3.1252 \(\int \sqrt{b d+2 c d x} \left (a+b x+c x^2\right ) \, dx\)

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]

-((b^2 - 4*a*c)*(b*d + 2*c*d*x)^(3/2))/(12*c^2*d) + (b*d + 2*c*d*x)^(7/2)/(28*c^
2*d^3)

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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]

-((b^2 - 4*a*c)*(b*d + 2*c*d*x)^(3/2))/(12*c^2*d) + (b*d + 2*c*d*x)^(7/2)/(28*c^
2*d^3)

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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]

-(-a*c + b**2/4)*(b*d + 2*c*d*x)**(3/2)/(3*c**2*d) + (b*d + 2*c*d*x)**(7/2)/(28*
c**2*d**3)

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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]

((d*(b + 2*c*x))^(3/2)*(-b^2 + 3*b*c*x + c*(7*a + 3*c*x^2)))/(21*c^2*d)

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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]

1/21*(2*c*x+b)*(3*c^2*x^2+3*b*c*x+7*a*c-b^2)*(2*c*d*x+b*d)^(1/2)/c^2

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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]

-1/84*(7*(2*c*d*x + b*d)^(3/2)*(b^2 - 4*a*c)*d^2 - 3*(2*c*d*x + b*d)^(7/2))/(c^2
*d^3)

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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]

1/21*(6*c^3*x^3 + 9*b*c^2*x^2 - b^3 + 7*a*b*c + (b^2*c + 14*a*c^2)*x)*sqrt(2*c*d
*x + b*d)/c^2

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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]

((4*a*c - b**2)*(b*d + 2*c*d*x)**(3/2)/(12*c) + (b*d + 2*c*d*x)**(7/2)/(28*c*d**
2))/(c*d)

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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]

1/420*(140*(2*c*d*x + b*d)^(3/2)*a - 14*(5*(2*c*d*x + b*d)^(3/2)*b*d - 3*(2*c*d*
x + b*d)^(5/2))*b/(c*d) + (35*(2*c*d*x + b*d)^(3/2)*b^2*c^12*d^14 - 42*(2*c*d*x
+ b*d)^(5/2)*b*c^12*d^13 + 15*(2*c*d*x + b*d)^(7/2)*c^12*d^12)/(c^13*d^14))/(c*d
)