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

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

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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 = {43} \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 43

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 = 0.03, size = 35, normalized size = 0.85 \begin {gather*} \frac {\frac {4 a^2}{b (a-b x)}+\frac {4 a \log (a-b x)}{b}+x}{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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^2}{(a c-b c x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.60, 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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giac [A]  time = 1.08, 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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maple [A]  time = 0.01, size = 44, normalized size = 1.07 \begin {gather*} -\frac {4 a^{2}}{\left (b x -a \right ) b \,c^{2}}+\frac {4 a \ln \left (b x -a \right )}{b \,c^{2}}+\frac {x}{c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.38, 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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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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sympy [A]  time = 0.20, 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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