3.11.44 \(\int \frac {(a+b x)^2}{(a c-b c x)^3} \, dx\) [1044]

Optimal. Leaf size=52 \[ \frac {2 a^2}{b c^3 (a-b x)^2}-\frac {4 a}{b c^3 (a-b x)}-\frac {\log (a-b x)}{b c^3} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {45} \begin {gather*} \frac {2 a^2}{b c^3 (a-b x)^2}-\frac {4 a}{b c^3 (a-b x)}-\frac {\log (a-b x)}{b c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/(a*c - b*c*x)^3,x]

[Out]

(2*a^2)/(b*c^3*(a - b*x)^2) - (4*a)/(b*c^3*(a - b*x)) - Log[a - b*x]/(b*c^3)

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*} \int \frac {(a+b x)^2}{(a c-b c x)^3} \, dx &=\int \left (\frac {4 a^2}{c^3 (a-b x)^3}-\frac {4 a}{c^3 (a-b x)^2}+\frac {1}{c^3 (a-b x)}\right ) \, dx\\ &=\frac {2 a^2}{b c^3 (a-b x)^2}-\frac {4 a}{b c^3 (a-b x)}-\frac {\log (a-b x)}{b c^3}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 33, normalized size = 0.63 \begin {gather*} -\frac {\frac {2 a (a-2 b x)}{(a-b x)^2}+\log (a-b x)}{b c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((2*a*(a - 2*b*x))/(a - b*x)^2 + Log[a - b*x])/(b*c^3))

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Maple [A]
time = 0.13, size = 48, normalized size = 0.92

method	result	size
risch	\(\frac {4 a x -\frac {2 a^{2}}{b}}{c^{3} \left (-b x +a \right )^{2}}-\frac {\ln \left (-b x +a \right )}{b \,c^{3}}\)	\(42\)
default	\(\frac {-\frac {4 a}{b \left (-b x +a \right )}-\frac {\ln \left (-b x +a \right )}{b}+\frac {2 a^{2}}{b \left (-b x +a \right )^{2}}}{c^{3}}\)	\(48\)
norman	\(\frac {-\frac {2 a^{2}}{b c}+\frac {4 a x}{c}}{c^{2} \left (-b x +a \right )^{2}}-\frac {\ln \left (-b x +a \right )}{b \,c^{3}}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2/(-b*c*x+a*c)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 61, normalized size = 1.17 \begin {gather*} \frac {2 \, {\left (2 \, a b x - a^{2}\right )}}{b^{3} c^{3} x^{2} - 2 \, a b^{2} c^{3} x + a^{2} b c^{3}} - \frac {\log \left (b x - a\right )}{b c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/(-b*c*x+a*c)^3,x, algorithm="maxima")

[Out]

2*(2*a*b*x - a^2)/(b^3*c^3*x^2 - 2*a*b^2*c^3*x + a^2*b*c^3) - log(b*x - a)/(b*c^3)

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Fricas [A]
time = 0.46, size = 69, normalized size = 1.33 \begin {gather*} \frac {4 \, a b x - 2 \, a^{2} - {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \log \left (b x - a\right )}{b^{3} c^{3} x^{2} - 2 \, a b^{2} c^{3} x + a^{2} b c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/(-b*c*x+a*c)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.13, size = 54, normalized size = 1.04 \begin {gather*} - \frac {2 a^{2} - 4 a b x}{a^{2} b c^{3} - 2 a b^{2} c^{3} x + b^{3} c^{3} x^{2}} - \frac {\log {\left (- a + b x \right )}}{b c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2/(-b*c*x+a*c)**3,x)

[Out]

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

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Giac [A]
time = 1.93, size = 46, normalized size = 0.88 \begin {gather*} -\frac {\log \left ({\left | b x - a \right |}\right )}{b c^{3}} + \frac {2 \, {\left (2 \, a b x - a^{2}\right )}}{{\left (b x - a\right )}^{2} b c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/(-b*c*x+a*c)^3,x, algorithm="giac")

[Out]

-log(abs(b*x - a))/(b*c^3) + 2*(2*a*b*x - a^2)/((b*x - a)^2*b*c^3)

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Mupad [B]
time = 0.17, size = 59, normalized size = 1.13 \begin {gather*} \frac {4\,a\,x-\frac {2\,a^2}{b}}{a^2\,c^3-2\,a\,b\,c^3\,x+b^2\,c^3\,x^2}-\frac {\ln \left (b\,x-a\right )}{b\,c^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^2/(a*c - b*c*x)^3,x)

[Out]

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

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