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

Optimal. Leaf size=198 \[ -\frac{3 b x \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{7/2} \sqrt{a x^2+b x^3+c x^4}}+\frac{3 b \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{128 c^3 x}-\frac{b (b+2 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{16 c^2 x^3}+\frac{\left (a x^2+b x^3+c x^4\right )^{5/2}}{5 c x^5} \]

[Out]

(3*b*(b^2 - 4*a*c)*(b + 2*c*x)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(128*c^3*x) - (b*(b
+ 2*c*x)*(a*x^2 + b*x^3 + c*x^4)^(3/2))/(16*c^2*x^3) + (a*x^2 + b*x^3 + c*x^4)^(
5/2)/(5*c*x^5) - (3*b*(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])])/(256*c^(7/2)*Sqrt[a*x^2 + b*x^3 + c*x^4])

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Rubi [A]  time = 0.297191, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{3 b x \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{7/2} \sqrt{a x^2+b x^3+c x^4}}+\frac{3 b \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a x^2+b x^3+c x^4}}{128 c^3 x}-\frac{b (b+2 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{16 c^2 x^3}+\frac{\left (a x^2+b x^3+c x^4\right )^{5/2}}{5 c x^5} \]

Antiderivative was successfully verified.

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

[Out]

(3*b*(b^2 - 4*a*c)*(b + 2*c*x)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(128*c^3*x) - (b*(b
+ 2*c*x)*(a*x^2 + b*x^3 + c*x^4)^(3/2))/(16*c^2*x^3) + (a*x^2 + b*x^3 + c*x^4)^(
5/2)/(5*c*x^5) - (3*b*(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])])/(256*c^(7/2)*Sqrt[a*x^2 + b*x^3 + c*x^4])

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Rubi in Sympy [A]  time = 40.0156, size = 187, normalized size = 0.94 \[ - \frac{b \left (b + 2 c x\right ) \left (a x^{2} + b x^{3} + c x^{4}\right )^{\frac{3}{2}}}{16 c^{2} x^{3}} + \frac{3 b \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \sqrt{a x^{2} + b x^{3} + c x^{4}}}{128 c^{3} x} - \frac{3 b x \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{256 c^{\frac{7}{2}} \sqrt{a x^{2} + b x^{3} + c x^{4}}} + \frac{\left (a x^{2} + b x^{3} + c x^{4}\right )^{\frac{5}{2}}}{5 c x^{5}} \]

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**2,x)

[Out]

-b*(b + 2*c*x)*(a*x**2 + b*x**3 + c*x**4)**(3/2)/(16*c**2*x**3) + 3*b*(b + 2*c*x
)*(-4*a*c + b**2)*sqrt(a*x**2 + b*x**3 + c*x**4)/(128*c**3*x) - 3*b*x*(-4*a*c +
b**2)**2*sqrt(a + b*x + c*x**2)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x*
*2)))/(256*c**(7/2)*sqrt(a*x**2 + b*x**3 + c*x**4)) + (a*x**2 + b*x**3 + c*x**4)
**(5/2)/(5*c*x**5)

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Mathematica [A]  time = 0.285907, size = 161, normalized size = 0.81 \[ \frac{x \sqrt{a+x (b+c x)} \left (2 \sqrt{c} \sqrt{a+x (b+c x)} \left (4 b^2 c \left (2 c x^2-25 a\right )+8 b c^2 x \left (7 a+22 c x^2\right )+128 c^2 \left (a+c x^2\right )^2+15 b^4-10 b^3 c x\right )-15 b \left (b^2-4 a c\right )^2 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )\right )}{1280 c^{7/2} \sqrt{x^2 (a+x (b+c x))}} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[a + x*(b + c*x)]*(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(15*b^4 - 10*b^3*c*x +
128*c^2*(a + c*x^2)^2 + 4*b^2*c*(-25*a + 2*c*x^2) + 8*b*c^2*x*(7*a + 22*c*x^2))
- 15*b*(b^2 - 4*a*c)^2*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]]))/(1280
*c^(7/2)*Sqrt[x^2*(a + x*(b + c*x))])

