3.1046 \(\int \frac{(a+b x)^2}{(a c-b c x)^5} \, dx\)

Optimal. Leaf size=56 \[ \frac{a^2}{b c^5 (a-b x)^4}-\frac{4 a}{3 b c^5 (a-b x)^3}+\frac{1}{2 b c^5 (a-b x)^2} \]

[Out]

a^2/(b*c^5*(a - b*x)^4) - (4*a)/(3*b*c^5*(a - b*x)^3) + 1/(2*b*c^5*(a - b*x)^2)

_______________________________________________________________________________________

Rubi [A]  time = 0.063221, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{a^2}{b c^5 (a-b x)^4}-\frac{4 a}{3 b c^5 (a-b x)^3}+\frac{1}{2 b c^5 (a-b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

a^2/(b*c^5*(a - b*x)^4) - (4*a)/(3*b*c^5*(a - b*x)^3) + 1/(2*b*c^5*(a - b*x)^2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 15.9722, size = 46, normalized size = 0.82 \[ \frac{a^{2}}{b c^{5} \left (a - b x\right )^{4}} - \frac{4 a}{3 b c^{5} \left (a - b x\right )^{3}} + \frac{1}{2 b c^{5} \left (a - b x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**2/(-b*c*x+a*c)**5,x)

[Out]

a**2/(b*c**5*(a - b*x)**4) - 4*a/(3*b*c**5*(a - b*x)**3) + 1/(2*b*c**5*(a - b*x)
**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0189852, size = 35, normalized size = 0.62 \[ \frac{a^2+2 a b x+3 b^2 x^2}{6 b c^5 (a-b x)^4} \]

Antiderivative was successfully verified.

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

[Out]

(a^2 + 2*a*b*x + 3*b^2*x^2)/(6*b*c^5*(a - b*x)^4)

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 51, normalized size = 0.9 \[{\frac{1}{{c}^{5}} \left ({\frac{4\,a}{3\,b \left ( bx-a \right ) ^{3}}}+{\frac{1}{2\,b \left ( bx-a \right ) ^{2}}}+{\frac{{a}^{2}}{b \left ( bx-a \right ) ^{4}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2/(-b*c*x+a*c)^5,x)

[Out]

1/c^5*(4/3*a/b/(b*x-a)^3+1/2/b/(b*x-a)^2+a^2/b/(b*x-a)^4)

_______________________________________________________________________________________

Maxima [A]  time = 1.34253, size = 105, normalized size = 1.88 \[ \frac{3 \, b^{2} x^{2} + 2 \, a b x + a^{2}}{6 \,{\left (b^{5} c^{5} x^{4} - 4 \, a b^{4} c^{5} x^{3} + 6 \, a^{2} b^{3} c^{5} x^{2} - 4 \, a^{3} b^{2} c^{5} x + a^{4} b c^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x + a)^2/(b*c*x - a*c)^5,x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.216227, size = 105, normalized size = 1.88 \[ \frac{3 \, b^{2} x^{2} + 2 \, a b x + a^{2}}{6 \,{\left (b^{5} c^{5} x^{4} - 4 \, a b^{4} c^{5} x^{3} + 6 \, a^{2} b^{3} c^{5} x^{2} - 4 \, a^{3} b^{2} c^{5} x + a^{4} b c^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x + a)^2/(b*c*x - a*c)^5,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 2.21737, size = 82, normalized size = 1.46 \[ \frac{a^{2} + 2 a b x + 3 b^{2} x^{2}}{6 a^{4} b c^{5} - 24 a^{3} b^{2} c^{5} x + 36 a^{2} b^{3} c^{5} x^{2} - 24 a b^{4} c^{5} x^{3} + 6 b^{5} c^{5} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**2/(-b*c*x+a*c)**5,x)

[Out]

(a**2 + 2*a*b*x + 3*b**2*x**2)/(6*a**4*b*c**5 - 24*a**3*b**2*c**5*x + 36*a**2*b*
*3*c**5*x**2 - 24*a*b**4*c**5*x**3 + 6*b**5*c**5*x**4)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.204342, size = 88, normalized size = 1.57 \[ \frac{\frac{6 \, a^{2} c^{3}}{{\left (b c x - a c\right )}^{4} b} + \frac{8 \, a c^{2}}{{\left (b c x - a c\right )}^{3} b} + \frac{3 \, c}{{\left (b c x - a c\right )}^{2} b}}{6 \, c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x + a)^2/(b*c*x - a*c)^5,x, algorithm="giac")

[Out]

1/6*(6*a^2*c^3/((b*c*x - a*c)^4*b) + 8*a*c^2/((b*c*x - a*c)^3*b) + 3*c/((b*c*x -
a*c)^2*b))/c^4