3.305 \(\int \frac{(a+b x)^{5/2}}{x^2} \, dx\)

Optimal. Leaf size=66 \[ -5 a^{3/2} b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-\frac{(a+b x)^{5/2}}{x}+\frac{5}{3} b (a+b x)^{3/2}+5 a b \sqrt{a+b x} \]

[Out]

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

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Rubi [A]  time = 0.0650279, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -5 a^{3/2} b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-\frac{(a+b x)^{5/2}}{x}+\frac{5}{3} b (a+b x)^{3/2}+5 a b \sqrt{a+b x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(5/2)/x^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.69868, size = 60, normalized size = 0.91 \[ - 5 a^{\frac{3}{2}} b \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )} + 5 a b \sqrt{a + b x} + \frac{5 b \left (a + b x\right )^{\frac{3}{2}}}{3} - \frac{\left (a + b x\right )^{\frac{5}{2}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0521342, size = 58, normalized size = 0.88 \[ \sqrt{a+b x} \left (-\frac{a^2}{x}+\frac{14 a b}{3}+\frac{2 b^2 x}{3}\right )-5 a^{3/2} b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[a + b*x]*((14*a*b)/3 - a^2/x + (2*b^2*x)/3) - 5*a^(3/2)*b*ArcTanh[Sqrt[a +
b*x]/Sqrt[a]]

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Maple [A]  time = 0.017, size = 61, normalized size = 0.9 \[ 2\,b \left ( 1/3\, \left ( bx+a \right ) ^{3/2}+2\,a\sqrt{bx+a}+{a}^{2} \left ( -1/2\,{\frac{\sqrt{bx+a}}{bx}}-5/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)/x^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226943, size = 1, normalized size = 0.02 \[ \left [\frac{15 \, a^{\frac{3}{2}} b x \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (2 \, b^{2} x^{2} + 14 \, a b x - 3 \, a^{2}\right )} \sqrt{b x + a}}{6 \, x}, -\frac{15 \, \sqrt{-a} a b x \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) -{\left (2 \, b^{2} x^{2} + 14 \, a b x - 3 \, a^{2}\right )} \sqrt{b x + a}}{3 \, x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(15*a^(3/2)*b*x*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(2*b^2*x^2
+ 14*a*b*x - 3*a^2)*sqrt(b*x + a))/x, -1/3*(15*sqrt(-a)*a*b*x*arctan(sqrt(b*x +
a)/sqrt(-a)) - (2*b^2*x^2 + 14*a*b*x - 3*a^2)*sqrt(b*x + a))/x]

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Sympy [A]  time = 11.5672, size = 99, normalized size = 1.5 \[ - \frac{a^{\frac{5}{2}} \sqrt{1 + \frac{b x}{a}}}{x} + \frac{14 a^{\frac{3}{2}} b \sqrt{1 + \frac{b x}{a}}}{3} + \frac{5 a^{\frac{3}{2}} b \log{\left (\frac{b x}{a} \right )}}{2} - 5 a^{\frac{3}{2}} b \log{\left (\sqrt{1 + \frac{b x}{a}} + 1 \right )} + \frac{2 \sqrt{a} b^{2} x \sqrt{1 + \frac{b x}{a}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(5/2)*sqrt(1 + b*x/a)/x + 14*a**(3/2)*b*sqrt(1 + b*x/a)/3 + 5*a**(3/2)*b*log
(b*x/a)/2 - 5*a**(3/2)*b*log(sqrt(1 + b*x/a) + 1) + 2*sqrt(a)*b**2*x*sqrt(1 + b*
x/a)/3

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GIAC/XCAS [A]  time = 0.207939, size = 100, normalized size = 1.52 \[ \frac{\frac{15 \, a^{2} b^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 2 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{2} + 12 \, \sqrt{b x + a} a b^{2} - \frac{3 \, \sqrt{b x + a} a^{2} b}{x}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(15*a^2*b^2*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a) + 2*(b*x + a)^(3/2)*b^2
+ 12*sqrt(b*x + a)*a*b^2 - 3*sqrt(b*x + a)*a^2*b/x)/b