Optimal. Leaf size=70 \[ \frac{2 \sqrt{b x+c x^2} (2 b B-A c)}{b c^2 \sqrt{x}}-\frac{2 x^{3/2} (b B-A c)}{b c \sqrt{b x+c x^2}} \]
[Out]
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Rubi [A] time = 0.138337, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 \sqrt{b x+c x^2} (2 b B-A c)}{b c^2 \sqrt{x}}-\frac{2 x^{3/2} (b B-A c)}{b c \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
[In] Int[(x^(3/2)*(A + B*x))/(b*x + c*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 9.83566, size = 60, normalized size = 0.86 \[ \frac{2 x^{\frac{3}{2}} \left (A c - B b\right )}{b c \sqrt{b x + c x^{2}}} - \frac{4 \left (\frac{A c}{2} - B b\right ) \sqrt{b x + c x^{2}}}{b c^{2} \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3/2)*(B*x+A)/(c*x**2+b*x)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0362186, size = 34, normalized size = 0.49 \[ \frac{2 \sqrt{x} (-A c+2 b B+B c x)}{c^2 \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^(3/2)*(A + B*x))/(b*x + c*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.005, size = 38, normalized size = 0.5 \[ -2\,{\frac{ \left ( cx+b \right ) \left ( -Bcx+Ac-2\,Bb \right ){x}^{3/2}}{{c}^{2} \left ( c{x}^{2}+bx \right ) ^{3/2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(3/2)*(B*x+A)/(c*x^2+b*x)^(3/2),x)
[Out]
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Maxima [A] time = 0.760242, size = 69, normalized size = 0.99 \[ 2 \, B{\left (\frac{\sqrt{c x + b} x}{c^{2} x + b c} + \frac{2 \, b}{\sqrt{c x + b} c^{2}}\right )} - \frac{2 \, A}{\sqrt{c x + b} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^(3/2)/(c*x^2 + b*x)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.286583, size = 50, normalized size = 0.71 \[ \frac{2 \,{\left (B c x^{2} +{\left (2 \, B b - A c\right )} x\right )}}{\sqrt{c x^{2} + b x} c^{2} \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^(3/2)/(c*x^2 + b*x)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{3}{2}} \left (A + B x\right )}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(3/2)*(B*x+A)/(c*x**2+b*x)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.273702, size = 66, normalized size = 0.94 \[ \frac{2 \,{\left (\sqrt{c x + b} B + \frac{B b - A c}{\sqrt{c x + b}}\right )}}{c^{2}} - \frac{2 \,{\left (2 \, B b - A c\right )}}{\sqrt{b} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^(3/2)/(c*x^2 + b*x)^(3/2),x, algorithm="giac")
[Out]