Optimal. Leaf size=55 \[ \frac{(b d+2 c d x)^{5/2}}{20 c^2 d^3}-\frac{\left (b^2-4 a c\right ) \sqrt{b d+2 c d x}}{4 c^2 d} \]
[Out]
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Rubi [A] time = 0.0727821, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{(b d+2 c d x)^{5/2}}{20 c^2 d^3}-\frac{\left (b^2-4 a c\right ) \sqrt{b d+2 c d x}}{4 c^2 d} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)/Sqrt[b*d + 2*c*d*x],x]
[Out]
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Rubi in Sympy [A] time = 14.2795, size = 48, normalized size = 0.87 \[ - \frac{\left (- a c + \frac{b^{2}}{4}\right ) \sqrt{b d + 2 c d x}}{c^{2} d} + \frac{\left (b d + 2 c d x\right )^{\frac{5}{2}}}{20 c^{2} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)/(2*c*d*x+b*d)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0359043, size = 43, normalized size = 0.78 \[ \frac{\left (c \left (5 a+c x^2\right )-b^2+b c x\right ) \sqrt{d (b+2 c x)}}{5 c^2 d} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)/Sqrt[b*d + 2*c*d*x],x]
[Out]
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Maple [A] time = 0.006, size = 44, normalized size = 0.8 \[{\frac{ \left ( 2\,cx+b \right ) \left ({c}^{2}{x}^{2}+bxc+5\,ac-{b}^{2} \right ) }{5\,{c}^{2}}{\frac{1}{\sqrt{2\,cdx+bd}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)/(2*c*d*x+b*d)^(1/2),x)
[Out]
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Maxima [A] time = 0.679925, size = 157, normalized size = 2.85 \[ \frac{60 \, \sqrt{2 \, c d x + b d} a - \frac{10 \,{\left (3 \, \sqrt{2 \, c d x + b d} b d -{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}\right )} b}{c d} + \frac{15 \, \sqrt{2 \, c d x + b d} b^{2} d^{2} - 10 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b d + 3 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}}{c d^{2}}}{60 \, c d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)/sqrt(2*c*d*x + b*d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.209329, size = 54, normalized size = 0.98 \[ \frac{{\left (c^{2} x^{2} + b c x - b^{2} + 5 \, a c\right )} \sqrt{2 \, c d x + b d}}{5 \, c^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)/sqrt(2*c*d*x + b*d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.55212, size = 258, normalized size = 4.69 \[ \begin{cases} - \frac{\frac{a b}{\sqrt{b d + 2 c d x}} + \frac{a \left (- \frac{b d}{\sqrt{b d + 2 c d x}} - \sqrt{b d + 2 c d x}\right )}{d} + \frac{b^{2} \left (- \frac{b d}{\sqrt{b d + 2 c d x}} - \sqrt{b d + 2 c d x}\right )}{2 c d} + \frac{3 b \left (\frac{b^{2} d^{2}}{\sqrt{b d + 2 c d x}} + 2 b d \sqrt{b d + 2 c d x} - \frac{\left (b d + 2 c d x\right )^{\frac{3}{2}}}{3}\right )}{4 c d^{2}} + \frac{- \frac{b^{3} d^{3}}{\sqrt{b d + 2 c d x}} - 3 b^{2} d^{2} \sqrt{b d + 2 c d x} + b d \left (b d + 2 c d x\right )^{\frac{3}{2}} - \frac{\left (b d + 2 c d x\right )^{\frac{5}{2}}}{5}}{4 c d^{3}}}{c} & \text{for}\: c \neq 0 \\\frac{a x + \frac{b x^{2}}{2}}{\sqrt{b d}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)/(2*c*d*x+b*d)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.224593, size = 176, normalized size = 3.2 \[ \frac{60 \, \sqrt{2 \, c d x + b d} a - \frac{10 \,{\left (3 \, \sqrt{2 \, c d x + b d} b d -{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}\right )} b}{c d} + \frac{15 \, \sqrt{2 \, c d x + b d} b^{2} c^{8} d^{10} - 10 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b c^{8} d^{9} + 3 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} c^{8} d^{8}}{c^{9} d^{10}}}{60 \, c d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)/sqrt(2*c*d*x + b*d),x, algorithm="giac")
[Out]