3.2.8 \(\int \frac {x}{(a+b x) (A+B x)} \, dx\) [108]

Optimal. Leaf size=35 \[ \frac {\frac {a \log (a+b x)}{b}-\frac {A \log (A+B x)}{B}}{-A b+a B} \]

[Out]

1/(-A*b+B*a)*(a/b*ln(b*x+a)-A/B*ln(B*x+A))

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Rubi [A]
time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.26, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {78} \begin {gather*} \frac {A \log (A+B x)}{B (A b-a B)}-\frac {a \log (a+b x)}{b (A b-a B)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/((a + b*x)*(A + B*x)),x]

[Out]

-((a*Log[a + b*x])/(b*(A*b - a*B))) + (A*Log[A + B*x])/(B*(A*b - a*B))

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\int \left (-\frac {a}{(A b-a B) (a+b x)}+\frac {A}{(A b-a B) (A+B x)}\right ) \, dx\\ &=-\frac {a \log (a+b x)}{b (A b-a B)}+\frac {A \log (A+B x)}{B (A b-a B)}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.01, size = 38, normalized size = 1.09 \begin {gather*} -\frac {a B \log (a+b x)-A b \log (A+B x)}{A b^2 B-a b B^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/((a + b*x)*(A + B*x)),x]

[Out]

-((a*B*Log[a + b*x] - A*b*Log[A + B*x])/(A*b^2*B - a*b*B^2))

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Maple [A]
time = 0.04, size = 45, normalized size = 1.29

method	result	size
parallelrisch	\(\frac {A \ln \left (B x +A \right ) b -a \ln \left (b x +a \right ) B}{B \left (A b -B a \right ) b}\)	\(38\)
default	\(\frac {A \ln \left (B x +A \right )}{\left (A b -B a \right ) B}-\frac {a \ln \left (b x +a \right )}{\left (A b -B a \right ) b}\)	\(45\)
norman	\(\frac {A \ln \left (B x +A \right )}{\left (A b -B a \right ) B}-\frac {a \ln \left (b x +a \right )}{\left (A b -B a \right ) b}\)	\(45\)
risch	\(\frac {A \ln \left (-B x -A \right )}{\left (A b -B a \right ) B}-\frac {a \ln \left (b x +a \right )}{\left (A b -B a \right ) b}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(b*x+a)/(B*x+A),x,method=_RETURNVERBOSE)

[Out]

A/(A*b-B*a)/B*ln(B*x+A)-a/(A*b-B*a)/b*ln(b*x+a)

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Maxima [A]
time = 0.34, size = 44, normalized size = 1.26 \begin {gather*} -\frac {A \log \left (B x + A\right )}{B^{2} a - A B b} + \frac {a \log \left (b x + a\right )}{B a b - A b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x+a)/(B*x+A),x, algorithm="maxima")

[Out]

-A*log(B*x + A)/(B^2*a - A*B*b) + a*log(b*x + a)/(B*a*b - A*b^2)

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Fricas [A]
time = 0.59, size = 38, normalized size = 1.09 \begin {gather*} -\frac {A b \log \left (B x + A\right ) - B a \log \left (b x + a\right )}{B^{2} a b - A B b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x+a)/(B*x+A),x, algorithm="fricas")

[Out]

-(A*b*log(B*x + A) - B*a*log(b*x + a))/(B^2*a*b - A*B*b^2)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (26) = 52\).
time = 0.40, size = 138, normalized size = 3.94 \begin {gather*} - \frac {A \log {\left (x + \frac {- \frac {A^{3} b^{2}}{B \left (- A b + B a\right )} + \frac {2 A^{2} a b}{- A b + B a} - \frac {A B a^{2}}{- A b + B a} + 2 A a}{A b + B a} \right )}}{B \left (- A b + B a\right )} + \frac {a \log {\left (x + \frac {\frac {A^{2} a b}{- A b + B a} - \frac {2 A B a^{2}}{- A b + B a} + 2 A a + \frac {B^{2} a^{3}}{b \left (- A b + B a\right )}}{A b + B a} \right )}}{b \left (- A b + B a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x+a)/(B*x+A),x)

[Out]

-A*log(x + (-A**3*b**2/(B*(-A*b + B*a)) + 2*A**2*a*b/(-A*b + B*a) - A*B*a**2/(-A*b + B*a) + 2*A*a)/(A*b + B*a)
)/(B*(-A*b + B*a)) + a*log(x + (A**2*a*b/(-A*b + B*a) - 2*A*B*a**2/(-A*b + B*a) + 2*A*a + B**2*a**3/(b*(-A*b +
B*a)))/(A*b + B*a))/(b*(-A*b + B*a))

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Giac [A]
time = 0.40, size = 46, normalized size = 1.31 \begin {gather*} -\frac {A \log \left ({\left | B x + A \right |}\right )}{B^{2} a - A B b} + \frac {a \log \left ({\left | b x + a \right |}\right )}{B a b - A b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x+a)/(B*x+A),x, algorithm="giac")

[Out]

-A*log(abs(B*x + A))/(B^2*a - A*B*b) + a*log(abs(b*x + a))/(B*a*b - A*b^2)

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Mupad [B]
time = 0.15, size = 37, normalized size = 1.06 \begin {gather*} \frac {A\,b\,\ln \left (A+B\,x\right )-B\,a\,\ln \left (a+b\,x\right )}{B\,b\,\left (A\,b-B\,a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((A + B*x)*(a + b*x)),x)

[Out]

(A*b*log(A + B*x) - B*a*log(a + b*x))/(B*b*(A*b - B*a))

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