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