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

Optimal. Leaf size=101 \[ -\frac{5 a^4 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{64 b^{3/2}}+\frac{5 a^2 (a+2 b x) \sqrt{a x+b x^2}}{64 b}+\frac{5}{24} a \left (a x+b x^2\right )^{3/2}+\frac{\left (a x+b x^2\right )^{5/2}}{4 x} \]

[Out]

(5*a^2*(a + 2*b*x)*Sqrt[a*x + b*x^2])/(64*b) + (5*a*(a*x + b*x^2)^(3/2))/24 + (a
*x + b*x^2)^(5/2)/(4*x) - (5*a^4*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(64*b^(
3/2))

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Rubi [A]  time = 0.116534, antiderivative size = 101, 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 \[ -\frac{5 a^4 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )}{64 b^{3/2}}+\frac{5 a^2 (a+2 b x) \sqrt{a x+b x^2}}{64 b}+\frac{5}{24} a \left (a x+b x^2\right )^{3/2}+\frac{\left (a x+b x^2\right )^{5/2}}{4 x} \]

Antiderivative was successfully verified.

[In]  Int[(a*x + b*x^2)^(5/2)/x^2,x]

[Out]

(5*a^2*(a + 2*b*x)*Sqrt[a*x + b*x^2])/(64*b) + (5*a*(a*x + b*x^2)^(3/2))/24 + (a
*x + b*x^2)^(5/2)/(4*x) - (5*a^4*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(64*b^(
3/2))

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Rubi in Sympy [A]  time = 12.8237, size = 90, normalized size = 0.89 \[ - \frac{5 a^{4} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a x + b x^{2}}} \right )}}{64 b^{\frac{3}{2}}} + \frac{5 a^{2} \left (a + 2 b x\right ) \sqrt{a x + b x^{2}}}{64 b} + \frac{5 a \left (a x + b x^{2}\right )^{\frac{3}{2}}}{24} + \frac{\left (a x + b x^{2}\right )^{\frac{5}{2}}}{4 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*a**4*atanh(sqrt(b)*x/sqrt(a*x + b*x**2))/(64*b**(3/2)) + 5*a**2*(a + 2*b*x)*s
qrt(a*x + b*x**2)/(64*b) + 5*a*(a*x + b*x**2)**(3/2)/24 + (a*x + b*x**2)**(5/2)/
(4*x)

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Mathematica [A]  time = 0.116319, size = 100, normalized size = 0.99 \[ \frac{\sqrt{x (a+b x)} \left (\sqrt{b} \left (15 a^3+118 a^2 b x+136 a b^2 x^2+48 b^3 x^3\right )-\frac{15 a^4 \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{\sqrt{x} \sqrt{a+b x}}\right )}{192 b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x*(a + b*x)]*(Sqrt[b]*(15*a^3 + 118*a^2*b*x + 136*a*b^2*x^2 + 48*b^3*x^3)
- (15*a^4*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[a + b*x]])/(Sqrt[x]*Sqrt[a + b*x])))/(192
*b^(3/2))

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Maple [A]  time = 0.007, size = 135, normalized size = 1.3 \[{\frac{2}{3\,a{x}^{2}} \left ( b{x}^{2}+ax \right ) ^{{\frac{7}{2}}}}-{\frac{2\,b}{3\,a} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}}-{\frac{5\,bx}{12} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}-{\frac{5\,a}{24} \left ( b{x}^{2}+ax \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}x}{32}\sqrt{b{x}^{2}+ax}}+{\frac{5\,{a}^{3}}{64\,b}\sqrt{b{x}^{2}+ax}}-{\frac{5\,{a}^{4}}{128}\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a*x)^(5/2)/x^2,x)

[Out]

2/3/a/x^2*(b*x^2+a*x)^(7/2)-2/3*b/a*(b*x^2+a*x)^(5/2)-5/12*b*(b*x^2+a*x)^(3/2)*x
-5/24*a*(b*x^2+a*x)^(3/2)+5/32*a^2*(b*x^2+a*x)^(1/2)*x+5/64/b*a^3*(b*x^2+a*x)^(1
/2)-5/128/b^(3/2)*a^4*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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239614, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{4} \log \left ({\left (2 \, b x + a\right )} \sqrt{b} - 2 \, \sqrt{b x^{2} + a x} b\right ) + 2 \,{\left (48 \, b^{3} x^{3} + 136 \, a b^{2} x^{2} + 118 \, a^{2} b x + 15 \, a^{3}\right )} \sqrt{b x^{2} + a x} \sqrt{b}}{384 \, b^{\frac{3}{2}}}, -\frac{15 \, a^{4} \arctan \left (\frac{\sqrt{b x^{2} + a x} \sqrt{-b}}{b x}\right ) -{\left (48 \, b^{3} x^{3} + 136 \, a b^{2} x^{2} + 118 \, a^{2} b x + 15 \, a^{3}\right )} \sqrt{b x^{2} + a x} \sqrt{-b}}{192 \, \sqrt{-b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/384*(15*a^4*log((2*b*x + a)*sqrt(b) - 2*sqrt(b*x^2 + a*x)*b) + 2*(48*b^3*x^3
+ 136*a*b^2*x^2 + 118*a^2*b*x + 15*a^3)*sqrt(b*x^2 + a*x)*sqrt(b))/b^(3/2), -1/1
92*(15*a^4*arctan(sqrt(b*x^2 + a*x)*sqrt(-b)/(b*x)) - (48*b^3*x^3 + 136*a*b^2*x^
2 + 118*a^2*b*x + 15*a^3)*sqrt(b*x^2 + a*x)*sqrt(-b))/(sqrt(-b)*b)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.223946, size = 113, normalized size = 1.12 \[ \frac{5 \, a^{4}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \sqrt{b} - a \right |}\right )}{128 \, b^{\frac{3}{2}}} + \frac{1}{192} \, \sqrt{b x^{2} + a x}{\left (\frac{15 \, a^{3}}{b} + 2 \,{\left (59 \, a^{2} + 4 \,{\left (6 \, b^{2} x + 17 \, a b\right )} x\right )} x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/128*a^4*ln(abs(-2*(sqrt(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) - a))/b^(3/2) + 1/19
2*sqrt(b*x^2 + a*x)*(15*a^3/b + 2*(59*a^2 + 4*(6*b^2*x + 17*a*b)*x)*x)