Optimal. Leaf size=100 \[ \frac{32 c^3 \sqrt{b x+c x^2}}{35 b^4 x}-\frac{16 c^2 \sqrt{b x+c x^2}}{35 b^3 x^2}+\frac{12 c \sqrt{b x+c x^2}}{35 b^2 x^3}-\frac{2 \sqrt{b x+c x^2}}{7 b x^4} \]
[Out]
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Rubi [A] time = 0.133149, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{32 c^3 \sqrt{b x+c x^2}}{35 b^4 x}-\frac{16 c^2 \sqrt{b x+c x^2}}{35 b^3 x^2}+\frac{12 c \sqrt{b x+c x^2}}{35 b^2 x^3}-\frac{2 \sqrt{b x+c x^2}}{7 b x^4} \]
Antiderivative was successfully verified.
[In] Int[1/(x^4*Sqrt[b*x + c*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 14.0888, size = 92, normalized size = 0.92 \[ - \frac{2 \sqrt{b x + c x^{2}}}{7 b x^{4}} + \frac{12 c \sqrt{b x + c x^{2}}}{35 b^{2} x^{3}} - \frac{16 c^{2} \sqrt{b x + c x^{2}}}{35 b^{3} x^{2}} + \frac{32 c^{3} \sqrt{b x + c x^{2}}}{35 b^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**4/(c*x**2+b*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0374799, size = 51, normalized size = 0.51 \[ \frac{2 \sqrt{x (b+c x)} \left (-5 b^3+6 b^2 c x-8 b c^2 x^2+16 c^3 x^3\right )}{35 b^4 x^4} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^4*Sqrt[b*x + c*x^2]),x]
[Out]
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Maple [A] time = 0.008, size = 55, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -16\,{x}^{3}{c}^{3}+8\,b{x}^{2}{c}^{2}-6\,{b}^{2}xc+5\,{b}^{3} \right ) }{35\,{x}^{3}{b}^{4}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^4/(c*x^2+b*x)^(1/2),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/(sqrt(c*x^2 + b*x)*x^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.216808, size = 66, normalized size = 0.66 \[ \frac{2 \,{\left (16 \, c^{3} x^{3} - 8 \, b c^{2} x^{2} + 6 \, b^{2} c x - 5 \, b^{3}\right )} \sqrt{c x^{2} + b x}}{35 \, b^{4} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^2 + b*x)*x^4),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} \sqrt{x \left (b + c x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**4/(c*x**2+b*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.219524, size = 144, normalized size = 1.44 \[ \frac{2 \,{\left (70 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} c^{\frac{3}{2}} + 84 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b c + 35 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{2} \sqrt{c} + 5 \, b^{3}\right )}}{35 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^2 + b*x)*x^4),x, algorithm="giac")
[Out]