Optimal. Leaf size=96 \[ \frac{4 b \sqrt{b x+c x^2} (4 b B-5 A c)}{15 c^3 \sqrt{x}}-\frac{2 \sqrt{x} \sqrt{b x+c x^2} (4 b B-5 A c)}{15 c^2}+\frac{2 B x^{3/2} \sqrt{b x+c x^2}}{5 c} \]
[Out]
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Rubi [A] time = 0.201713, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{4 b \sqrt{b x+c x^2} (4 b B-5 A c)}{15 c^3 \sqrt{x}}-\frac{2 \sqrt{x} \sqrt{b x+c x^2} (4 b B-5 A c)}{15 c^2}+\frac{2 B x^{3/2} \sqrt{b x+c x^2}}{5 c} \]
Antiderivative was successfully verified.
[In] Int[(x^(3/2)*(A + B*x))/Sqrt[b*x + c*x^2],x]
[Out]
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Rubi in Sympy [A] time = 11.8392, size = 92, normalized size = 0.96 \[ \frac{2 B x^{\frac{3}{2}} \sqrt{b x + c x^{2}}}{5 c} - \frac{4 b \left (5 A c - 4 B b\right ) \sqrt{b x + c x^{2}}}{15 c^{3} \sqrt{x}} + \frac{2 \sqrt{x} \left (5 A c - 4 B b\right ) \sqrt{b x + c x^{2}}}{15 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3/2)*(B*x+A)/(c*x**2+b*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0668697, size = 55, normalized size = 0.57 \[ \frac{2 \sqrt{x (b+c x)} \left (-2 b c (5 A+2 B x)+c^2 x (5 A+3 B x)+8 b^2 B\right )}{15 c^3 \sqrt{x}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^(3/2)*(A + B*x))/Sqrt[b*x + c*x^2],x]
[Out]
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Maple [A] time = 0.006, size = 59, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -3\,B{c}^{2}{x}^{2}-5\,Ax{c}^{2}+4\,Bxbc+10\,Abc-8\,{b}^{2}B \right ) }{15\,{c}^{3}}\sqrt{x}{\frac{1}{\sqrt{c{x}^{2}+bx}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(3/2)*(B*x+A)/(c*x^2+b*x)^(1/2),x)
[Out]
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Maxima [A] time = 0.702023, size = 101, normalized size = 1.05 \[ \frac{2 \,{\left (c^{2} x^{2} - b c x - 2 \, b^{2}\right )} A}{3 \, \sqrt{c x + b} c^{2}} + \frac{2 \,{\left (3 \, c^{3} x^{3} - b c^{2} x^{2} + 4 \, b^{2} c x + 8 \, b^{3}\right )} B}{15 \, \sqrt{c x + b} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^(3/2)/sqrt(c*x^2 + b*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.292595, size = 112, normalized size = 1.17 \[ \frac{2 \,{\left (3 \, B c^{3} x^{4} -{\left (B b c^{2} - 5 \, A c^{3}\right )} x^{3} +{\left (4 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{2} + 2 \,{\left (4 \, B b^{3} - 5 \, A b^{2} c\right )} x\right )}}{15 \, \sqrt{c x^{2} + b x} c^{3} \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^(3/2)/sqrt(c*x^2 + b*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{3}{2}} \left (A + B x\right )}{\sqrt{x \left (b + c x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(3/2)*(B*x+A)/(c*x**2+b*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.272768, size = 112, normalized size = 1.17 \[ \frac{2 \,{\left (3 \,{\left (c x + b\right )}^{\frac{5}{2}} B - 10 \,{\left (c x + b\right )}^{\frac{3}{2}} B b + 15 \, \sqrt{c x + b} B b^{2} + 5 \,{\left (c x + b\right )}^{\frac{3}{2}} A c - 15 \, \sqrt{c x + b} A b c\right )}}{15 \, c^{3}} - \frac{4 \,{\left (4 \, B b^{\frac{5}{2}} - 5 \, A b^{\frac{3}{2}} c\right )}}{15 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^(3/2)/sqrt(c*x^2 + b*x),x, algorithm="giac")
[Out]