Optimal. Leaf size=136 \[ -\frac{256 b^4 \sqrt{x}}{35 c^5 \sqrt{b x+c x^2}}-\frac{128 b^3 x^{3/2}}{35 c^4 \sqrt{b x+c x^2}}+\frac{32 b^2 x^{5/2}}{35 c^3 \sqrt{b x+c x^2}}-\frac{16 b x^{7/2}}{35 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{9/2}}{7 c \sqrt{b x+c x^2}} \]
[Out]
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Rubi [A] time = 0.170437, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{256 b^4 \sqrt{x}}{35 c^5 \sqrt{b x+c x^2}}-\frac{128 b^3 x^{3/2}}{35 c^4 \sqrt{b x+c x^2}}+\frac{32 b^2 x^{5/2}}{35 c^3 \sqrt{b x+c x^2}}-\frac{16 b x^{7/2}}{35 c^2 \sqrt{b x+c x^2}}+\frac{2 x^{9/2}}{7 c \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
[In] Int[x^(11/2)/(b*x + c*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 18.4632, size = 128, normalized size = 0.94 \[ - \frac{256 b^{4} \sqrt{x}}{35 c^{5} \sqrt{b x + c x^{2}}} - \frac{128 b^{3} x^{\frac{3}{2}}}{35 c^{4} \sqrt{b x + c x^{2}}} + \frac{32 b^{2} x^{\frac{5}{2}}}{35 c^{3} \sqrt{b x + c x^{2}}} - \frac{16 b x^{\frac{7}{2}}}{35 c^{2} \sqrt{b x + c x^{2}}} + \frac{2 x^{\frac{9}{2}}}{7 c \sqrt{b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(11/2)/(c*x**2+b*x)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0340494, size = 64, normalized size = 0.47 \[ \frac{2 \sqrt{x} \left (-128 b^4-64 b^3 c x+16 b^2 c^2 x^2-8 b c^3 x^3+5 c^4 x^4\right )}{35 c^5 \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(11/2)/(b*x + c*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.008, size = 66, normalized size = 0.5 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -5\,{x}^{4}{c}^{4}+8\,b{x}^{3}{c}^{3}-16\,{b}^{2}{x}^{2}{c}^{2}+64\,{b}^{3}xc+128\,{b}^{4} \right ) }{35\,{c}^{5}}{x}^{{\frac{3}{2}}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(11/2)/(c*x^2+b*x)^(3/2),x)
[Out]
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Maxima [A] time = 0.740722, size = 289, normalized size = 2.12 \[ \frac{2 \,{\left (3 \,{\left (5 \, c^{5} x^{4} - b c^{4} x^{3} + 2 \, b^{2} c^{3} x^{2} - 8 \, b^{3} c^{2} x - 16 \, b^{4} c\right )} x^{4} - 2 \,{\left (3 \, b c^{4} x^{4} - 2 \, b^{2} c^{3} x^{3} + 11 \, b^{3} c^{2} x^{2} + 40 \, b^{4} c x + 24 \, b^{5}\right )} x^{3} + 14 \,{\left (b^{2} c^{3} x^{4} - 2 \, b^{3} c^{2} x^{3} - 7 \, b^{4} c x^{2} - 4 \, b^{5} x\right )} x^{2} - 70 \,{\left (b^{3} c^{2} x^{4} + 2 \, b^{4} c x^{3} + b^{5} x^{2}\right )} x\right )}}{105 \,{\left (c^{6} x^{4} + b c^{5} x^{3}\right )} \sqrt{c x + b}} - \frac{4 \, b^{4}}{\sqrt{c x + b} c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(11/2)/(c*x^2 + b*x)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.218607, size = 85, normalized size = 0.62 \[ \frac{2 \,{\left (5 \, c^{4} x^{5} - 8 \, b c^{3} x^{4} + 16 \, b^{2} c^{2} x^{3} - 64 \, b^{3} c x^{2} - 128 \, b^{4} x\right )}}{35 \, \sqrt{c x^{2} + b x} c^{5} \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(11/2)/(c*x^2 + b*x)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(11/2)/(c*x**2+b*x)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.212533, size = 95, normalized size = 0.7 \[ \frac{256 \, b^{\frac{7}{2}}}{35 \, c^{5}} + \frac{2 \,{\left (5 \,{\left (c x + b\right )}^{\frac{7}{2}} - 28 \,{\left (c x + b\right )}^{\frac{5}{2}} b + 70 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{2} - 140 \, \sqrt{c x + b} b^{3} - \frac{35 \, b^{4}}{\sqrt{c x + b}}\right )}}{35 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(11/2)/(c*x^2 + b*x)^(3/2),x, algorithm="giac")
[Out]