3.108 \(\int \frac{\sqrt{a x+b x^3+c x^5}}{x^{3/2}} \, dx\)

Optimal. Leaf size=194 \[ \frac{\sqrt{a x+b x^3+c x^5}}{2 \sqrt{x}}-\frac{\sqrt{a} \sqrt{x} \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 \sqrt{a x+b x^3+c x^5}}+\frac{b \sqrt{x} \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{4 \sqrt{c} \sqrt{a x+b x^3+c x^5}} \]

[Out]

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

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Rubi [A]  time = 0.444921, antiderivative size = 194, 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 \[ \frac{\sqrt{a x+b x^3+c x^5}}{2 \sqrt{x}}-\frac{\sqrt{a} \sqrt{x} \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 \sqrt{a x+b x^3+c x^5}}+\frac{b \sqrt{x} \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{4 \sqrt{c} \sqrt{a x+b x^3+c x^5}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 46.3928, size = 177, normalized size = 0.91 \[ - \frac{\sqrt{a} \sqrt{x} \sqrt{a + b x^{2} + c x^{4}} \operatorname{atanh}{\left (\frac{2 a + b x^{2}}{2 \sqrt{a} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{2 \sqrt{a x + b x^{3} + c x^{5}}} + \frac{b \sqrt{x} \sqrt{a + b x^{2} + c x^{4}} \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{4 \sqrt{c} \sqrt{a x + b x^{3} + c x^{5}}} + \frac{\sqrt{a x + b x^{3} + c x^{5}}}{2 \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**5+b*x**3+a*x)**(1/2)/x**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.325108, size = 165, normalized size = 0.85 \[ \frac{\sqrt{x} \sqrt{a+b x^2+c x^4} \left (2 \sqrt{c} \sqrt{a+b x^2+c x^4}-2 \sqrt{a} \sqrt{c} \log \left (2 \sqrt{a} \sqrt{a+b x^2+c x^4}+2 a+b x^2\right )+b \log \left (2 \sqrt{c} \sqrt{a+b x^2+c x^4}+b+2 c x^2\right )+4 \sqrt{a} \sqrt{c} \log (x)\right )}{4 \sqrt{c} \sqrt{x \left (a+b x^2+c x^4\right )}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 136, normalized size = 0.7 \[ -{\frac{1}{4}\sqrt{x \left ( c{x}^{4}+b{x}^{2}+a \right ) } \left ( 2\,\sqrt{a}\ln \left ({\frac{2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a}}{{x}^{2}}} \right ) \sqrt{c}-2\,\sqrt{c{x}^{4}+b{x}^{2}+a}\sqrt{c}-b\ln \left ({\frac{1}{2} \left ( 2\,c{x}^{2}+2\,\sqrt{c{x}^{4}+b{x}^{2}+a}\sqrt{c}+b \right ){\frac{1}{\sqrt{c}}}} \right ) \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}{\frac{1}{\sqrt{c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^5+b*x^3+a*x)^(1/2)/x^(3/2),x)

[Out]

-1/4*(x*(c*x^4+b*x^2+a))^(1/2)/x^(1/2)*(2*a^(1/2)*ln((2*a+b*x^2+2*a^(1/2)*(c*x^4
+b*x^2+a)^(1/2))/x^2)*c^(1/2)-2*(c*x^4+b*x^2+a)^(1/2)*c^(1/2)-b*ln(1/2*(2*c*x^2+
2*(c*x^4+b*x^2+a)^(1/2)*c^(1/2)+b)/c^(1/2)))/(c*x^4+b*x^2+a)^(1/2)/c^(1/2)

