3.1181 \(\int (b d+2 c d x)^3 \sqrt{a+b x+c x^2} \, dx\)

Optimal. Leaf size=59 \[ \frac{4}{15} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}+\frac{2}{5} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{3/2} \]

[Out]

(4*(b^2 - 4*a*c)*d^3*(a + b*x + c*x^2)^(3/2))/15 + (2*d^3*(b + 2*c*x)^2*(a + b*x
+ c*x^2)^(3/2))/5

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Rubi [A]  time = 0.0801714, 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 \[ \frac{4}{15} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}+\frac{2}{5} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{3/2} \]

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*(a + b*x + c*x^2)^(3/2))/15 + (2*d^3*(b + 2*c*x)^2*(a + b*x
+ c*x^2)^(3/2))/5

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Rubi in Sympy [A]  time = 17.5201, size = 56, normalized size = 0.95 \[ \frac{2 d^{3} \left (b + 2 c x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{5} + \frac{4 d^{3} \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{15} \]

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*(a + b*x + c*x**2)**(3/2)/5 + 4*d**3*(-4*a*c + b**2)*(a +
b*x + c*x**2)**(3/2)/15

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Mathematica [A]  time = 0.0697064, size = 82, normalized size = 1.39 \[ -\frac{2}{15} d^3 \sqrt{a+x (b+c x)} \left (8 a^2 c-a \left (5 b^2+4 b c x+4 c^2 x^2\right )-x \left (5 b^3+17 b^2 c x+24 b c^2 x^2+12 c^3 x^3\right )\right ) \]

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)]*(8*a^2*c - a*(5*b^2 + 4*b*c*x + 4*c^2*x^2) - x*(5*
b^3 + 17*b^2*c*x + 24*b*c^2*x^2 + 12*c^3*x^3)))/15

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Maple [A]  time = 0.01, size = 41, normalized size = 0.7 \[ -{\frac{ \left ( -24\,{c}^{2}{x}^{2}-24\,bxc+16\,ac-10\,{b}^{2} \right ){d}^{3}}{15} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}} \]

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]

-2/15*(c*x^2+b*x+a)^(3/2)*(-12*c^2*x^2-12*b*c*x+8*a*c-5*b^2)*d^3

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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 = 0.233338, size = 123, normalized size = 2.08 \[ \frac{2}{15} \,{\left (12 \, c^{3} d^{3} x^{4} + 24 \, b c^{2} d^{3} x^{3} +{\left (17 \, b^{2} c + 4 \, a c^{2}\right )} d^{3} x^{2} +{\left (5 \, b^{3} + 4 \, a b c\right )} d^{3} x +{\left (5 \, a b^{2} - 8 \, a^{2} c\right )} d^{3}\right )} \sqrt{c x^{2} + b x + a} \]

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]

2/15*(12*c^3*d^3*x^4 + 24*b*c^2*d^3*x^3 + (17*b^2*c + 4*a*c^2)*d^3*x^2 + (5*b^3
+ 4*a*b*c)*d^3*x + (5*a*b^2 - 8*a^2*c)*d^3)*sqrt(c*x^2 + b*x + a)

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Sympy [A]  time = 0.901535, size = 216, normalized size = 3.66 \[ - \frac{16 a^{2} c d^{3} \sqrt{a + b x + c x^{2}}}{15} + \frac{2 a b^{2} d^{3} \sqrt{a + b x + c x^{2}}}{3} + \frac{8 a b c d^{3} x \sqrt{a + b x + c x^{2}}}{15} + \frac{8 a c^{2} d^{3} x^{2} \sqrt{a + b x + c x^{2}}}{15} + \frac{2 b^{3} d^{3} x \sqrt{a + b x + c x^{2}}}{3} + \frac{34 b^{2} c d^{3} x^{2} \sqrt{a + b x + c x^{2}}}{15} + \frac{16 b c^{2} d^{3} x^{3} \sqrt{a + b x + c x^{2}}}{5} + \frac{8 c^{3} d^{3} x^{4} \sqrt{a + b x + c x^{2}}}{5} \]

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**2*c*d**3*sqrt(a + b*x + c*x**2)/15 + 2*a*b**2*d**3*sqrt(a + b*x + c*x**2)
/3 + 8*a*b*c*d**3*x*sqrt(a + b*x + c*x**2)/15 + 8*a*c**2*d**3*x**2*sqrt(a + b*x
+ c*x**2)/15 + 2*b**3*d**3*x*sqrt(a + b*x + c*x**2)/3 + 34*b**2*c*d**3*x**2*sqrt
(a + b*x + c*x**2)/15 + 16*b*c**2*d**3*x**3*sqrt(a + b*x + c*x**2)/5 + 8*c**3*d*
*3*x**4*sqrt(a + b*x + c*x**2)/5

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GIAC/XCAS [A]  time = 0.221836, size = 163, normalized size = 2.76 \[ \frac{2}{15} \, \sqrt{c x^{2} + b x + a}{\left ({\left ({\left (12 \,{\left (c^{3} d^{3} x + 2 \, b c^{2} d^{3}\right )} x + \frac{17 \, b^{2} c^{5} d^{3} + 4 \, a c^{6} d^{3}}{c^{4}}\right )} x + \frac{5 \, b^{3} c^{4} d^{3} + 4 \, a b c^{5} d^{3}}{c^{4}}\right )} x + \frac{5 \, a b^{2} c^{4} d^{3} - 8 \, a^{2} c^{5} d^{3}}{c^{4}}\right )} \]

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]

2/15*sqrt(c*x^2 + b*x + a)*(((12*(c^3*d^3*x + 2*b*c^2*d^3)*x + (17*b^2*c^5*d^3 +
4*a*c^6*d^3)/c^4)*x + (5*b^3*c^4*d^3 + 4*a*b*c^5*d^3)/c^4)*x + (5*a*b^2*c^4*d^3
- 8*a^2*c^5*d^3)/c^4)