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

Optimal. Leaf size=107 \[ \frac{3 a^5 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{128 b^{5/2}}-\frac{3 a^3 (a+2 b x) \sqrt{a x+b x^2}}{128 b^2}+\frac{a (a+2 b x) \left (a x+b x^2\right )^{3/2}}{16 b}+\frac{1}{5} \left (a x+b x^2\right )^{5/2} \]

[Out]

(-3*a^3*(a + 2*b*x)*Sqrt[a*x + b*x^2])/(128*b^2) + (a*(a + 2*b*x)*(a*x + b*x^2)^
(3/2))/(16*b) + (a*x + b*x^2)^(5/2)/5 + (3*a^5*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*
x^2]])/(128*b^(5/2))

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Rubi [A]  time = 0.108324, antiderivative size = 107, 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 \[ \frac{3 a^5 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{128 b^{5/2}}-\frac{3 a^3 (a+2 b x) \sqrt{a x+b x^2}}{128 b^2}+\frac{a (a+2 b x) \left (a x+b x^2\right )^{3/2}}{16 b}+\frac{1}{5} \left (a x+b x^2\right )^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[(a*x + b*x^2)^(5/2)/x,x]

[Out]

(-3*a^3*(a + 2*b*x)*Sqrt[a*x + b*x^2])/(128*b^2) + (a*(a + 2*b*x)*(a*x + b*x^2)^
(3/2))/(16*b) + (a*x + b*x^2)^(5/2)/5 + (3*a^5*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*
x^2]])/(128*b^(5/2))

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Rubi in Sympy [A]  time = 12.0467, size = 97, normalized size = 0.91 \[ \frac{3 a^{5} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a x + b x^{2}}} \right )}}{128 b^{\frac{5}{2}}} - \frac{3 a^{3} \left (a + 2 b x\right ) \sqrt{a x + b x^{2}}}{128 b^{2}} + \frac{a \left (a + 2 b x\right ) \left (a x + b x^{2}\right )^{\frac{3}{2}}}{16 b} + \frac{\left (a x + b x^{2}\right )^{\frac{5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a**5*atanh(sqrt(b)*x/sqrt(a*x + b*x**2))/(128*b**(5/2)) - 3*a**3*(a + 2*b*x)*s
qrt(a*x + b*x**2)/(128*b**2) + a*(a + 2*b*x)*(a*x + b*x**2)**(3/2)/(16*b) + (a*x
+ b*x**2)**(5/2)/5

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Mathematica [A]  time = 0.123987, size = 111, normalized size = 1.04 \[ \frac{\sqrt{x (a+b x)} \left (\frac{15 a^5 \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{\sqrt{x} \sqrt{a+b x}}+\sqrt{b} \left (-15 a^4+10 a^3 b x+248 a^2 b^2 x^2+336 a b^3 x^3+128 b^4 x^4\right )\right )}{640 b^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x*(a + b*x)]*(Sqrt[b]*(-15*a^4 + 10*a^3*b*x + 248*a^2*b^2*x^2 + 336*a*b^3*
x^3 + 128*b^4*x^4) + (15*a^5*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[a + b*x]])/(Sqrt[x]*Sq
rt[a + b*x])))/(640*b^(5/2))

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Maple [A]  time = 0.007, size = 120, normalized size = 1.1 \[{\frac{1}{5} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}}+{\frac{ax}{8} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}}{16\,b} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{a}^{3}x}{64\,b}\sqrt{b{x}^{2}+ax}}-{\frac{3\,{a}^{4}}{128\,{b}^{2}}\sqrt{b{x}^{2}+ax}}+{\frac{3\,{a}^{5}}{256}\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a*x)^(5/2)/x,x)

[Out]

1/5*(b*x^2+a*x)^(5/2)+1/8*a*(b*x^2+a*x)^(3/2)*x+1/16/b*(b*x^2+a*x)^(3/2)*a^2-3/6
4/b*a^3*(b*x^2+a*x)^(1/2)*x-3/128/b^2*a^4*(b*x^2+a*x)^(1/2)+3/256/b^(5/2)*a^5*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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233068, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{5} \log \left ({\left (2 \, b x + a\right )} \sqrt{b} + 2 \, \sqrt{b x^{2} + a x} b\right ) + 2 \,{\left (128 \, b^{4} x^{4} + 336 \, a b^{3} x^{3} + 248 \, a^{2} b^{2} x^{2} + 10 \, a^{3} b x - 15 \, a^{4}\right )} \sqrt{b x^{2} + a x} \sqrt{b}}{1280 \, b^{\frac{5}{2}}}, \frac{15 \, a^{5} \arctan \left (\frac{\sqrt{b x^{2} + a x} \sqrt{-b}}{b x}\right ) +{\left (128 \, b^{4} x^{4} + 336 \, a b^{3} x^{3} + 248 \, a^{2} b^{2} x^{2} + 10 \, a^{3} b x - 15 \, a^{4}\right )} \sqrt{b x^{2} + a x} \sqrt{-b}}{640 \, \sqrt{-b} b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/1280*(15*a^5*log((2*b*x + a)*sqrt(b) + 2*sqrt(b*x^2 + a*x)*b) + 2*(128*b^4*x^
4 + 336*a*b^3*x^3 + 248*a^2*b^2*x^2 + 10*a^3*b*x - 15*a^4)*sqrt(b*x^2 + a*x)*sqr
t(b))/b^(5/2), 1/640*(15*a^5*arctan(sqrt(b*x^2 + a*x)*sqrt(-b)/(b*x)) + (128*b^4
*x^4 + 336*a*b^3*x^3 + 248*a^2*b^2*x^2 + 10*a^3*b*x - 15*a^4)*sqrt(b*x^2 + a*x)*
sqrt(-b))/(sqrt(-b)*b^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((x*(a + b*x))**(5/2)/x, x)

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GIAC/XCAS [A]  time = 0.223699, size = 130, normalized size = 1.21 \[ -\frac{3 \, a^{5}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \sqrt{b} - a \right |}\right )}{256 \, b^{\frac{5}{2}}} - \frac{1}{640} \, \sqrt{b x^{2} + a x}{\left (\frac{15 \, a^{4}}{b^{2}} - 2 \,{\left (\frac{5 \, a^{3}}{b} + 4 \,{\left (31 \, a^{2} + 2 \,{\left (8 \, b^{2} x + 21 \, a b\right )} x\right )} x\right )} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/256*a^5*ln(abs(-2*(sqrt(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) - a))/b^(5/2) - 1/6
40*sqrt(b*x^2 + a*x)*(15*a^4/b^2 - 2*(5*a^3/b + 4*(31*a^2 + 2*(8*b^2*x + 21*a*b)
*x)*x)*x)