Optimal. Leaf size=108 \[ -\frac{32 b^3 \sqrt{b x+c x^2}}{35 c^4 \sqrt{x}}+\frac{16 b^2 \sqrt{x} \sqrt{b x+c x^2}}{35 c^3}-\frac{12 b x^{3/2} \sqrt{b x+c x^2}}{35 c^2}+\frac{2 x^{5/2} \sqrt{b x+c x^2}}{7 c} \]
[Out]
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Rubi [A] time = 0.129986, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{32 b^3 \sqrt{b x+c x^2}}{35 c^4 \sqrt{x}}+\frac{16 b^2 \sqrt{x} \sqrt{b x+c x^2}}{35 c^3}-\frac{12 b x^{3/2} \sqrt{b x+c x^2}}{35 c^2}+\frac{2 x^{5/2} \sqrt{b x+c x^2}}{7 c} \]
Antiderivative was successfully verified.
[In] Int[x^(7/2)/Sqrt[b*x + c*x^2],x]
[Out]
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Rubi in Sympy [A] time = 13.6826, size = 100, normalized size = 0.93 \[ - \frac{32 b^{3} \sqrt{b x + c x^{2}}}{35 c^{4} \sqrt{x}} + \frac{16 b^{2} \sqrt{x} \sqrt{b x + c x^{2}}}{35 c^{3}} - \frac{12 b x^{\frac{3}{2}} \sqrt{b x + c x^{2}}}{35 c^{2}} + \frac{2 x^{\frac{5}{2}} \sqrt{b x + c x^{2}}}{7 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(7/2)/(c*x**2+b*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0370991, size = 53, normalized size = 0.49 \[ \frac{2 \sqrt{x (b+c x)} \left (-16 b^3+8 b^2 c x-6 b c^2 x^2+5 c^3 x^3\right )}{35 c^4 \sqrt{x}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(7/2)/Sqrt[b*x + c*x^2],x]
[Out]
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Maple [A] time = 0.009, size = 55, normalized size = 0.5 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -5\,{x}^{3}{c}^{3}+6\,b{x}^{2}{c}^{2}-8\,{b}^{2}xc+16\,{b}^{3} \right ) }{35\,{c}^{4}}\sqrt{x}{\frac{1}{\sqrt{c{x}^{2}+bx}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(7/2)/(c*x^2+b*x)^(1/2),x)
[Out]
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Maxima [A] time = 0.707242, size = 72, normalized size = 0.67 \[ \frac{2 \,{\left (5 \, c^{4} x^{4} - b c^{3} x^{3} + 2 \, b^{2} c^{2} x^{2} - 8 \, b^{3} c x - 16 \, b^{4}\right )}}{35 \, \sqrt{c x + b} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(7/2)/sqrt(c*x^2 + b*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.221352, size = 85, normalized size = 0.79 \[ \frac{2 \,{\left (5 \, c^{4} x^{5} - b c^{3} x^{4} + 2 \, b^{2} c^{2} x^{3} - 8 \, b^{3} c x^{2} - 16 \, b^{4} x\right )}}{35 \, \sqrt{c x^{2} + b x} c^{4} \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(7/2)/sqrt(c*x^2 + b*x),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**(7/2)/(c*x**2+b*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.210632, size = 78, normalized size = 0.72 \[ \frac{32 \, b^{\frac{7}{2}}}{35 \, c^{4}} + \frac{2 \,{\left (5 \,{\left (c x + b\right )}^{\frac{7}{2}} - 21 \,{\left (c x + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{2} - 35 \, \sqrt{c x + b} b^{3}\right )}}{35 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(7/2)/sqrt(c*x^2 + b*x),x, algorithm="giac")
[Out]