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

Optimal. Leaf size=91 \[ -\frac{5 a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{7/2}}+\frac{5 \sqrt{x} \sqrt{a+b x}}{b^3}-\frac{10 x^{3/2}}{3 b^2 \sqrt{a+b x}}-\frac{2 x^{5/2}}{3 b (a+b x)^{3/2}} \]

[Out]

(-2*x^(5/2))/(3*b*(a + b*x)^(3/2)) - (10*x^(3/2))/(3*b^2*Sqrt[a + b*x]) + (5*Sqr
t[x]*Sqrt[a + b*x])/b^3 - (5*a*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/b^(7/2)

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Rubi [A]  time = 0.0718013, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{5 a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{7/2}}+\frac{5 \sqrt{x} \sqrt{a+b x}}{b^3}-\frac{10 x^{3/2}}{3 b^2 \sqrt{a+b x}}-\frac{2 x^{5/2}}{3 b (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)/(a + b*x)^(5/2),x]

[Out]

(-2*x^(5/2))/(3*b*(a + b*x)^(3/2)) - (10*x^(3/2))/(3*b^2*Sqrt[a + b*x]) + (5*Sqr
t[x]*Sqrt[a + b*x])/b^3 - (5*a*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/b^(7/2)

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Rubi in Sympy [A]  time = 10.9681, size = 85, normalized size = 0.93 \[ - \frac{5 a \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )}}{b^{\frac{7}{2}}} - \frac{2 x^{\frac{5}{2}}}{3 b \left (a + b x\right )^{\frac{3}{2}}} - \frac{10 x^{\frac{3}{2}}}{3 b^{2} \sqrt{a + b x}} + \frac{5 \sqrt{x} \sqrt{a + b x}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*a*atanh(sqrt(a + b*x)/(sqrt(b)*sqrt(x)))/b**(7/2) - 2*x**(5/2)/(3*b*(a + b*x)
**(3/2)) - 10*x**(3/2)/(3*b**2*sqrt(a + b*x)) + 5*sqrt(x)*sqrt(a + b*x)/b**3

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Mathematica [A]  time = 0.119146, size = 73, normalized size = 0.8 \[ \frac{\sqrt{x} \left (15 a^2+20 a b x+3 b^2 x^2\right )}{3 b^3 (a+b x)^{3/2}}-\frac{5 a \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*(15*a^2 + 20*a*b*x + 3*b^2*x^2))/(3*b^3*(a + b*x)^(3/2)) - (5*a*Log[b*S
qrt[x] + Sqrt[b]*Sqrt[a + b*x]])/b^(7/2)

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Maple [B]  time = 0.26, size = 147, normalized size = 1.6 \[{\frac{1}{{b}^{3}}\sqrt{x}\sqrt{bx+a}}+{1 \left ( -{\frac{5\,a}{2}\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{7}{2}}}}+{\frac{14\,a}{3\,{b}^{4}}\sqrt{b \left ( x+{\frac{a}{b}} \right ) ^{2}- \left ( x+{\frac{a}{b}} \right ) a} \left ( x+{\frac{a}{b}} \right ) ^{-1}}-{\frac{2\,{a}^{2}}{3\,{b}^{5}}\sqrt{b \left ( x+{\frac{a}{b}} \right ) ^{2}- \left ( x+{\frac{a}{b}} \right ) a} \left ( x+{\frac{a}{b}} \right ) ^{-2}} \right ) \sqrt{x \left ( bx+a \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)/(b*x+a)^(5/2),x)

[Out]

x^(1/2)*(b*x+a)^(1/2)/b^3+(-5/2/b^(7/2)*a*ln((1/2*a+b*x)/b^(1/2)+(b*x^2+a*x)^(1/
2))+14/3/b^4*a/(x+a/b)*(b*(x+a/b)^2-(x+a/b)*a)^(1/2)-2/3/b^5*a^2/(x+a/b)^2*(b*(x
+a/b)^2-(x+a/b)*a)^(1/2))*(x*(b*x+a))^(1/2)/x^(1/2)/(b*x+a)^(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(x^(5/2)/(b*x + a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225337, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (a b x + a^{2}\right )} \sqrt{b x + a} \sqrt{x} \log \left (-2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) + 2 \,{\left (3 \, b^{2} x^{3} + 20 \, a b x^{2} + 15 \, a^{2} x\right )} \sqrt{b}}{6 \,{\left (b^{4} x + a b^{3}\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x}}, -\frac{15 \,{\left (a b x + a^{2}\right )} \sqrt{b x + a} \sqrt{x} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (3 \, b^{2} x^{3} + 20 \, a b x^{2} + 15 \, a^{2} x\right )} \sqrt{-b}}{3 \,{\left (b^{4} x + a b^{3}\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(15*(a*b*x + a^2)*sqrt(b*x + a)*sqrt(x)*log(-2*sqrt(b*x + a)*b*sqrt(x) + (2
*b*x + a)*sqrt(b)) + 2*(3*b^2*x^3 + 20*a*b*x^2 + 15*a^2*x)*sqrt(b))/((b^4*x + a*
b^3)*sqrt(b*x + a)*sqrt(b)*sqrt(x)), -1/3*(15*(a*b*x + a^2)*sqrt(b*x + a)*sqrt(x
)*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) - (3*b^2*x^3 + 20*a*b*x^2 + 15*a^2*
x)*sqrt(-b))/((b^4*x + a*b^3)*sqrt(b*x + a)*sqrt(-b)*sqrt(x))]

