\(\int \frac {(a+b x)^2}{(a c-b c x)^5} \, dx\) [1046]

Optimal result

Integrand size = 19, antiderivative size = 56 \[ \int \frac {(a+b x)^2}{(a c-b c x)^5} \, dx=\frac {a^2}{b c^5 (a-b x)^4}-\frac {4 a}{3 b c^5 (a-b x)^3}+\frac {1}{2 b c^5 (a-b x)^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {45} \[ \int \frac {(a+b x)^2}{(a c-b c x)^5} \, dx=\frac {a^2}{b c^5 (a-b x)^4}-\frac {4 a}{3 b c^5 (a-b x)^3}+\frac {1}{2 b c^5 (a-b x)^2} \]

[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)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {4 a^2}{c^5 (a-b x)^5}-\frac {4 a}{c^5 (a-b x)^4}+\frac {1}{c^5 (a-b x)^3}\right ) \, dx \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.62 \[ \int \frac {(a+b x)^2}{(a c-b c x)^5} \, dx=\frac {a^2+2 a b x+3 b^2 x^2}{6 b c^5 (a-b x)^4} \]

[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] (verified)

Time = 0.16 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.57

method	result	size
risch	\(\frac {\frac {b \,x^{2}}{2}+\frac {a x}{3}+\frac {a^{2}}{6 b}}{c^{5} \left (-b x +a \right )^{4}}\)	\(32\)
gosper	\(\frac {3 b^{2} x^{2}+2 a b x +a^{2}}{6 \left (-b x +a \right )^{4} c^{5} b}\)	\(34\)
norman	\(\frac {\frac {a^{2}}{6 b c}+\frac {b \,x^{2}}{2 c}+\frac {a x}{3 c}}{c^{4} \left (-b x +a \right )^{4}}\)	\(41\)
parallelrisch	\(\frac {3 b^{5} x^{2}+2 a \,b^{4} x +a^{2} b^{3}}{6 b^{4} c^{5} \left (b x -a \right )^{4}}\)	\(41\)
default	\(\frac {\frac {1}{2 b \left (-b x +a \right )^{2}}+\frac {a^{2}}{b \left (-b x +a \right )^{4}}-\frac {4 a}{3 b \left (-b x +a \right )^{3}}}{c^{5}}\)	\(48\)

[In]

int((b*x+a)^2/(-b*c*x+a*c)^5,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.39 \[ \int \frac {(a+b x)^2}{(a c-b c x)^5} \, dx=\frac {3 \, b^{2} x^{2} + 2 \, a b x + a^{2}}{6 \, {\left (b^{5} c^{5} x^{4} - 4 \, a b^{4} c^{5} x^{3} + 6 \, a^{2} b^{3} c^{5} x^{2} - 4 \, a^{3} b^{2} c^{5} x + a^{4} b c^{5}\right )}} \]

[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] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.52 \[ \int \frac {(a+b x)^2}{(a c-b c x)^5} \, dx=- \frac {- a^{2} - 2 a b x - 3 b^{2} x^{2}}{6 a^{4} b c^{5} - 24 a^{3} b^{2} c^{5} x + 36 a^{2} b^{3} c^{5} x^{2} - 24 a b^{4} c^{5} x^{3} + 6 b^{5} c^{5} x^{4}} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.39 \[ \int \frac {(a+b x)^2}{(a c-b c x)^5} \, dx=\frac {3 \, b^{2} x^{2} + 2 \, a b x + a^{2}}{6 \, {\left (b^{5} c^{5} x^{4} - 4 \, a b^{4} c^{5} x^{3} + 6 \, a^{2} b^{3} c^{5} x^{2} - 4 \, a^{3} b^{2} c^{5} x + a^{4} b c^{5}\right )}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.14 \[ \int \frac {(a+b x)^2}{(a c-b c x)^5} \, dx=\frac {\frac {6 \, a^{2}}{{\left (b c x - a c\right )}^{4} b} + \frac {8 \, a}{{\left (b c x - a c\right )}^{3} b c} + \frac {3}{{\left (b c x - a c\right )}^{2} b c^{2}}}{6 \, c} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.36 \[ \int \frac {(a+b x)^2}{(a c-b c x)^5} \, dx=\frac {\frac {a\,x}{3}+\frac {b\,x^2}{2}+\frac {a^2}{6\,b}}{a^4\,c^5-4\,a^3\,b\,c^5\,x+6\,a^2\,b^2\,c^5\,x^2-4\,a\,b^3\,c^5\,x^3+b^4\,c^5\,x^4} \]

[In]

int((a + b*x)^2/(a*c - b*c*x)^5,x)

[Out]

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