Optimal. Leaf size=106 \[ \frac{256 b^3 (a+2 b x)}{21 a^6 \sqrt{a x+b x^2}}-\frac{32 b^2 (a+2 b x)}{21 a^4 \left (a x+b x^2\right )^{3/2}}+\frac{4 b}{7 a^2 x \left (a x+b x^2\right )^{3/2}}-\frac{2}{7 a x^2 \left (a x+b x^2\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.114943, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{256 b^3 (a+2 b x)}{21 a^6 \sqrt{a x+b x^2}}-\frac{32 b^2 (a+2 b x)}{21 a^4 \left (a x+b x^2\right )^{3/2}}+\frac{4 b}{7 a^2 x \left (a x+b x^2\right )^{3/2}}-\frac{2}{7 a x^2 \left (a x+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*(a*x + b*x^2)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 11.8109, size = 100, normalized size = 0.94 \[ - \frac{2}{7 a x^{2} \left (a x + b x^{2}\right )^{\frac{3}{2}}} + \frac{4 b}{7 a^{2} x \left (a x + b x^{2}\right )^{\frac{3}{2}}} - \frac{32 b^{2} \left (a + 2 b x\right )}{21 a^{4} \left (a x + b x^{2}\right )^{\frac{3}{2}}} + \frac{128 b^{3} \left (2 a + 4 b x\right )}{21 a^{6} \sqrt{a x + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(b*x**2+a*x)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0600019, size = 73, normalized size = 0.69 \[ \frac{2 \left (-3 a^5+6 a^4 b x-16 a^3 b^2 x^2+96 a^2 b^3 x^3+384 a b^4 x^4+256 b^5 x^5\right )}{21 a^6 x^2 (x (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*(a*x + b*x^2)^(5/2)),x]
[Out]
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Maple [A] time = 0.008, size = 77, normalized size = 0.7 \[ -{\frac{ \left ( 2\,bx+2\,a \right ) \left ( -256\,{b}^{5}{x}^{5}-384\,{b}^{4}{x}^{4}a-96\,{b}^{3}{x}^{3}{a}^{2}+16\,{b}^{2}{x}^{2}{a}^{3}-6\,bx{a}^{4}+3\,{a}^{5} \right ) }{21\,x{a}^{6}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(b*x^2+a*x)^(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(1/((b*x^2 + a*x)^(5/2)*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.221299, size = 112, normalized size = 1.06 \[ \frac{2 \,{\left (256 \, b^{5} x^{5} + 384 \, a b^{4} x^{4} + 96 \, a^{2} b^{3} x^{3} - 16 \, a^{3} b^{2} x^{2} + 6 \, a^{4} b x - 3 \, a^{5}\right )}}{21 \,{\left (a^{6} b x^{4} + a^{7} x^{3}\right )} \sqrt{b x^{2} + a x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a*x)^(5/2)*x^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \left (x \left (a + b x\right )\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(b*x**2+a*x)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a x\right )}^{\frac{5}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a*x)^(5/2)*x^2),x, algorithm="giac")
[Out]