x^3 + c*x^4])

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Rubi in Sympy [A]  time = 57.1001, size = 206, normalized size = 0.94 \[ - \frac{3 \sqrt{a} b x \sqrt{a + b x + c x^{2}} \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{2 \sqrt{a x^{2} + b x^{3} + c x^{4}}} + \frac{3 \left (3 b + 2 c x\right ) \sqrt{a x^{2} + b x^{3} + c x^{4}}}{4 x} - \frac{\left (a x^{2} + b x^{3} + c x^{4}\right )^{\frac{3}{2}}}{x^{4}} + \frac{3 x \left (4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{8 \sqrt{c} \sqrt{a x^{2} + b x^{3} + c x^{4}}} \]

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**5,x)

[Out]

-3*sqrt(a)*b*x*sqrt(a + b*x + c*x**2)*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x
+ c*x**2)))/(2*sqrt(a*x**2 + b*x**3 + c*x**4)) + 3*(3*b + 2*c*x)*sqrt(a*x**2 + b
*x**3 + c*x**4)/(4*x) - (a*x**2 + b*x**3 + c*x**4)**(3/2)/x**4 + 3*x*(4*a*c + b*
*2)*sqrt(a + b*x + c*x**2)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))
/(8*sqrt(c)*sqrt(a*x**2 + b*x**3 + c*x**4))

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Mathematica [A]  time = 0.258069, size = 229, normalized size = 1.05 \[ \frac{\sqrt{a+x (b+c x)} \left (3 b^2 x \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )+4 c^{3/2} x^2 \sqrt{a+x (b+c x)}+10 b \sqrt{c} x \sqrt{a+x (b+c x)}-8 a \sqrt{c} \sqrt{a+x (b+c x)}+12 \sqrt{a} b \sqrt{c} x \log (x)-12 \sqrt{a} b \sqrt{c} x \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )+12 a c x \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )\right )}{8 \sqrt{c} \sqrt{x^2 (a+x (b+c x))}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + x*(b + c*x)]*(-8*a*Sqrt[c]*Sqrt[a + x*(b + c*x)] + 10*b*Sqrt[c]*x*Sqrt
[a + x*(b + c*x)] + 4*c^(3/2)*x^2*Sqrt[a + x*(b + c*x)] + 12*Sqrt[a]*b*Sqrt[c]*x
*Log[x] - 12*Sqrt[a]*b*Sqrt[c]*x*Log[2*a + b*x + 2*Sqrt[a]*Sqrt[a + x*(b + c*x)]
] + 3*b^2*x*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]] + 12*a*c*x*Log[b +
2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]]))/(8*Sqrt[c]*Sqrt[x^2*(a + x*(b + c*x))
])

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Maple [A]  time = 0.009, size = 254, normalized size = 1.2 \[ -{\frac{1}{8\,a{x}^{4}} \left ( c{x}^{4}+b{x}^{3}+a{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( -8\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{5/2}{x}^{2}+8\, \left ( c{x}^{2}+bx+a \right ) ^{5/2}{c}^{3/2}-8\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{3/2}xb-12\,\sqrt{c{x}^{2}+bx+a}{c}^{5/2}{x}^{2}a+12\,{a}^{3/2}\ln \left ({\frac{2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a}}{x}} \right ){c}^{3/2}xb-18\,\sqrt{c{x}^{2}+bx+a}{c}^{3/2}xab-3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ) cxa{b}^{2}-12\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ) x{a}^{2}{c}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(c*x^4+b*x^3+a*x^2)^(3/2)*(-8*(c*x^2+b*x+a)^(3/2)*c^(5/2)*x^2+8*(c*x^2+b*x+
a)^(5/2)*c^(3/2)-8*(c*x^2+b*x+a)^(3/2)*c^(3/2)*x*b-12*(c*x^2+b*x+a)^(1/2)*c^(5/2
)*x^2*a+12*a^(3/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)*c^(3/2)*x*b-18*
(c*x^2+b*x+a)^(1/2)*c^(3/2)*x*a*b-3*ln(1/2*(2*(c*x^2+b*x+a)^(1/2)*c^(1/2)+2*c*x+
b)/c^(1/2))*c*x*a*b^2-12*ln(1/2*(2*(c*x^2+b*x+a)^(1/2)*c^(1/2)+2*c*x+b)/c^(1/2))
*x*a^2*c^2)/x^4/(c*x^2+b*x+a)^(3/2)/a/c^(3/2)

