Optimal. Leaf size=38 \[ \frac{a}{2 b^2 \left (a+b \sqrt{x}\right )^4}-\frac{2}{3 b^2 \left (a+b \sqrt{x}\right )^3} \]
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Rubi [A] time = 0.0522011, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{a}{2 b^2 \left (a+b \sqrt{x}\right )^4}-\frac{2}{3 b^2 \left (a+b \sqrt{x}\right )^3} \]
Antiderivative was successfully verified.
[In] Int[(a + b*Sqrt[x])^(-5),x]
[Out]
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Rubi in Sympy [A] time = 3.36315, size = 48, normalized size = 1.26 \[ \frac{x}{2 a \left (a + b \sqrt{x}\right )^{4}} + \frac{x}{3 a^{2} \left (a + b \sqrt{x}\right )^{3}} + \frac{x}{6 a^{3} \left (a + b \sqrt{x}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b*x**(1/2))**5,x)
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Mathematica [A] time = 0.0159688, size = 28, normalized size = 0.74 \[ -\frac{a+4 b \sqrt{x}}{6 b^2 \left (a+b \sqrt{x}\right )^4} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*Sqrt[x])^(-5),x]
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Maple [B] time = 0.063, size = 200, normalized size = 5.3 \[ -{\frac{1}{3\,{b}^{2}} \left ( b\sqrt{x}-a \right ) ^{-3}}-{\frac{a}{4\,{b}^{2}} \left ( b\sqrt{x}-a \right ) ^{-4}}-{\frac{1}{3\,{b}^{2}} \left ( a+b\sqrt{x} \right ) ^{-3}}+{\frac{a}{4\,{b}^{2}} \left ( a+b\sqrt{x} \right ) ^{-4}}+{\frac{{a}^{5}}{4\, \left ({b}^{2}x-{a}^{2} \right ) ^{4}{b}^{2}}}-5\,a{b}^{4} \left ( -1/4\,{\frac{{a}^{4}}{{b}^{6} \left ({b}^{2}x-{a}^{2} \right ) ^{4}}}-2/3\,{\frac{{a}^{2}}{{b}^{6} \left ({b}^{2}x-{a}^{2} \right ) ^{3}}}-1/2\,{\frac{1}{{b}^{6} \left ({b}^{2}x-{a}^{2} \right ) ^{2}}} \right ) -10\,{a}^{3}{b}^{2} \left ( -1/4\,{\frac{{a}^{2}}{{b}^{4} \left ({b}^{2}x-{a}^{2} \right ) ^{4}}}-1/3\,{\frac{1}{ \left ({b}^{2}x-{a}^{2} \right ) ^{3}{b}^{4}}} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b*x^(1/2))^5,x)
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Maxima [A] time = 1.43151, size = 41, normalized size = 1.08 \[ -\frac{2}{3 \,{\left (b \sqrt{x} + a\right )}^{3} b^{2}} + \frac{a}{2 \,{\left (b \sqrt{x} + a\right )}^{4} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*sqrt(x) + a)^(-5),x, algorithm="maxima")
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Fricas [A] time = 0.239021, size = 74, normalized size = 1.95 \[ -\frac{4 \, b \sqrt{x} + a}{6 \,{\left (b^{6} x^{2} + 6 \, a^{2} b^{4} x + a^{4} b^{2} + 4 \,{\left (a b^{5} x + a^{3} b^{3}\right )} \sqrt{x}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*sqrt(x) + a)^(-5),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.15061, size = 121, normalized size = 3.18 \[ \begin{cases} - \frac{a}{6 a^{4} b^{2} + 24 a^{3} b^{3} \sqrt{x} + 36 a^{2} b^{4} x + 24 a b^{5} x^{\frac{3}{2}} + 6 b^{6} x^{2}} - \frac{4 b \sqrt{x}}{6 a^{4} b^{2} + 24 a^{3} b^{3} \sqrt{x} + 36 a^{2} b^{4} x + 24 a b^{5} x^{\frac{3}{2}} + 6 b^{6} x^{2}} & \text{for}\: b \neq 0 \\\frac{x}{a^{5}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b*x**(1/2))**5,x)
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GIAC/XCAS [A] time = 0.263334, size = 30, normalized size = 0.79 \[ -\frac{4 \, b \sqrt{x} + a}{6 \,{\left (b \sqrt{x} + a\right )}^{4} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*sqrt(x) + a)^(-5),x, algorithm="giac")
[Out]