3.2872 \(\int \frac{(c+d x)^2}{\left (a+b (c+d x)^3\right )^3} \, dx\)

Optimal. Leaf size=23 \[ -\frac{1}{6 b d \left (a+b (c+d x)^3\right )^2} \]

[Out]

-1/(6*b*d*(a + b*(c + d*x)^3)^2)

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Rubi [A]  time = 0.0170314, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ -\frac{1}{6 b d \left (a+b (c+d x)^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-1/(6*b*d*(a + b*(c + d*x)^3)^2)

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Rubi in Sympy [A]  time = 4.04355, size = 19, normalized size = 0.83 \[ - \frac{1}{6 b d \left (a + b \left (c + d x\right )^{3}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(6*b*d*(a + b*(c + d*x)**3)**2)

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Mathematica [A]  time = 0.0238455, size = 23, normalized size = 1. \[ -\frac{1}{6 b d \left (a+b (c+d x)^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-1/(6*b*d*(a + b*(c + d*x)^3)^2)

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Maple [B]  time = 0.001, size = 44, normalized size = 1.9 \[ -{\frac{1}{6\,bd \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6/b/d/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)^2

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Maxima [A]  time = 1.3381, size = 28, normalized size = 1.22 \[ -\frac{1}{6 \,{\left ({\left (d x + c\right )}^{3} b + a\right )}^{2} b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6/(((d*x + c)^3*b + a)^2*b*d)

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Fricas [A]  time = 0.227018, size = 180, normalized size = 7.83 \[ -\frac{1}{6 \,{\left (b^{3} d^{7} x^{6} + 6 \, b^{3} c d^{6} x^{5} + 15 \, b^{3} c^{2} d^{5} x^{4} + 2 \,{\left (10 \, b^{3} c^{3} + a b^{2}\right )} d^{4} x^{3} + 3 \,{\left (5 \, b^{3} c^{4} + 2 \, a b^{2} c\right )} d^{3} x^{2} + 6 \,{\left (b^{3} c^{5} + a b^{2} c^{2}\right )} d^{2} x +{\left (b^{3} c^{6} + 2 \, a b^{2} c^{3} + a^{2} b\right )} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6/(b^3*d^7*x^6 + 6*b^3*c*d^6*x^5 + 15*b^3*c^2*d^5*x^4 + 2*(10*b^3*c^3 + a*b^2
)*d^4*x^3 + 3*(5*b^3*c^4 + 2*a*b^2*c)*d^3*x^2 + 6*(b^3*c^5 + a*b^2*c^2)*d^2*x +
(b^3*c^6 + 2*a*b^2*c^3 + a^2*b)*d)

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Sympy [A]  time = 84.02, size = 153, normalized size = 6.65 \[ - \frac{1}{6 a^{2} b d + 12 a b^{2} c^{3} d + 6 b^{3} c^{6} d + 90 b^{3} c^{2} d^{5} x^{4} + 36 b^{3} c d^{6} x^{5} + 6 b^{3} d^{7} x^{6} + x^{3} \left (12 a b^{2} d^{4} + 120 b^{3} c^{3} d^{4}\right ) + x^{2} \left (36 a b^{2} c d^{3} + 90 b^{3} c^{4} d^{3}\right ) + x \left (36 a b^{2} c^{2} d^{2} + 36 b^{3} c^{5} d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(6*a**2*b*d + 12*a*b**2*c**3*d + 6*b**3*c**6*d + 90*b**3*c**2*d**5*x**4 + 36*
b**3*c*d**6*x**5 + 6*b**3*d**7*x**6 + x**3*(12*a*b**2*d**4 + 120*b**3*c**3*d**4)
+ x**2*(36*a*b**2*c*d**3 + 90*b**3*c**4*d**3) + x*(36*a*b**2*c**2*d**2 + 36*b**
3*c**5*d**2))

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GIAC/XCAS [A]  time = 0.214606, size = 28, normalized size = 1.22 \[ -\frac{1}{6 \,{\left ({\left (d x + c\right )}^{3} b + a\right )}^{2} b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6/(((d*x + c)^3*b + a)^2*b*d)