3.1224 \(\int \frac{(b d+2 c d x)^3}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=59 \[ \frac{4}{3} d^3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}+\frac{2}{3} d^3 (b+2 c x)^2 \sqrt{a+b x+c x^2} \]

[Out]

(4*(b^2 - 4*a*c)*d^3*Sqrt[a + b*x + c*x^2])/3 + (2*d^3*(b + 2*c*x)^2*Sqrt[a + b*
x + c*x^2])/3

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Rubi [A]  time = 0.0800732, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{4}{3} d^3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}+\frac{2}{3} d^3 (b+2 c x)^2 \sqrt{a+b x+c x^2} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^3/Sqrt[a + b*x + c*x^2],x]

[Out]

(4*(b^2 - 4*a*c)*d^3*Sqrt[a + b*x + c*x^2])/3 + (2*d^3*(b + 2*c*x)^2*Sqrt[a + b*
x + c*x^2])/3

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Rubi in Sympy [A]  time = 17.5402, size = 56, normalized size = 0.95 \[ \frac{2 d^{3} \left (b + 2 c x\right )^{2} \sqrt{a + b x + c x^{2}}}{3} + \frac{4 d^{3} \left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**3/(c*x**2+b*x+a)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0689426, size = 43, normalized size = 0.73 \[ \frac{2}{3} d^3 \sqrt{a+x (b+c x)} \left (4 c \left (c x^2-2 a\right )+3 b^2+4 b c x\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^3/Sqrt[a + b*x + c*x^2],x]

[Out]

(2*d^3*Sqrt[a + x*(b + c*x)]*(3*b^2 + 4*b*c*x + 4*c*(-2*a + c*x^2)))/3

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Maple [A]  time = 0.008, size = 41, normalized size = 0.7 \[ -{\frac{2\,{d}^{3} \left ( -4\,{c}^{2}{x}^{2}-4\,bxc+8\,ac-3\,{b}^{2} \right ) }{3}\sqrt{c{x}^{2}+bx+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^3/(c*x^2+b*x+a)^(1/2),x)

[Out]

-2/3*d^3*(-4*c^2*x^2-4*b*c*x+8*a*c-3*b^2)*(c*x^2+b*x+a)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^3/sqrt(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.266223, size = 65, normalized size = 1.1 \[ \frac{2}{3} \,{\left (4 \, c^{2} d^{3} x^{2} + 4 \, b c d^{3} x +{\left (3 \, b^{2} - 8 \, a c\right )} d^{3}\right )} \sqrt{c x^{2} + b x + a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^3/sqrt(c*x^2 + b*x + a),x, algorithm="fricas")

[Out]

2/3*(4*c^2*d^3*x^2 + 4*b*c*d^3*x + (3*b^2 - 8*a*c)*d^3)*sqrt(c*x^2 + b*x + a)

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Sympy [A]  time = 0.711822, size = 97, normalized size = 1.64 \[ - \frac{16 a c d^{3} \sqrt{a + b x + c x^{2}}}{3} + 2 b^{2} d^{3} \sqrt{a + b x + c x^{2}} + \frac{8 b c d^{3} x \sqrt{a + b x + c x^{2}}}{3} + \frac{8 c^{2} d^{3} x^{2} \sqrt{a + b x + c x^{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x+b*d)**3/(c*x**2+b*x+a)**(1/2),x)

[Out]

-16*a*c*d**3*sqrt(a + b*x + c*x**2)/3 + 2*b**2*d**3*sqrt(a + b*x + c*x**2) + 8*b
*c*d**3*x*sqrt(a + b*x + c*x**2)/3 + 8*c**2*d**3*x**2*sqrt(a + b*x + c*x**2)/3

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GIAC/XCAS [A]  time = 0.228044, size = 78, normalized size = 1.32 \[ \frac{2}{3} \, \sqrt{c x^{2} + b x + a}{\left (4 \,{\left (c^{2} d^{3} x + b c d^{3}\right )} x + \frac{3 \, b^{2} c^{2} d^{3} - 8 \, a c^{3} d^{3}}{c^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^3/sqrt(c*x^2 + b*x + a),x, algorithm="giac")

[Out]

2/3*sqrt(c*x^2 + b*x + a)*(4*(c^2*d^3*x + b*c*d^3)*x + (3*b^2*c^2*d^3 - 8*a*c^3*
d^3)/c^2)