Optimal. Leaf size=55 \[ \frac{(b d+2 c d x)^{7/2}}{28 c^2 d^3}-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}{12 c^2 d} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0715917, 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 \[ \frac{(b d+2 c d x)^{7/2}}{28 c^2 d^3}-\frac{\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}{12 c^2 d} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[b*d + 2*c*d*x]*(a + b*x + c*x^2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 14.2457, size = 49, normalized size = 0.89 \[ - \frac{\left (- a c + \frac{b^{2}}{4}\right ) \left (b d + 2 c d x\right )^{\frac{3}{2}}}{3 c^{2} d} + \frac{\left (b d + 2 c d x\right )^{\frac{7}{2}}}{28 c^{2} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*d*x+b*d)**(1/2)*(c*x**2+b*x+a),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0359155, size = 45, normalized size = 0.82 \[ \frac{\left (c \left (7 a+3 c x^2\right )-b^2+3 b c x\right ) (d (b+2 c x))^{3/2}}{21 c^2 d} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[b*d + 2*c*d*x]*(a + b*x + c*x^2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.005, size = 46, normalized size = 0.8 \[{\frac{ \left ( 2\,cx+b \right ) \left ( 3\,{c}^{2}{x}^{2}+3\,bxc+7\,ac-{b}^{2} \right ) }{21\,{c}^{2}}\sqrt{2\,cdx+bd}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.686893, size = 62, normalized size = 1.13 \[ -\frac{7 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}{\left (b^{2} - 4 \, a c\right )} d^{2} - 3 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}}}{84 \, c^{2} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(2*c*d*x + b*d)*(c*x^2 + b*x + a),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.208097, size = 78, normalized size = 1.42 \[ \frac{{\left (6 \, c^{3} x^{3} + 9 \, b c^{2} x^{2} - b^{3} + 7 \, a b c +{\left (b^{2} c + 14 \, a c^{2}\right )} x\right )} \sqrt{2 \, c d x + b d}}{21 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(2*c*d*x + b*d)*(c*x^2 + b*x + a),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 3.85705, size = 48, normalized size = 0.87 \[ \frac{\frac{\left (4 a c - b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{3}{2}}}{12 c} + \frac{\left (b d + 2 c d x\right )^{\frac{7}{2}}}{28 c d^{2}}}{c d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x+b*d)**(1/2)*(c*x**2+b*x+a),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.23289, size = 176, normalized size = 3.2 \[ \frac{140 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} a - \frac{14 \,{\left (5 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b d - 3 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}\right )} b}{c d} + \frac{35 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{2} c^{12} d^{14} - 42 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} b c^{12} d^{13} + 15 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} c^{12} d^{12}}{c^{13} d^{14}}}{420 \, c d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(2*c*d*x + b*d)*(c*x^2 + b*x + a),x, algorithm="giac")
[Out]