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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(sqrt(c*x^5 + b*x^3 + a*x)/x^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.323933, size = 1, normalized size = 0.01 \[ \left [\frac{b x \log \left (-\frac{4 \, \sqrt{c x^{5} + b x^{3} + a x}{\left (2 \, c^{2} x^{2} + b c\right )} \sqrt{x} +{\left (8 \, c^{2} x^{5} + 8 \, b c x^{3} +{\left (b^{2} + 4 \, a c\right )} x\right )} \sqrt{c}}{x}\right ) + 2 \, \sqrt{a} \sqrt{c} x \log \left (-\frac{{\left (b^{2} + 4 \, a c\right )} x^{5} + 8 \, a b x^{3} + 8 \, a^{2} x - 4 \, \sqrt{c x^{5} + b x^{3} + a x}{\left (b x^{2} + 2 \, a\right )} \sqrt{a} \sqrt{x}}{x^{5}}\right ) + 4 \, \sqrt{c x^{5} + b x^{3} + a x} \sqrt{c} \sqrt{x}}{8 \, \sqrt{c} x}, \frac{b x \arctan \left (\frac{{\left (2 \, c x^{3} + b x\right )} \sqrt{-c}}{2 \, \sqrt{c x^{5} + b x^{3} + a x} c \sqrt{x}}\right ) + \sqrt{a} \sqrt{-c} x \log \left (-\frac{{\left (b^{2} + 4 \, a c\right )} x^{5} + 8 \, a b x^{3} + 8 \, a^{2} x - 4 \, \sqrt{c x^{5} + b x^{3} + a x}{\left (b x^{2} + 2 \, a\right )} \sqrt{a} \sqrt{x}}{x^{5}}\right ) + 2 \, \sqrt{c x^{5} + b x^{3} + a x} \sqrt{-c} \sqrt{x}}{4 \, \sqrt{-c} x}, -\frac{4 \, \sqrt{-a} \sqrt{c} x \arctan \left (\frac{b x^{3} + 2 \, a x}{2 \, \sqrt{c x^{5} + b x^{3} + a x} \sqrt{-a} \sqrt{x}}\right ) - b x \log \left (-\frac{4 \, \sqrt{c x^{5} + b x^{3} + a x}{\left (2 \, c^{2} x^{2} + b c\right )} \sqrt{x} +{\left (8 \, c^{2} x^{5} + 8 \, b c x^{3} +{\left (b^{2} + 4 \, a c\right )} x\right )} \sqrt{c}}{x}\right ) - 4 \, \sqrt{c x^{5} + b x^{3} + a x} \sqrt{c} \sqrt{x}}{8 \, \sqrt{c} x}, -\frac{2 \, \sqrt{-a} \sqrt{-c} x \arctan \left (\frac{b x^{3} + 2 \, a x}{2 \, \sqrt{c x^{5} + b x^{3} + a x} \sqrt{-a} \sqrt{x}}\right ) - b x \arctan \left (\frac{{\left (2 \, c x^{3} + b x\right )} \sqrt{-c}}{2 \, \sqrt{c x^{5} + b x^{3} + a x} c \sqrt{x}}\right ) - 2 \, \sqrt{c x^{5} + b x^{3} + a x} \sqrt{-c} \sqrt{x}}{4 \, \sqrt{-c} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(b*x*log(-(4*sqrt(c*x^5 + b*x^3 + a*x)*(2*c^2*x^2 + b*c)*sqrt(x) + (8*c^2*x
^5 + 8*b*c*x^3 + (b^2 + 4*a*c)*x)*sqrt(c))/x) + 2*sqrt(a)*sqrt(c)*x*log(-((b^2 +
4*a*c)*x^5 + 8*a*b*x^3 + 8*a^2*x - 4*sqrt(c*x^5 + b*x^3 + a*x)*(b*x^2 + 2*a)*sq
rt(a)*sqrt(x))/x^5) + 4*sqrt(c*x^5 + b*x^3 + a*x)*sqrt(c)*sqrt(x))/(sqrt(c)*x),
1/4*(b*x*arctan(1/2*(2*c*x^3 + b*x)*sqrt(-c)/(sqrt(c*x^5 + b*x^3 + a*x)*c*sqrt(x
))) + sqrt(a)*sqrt(-c)*x*log(-((b^2 + 4*a*c)*x^5 + 8*a*b*x^3 + 8*a^2*x - 4*sqrt(
c*x^5 + b*x^3 + a*x)*(b*x^2 + 2*a)*sqrt(a)*sqrt(x))/x^5) + 2*sqrt(c*x^5 + b*x^3
+ a*x)*sqrt(-c)*sqrt(x))/(sqrt(-c)*x), -1/8*(4*sqrt(-a)*sqrt(c)*x*arctan(1/2*(b*
x^3 + 2*a*x)/(sqrt(c*x^5 + b*x^3 + a*x)*sqrt(-a)*sqrt(x))) - b*x*log(-(4*sqrt(c*
x^5 + b*x^3 + a*x)*(2*c^2*x^2 + b*c)*sqrt(x) + (8*c^2*x^5 + 8*b*c*x^3 + (b^2 + 4
*a*c)*x)*sqrt(c))/x) - 4*sqrt(c*x^5 + b*x^3 + a*x)*sqrt(c)*sqrt(x))/(sqrt(c)*x),
-1/4*(2*sqrt(-a)*sqrt(-c)*x*arctan(1/2*(b*x^3 + 2*a*x)/(sqrt(c*x^5 + b*x^3 + a*
x)*sqrt(-a)*sqrt(x))) - b*x*arctan(1/2*(2*c*x^3 + b*x)*sqrt(-c)/(sqrt(c*x^5 + b*
x^3 + a*x)*c*sqrt(x))) - 2*sqrt(c*x^5 + b*x^3 + a*x)*sqrt(-c)*sqrt(x))/(sqrt(-c)
*x)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (a + b x^{2} + c x^{4}\right )}}{x^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**5+b*x**3+a*x)**(1/2)/x**(3/2),x)

[Out]

Integral(sqrt(x*(a + b*x**2 + c*x**4))/x**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{5} + b x^{3} + a x}}{x^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(c*x^5 + b*x^3 + a*x)/x^(3/2), x)