3.11.43 \(\int \frac {(a+b x)^2}{(a c-b c x)^2} \, dx\) [1043]

Optimal. Leaf size=41 \[ \frac {x}{c^2}+\frac {4 a^2}{b c^2 (a-b x)}+\frac {4 a \log (a-b x)}{b c^2} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 41, 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 {4 a^2}{b c^2 (a-b x)}+\frac {4 a \log (a-b x)}{b c^2}+\frac {x}{c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/(a*c - b*c*x)^2,x]

[Out]

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

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Mathematica [A]
time = 0.02, size = 35, normalized size = 0.85 \begin {gather*} \frac {x+\frac {4 a^2}{b (a-b x)}+\frac {4 a \log (a-b x)}{b}}{c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.16, size = 36, normalized size = 0.88

method	result	size
default	\(\frac {x +\frac {4 a^{2}}{b \left (-b x +a \right )}+\frac {4 a \ln \left (-b x +a \right )}{b}}{c^{2}}\)	\(36\)
risch	\(\frac {x}{c^{2}}+\frac {4 a^{2}}{b \,c^{2} \left (-b x +a \right )}+\frac {4 a \ln \left (-b x +a \right )}{b \,c^{2}}\)	\(42\)
norman	\(\frac {\frac {5 a^{2}}{b c}-\frac {b \,x^{2}}{c}}{c \left (-b x +a \right )}+\frac {4 a \ln \left (-b x +a \right )}{b \,c^{2}}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 46, normalized size = 1.12 \begin {gather*} -\frac {4 \, a^{2}}{b^{2} c^{2} x - a b c^{2}} + \frac {x}{c^{2}} + \frac {4 \, a \log \left (b x - a\right )}{b c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.55, size = 57, normalized size = 1.39 \begin {gather*} \frac {b^{2} x^{2} - a b x - 4 \, a^{2} + 4 \, {\left (a b x - a^{2}\right )} \log \left (b x - a\right )}{b^{2} c^{2} x - a b c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.08, size = 39, normalized size = 0.95 \begin {gather*} - \frac {4 a^{2}}{- a b c^{2} + b^{2} c^{2} x} + \frac {4 a \log {\left (- a + b x \right )}}{b c^{2}} + \frac {x}{c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.92, size = 79, normalized size = 1.93 \begin {gather*} -\frac {4 \, a^{2}}{{\left (b c x - a c\right )} b c} - \frac {4 \, a \log \left (\frac {{\left | b c x - a c \right |}}{{\left (b c x - a c\right )}^{2} {\left | b \right |} {\left | c \right |}}\right )}{b c^{2}} + \frac {b c x - a c}{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)^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.15, size = 46, normalized size = 1.12 \begin {gather*} \frac {x}{c^2}+\frac {4\,a^2}{b\,\left (a\,c^2-b\,c^2\,x\right )}+\frac {4\,a\,\ln \left (b\,x-a\right )}{b\,c^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^2/(a*c - b*c*x)^2,x)

[Out]

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

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