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

Optimal. Leaf size=65 \[ -2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+2 a^2 \sqrt{a+b x}+\frac{2}{3} a (a+b x)^{3/2}+\frac{2}{5} (a+b x)^{5/2} \]

[Out]

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

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Rubi [A]  time = 0.0628168, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+2 a^2 \sqrt{a+b x}+\frac{2}{3} a (a+b x)^{3/2}+\frac{2}{5} (a+b x)^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(5/2)/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.56877, size = 60, normalized size = 0.92 \[ - 2 a^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )} + 2 a^{2} \sqrt{a + b x} + \frac{2 a \left (a + b x\right )^{\frac{3}{2}}}{3} + \frac{2 \left (a + b x\right )^{\frac{5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0361136, size = 56, normalized size = 0.86 \[ \frac{2}{15} \sqrt{a+b x} \left (23 a^2+11 a b x+3 b^2 x^2\right )-2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^(5/2)/x,x]

[Out]

(2*Sqrt[a + b*x]*(23*a^2 + 11*a*b*x + 3*b^2*x^2))/15 - 2*a^(5/2)*ArcTanh[Sqrt[a
+ b*x]/Sqrt[a]]

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Maple [A]  time = 0.011, size = 50, normalized size = 0.8 \[{\frac{2\,a}{3} \left ( bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{2}{5} \left ( bx+a \right ) ^{{\frac{5}{2}}}}-2\,{a}^{5/2}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) +2\,{a}^{2}\sqrt{bx+a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)/x,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220695, size = 1, normalized size = 0.02 \[ \left [a^{\frac{5}{2}} \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + \frac{2}{15} \,{\left (3 \, b^{2} x^{2} + 11 \, a b x + 23 \, a^{2}\right )} \sqrt{b x + a}, -2 \, \sqrt{-a} a^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) + \frac{2}{15} \,{\left (3 \, b^{2} x^{2} + 11 \, a b x + 23 \, a^{2}\right )} \sqrt{b x + a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[a^(5/2)*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2/15*(3*b^2*x^2 + 11*a*b
*x + 23*a^2)*sqrt(b*x + a), -2*sqrt(-a)*a^2*arctan(sqrt(b*x + a)/sqrt(-a)) + 2/1
5*(3*b^2*x^2 + 11*a*b*x + 23*a^2)*sqrt(b*x + a)]

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Sympy [A]  time = 11.7631, size = 97, normalized size = 1.49 \[ \frac{46 a^{\frac{5}{2}} \sqrt{1 + \frac{b x}{a}}}{15} + a^{\frac{5}{2}} \log{\left (\frac{b x}{a} \right )} - 2 a^{\frac{5}{2}} \log{\left (\sqrt{1 + \frac{b x}{a}} + 1 \right )} + \frac{22 a^{\frac{3}{2}} b x \sqrt{1 + \frac{b x}{a}}}{15} + \frac{2 \sqrt{a} b^{2} x^{2} \sqrt{1 + \frac{b x}{a}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(5/2)/x,x)

[Out]

46*a**(5/2)*sqrt(1 + b*x/a)/15 + a**(5/2)*log(b*x/a) - 2*a**(5/2)*log(sqrt(1 + b
*x/a) + 1) + 22*a**(3/2)*b*x*sqrt(1 + b*x/a)/15 + 2*sqrt(a)*b**2*x**2*sqrt(1 + b
*x/a)/5

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GIAC/XCAS [A]  time = 0.208983, size = 76, normalized size = 1.17 \[ \frac{2 \, a^{3} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \frac{2}{5} \,{\left (b x + a\right )}^{\frac{5}{2}} + \frac{2}{3} \,{\left (b x + a\right )}^{\frac{3}{2}} a + 2 \, \sqrt{b x + a} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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