3.88 \(\int \frac{\left (b x+c x^2\right )^{3/2}}{x^{5/2}} \, dx\)

Optimal. Leaf size=76 \[ -2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )+\frac{2 b \sqrt{b x+c x^2}}{\sqrt{x}}+\frac{2 \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.101065, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -2 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )+\frac{2 b \sqrt{b x+c x^2}}{\sqrt{x}}+\frac{2 \left (b x+c x^2\right )^{3/2}}{3 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.6382, size = 70, normalized size = 0.92 \[ - 2 b^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )} + \frac{2 b \sqrt{b x + c x^{2}}}{\sqrt{x}} + \frac{2 \left (b x + c x^{2}\right )^{\frac{3}{2}}}{3 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0858002, size = 70, normalized size = 0.92 \[ \frac{2 \sqrt{x} \sqrt{b+c x} \left (\sqrt{b+c x} (4 b+c x)-3 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )\right )}{3 \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 61, normalized size = 0.8 \[ -{\frac{2}{3}\sqrt{x \left ( cx+b \right ) } \left ( 3\,{b}^{3/2}{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) -xc\sqrt{cx+b}-4\,\sqrt{cx+b}b \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{cx+b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241969, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, c^{2} x^{3} + 10 \, b c x^{2} + 3 \, \sqrt{c x^{2} + b x} b^{\frac{3}{2}} \sqrt{x} \log \left (-\frac{c x^{2} + 2 \, b x - 2 \, \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{x^{2}}\right ) + 8 \, b^{2} x}{3 \, \sqrt{c x^{2} + b x} \sqrt{x}}, \frac{2 \,{\left (c^{2} x^{3} + 5 \, b c x^{2} - 3 \, \sqrt{c x^{2} + b x} \sqrt{-b} b \sqrt{x} \arctan \left (\frac{b \sqrt{x}}{\sqrt{c x^{2} + b x} \sqrt{-b}}\right ) + 4 \, b^{2} x\right )}}{3 \, \sqrt{c x^{2} + b x} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/3*(2*c^2*x^3 + 10*b*c*x^2 + 3*sqrt(c*x^2 + b*x)*b^(3/2)*sqrt(x)*log(-(c*x^2 +
2*b*x - 2*sqrt(c*x^2 + b*x)*sqrt(b)*sqrt(x))/x^2) + 8*b^2*x)/(sqrt(c*x^2 + b*x)
*sqrt(x)), 2/3*(c^2*x^3 + 5*b*c*x^2 - 3*sqrt(c*x^2 + b*x)*sqrt(-b)*b*sqrt(x)*arc
tan(b*sqrt(x)/(sqrt(c*x^2 + b*x)*sqrt(-b))) + 4*b^2*x)/(sqrt(c*x^2 + b*x)*sqrt(x
))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}{x^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((x*(b + c*x))**(3/2)/x**(5/2), x)

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GIAC/XCAS [A]  time = 0.215876, size = 104, normalized size = 1.37 \[ \frac{2 \, b^{2} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} + \frac{2}{3} \,{\left (c x + b\right )}^{\frac{3}{2}} + 2 \, \sqrt{c x + b} b - \frac{2 \,{\left (3 \, b^{2} \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + 4 \, \sqrt{-b} b^{\frac{3}{2}}\right )}}{3 \, \sqrt{-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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