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

Optimal. Leaf size=89 \[ 5 a b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )+5 b^2 \sqrt{a x+b x^2}-\frac{10 b \left (a x+b x^2\right )^{3/2}}{3 x^2}-\frac{2 \left (a x+b x^2\right )^{5/2}}{3 x^4} \]

[Out]

5*b^2*Sqrt[a*x + b*x^2] - (10*b*(a*x + b*x^2)^(3/2))/(3*x^2) - (2*(a*x + b*x^2)^
(5/2))/(3*x^4) + 5*a*b^(3/2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]]

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Rubi [A]  time = 0.120631, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ 5 a b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )+5 b^2 \sqrt{a x+b x^2}-\frac{10 b \left (a x+b x^2\right )^{3/2}}{3 x^2}-\frac{2 \left (a x+b x^2\right )^{5/2}}{3 x^4} \]

Antiderivative was successfully verified.

[In]  Int[(a*x + b*x^2)^(5/2)/x^5,x]

[Out]

5*b^2*Sqrt[a*x + b*x^2] - (10*b*(a*x + b*x^2)^(3/2))/(3*x^2) - (2*(a*x + b*x^2)^
(5/2))/(3*x^4) + 5*a*b^(3/2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]]

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Rubi in Sympy [A]  time = 13.3156, size = 83, normalized size = 0.93 \[ 5 a b^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a x + b x^{2}}} \right )} + 5 b^{2} \sqrt{a x + b x^{2}} - \frac{10 b \left (a x + b x^{2}\right )^{\frac{3}{2}}}{3 x^{2}} - \frac{2 \left (a x + b x^{2}\right )^{\frac{5}{2}}}{3 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*a*b**(3/2)*atanh(sqrt(b)*x/sqrt(a*x + b*x**2)) + 5*b**2*sqrt(a*x + b*x**2) - 1
0*b*(a*x + b*x**2)**(3/2)/(3*x**2) - 2*(a*x + b*x**2)**(5/2)/(3*x**4)

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Mathematica [A]  time = 0.0734396, size = 94, normalized size = 1.06 \[ \frac{\sqrt{x (a+b x)} \left (\sqrt{a+b x} \left (-2 a^2-14 a b x+3 b^2 x^2\right )+15 a b^{3/2} x^{3/2} \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )\right )}{3 x^2 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x*(a + b*x)]*(Sqrt[a + b*x]*(-2*a^2 - 14*a*b*x + 3*b^2*x^2) + 15*a*b^(3/2)
*x^(3/2)*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[a + b*x]]))/(3*x^2*Sqrt[a + b*x])

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Maple [B]  time = 0.007, size = 209, normalized size = 2.4 \[ -{\frac{2}{3\,a{x}^{5}} \left ( b{x}^{2}+ax \right ) ^{{\frac{7}{2}}}}-{\frac{8\,b}{3\,{a}^{2}{x}^{4}} \left ( b{x}^{2}+ax \right ) ^{{\frac{7}{2}}}}+16\,{\frac{{b}^{2} \left ( b{x}^{2}+ax \right ) ^{7/2}}{{a}^{3}{x}^{3}}}-{\frac{128\,{b}^{3}}{3\,{a}^{4}{x}^{2}} \left ( b{x}^{2}+ax \right ) ^{{\frac{7}{2}}}}+{\frac{128\,{b}^{4}}{3\,{a}^{4}} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}}+{\frac{80\,{b}^{4}x}{3\,{a}^{3}} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}+{\frac{40\,{b}^{3}}{3\,{a}^{2}} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}-10\,{\frac{{b}^{3}\sqrt{b{x}^{2}+ax}x}{a}}-5\,{b}^{2}\sqrt{b{x}^{2}+ax}+{\frac{5\,a}{2}{b}^{{\frac{3}{2}}}\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a*x)^(5/2)/x^5,x)

[Out]

-2/3/a/x^5*(b*x^2+a*x)^(7/2)-8/3*b/a^2/x^4*(b*x^2+a*x)^(7/2)+16*b^2/a^3/x^3*(b*x
^2+a*x)^(7/2)-128/3*b^3/a^4/x^2*(b*x^2+a*x)^(7/2)+128/3*b^4/a^4*(b*x^2+a*x)^(5/2
)+80/3*b^4/a^3*(b*x^2+a*x)^(3/2)*x+40/3*b^3/a^2*(b*x^2+a*x)^(3/2)-10*b^3/a*(b*x^
2+a*x)^(1/2)*x-5*b^2*(b*x^2+a*x)^(1/2)+5/2*b^(3/2)*a*ln((1/2*a+b*x)/b^(1/2)+(b*x
^2+a*x)^(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^2 + a*x)^(5/2)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232114, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a b^{\frac{3}{2}} x^{2} \log \left (2 \, b x + a + 2 \, \sqrt{b x^{2} + a x} \sqrt{b}\right ) + 2 \,{\left (3 \, b^{2} x^{2} - 14 \, a b x - 2 \, a^{2}\right )} \sqrt{b x^{2} + a x}}{6 \, x^{2}}, \frac{15 \, a \sqrt{-b} b x^{2} \arctan \left (\frac{\sqrt{b x^{2} + a x}}{\sqrt{-b} x}\right ) +{\left (3 \, b^{2} x^{2} - 14 \, a b x - 2 \, a^{2}\right )} \sqrt{b x^{2} + a x}}{3 \, x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((x*(a + b*x))**(5/2)/x**5, x)

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GIAC/XCAS [A]  time = 0.229356, size = 180, normalized size = 2.02 \[ -\frac{5}{2} \, a b^{\frac{3}{2}}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \sqrt{b} - a \right |}\right ) + \sqrt{b x^{2} + a x} b^{2} + \frac{2 \,{\left (9 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{2} a^{2} b + 3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} a^{3} \sqrt{b} + a^{4}\right )}}{3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/2*a*b^(3/2)*ln(abs(-2*(sqrt(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) - a)) + sqrt(b*
x^2 + a*x)*b^2 + 2/3*(9*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2*a^2*b + 3*(sqrt(b)*x -
sqrt(b*x^2 + a*x))*a^3*sqrt(b) + a^4)/(sqrt(b)*x - sqrt(b*x^2 + a*x))^3