3.30 \(\int \frac{\left (a x+b x^2\right )^{5/2}}{x^4} \, dx\)

Optimal. Leaf size=92 \[ \frac{15}{4} a^2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )+\frac{5 b \left (a x+b x^2\right )^{3/2}}{2 x}+\frac{15}{4} a b \sqrt{a x+b x^2}-\frac{2 \left (a x+b x^2\right )^{5/2}}{x^3} \]

[Out]

(15*a*b*Sqrt[a*x + b*x^2])/4 + (5*b*(a*x + b*x^2)^(3/2))/(2*x) - (2*(a*x + b*x^2
)^(5/2))/x^3 + (15*a^2*Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/4

_______________________________________________________________________________________

Rubi [A]  time = 0.124471, antiderivative size = 92, 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 \[ \frac{15}{4} a^2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a x+b x^2}}\right )+\frac{5 b \left (a x+b x^2\right )^{3/2}}{2 x}+\frac{15}{4} a b \sqrt{a x+b x^2}-\frac{2 \left (a x+b x^2\right )^{5/2}}{x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a*x + b*x^2)^(5/2)/x^4,x]

[Out]

(15*a*b*Sqrt[a*x + b*x^2])/4 + (5*b*(a*x + b*x^2)^(3/2))/(2*x) - (2*(a*x + b*x^2
)^(5/2))/x^3 + (15*a^2*Sqrt[b]*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/4

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.6052, size = 85, normalized size = 0.92 \[ \frac{15 a^{2} \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a x + b x^{2}}} \right )}}{4} + \frac{15 a b \sqrt{a x + b x^{2}}}{4} + \frac{5 b \left (a x + b x^{2}\right )^{\frac{3}{2}}}{2 x} - \frac{2 \left (a x + b x^{2}\right )^{\frac{5}{2}}}{x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a*x)**(5/2)/x**4,x)

[Out]

15*a**2*sqrt(b)*atanh(sqrt(b)*x/sqrt(a*x + b*x**2))/4 + 15*a*b*sqrt(a*x + b*x**2
)/4 + 5*b*(a*x + b*x**2)**(3/2)/(2*x) - 2*(a*x + b*x**2)**(5/2)/x**3

_______________________________________________________________________________________

Mathematica [A]  time = 0.0917801, size = 93, normalized size = 1.01 \[ \frac{\sqrt{a+b x} \left (\sqrt{a+b x} \left (-8 a^2+9 a b x+2 b^2 x^2\right )+15 a^2 \sqrt{b} \sqrt{x} \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )\right )}{4 \sqrt{x (a+b x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*x + b*x^2)^(5/2)/x^4,x]

[Out]

(Sqrt[a + b*x]*(Sqrt[a + b*x]*(-8*a^2 + 9*a*b*x + 2*b^2*x^2) + 15*a^2*Sqrt[b]*Sq
rt[x]*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[a + b*x]]))/(4*Sqrt[x*(a + b*x)])

_______________________________________________________________________________________

Maple [B]  time = 0.008, size = 185, normalized size = 2. \[ -2\,{\frac{ \left ( b{x}^{2}+ax \right ) ^{7/2}}{a{x}^{4}}}+12\,{\frac{b \left ( b{x}^{2}+ax \right ) ^{7/2}}{{a}^{2}{x}^{3}}}-32\,{\frac{{b}^{2} \left ( b{x}^{2}+ax \right ) ^{7/2}}{{x}^{2}{a}^{3}}}+32\,{\frac{{b}^{3} \left ( b{x}^{2}+ax \right ) ^{5/2}}{{a}^{3}}}+20\,{\frac{{b}^{3} \left ( b{x}^{2}+ax \right ) ^{3/2}x}{{a}^{2}}}+10\,{\frac{{b}^{2} \left ( b{x}^{2}+ax \right ) ^{3/2}}{a}}-{\frac{15\,{b}^{2}x}{2}\sqrt{b{x}^{2}+ax}}-{\frac{15\,ab}{4}\sqrt{b{x}^{2}+ax}}+{\frac{15\,{a}^{2}}{8}\sqrt{b}\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a*x)^(5/2)/x^4,x)

[Out]

-2/a/x^4*(b*x^2+a*x)^(7/2)+12*b/a^2/x^3*(b*x^2+a*x)^(7/2)-32*b^2/a^3/x^2*(b*x^2+
a*x)^(7/2)+32*b^3/a^3*(b*x^2+a*x)^(5/2)+20*b^3/a^2*(b*x^2+a*x)^(3/2)*x+10*b^2/a*
(b*x^2+a*x)^(3/2)-15/2*b^2*(b*x^2+a*x)^(1/2)*x-15/4*a*b*(b*x^2+a*x)^(1/2)+15/8*b
^(1/2)*a^2*ln((1/2*a+b*x)/b^(1/2)+(b*x^2+a*x)^(1/2))

_______________________________________________________________________________________

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^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.232924, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{2} \sqrt{b} x \log \left (2 \, b x + a + 2 \, \sqrt{b x^{2} + a x} \sqrt{b}\right ) + 2 \,{\left (2 \, b^{2} x^{2} + 9 \, a b x - 8 \, a^{2}\right )} \sqrt{b x^{2} + a x}}{8 \, x}, \frac{15 \, a^{2} \sqrt{-b} x \arctan \left (\frac{\sqrt{b x^{2} + a x}}{\sqrt{-b} x}\right ) +{\left (2 \, b^{2} x^{2} + 9 \, a b x - 8 \, a^{2}\right )} \sqrt{b x^{2} + a x}}{4 \, x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a*x)^(5/2)/x^4,x, algorithm="fricas")

[Out]

[1/8*(15*a^2*sqrt(b)*x*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b)) + 2*(2*b^2*x
^2 + 9*a*b*x - 8*a^2)*sqrt(b*x^2 + a*x))/x, 1/4*(15*a^2*sqrt(-b)*x*arctan(sqrt(b
*x^2 + a*x)/(sqrt(-b)*x)) + (2*b^2*x^2 + 9*a*b*x - 8*a^2)*sqrt(b*x^2 + a*x))/x]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (a + b x\right )\right )^{\frac{5}{2}}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a*x)**(5/2)/x**4,x)

[Out]

Integral((x*(a + b*x))**(5/2)/x**4, x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.226964, size = 120, normalized size = 1.3 \[ -\frac{15}{8} \, a^{2} \sqrt{b}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \sqrt{b} - a \right |}\right ) + \frac{2 \, a^{3}}{\sqrt{b} x - \sqrt{b x^{2} + a x}} + \frac{1}{4} \,{\left (2 \, b^{2} x + 9 \, a b\right )} \sqrt{b x^{2} + a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a*x)^(5/2)/x^4,x, algorithm="giac")

[Out]

-15/8*a^2*sqrt(b)*ln(abs(-2*(sqrt(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) - a)) + 2*a^
3/(sqrt(b)*x - sqrt(b*x^2 + a*x)) + 1/4*(2*b^2*x + 9*a*b)*sqrt(b*x^2 + a*x)