3.206 \(\int \frac{\left (a+b x^3+c x^6\right )^{3/2}}{x} \, dx\)

Optimal. Leaf size=155 \[ -\frac{1}{3} a^{3/2} \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )-\frac{b \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{48 c^{3/2}}+\frac{\left (8 a c+b^2+2 b c x^3\right ) \sqrt{a+b x^3+c x^6}}{24 c}+\frac{1}{9} \left (a+b x^3+c x^6\right )^{3/2} \]

[Out]

((b^2 + 8*a*c + 2*b*c*x^3)*Sqrt[a + b*x^3 + c*x^6])/(24*c) + (a + b*x^3 + c*x^6)
^(3/2)/9 - (a^(3/2)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/
3 - (b*(b^2 - 12*a*c)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])]
)/(48*c^(3/2))

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Rubi [A]  time = 0.427794, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{1}{3} a^{3/2} \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )-\frac{b \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{48 c^{3/2}}+\frac{\left (8 a c+b^2+2 b c x^3\right ) \sqrt{a+b x^3+c x^6}}{24 c}+\frac{1}{9} \left (a+b x^3+c x^6\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3 + c*x^6)^(3/2)/x,x]

[Out]

((b^2 + 8*a*c + 2*b*c*x^3)*Sqrt[a + b*x^3 + c*x^6])/(24*c) + (a + b*x^3 + c*x^6)
^(3/2)/9 - (a^(3/2)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/
3 - (b*(b^2 - 12*a*c)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])]
)/(48*c^(3/2))

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Rubi in Sympy [A]  time = 46.631, size = 139, normalized size = 0.9 \[ - \frac{a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{2 a + b x^{3}}{2 \sqrt{a} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{3} - \frac{b \left (- 12 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{3}}{2 \sqrt{c} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{48 c^{\frac{3}{2}}} + \frac{\left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}{9} + \frac{\sqrt{a + b x^{3} + c x^{6}} \left (4 a c + \frac{b^{2}}{2} + b c x^{3}\right )}{12 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**6+b*x**3+a)**(3/2)/x,x)

[Out]

-a**(3/2)*atanh((2*a + b*x**3)/(2*sqrt(a)*sqrt(a + b*x**3 + c*x**6)))/3 - b*(-12
*a*c + b**2)*atanh((b + 2*c*x**3)/(2*sqrt(c)*sqrt(a + b*x**3 + c*x**6)))/(48*c**
(3/2)) + (a + b*x**3 + c*x**6)**(3/2)/9 + sqrt(a + b*x**3 + c*x**6)*(4*a*c + b**
2/2 + b*c*x**3)/(12*c)

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Mathematica [A]  time = 0.473632, size = 181, normalized size = 1.17 \[ \frac{1}{144} \left (48 a^{3/2} \left (\log \left (x^3\right )-\log \left (2 \sqrt{a} \sqrt{a+b x^3+c x^6}+2 a+b x^3\right )\right )-\frac{3 b^3 \log \left (2 \sqrt{c} \sqrt{a+b x^3+c x^6}+b+2 c x^3\right )}{c^{3/2}}+\frac{2 \sqrt{a+b x^3+c x^6} \left (8 c \left (4 a+c x^6\right )+3 b^2+14 b c x^3\right )}{c}+\frac{36 a b \log \left (2 \sqrt{c} \sqrt{a+b x^3+c x^6}+b+2 c x^3\right )}{\sqrt{c}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3 + c*x^6)^(3/2)/x,x]

[Out]

((2*Sqrt[a + b*x^3 + c*x^6]*(3*b^2 + 14*b*c*x^3 + 8*c*(4*a + c*x^6)))/c + 48*a^(
3/2)*(Log[x^3] - Log[2*a + b*x^3 + 2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6]]) - (3*b^3*
Log[b + 2*c*x^3 + 2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6]])/c^(3/2) + (36*a*b*Log[b +
2*c*x^3 + 2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6]])/Sqrt[c])/144

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Maple [F]  time = 0.022, size = 0, normalized size = 0. \[ \int{\frac{1}{x} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^6+b*x^3+a)^(3/2)/x,x)

[Out]

int((c*x^6+b*x^3+a)^(3/2)/x,x)

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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^6 + b*x^3 + a)^(3/2)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.395202, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^(3/2)/x,x, algorithm="fricas")

[Out]

[1/288*(48*a^(3/2)*c^(3/2)*log(-((b^2 + 4*a*c)*x^6 + 8*a*b*x^3 - 4*sqrt(c*x^6 +
b*x^3 + a)*(b*x^3 + 2*a)*sqrt(a) + 8*a^2)/x^6) + 4*(8*c^2*x^6 + 14*b*c*x^3 + 3*b
^2 + 32*a*c)*sqrt(c*x^6 + b*x^3 + a)*sqrt(c) - 3*(b^3 - 12*a*b*c)*log(-4*sqrt(c*
x^6 + b*x^3 + a)*(2*c^2*x^3 + b*c) - (8*c^2*x^6 + 8*b*c*x^3 + b^2 + 4*a*c)*sqrt(
c)))/c^(3/2), 1/144*(24*a^(3/2)*sqrt(-c)*c*log(-((b^2 + 4*a*c)*x^6 + 8*a*b*x^3 -
4*sqrt(c*x^6 + b*x^3 + a)*(b*x^3 + 2*a)*sqrt(a) + 8*a^2)/x^6) + 2*(8*c^2*x^6 +
14*b*c*x^3 + 3*b^2 + 32*a*c)*sqrt(c*x^6 + b*x^3 + a)*sqrt(-c) - 3*(b^3 - 12*a*b*
c)*arctan(1/2*(2*c*x^3 + b)*sqrt(-c)/(sqrt(c*x^6 + b*x^3 + a)*c)))/(sqrt(-c)*c),
-1/288*(96*sqrt(-a)*a*c^(3/2)*arctan(1/2*(b*x^3 + 2*a)/(sqrt(c*x^6 + b*x^3 + a)
*sqrt(-a))) - 4*(8*c^2*x^6 + 14*b*c*x^3 + 3*b^2 + 32*a*c)*sqrt(c*x^6 + b*x^3 + a
)*sqrt(c) + 3*(b^3 - 12*a*b*c)*log(-4*sqrt(c*x^6 + b*x^3 + a)*(2*c^2*x^3 + b*c)
- (8*c^2*x^6 + 8*b*c*x^3 + b^2 + 4*a*c)*sqrt(c)))/c^(3/2), -1/144*(48*sqrt(-a)*a
*sqrt(-c)*c*arctan(1/2*(b*x^3 + 2*a)/(sqrt(c*x^6 + b*x^3 + a)*sqrt(-a))) - 2*(8*
c^2*x^6 + 14*b*c*x^3 + 3*b^2 + 32*a*c)*sqrt(c*x^6 + b*x^3 + a)*sqrt(-c) + 3*(b^3
- 12*a*b*c)*arctan(1/2*(2*c*x^3 + b)*sqrt(-c)/(sqrt(c*x^6 + b*x^3 + a)*c)))/(sq
rt(-c)*c)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**6+b*x**3+a)**(3/2)/x,x)

[Out]

Integral((a + b*x**3 + c*x**6)**(3/2)/x, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{6} + b x^{3} + a\right )}^{\frac{3}{2}}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^(3/2)/x,x, algorithm="giac")

[Out]

integrate((c*x^6 + b*x^3 + a)^(3/2)/x, x)