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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^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.358942, size = 1, normalized size = 0. \[ \left [\frac{12 \, \sqrt{a} b c x^{2} \log \left (-\frac{8 \, a b x^{2} +{\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x - 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (b x + 2 \, a\right )} \sqrt{a}}{x^{3}}\right ) + 3 \,{\left (b^{2} + 4 \, a c\right )} \sqrt{c} x^{2} \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 (2 \, c^{2} x^{2} + 5 \, b c x - 4 \, a c\right )}}{16 \, c x^{2}}, \frac{6 \, \sqrt{a} b c x^{2} \log \left (-\frac{8 \, a b x^{2} +{\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x - 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (b x + 2 \, a\right )} \sqrt{a}}{x^{3}}\right ) - 3 \,{\left (b^{2} + 4 \, a c\right )} \sqrt{-c} x^{2} \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 (2 \, c^{2} x^{2} + 5 \, b c x - 4 \, a c\right )}}{8 \, c x^{2}}, -\frac{24 \, \sqrt{-a} b c x^{2} \arctan \left (\frac{b x^{2} + 2 \, a x}{2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} \sqrt{-a}}\right ) - 3 \,{\left (b^{2} + 4 \, a c\right )} \sqrt{c} x^{2} \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 (2 \, c^{2} x^{2} + 5 \, b c x - 4 \, a c\right )}}{16 \, c x^{2}}, -\frac{12 \, \sqrt{-a} b c x^{2} \arctan \left (\frac{b x^{2} + 2 \, a x}{2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} \sqrt{-a}}\right ) + 3 \,{\left (b^{2} + 4 \, a c\right )} \sqrt{-c} x^{2} \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 (2 \, c^{2} x^{2} + 5 \, b c x - 4 \, a c\right )}}{8 \, c x^{2}}\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^5,x, algorithm="fricas")

[Out]

[1/16*(12*sqrt(a)*b*c*x^2*log(-(8*a*b*x^2 + (b^2 + 4*a*c)*x^3 + 8*a^2*x - 4*sqrt
(c*x^4 + b*x^3 + a*x^2)*(b*x + 2*a)*sqrt(a))/x^3) + 3*(b^2 + 4*a*c)*sqrt(c)*x^2*
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)*(2*c^2*x^2 + 5*b*c*x
- 4*a*c))/(c*x^2), 1/8*(6*sqrt(a)*b*c*x^2*log(-(8*a*b*x^2 + (b^2 + 4*a*c)*x^3 +
8*a^2*x - 4*sqrt(c*x^4 + b*x^3 + a*x^2)*(b*x + 2*a)*sqrt(a))/x^3) - 3*(b^2 + 4*
a*c)*sqrt(-c)*x^2*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)*(2*c^2*x^2 + 5*b*c*x - 4*a*c))/(c*x^2), -
1/16*(24*sqrt(-a)*b*c*x^2*arctan(1/2*(b*x^2 + 2*a*x)/(sqrt(c*x^4 + b*x^3 + a*x^2
)*sqrt(-a))) - 3*(b^2 + 4*a*c)*sqrt(c)*x^2*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)*(2*c^2*x^2 + 5*b*c*x - 4*a*c))/(c*x^2), -1/8*(12*sqrt(-a)
*b*c*x^2*arctan(1/2*(b*x^2 + 2*a*x)/(sqrt(c*x^4 + b*x^3 + a*x^2)*sqrt(-a))) + 3*
(b^2 + 4*a*c)*sqrt(-c)*x^2*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)*(2*c^2*x^2 + 5*b*c*x - 4*a*c))/(
c*x^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError