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

Optimal. Leaf size=94 \[ \frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{8 \sqrt{b}}-\frac{5}{8} a (a+2 b x) \sqrt{a x+b x^2}+\frac{2 \left (a x+b x^2\right )^{5/2}}{x^2}-\frac{5}{3} b \left (a x+b x^2\right )^{3/2} \]

[Out]

(-5*a*(a + 2*b*x)*Sqrt[a*x + b*x^2])/8 - (5*b*(a*x + b*x^2)^(3/2))/3 + (2*(a*x +
b*x^2)^(5/2))/x^2 + (5*a^3*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(8*Sqrt[b])

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Rubi [A]  time = 0.114336, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{8 \sqrt{b}}-\frac{5}{8} a (a+2 b x) \sqrt{a x+b x^2}+\frac{2 \left (a x+b x^2\right )^{5/2}}{x^2}-\frac{5}{3} b \left (a x+b x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(a*x + b*x^2)^(5/2)/x^3,x]

[Out]

(-5*a*(a + 2*b*x)*Sqrt[a*x + b*x^2])/8 - (5*b*(a*x + b*x^2)^(3/2))/3 + (2*(a*x +
b*x^2)^(5/2))/x^2 + (5*a^3*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(8*Sqrt[b])

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Rubi in Sympy [A]  time = 12.3092, size = 88, normalized size = 0.94 \[ \frac{5 a^{3} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a x + b x^{2}}} \right )}}{8 \sqrt{b}} - \frac{5 a \left (a + 2 b x\right ) \sqrt{a x + b x^{2}}}{8} - \frac{5 b \left (a x + b x^{2}\right )^{\frac{3}{2}}}{3} + \frac{2 \left (a x + b x^{2}\right )^{\frac{5}{2}}}{x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*a**3*atanh(sqrt(b)*x/sqrt(a*x + b*x**2))/(8*sqrt(b)) - 5*a*(a + 2*b*x)*sqrt(a*
x + b*x**2)/8 - 5*b*(a*x + b*x**2)**(3/2)/3 + 2*(a*x + b*x**2)**(5/2)/x**2

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Mathematica [A]  time = 0.0940536, size = 82, normalized size = 0.87 \[ \frac{1}{24} \sqrt{x (a+b x)} \left (\frac{15 a^3 \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{\sqrt{b} \sqrt{x} \sqrt{a+b x}}+33 a^2+26 a b x+8 b^2 x^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x*(a + b*x)]*(33*a^2 + 26*a*b*x + 8*b^2*x^2 + (15*a^3*Log[b*Sqrt[x] + Sqrt
[b]*Sqrt[a + b*x]])/(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x])))/24

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Maple [B]  time = 0.007, size = 158, normalized size = 1.7 \[ 2\,{\frac{ \left ( b{x}^{2}+ax \right ) ^{7/2}}{a{x}^{3}}}-{\frac{16\,b}{3\,{a}^{2}{x}^{2}} \left ( b{x}^{2}+ax \right ) ^{{\frac{7}{2}}}}+{\frac{16\,{b}^{2}}{3\,{a}^{2}} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}}+{\frac{10\,{b}^{2}x}{3\,a} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}+{\frac{5\,b}{3} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}-{\frac{5\,abx}{4}\sqrt{b{x}^{2}+ax}}-{\frac{5\,{a}^{2}}{8}\sqrt{b{x}^{2}+ax}}+{\frac{5\,{a}^{3}}{16}\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a*x)^(5/2)/x^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237574, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{3} \log \left ({\left (2 \, b x + a\right )} \sqrt{b} + 2 \, \sqrt{b x^{2} + a x} b\right ) + 2 \,{\left (8 \, b^{2} x^{2} + 26 \, a b x + 33 \, a^{2}\right )} \sqrt{b x^{2} + a x} \sqrt{b}}{48 \, \sqrt{b}}, \frac{15 \, a^{3} \arctan \left (\frac{\sqrt{b x^{2} + a x} \sqrt{-b}}{b x}\right ) +{\left (8 \, b^{2} x^{2} + 26 \, a b x + 33 \, a^{2}\right )} \sqrt{b x^{2} + a x} \sqrt{-b}}{24 \, \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(15*a^3*log((2*b*x + a)*sqrt(b) + 2*sqrt(b*x^2 + a*x)*b) + 2*(8*b^2*x^2 +
26*a*b*x + 33*a^2)*sqrt(b*x^2 + a*x)*sqrt(b))/sqrt(b), 1/24*(15*a^3*arctan(sqrt(
b*x^2 + a*x)*sqrt(-b)/(b*x)) + (8*b^2*x^2 + 26*a*b*x + 33*a^2)*sqrt(b*x^2 + a*x)
*sqrt(-b))/sqrt(-b)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.222701, size = 97, normalized size = 1.03 \[ -\frac{5 \, a^{3}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \sqrt{b} - a \right |}\right )}{16 \, \sqrt{b}} + \frac{1}{24} \, \sqrt{b x^{2} + a x}{\left (33 \, a^{2} + 2 \,{\left (4 \, b^{2} x + 13 \, a b\right )} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/16*a^3*ln(abs(-2*(sqrt(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) - a))/sqrt(b) + 1/24
*sqrt(b*x^2 + a*x)*(33*a^2 + 2*(4*b^2*x + 13*a*b)*x)