3.10.78 \(\int \frac {A+B x}{(a+b x) (d+e x)} \, dx\)

Optimal. Leaf size=57 \[ \frac {(A b-a B) \log (a+b x)}{b (b d-a e)}+\frac {(B d-A e) \log (d+e x)}{e (b d-a e)} \]

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {72} \begin {gather*} \frac {(A b-a B) \log (a+b x)}{b (b d-a e)}+\frac {(B d-A e) \log (d+e x)}{e (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((a + b*x)*(d + e*x)),x]

[Out]

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

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {A+B x}{(a+b x) (d+e x)} \, dx &=\int \left (\frac {A b-a B}{(b d-a e) (a+b x)}+\frac {B d-A e}{(b d-a e) (d+e x)}\right ) \, dx\\ &=\frac {(A b-a B) \log (a+b x)}{b (b d-a e)}+\frac {(B d-A e) \log (d+e x)}{e (b d-a e)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 50, normalized size = 0.88 \begin {gather*} \frac {e (A b-a B) \log (a+b x)+b (B d-A e) \log (d+e x)}{b e (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((a + b*x)*(d + e*x)),x]

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A+B x}{(a+b x) (d+e x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(A + B*x)/((a + b*x)*(d + e*x)),x]

[Out]

IntegrateAlgebraic[(A + B*x)/((a + b*x)*(d + e*x)), x]

________________________________________________________________________________________

fricas [A]  time = 1.10, size = 53, normalized size = 0.93 \begin {gather*} -\frac {{\left (B a - A b\right )} e \log \left (b x + a\right ) - {\left (B b d - A b e\right )} \log \left (e x + d\right )}{b^{2} d e - a b e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 1.17, size = 63, normalized size = 1.11 \begin {gather*} -\frac {{\left (B a - A b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{2} d - a b e} + \frac {{\left (B d - A e\right )} \log \left ({\left | x e + d \right |}\right )}{b d e - a e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.01, size = 84, normalized size = 1.47 \begin {gather*} -\frac {A \ln \left (b x +a \right )}{a e -b d}+\frac {A \ln \left (e x +d \right )}{a e -b d}+\frac {B a \ln \left (b x +a \right )}{\left (a e -b d \right ) b}-\frac {B d \ln \left (e x +d \right )}{\left (a e -b d \right ) e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(b*x+a)/(e*x+d),x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.61, size = 58, normalized size = 1.02 \begin {gather*} -\frac {{\left (B a - A b\right )} \log \left (b x + a\right )}{b^{2} d - a b e} + \frac {{\left (B d - A e\right )} \log \left (e x + d\right )}{b d e - a e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.25, size = 57, normalized size = 1.00 \begin {gather*} \frac {\ln \left (d+e\,x\right )\,\left (A\,e-B\,d\right )}{a\,e^2-b\,d\,e}+\frac {\ln \left (a+b\,x\right )\,\left (A\,b-B\,a\right )}{b^2\,d-a\,b\,e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/((a + b*x)*(d + e*x)),x)

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 1.56, size = 226, normalized size = 3.96 \begin {gather*} - \frac {\left (- A e + B d\right ) \log {\left (x + \frac {- A a e - A b d + 2 B a d - \frac {a^{2} e \left (- A e + B d\right )}{a e - b d} + \frac {2 a b d \left (- A e + B d\right )}{a e - b d} - \frac {b^{2} d^{2} \left (- A e + B d\right )}{e \left (a e - b d\right )}}{- 2 A b e + B a e + B b d} \right )}}{e \left (a e - b d\right )} + \frac {\left (- A b + B a\right ) \log {\left (x + \frac {- A a e - A b d + 2 B a d + \frac {a^{2} e^{2} \left (- A b + B a\right )}{b \left (a e - b d\right )} - \frac {2 a d e \left (- A b + B a\right )}{a e - b d} + \frac {b d^{2} \left (- A b + B a\right )}{a e - b d}}{- 2 A b e + B a e + B b d} \right )}}{b \left (a e - b d\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-A*e + B*d)*log(x + (-A*a*e - A*b*d + 2*B*a*d - a**2*e*(-A*e + B*d)/(a*e - b*d) + 2*a*b*d*(-A*e + B*d)/(a*e
- b*d) - b**2*d**2*(-A*e + B*d)/(e*(a*e - b*d)))/(-2*A*b*e + B*a*e + B*b*d))/(e*(a*e - b*d)) + (-A*b + B*a)*lo
g(x + (-A*a*e - A*b*d + 2*B*a*d + a**2*e**2*(-A*b + B*a)/(b*(a*e - b*d)) - 2*a*d*e*(-A*b + B*a)/(a*e - b*d) +
b*d**2*(-A*b + B*a)/(a*e - b*d))/(-2*A*b*e + B*a*e + B*b*d))/(b*(a*e - b*d))

________________________________________________________________________________________