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Sympy [A]  time = 84.0961, size = 396, normalized size = 4.35 \[ - \frac{15 a^{\frac{81}{2}} b^{22} x^{\frac{51}{2}} \sqrt{1 + \frac{b x}{a}} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{3 a^{\frac{79}{2}} b^{\frac{51}{2}} x^{\frac{51}{2}} \sqrt{1 + \frac{b x}{a}} + 3 a^{\frac{77}{2}} b^{\frac{53}{2}} x^{\frac{53}{2}} \sqrt{1 + \frac{b x}{a}}} - \frac{15 a^{\frac{79}{2}} b^{23} x^{\frac{53}{2}} \sqrt{1 + \frac{b x}{a}} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{3 a^{\frac{79}{2}} b^{\frac{51}{2}} x^{\frac{51}{2}} \sqrt{1 + \frac{b x}{a}} + 3 a^{\frac{77}{2}} b^{\frac{53}{2}} x^{\frac{53}{2}} \sqrt{1 + \frac{b x}{a}}} + \frac{15 a^{40} b^{\frac{45}{2}} x^{26}}{3 a^{\frac{79}{2}} b^{\frac{51}{2}} x^{\frac{51}{2}} \sqrt{1 + \frac{b x}{a}} + 3 a^{\frac{77}{2}} b^{\frac{53}{2}} x^{\frac{53}{2}} \sqrt{1 + \frac{b x}{a}}} + \frac{20 a^{39} b^{\frac{47}{2}} x^{27}}{3 a^{\frac{79}{2}} b^{\frac{51}{2}} x^{\frac{51}{2}} \sqrt{1 + \frac{b x}{a}} + 3 a^{\frac{77}{2}} b^{\frac{53}{2}} x^{\frac{53}{2}} \sqrt{1 + \frac{b x}{a}}} + \frac{3 a^{38} b^{\frac{49}{2}} x^{28}}{3 a^{\frac{79}{2}} b^{\frac{51}{2}} x^{\frac{51}{2}} \sqrt{1 + \frac{b x}{a}} + 3 a^{\frac{77}{2}} b^{\frac{53}{2}} x^{\frac{53}{2}} \sqrt{1 + \frac{b x}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15*a**(81/2)*b**22*x**(51/2)*sqrt(1 + b*x/a)*asinh(sqrt(b)*sqrt(x)/sqrt(a))/(3*
a**(79/2)*b**(51/2)*x**(51/2)*sqrt(1 + b*x/a) + 3*a**(77/2)*b**(53/2)*x**(53/2)*
sqrt(1 + b*x/a)) - 15*a**(79/2)*b**23*x**(53/2)*sqrt(1 + b*x/a)*asinh(sqrt(b)*sq
rt(x)/sqrt(a))/(3*a**(79/2)*b**(51/2)*x**(51/2)*sqrt(1 + b*x/a) + 3*a**(77/2)*b*
*(53/2)*x**(53/2)*sqrt(1 + b*x/a)) + 15*a**40*b**(45/2)*x**26/(3*a**(79/2)*b**(5
1/2)*x**(51/2)*sqrt(1 + b*x/a) + 3*a**(77/2)*b**(53/2)*x**(53/2)*sqrt(1 + b*x/a)
) + 20*a**39*b**(47/2)*x**27/(3*a**(79/2)*b**(51/2)*x**(51/2)*sqrt(1 + b*x/a) +
3*a**(77/2)*b**(53/2)*x**(53/2)*sqrt(1 + b*x/a)) + 3*a**38*b**(49/2)*x**28/(3*a*
*(79/2)*b**(51/2)*x**(51/2)*sqrt(1 + b*x/a) + 3*a**(77/2)*b**(53/2)*x**(53/2)*sq
rt(1 + b*x/a))

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GIAC/XCAS [A]  time = 0.22866, size = 266, normalized size = 2.92 \[ \frac{{\left (\frac{15 \, a{\rm ln}\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{b^{\frac{5}{2}}} + \frac{6 \, \sqrt{{\left (b x + a\right )} b - a b} \sqrt{b x + a}}{b^{3}} + \frac{8 \,{\left (9 \, a^{2}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{4} \sqrt{b} + 12 \, a^{3}{\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac{3}{2}} + 7 \, a^{4} b^{\frac{5}{2}}\right )}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} b^{2}}\right )}{\left | b \right |}}{6 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(15*a*ln((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2)/b^(5/2) + 6*sq
rt((b*x + a)*b - a*b)*sqrt(b*x + a)/b^3 + 8*(9*a^2*(sqrt(b*x + a)*sqrt(b) - sqrt
((b*x + a)*b - a*b))^4*sqrt(b) + 12*a^3*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*
b - a*b))^2*b^(3/2) + 7*a^4*b^(5/2))/(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b
- a*b))^2 + a*b)^3*b^2))*abs(b)/b^2