3.80 \(\int \frac{\sqrt{b x+c x^2}}{x^{9/2}} \, dx\)

Optimal. Leaf size=114 \[ -\frac{c^3 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{5/2}}+\frac{c^2 \sqrt{b x+c x^2}}{8 b^2 x^{3/2}}-\frac{c \sqrt{b x+c x^2}}{12 b x^{5/2}}-\frac{\sqrt{b x+c x^2}}{3 x^{7/2}} \]

[Out]

-Sqrt[b*x + c*x^2]/(3*x^(7/2)) - (c*Sqrt[b*x + c*x^2])/(12*b*x^(5/2)) + (c^2*Sqr
t[b*x + c*x^2])/(8*b^2*x^(3/2)) - (c^3*ArcTanh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x
])])/(8*b^(5/2))

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Rubi [A]  time = 0.146341, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{c^3 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{5/2}}+\frac{c^2 \sqrt{b x+c x^2}}{8 b^2 x^{3/2}}-\frac{c \sqrt{b x+c x^2}}{12 b x^{5/2}}-\frac{\sqrt{b x+c x^2}}{3 x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x + c*x^2]/x^(9/2),x]

[Out]

-Sqrt[b*x + c*x^2]/(3*x^(7/2)) - (c*Sqrt[b*x + c*x^2])/(12*b*x^(5/2)) + (c^2*Sqr
t[b*x + c*x^2])/(8*b^2*x^(3/2)) - (c^3*ArcTanh[Sqrt[b*x + c*x^2]/(Sqrt[b]*Sqrt[x
])])/(8*b^(5/2))

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Rubi in Sympy [A]  time = 16.4844, size = 99, normalized size = 0.87 \[ - \frac{\sqrt{b x + c x^{2}}}{3 x^{\frac{7}{2}}} - \frac{c \sqrt{b x + c x^{2}}}{12 b x^{\frac{5}{2}}} + \frac{c^{2} \sqrt{b x + c x^{2}}}{8 b^{2} x^{\frac{3}{2}}} - \frac{c^{3} \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{8 b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(1/2)/x**(9/2),x)

[Out]

-sqrt(b*x + c*x**2)/(3*x**(7/2)) - c*sqrt(b*x + c*x**2)/(12*b*x**(5/2)) + c**2*s
qrt(b*x + c*x**2)/(8*b**2*x**(3/2)) - c**3*atanh(sqrt(b*x + c*x**2)/(sqrt(b)*sqr
t(x)))/(8*b**(5/2))

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Mathematica [A]  time = 0.0780781, size = 93, normalized size = 0.82 \[ -\frac{\sqrt{x (b+c x)} \left (\sqrt{b} \sqrt{b+c x} \left (8 b^2+2 b c x-3 c^2 x^2\right )+3 c^3 x^3 \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )\right )}{24 b^{5/2} x^{7/2} \sqrt{b+c x}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[b*x + c*x^2]/x^(9/2),x]

[Out]

-(Sqrt[x*(b + c*x)]*(Sqrt[b]*Sqrt[b + c*x]*(8*b^2 + 2*b*c*x - 3*c^2*x^2) + 3*c^3
*x^3*ArcTanh[Sqrt[b + c*x]/Sqrt[b]]))/(24*b^(5/2)*x^(7/2)*Sqrt[b + c*x])

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Maple [A]  time = 0.015, size = 90, normalized size = 0.8 \[ -{\frac{1}{24}\sqrt{x \left ( cx+b \right ) } \left ( 3\,{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{3}{c}^{3}-3\,{x}^{2}{c}^{2}\sqrt{b}\sqrt{cx+b}+2\,x{b}^{3/2}c\sqrt{cx+b}+8\,{b}^{5/2}\sqrt{cx+b} \right ){b}^{-{\frac{5}{2}}}{x}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{cx+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(1/2)/x^(9/2),x)

[Out]

-1/24*(x*(c*x+b))^(1/2)/b^(5/2)*(3*arctanh((c*x+b)^(1/2)/b^(1/2))*x^3*c^3-3*x^2*
c^2*b^(1/2)*(c*x+b)^(1/2)+2*x*b^(3/2)*c*(c*x+b)^(1/2)+8*b^(5/2)*(c*x+b)^(1/2))/x
^(7/2)/(c*x+b)^(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^2 + b*x)/x^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228379, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, c^{3} x^{4} \log \left (\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} -{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right ) + 2 \,{\left (3 \, c^{2} x^{2} - 2 \, b c x - 8 \, b^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{48 \, b^{\frac{5}{2}} x^{4}}, -\frac{3 \, c^{3} x^{4} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) -{\left (3 \, c^{2} x^{2} - 2 \, b c x - 8 \, b^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{-b} \sqrt{x}}{24 \, \sqrt{-b} b^{2} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)/x^(9/2),x, algorithm="fricas")

[Out]

[1/48*(3*c^3*x^4*log((2*sqrt(c*x^2 + b*x)*b*sqrt(x) - (c*x^2 + 2*b*x)*sqrt(b))/x
^2) + 2*(3*c^2*x^2 - 2*b*c*x - 8*b^2)*sqrt(c*x^2 + b*x)*sqrt(b)*sqrt(x))/(b^(5/2
)*x^4), -1/24*(3*c^3*x^4*arctan(sqrt(-b)*sqrt(x)/sqrt(c*x^2 + b*x)) - (3*c^2*x^2
- 2*b*c*x - 8*b^2)*sqrt(c*x^2 + b*x)*sqrt(-b)*sqrt(x))/(sqrt(-b)*b^2*x^4)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**(1/2)/x**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.242398, size = 97, normalized size = 0.85 \[ \frac{1}{24} \, c^{3}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{2}} + \frac{3 \,{\left (c x + b\right )}^{\frac{5}{2}} - 8 \,{\left (c x + b\right )}^{\frac{3}{2}} b - 3 \, \sqrt{c x + b} b^{2}}{b^{2} c^{3} x^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)/x^(9/2),x, algorithm="giac")

[Out]

1/24*c^3*(3*arctan(sqrt(c*x + b)/sqrt(-b))/(sqrt(-b)*b^2) + (3*(c*x + b)^(5/2) -
8*(c*x + b)^(3/2)*b - 3*sqrt(c*x + b)*b^2)/(b^2*c^3*x^3))