Optimal. Leaf size=63 \[ \frac{16 b \sqrt{a+b x}}{3 a^3 \sqrt{x}}-\frac{8 \sqrt{a+b x}}{3 a^2 x^{3/2}}+\frac{2}{a x^{3/2} \sqrt{a+b x}} \]
[Out]
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Rubi [A] time = 0.0416455, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{16 b \sqrt{a+b x}}{3 a^3 \sqrt{x}}-\frac{8 \sqrt{a+b x}}{3 a^2 x^{3/2}}+\frac{2}{a x^{3/2} \sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^(5/2)*(a + b*x)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 6.26241, size = 58, normalized size = 0.92 \[ \frac{2}{a x^{\frac{3}{2}} \sqrt{a + b x}} - \frac{8 \sqrt{a + b x}}{3 a^{2} x^{\frac{3}{2}}} + \frac{16 b \sqrt{a + b x}}{3 a^{3} \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**(5/2)/(b*x+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0280075, size = 38, normalized size = 0.6 \[ -\frac{2 \left (a^2-4 a b x-8 b^2 x^2\right )}{3 a^3 x^{3/2} \sqrt{a+b x}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^(5/2)*(a + b*x)^(3/2)),x]
[Out]
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Maple [A] time = 0.006, size = 33, normalized size = 0.5 \[ -{\frac{-16\,{b}^{2}{x}^{2}-8\,abx+2\,{a}^{2}}{3\,{a}^{3}}{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{bx+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^(5/2)/(b*x+a)^(3/2),x)
[Out]
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Maxima [A] time = 1.34735, size = 68, normalized size = 1.08 \[ \frac{2 \, b^{2} \sqrt{x}}{\sqrt{b x + a} a^{3}} + \frac{2 \,{\left (\frac{6 \, \sqrt{b x + a} b}{\sqrt{x}} - \frac{{\left (b x + a\right )}^{\frac{3}{2}}}{x^{\frac{3}{2}}}\right )}}{3 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(3/2)*x^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.209284, size = 46, normalized size = 0.73 \[ \frac{2 \,{\left (8 \, b^{2} x^{2} + 4 \, a b x - a^{2}\right )}}{3 \, \sqrt{b x + a} a^{3} x^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(3/2)*x^(5/2)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 103.614, size = 219, normalized size = 3.48 \[ - \frac{2 a^{3} b^{\frac{9}{2}} \sqrt{\frac{a}{b x} + 1}}{3 a^{5} b^{4} x + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} + \frac{6 a^{2} b^{\frac{11}{2}} x \sqrt{\frac{a}{b x} + 1}}{3 a^{5} b^{4} x + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} + \frac{24 a b^{\frac{13}{2}} x^{2} \sqrt{\frac{a}{b x} + 1}}{3 a^{5} b^{4} x + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} + \frac{16 b^{\frac{15}{2}} x^{3} \sqrt{\frac{a}{b x} + 1}}{3 a^{5} b^{4} x + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**(5/2)/(b*x+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.216189, size = 126, normalized size = 2. \[ -\frac{\sqrt{b x + a}{\left (\frac{5 \,{\left (b x + a\right )}{\left | b \right |}}{b^{2}} - \frac{6 \, a{\left | b \right |}}{b^{2}}\right )}}{24 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{3}{2}}} + \frac{4 \, b^{\frac{7}{2}}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} a^{2}{\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + a)^(3/2)*x^(5/2)),x, algorithm="giac")
[Out]