3.79 \(\int \frac{\sqrt{b x+c x^2}}{x^{7/2}} \, dx\)

Optimal. Leaf size=86 \[ \frac{c^2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{3/2}}-\frac{c \sqrt{b x+c x^2}}{4 b x^{3/2}}-\frac{\sqrt{b x+c x^2}}{2 x^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.10688, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{c^2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 b^{3/2}}-\frac{c \sqrt{b x+c x^2}}{4 b x^{3/2}}-\frac{\sqrt{b x+c x^2}}{2 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x + c*x^2]/x^(7/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 13.0746, size = 73, normalized size = 0.85 \[ - \frac{\sqrt{b x + c x^{2}}}{2 x^{\frac{5}{2}}} - \frac{c \sqrt{b x + c x^{2}}}{4 b x^{\frac{3}{2}}} + \frac{c^{2} \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{4 b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(1/2)/x**(7/2),x)

[Out]

-sqrt(b*x + c*x**2)/(2*x**(5/2)) - c*sqrt(b*x + c*x**2)/(4*b*x**(3/2)) + c**2*at
anh(sqrt(b*x + c*x**2)/(sqrt(b)*sqrt(x)))/(4*b**(3/2))

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Mathematica [A]  time = 0.0628863, size = 81, normalized size = 0.94 \[ \frac{\sqrt{x (b+c x)} \left (c^2 x^2 \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )-\sqrt{b} \sqrt{b+c x} (2 b+c x)\right )}{4 b^{3/2} x^{5/2} \sqrt{b+c x}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[b*x + c*x^2]/x^(7/2),x]

[Out]

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

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Maple [A]  time = 0.013, size = 71, normalized size = 0.8 \[{\frac{1}{4}\sqrt{x \left ( cx+b \right ) } \left ({\it Artanh} \left ({1\sqrt{cx+b}{\frac{1}{\sqrt{b}}}} \right ){c}^{2}{x}^{2}-xc\sqrt{cx+b}\sqrt{b}-2\,{b}^{3/2}\sqrt{cx+b} \right ){b}^{-{\frac{3}{2}}}{x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{cx+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(1/2)/x^(7/2),x)

[Out]

1/4*(x*(c*x+b))^(1/2)/b^(3/2)*(arctanh((c*x+b)^(1/2)/b^(1/2))*c^2*x^2-x*c*(c*x+b
)^(1/2)*b^(1/2)-2*b^(3/2)*(c*x+b)^(1/2))/x^(5/2)/(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^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230496, size = 1, normalized size = 0.01 \[ \left [\frac{c^{2} x^{3} \log \left (-\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} +{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right ) - 2 \, \sqrt{c x^{2} + b x}{\left (c x + 2 \, b\right )} \sqrt{b} \sqrt{x}}{8 \, b^{\frac{3}{2}} x^{3}}, \frac{c^{2} x^{3} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) - \sqrt{c x^{2} + b x}{\left (c x + 2 \, b\right )} \sqrt{-b} \sqrt{x}}{4 \, \sqrt{-b} b x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)/x^(7/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )}}{x^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**(1/2)/x**(7/2),x)

[Out]

Integral(sqrt(x*(b + c*x))/x**(7/2), x)

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GIAC/XCAS [A]  time = 0.22872, size = 76, normalized size = 0.88 \[ -\frac{1}{4} \, c^{2}{\left (\frac{\arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} + \frac{{\left (c x + b\right )}^{\frac{3}{2}} + \sqrt{c x + b} b}{b c^{2} x^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)/x^(7/2),x, algorithm="giac")

[Out]

-1/4*c^2*(arctan(sqrt(c*x + b)/sqrt(-b))/(sqrt(-b)*b) + ((c*x + b)^(3/2) + sqrt(
c*x + b)*b)/(b*c^2*x^2))