Optimal. Leaf size=80 \[ \frac{16 b^2 \left (b x+c x^2\right )^{5/2}}{315 c^3 x^{5/2}}-\frac{8 b \left (b x+c x^2\right )^{5/2}}{63 c^2 x^{3/2}}+\frac{2 \left (b x+c x^2\right )^{5/2}}{9 c \sqrt{x}} \]
[Out]
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Rubi [A] time = 0.0921036, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{16 b^2 \left (b x+c x^2\right )^{5/2}}{315 c^3 x^{5/2}}-\frac{8 b \left (b x+c x^2\right )^{5/2}}{63 c^2 x^{3/2}}+\frac{2 \left (b x+c x^2\right )^{5/2}}{9 c \sqrt{x}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]*(b*x + c*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 9.23781, size = 73, normalized size = 0.91 \[ \frac{16 b^{2} \left (b x + c x^{2}\right )^{\frac{5}{2}}}{315 c^{3} x^{\frac{5}{2}}} - \frac{8 b \left (b x + c x^{2}\right )^{\frac{5}{2}}}{63 c^{2} x^{\frac{3}{2}}} + \frac{2 \left (b x + c x^{2}\right )^{\frac{5}{2}}}{9 c \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(1/2)*(c*x**2+b*x)**(3/2),x)
[Out]
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Mathematica [A] time = 0.037573, size = 42, normalized size = 0.52 \[ \frac{2 (x (b+c x))^{5/2} \left (8 b^2-20 b c x+35 c^2 x^2\right )}{315 c^3 x^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x]*(b*x + c*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.006, size = 44, normalized size = 0.6 \[{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 35\,{c}^{2}{x}^{2}-20\,bcx+8\,{b}^{2} \right ) }{315\,{c}^{3}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(1/2)*(c*x^2+b*x)^(3/2),x)
[Out]
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Maxima [A] time = 0.713063, size = 138, normalized size = 1.72 \[ \frac{2 \,{\left ({\left (35 \, c^{4} x^{4} + 5 \, b c^{3} x^{3} - 6 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x - 16 \, b^{4}\right )} x^{3} + 3 \,{\left (15 \, b c^{3} x^{4} + 3 \, b^{2} c^{2} x^{3} - 4 \, b^{3} c x^{2} + 8 \, b^{4} x\right )} x^{2}\right )} \sqrt{c x + b}}{315 \, c^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(3/2)*sqrt(x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.217578, size = 100, normalized size = 1.25 \[ \frac{2 \,{\left (35 \, c^{5} x^{6} + 85 \, b c^{4} x^{5} + 53 \, b^{2} c^{3} x^{4} - b^{3} c^{2} x^{3} + 4 \, b^{4} c x^{2} + 8 \, b^{5} x\right )}}{315 \, \sqrt{c x^{2} + b x} c^{3} \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(3/2)*sqrt(x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x} \left (x \left (b + c x\right )\right )^{\frac{3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(1/2)*(c*x**2+b*x)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.214183, size = 149, normalized size = 1.86 \[ \frac{2}{315} \, c{\left (\frac{16 \, b^{\frac{9}{2}}}{c^{4}} + \frac{35 \,{\left (c x + b\right )}^{\frac{9}{2}} - 135 \,{\left (c x + b\right )}^{\frac{7}{2}} b + 189 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{2} - 105 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{3}}{c^{4}}\right )} - \frac{2}{105} \, b{\left (\frac{8 \, b^{\frac{7}{2}}}{c^{3}} - \frac{15 \,{\left (c x + b\right )}^{\frac{7}{2}} - 42 \,{\left (c x + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{2}}{c^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(3/2)*sqrt(x),x, algorithm="giac")
[Out]