3.1242 \(\int \frac{(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=52 \[ -\frac{16 c d^3}{3 \sqrt{a+b x+c x^2}}-\frac{2 d^3 (b+2 c x)^2}{3 \left (a+b x+c x^2\right )^{3/2}} \]

[Out]

(-2*d^3*(b + 2*c*x)^2)/(3*(a + b*x + c*x^2)^(3/2)) - (16*c*d^3)/(3*Sqrt[a + b*x
+ c*x^2])

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Rubi [A]  time = 0.0720874, antiderivative size = 52, 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 \[ -\frac{16 c d^3}{3 \sqrt{a+b x+c x^2}}-\frac{2 d^3 (b+2 c x)^2}{3 \left (a+b x+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^3/(a + b*x + c*x^2)^(5/2),x]

[Out]

(-2*d^3*(b + 2*c*x)^2)/(3*(a + b*x + c*x^2)^(3/2)) - (16*c*d^3)/(3*Sqrt[a + b*x
+ c*x^2])

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Rubi in Sympy [A]  time = 17.9931, size = 51, normalized size = 0.98 \[ - \frac{16 c d^{3}}{3 \sqrt{a + b x + c x^{2}}} - \frac{2 d^{3} \left (b + 2 c x\right )^{2}}{3 \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} \]

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)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.075796, size = 42, normalized size = 0.81 \[ -\frac{2 d^3 \left (4 c \left (2 a+3 c x^2\right )+b^2+12 b c x\right )}{3 (a+x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^3/(a + b*x + c*x^2)^(5/2),x]

[Out]

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

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Maple [A]  time = 0.01, size = 39, normalized size = 0.8 \[ -{\frac{2\,{d}^{3} \left ( 12\,{c}^{2}{x}^{2}+12\,bxc+8\,ac+{b}^{2} \right ) }{3} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}} \]

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)^(5/2),x)

[Out]

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

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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/(c*x^2 + b*x + a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.302502, size = 112, normalized size = 2.15 \[ -\frac{2 \,{\left (12 \, c^{2} d^{3} x^{2} + 12 \, b c d^{3} x +{\left (b^{2} + 8 \, a c\right )} d^{3}\right )} \sqrt{c x^{2} + b x + a}}{3 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \]

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)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.5421, size = 264, normalized size = 5.08 \[ - \frac{16 a c d^{3}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} - \frac{2 b^{2} d^{3}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} - \frac{24 b c d^{3} x}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} - \frac{24 c^{2} d^{3} x^{2}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} \]

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)**(5/2),x)

[Out]

-16*a*c*d**3/(3*a*sqrt(a + b*x + c*x**2) + 3*b*x*sqrt(a + b*x + c*x**2) + 3*c*x*
*2*sqrt(a + b*x + c*x**2)) - 2*b**2*d**3/(3*a*sqrt(a + b*x + c*x**2) + 3*b*x*sqr
t(a + b*x + c*x**2) + 3*c*x**2*sqrt(a + b*x + c*x**2)) - 24*b*c*d**3*x/(3*a*sqrt
(a + b*x + c*x**2) + 3*b*x*sqrt(a + b*x + c*x**2) + 3*c*x**2*sqrt(a + b*x + c*x*
*2)) - 24*c**2*d**3*x**2/(3*a*sqrt(a + b*x + c*x**2) + 3*b*x*sqrt(a + b*x + c*x*
*2) + 3*c*x**2*sqrt(a + b*x + c*x**2))

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GIAC/XCAS [A]  time = 0.230525, size = 275, normalized size = 5.29 \[ -\frac{12 \,{\left (\frac{{\left (b^{4} c^{2} d^{3} - 8 \, a b^{2} c^{3} d^{3} + 16 \, a^{2} c^{4} d^{3}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{b^{5} c d^{3} - 8 \, a b^{3} c^{2} d^{3} + 16 \, a^{2} b c^{3} d^{3}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{b^{6} d^{3} - 48 \, a^{2} b^{2} c^{2} d^{3} + 128 \, a^{3} c^{3} d^{3}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} \]

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)^(5/2),x, algorithm="giac")

[Out]

-1/3*(12*((b^4*c^2*d^3 - 8*a*b^2*c^3*d^3 + 16*a^2*c^4*d^3)*x/(b^4*c^2 - 8*a*b^2*
c^3 + 16*a^2*c^4) + (b^5*c*d^3 - 8*a*b^3*c^2*d^3 + 16*a^2*b*c^3*d^3)/(b^4*c^2 -
8*a*b^2*c^3 + 16*a^2*c^4))*x + (b^6*d^3 - 48*a^2*b^2*c^2*d^3 + 128*a^3*c^3*d^3)/
(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))/(c*x^2 + b*x + a)^(3/2)