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Maple [A]  time = 0.008, size = 289, normalized size = 1.5 \[{\frac{1}{1280\,{x}^{3}} \left ( c{x}^{4}+b{x}^{3}+a{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 256\, \left ( c{x}^{2}+bx+a \right ) ^{5/2}{c}^{7/2}-160\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{7/2}xb-80\, \left ( c{x}^{2}+bx+a \right ) ^{3/2}{c}^{5/2}{b}^{2}-240\,\sqrt{c{x}^{2}+bx+a}{c}^{7/2}xab+60\,\sqrt{c{x}^{2}+bx+a}{c}^{5/2}x{b}^{3}-120\,\sqrt{c{x}^{2}+bx+a}{c}^{5/2}a{b}^{2}+30\,\sqrt{c{x}^{2}+bx+a}{c}^{3/2}{b}^{4}-240\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){a}^{2}b{c}^{3}+120\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ) a{b}^{3}{c}^{2}-15\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){b}^{5}c \right ) \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1280*(c*x^4+b*x^3+a*x^2)^(3/2)*(256*(c*x^2+b*x+a)^(5/2)*c^(7/2)-160*(c*x^2+b*x
+a)^(3/2)*c^(7/2)*x*b-80*(c*x^2+b*x+a)^(3/2)*c^(5/2)*b^2-240*(c*x^2+b*x+a)^(1/2)
*c^(7/2)*x*a*b+60*(c*x^2+b*x+a)^(1/2)*c^(5/2)*x*b^3-120*(c*x^2+b*x+a)^(1/2)*c^(5
/2)*a*b^2+30*(c*x^2+b*x+a)^(1/2)*c^(3/2)*b^4-240*ln(1/2*(2*(c*x^2+b*x+a)^(1/2)*c
^(1/2)+2*c*x+b)/c^(1/2))*a^2*b*c^3+120*ln(1/2*(2*(c*x^2+b*x+a)^(1/2)*c^(1/2)+2*c
*x+b)/c^(1/2))*a*b^3*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^5*c)/x^3/(c*x^2+b*x+a)^(3/2)/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((c*x^4 + b*x^3 + a*x^2)^(3/2)/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.337554, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \sqrt{c} x \log \left (\frac{4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c^{2} x + b c\right )} -{\left (8 \, c^{2} x^{3} + 8 \, b c x^{2} +{\left (b^{2} + 4 \, a c\right )} x\right )} \sqrt{c}}{x}\right ) + 4 \,{\left (128 \, c^{5} x^{4} + 176 \, b c^{4} x^{3} + 15 \, b^{4} c - 100 \, a b^{2} c^{2} + 128 \, a^{2} c^{3} + 8 \,{\left (b^{2} c^{3} + 32 \, a c^{4}\right )} x^{2} - 2 \,{\left (5 \, b^{3} c^{2} - 28 \, a b c^{3}\right )} x\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{2560 \, c^{4} x}, \frac{15 \,{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} \sqrt{-c} x \arctan \left (\frac{{\left (2 \, c x^{2} + b x\right )} \sqrt{-c}}{2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} c}\right ) + 2 \,{\left (128 \, c^{5} x^{4} + 176 \, b c^{4} x^{3} + 15 \, b^{4} c - 100 \, a b^{2} c^{2} + 128 \, a^{2} c^{3} + 8 \,{\left (b^{2} c^{3} + 32 \, a c^{4}\right )} x^{2} - 2 \,{\left (5 \, b^{3} c^{2} - 28 \, a b c^{3}\right )} x\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{1280 \, c^{4} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2560*(15*(b^5 - 8*a*b^3*c + 16*a^2*b*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*(128*c^5*x^4 + 176*b*c^4*x^3 + 15*b^4*c - 100*a*b^2*c^2 + 128*a^2*c^3 + 8*
(b^2*c^3 + 32*a*c^4)*x^2 - 2*(5*b^3*c^2 - 28*a*b*c^3)*x)*sqrt(c*x^4 + b*x^3 + a*
x^2))/(c^4*x), 1/1280*(15*(b^5 - 8*a*b^3*c + 16*a^2*b*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*(128*c^5*x^4 + 17
6*b*c^4*x^3 + 15*b^4*c - 100*a*b^2*c^2 + 128*a^2*c^3 + 8*(b^2*c^3 + 32*a*c^4)*x^
2 - 2*(5*b^3*c^2 - 28*a*b*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. \[ \int \frac{\left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac{3}{2}}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.315092, size = 383, normalized size = 1.93 \[ \frac{1}{640} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \, c x{\rm sign}\left (x\right ) + 11 \, b{\rm sign}\left (x\right )\right )} x + \frac{b^{2} c^{3}{\rm sign}\left (x\right ) + 32 \, a c^{4}{\rm sign}\left (x\right )}{c^{4}}\right )} x - \frac{5 \, b^{3} c^{2}{\rm sign}\left (x\right ) - 28 \, a b c^{3}{\rm sign}\left (x\right )}{c^{4}}\right )} x + \frac{15 \, b^{4} c{\rm sign}\left (x\right ) - 100 \, a b^{2} c^{2}{\rm sign}\left (x\right ) + 128 \, a^{2} c^{3}{\rm sign}\left (x\right )}{c^{4}}\right )} + \frac{3 \,{\left (b^{5}{\rm sign}\left (x\right ) - 8 \, a b^{3} c{\rm sign}\left (x\right ) + 16 \, a^{2} b c^{2}{\rm sign}\left (x\right )\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{256 \, c^{\frac{7}{2}}} - \frac{{\left (15 \, b^{5}{\rm ln}\left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 120 \, a b^{3} c{\rm ln}\left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 240 \, a^{2} b c^{2}{\rm ln}\left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 30 \, \sqrt{a} b^{4} \sqrt{c} - 200 \, a^{\frac{3}{2}} b^{2} c^{\frac{3}{2}} + 256 \, a^{\frac{5}{2}} c^{\frac{5}{2}}\right )}{\rm sign}\left (x\right )}{1280 \, c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/640*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*c*x*sign(x) + 11*b*sign(x))*x + (b^2*c^3
*sign(x) + 32*a*c^4*sign(x))/c^4)*x - (5*b^3*c^2*sign(x) - 28*a*b*c^3*sign(x))/c
^4)*x + (15*b^4*c*sign(x) - 100*a*b^2*c^2*sign(x) + 128*a^2*c^3*sign(x))/c^4) +
3/256*(b^5*sign(x) - 8*a*b^3*c*sign(x) + 16*a^2*b*c^2*sign(x))*ln(abs(-2*(sqrt(c
)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(7/2) - 1/1280*(15*b^5*ln(abs(-b +
2*sqrt(a)*sqrt(c))) - 120*a*b^3*c*ln(abs(-b + 2*sqrt(a)*sqrt(c))) + 240*a^2*b*c^
2*ln(abs(-b + 2*sqrt(a)*sqrt(c))) + 30*sqrt(a)*b^4*sqrt(c) - 200*a^(3/2)*b^2*c^(
3/2) + 256*a^(5/2)*c^(5/2))*sign(x)/c^(7/2)