Optimal. Leaf size=117 \[ -\frac {A b-a B}{(a+b x) (b d-a e)^2}+\frac {B d-A e}{(d+e x) (b d-a e)^2}+\frac {\log (a+b x) (a B e-2 A b e+b B d)}{(b d-a e)^3}-\frac {\log (d+e x) (a B e-2 A b e+b B d)}{(b d-a e)^3} \]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 77} \begin {gather*} -\frac {A b-a B}{(a+b x) (b d-a e)^2}+\frac {B d-A e}{(d+e x) (b d-a e)^2}+\frac {\log (a+b x) (a B e-2 A b e+b B d)}{(b d-a e)^3}-\frac {\log (d+e x) (a B e-2 A b e+b B d)}{(b d-a e)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 77
Rubi steps
\begin {align*} \int \frac {A+B x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac {A+B x}{(a+b x)^2 (d+e x)^2} \, dx\\ &=\int \left (\frac {b (A b-a B)}{(b d-a e)^2 (a+b x)^2}+\frac {b (b B d-2 A b e+a B e)}{(b d-a e)^3 (a+b x)}+\frac {e (-B d+A e)}{(b d-a e)^2 (d+e x)^2}+\frac {e (-b B d+2 A b e-a B e)}{(b d-a e)^3 (d+e x)}\right ) \, dx\\ &=-\frac {A b-a B}{(b d-a e)^2 (a+b x)}+\frac {B d-A e}{(b d-a e)^2 (d+e x)}+\frac {(b B d-2 A b e+a B e) \log (a+b x)}{(b d-a e)^3}-\frac {(b B d-2 A b e+a B e) \log (d+e x)}{(b d-a e)^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 103, normalized size = 0.88 \begin {gather*} \frac {\frac {(a B-A b) (b d-a e)}{a+b x}+\frac {(b d-a e) (B d-A e)}{d+e x}+\log (a+b x) (a B e-2 A b e+b B d)-\log (d+e x) (a B e-2 A b e+b B d)}{(b d-a e)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A+B x}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.44, size = 396, normalized size = 3.38 \begin {gather*} -\frac {2 \, B a^{2} d e - A a^{2} e^{2} - {\left (2 \, B a b - A b^{2}\right )} d^{2} - {\left (B b^{2} d^{2} - 2 \, A b^{2} d e - {\left (B a^{2} - 2 \, A a b\right )} e^{2}\right )} x - {\left (B a b d^{2} + {\left (B a^{2} - 2 \, A a b\right )} d e + {\left (B b^{2} d e + {\left (B a b - 2 \, A b^{2}\right )} e^{2}\right )} x^{2} + {\left (B b^{2} d^{2} + 2 \, {\left (B a b - A b^{2}\right )} d e + {\left (B a^{2} - 2 \, A a b\right )} e^{2}\right )} x\right )} \log \left (b x + a\right ) + {\left (B a b d^{2} + {\left (B a^{2} - 2 \, A a b\right )} d e + {\left (B b^{2} d e + {\left (B a b - 2 \, A b^{2}\right )} e^{2}\right )} x^{2} + {\left (B b^{2} d^{2} + 2 \, {\left (B a b - A b^{2}\right )} d e + {\left (B a^{2} - 2 \, A a b\right )} e^{2}\right )} x\right )} \log \left (e x + d\right )}{a b^{3} d^{4} - 3 \, a^{2} b^{2} d^{3} e + 3 \, a^{3} b d^{2} e^{2} - a^{4} d e^{3} + {\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x^{2} + {\left (b^{4} d^{4} - 2 \, a b^{3} d^{3} e + 2 \, a^{3} b d e^{3} - a^{4} e^{4}\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.22, size = 195, normalized size = 1.67 \begin {gather*} \frac {{\left (B b d e + B a e^{2} - 2 \, A b e^{2}\right )} \log \left ({\left | b - \frac {b d}{x e + d} + \frac {a e}{x e + d} \right |}\right )}{b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}} + \frac {\frac {B d e^{2}}{x e + d} - \frac {A e^{3}}{x e + d}}{b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}} + \frac {B a b e - A b^{2} e}{{\left (b d - a e\right )}^{3} {\left (b - \frac {b d}{x e + d} + \frac {a e}{x e + d}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 208, normalized size = 1.78 \begin {gather*} \frac {2 A b e \ln \left (b x +a \right )}{\left (a e -b d \right )^{3}}-\frac {2 A b e \ln \left (e x +d \right )}{\left (a e -b d \right )^{3}}-\frac {B a e \ln \left (b x +a \right )}{\left (a e -b d \right )^{3}}+\frac {B a e \ln \left (e x +d \right )}{\left (a e -b d \right )^{3}}-\frac {B b d \ln \left (b x +a \right )}{\left (a e -b d \right )^{3}}+\frac {B b d \ln \left (e x +d \right )}{\left (a e -b d \right )^{3}}-\frac {A b}{\left (a e -b d \right )^{2} \left (b x +a \right )}-\frac {A e}{\left (a e -b d \right )^{2} \left (e x +d \right )}+\frac {B a}{\left (a e -b d \right )^{2} \left (b x +a \right )}+\frac {B d}{\left (a e -b d \right )^{2} \left (e x +d \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.54, size = 256, normalized size = 2.19 \begin {gather*} \frac {{\left (B b d + {\left (B a - 2 \, A b\right )} e\right )} \log \left (b x + a\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} - \frac {{\left (B b d + {\left (B a - 2 \, A b\right )} e\right )} \log \left (e x + d\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} - \frac {A a e - {\left (2 \, B a - A b\right )} d - {\left (B b d + {\left (B a - 2 \, A b\right )} e\right )} x}{a b^{2} d^{3} - 2 \, a^{2} b d^{2} e + a^{3} d e^{2} + {\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x^{2} + {\left (b^{3} d^{3} - a b^{2} d^{2} e - a^{2} b d e^{2} + a^{3} e^{3}\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.24, size = 263, normalized size = 2.25 \begin {gather*} -\frac {\frac {A\,a\,e+A\,b\,d-2\,B\,a\,d}{a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2}-\frac {x\,\left (B\,a\,e-2\,A\,b\,e+B\,b\,d\right )}{a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2}}{b\,e\,x^2+\left (a\,e+b\,d\right )\,x+a\,d}-\frac {2\,\mathrm {atanh}\left (\frac {\left (\frac {a^3\,e^3-a^2\,b\,d\,e^2-a\,b^2\,d^2\,e+b^3\,d^3}{a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2}+2\,b\,e\,x\right )\,\left (e\,\left (2\,A\,b-B\,a\right )-B\,b\,d\right )\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}{{\left (a\,e-b\,d\right )}^3\,\left (B\,a\,e-2\,A\,b\,e+B\,b\,d\right )}\right )\,\left (e\,\left (2\,A\,b-B\,a\right )-B\,b\,d\right )}{{\left (a\,e-b\,d\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 2.29, size = 706, normalized size = 6.03 \begin {gather*} \frac {- A a e - A b d + 2 B a d + x \left (- 2 A b e + B a e + B b d\right )}{a^{3} d e^{2} - 2 a^{2} b d^{2} e + a b^{2} d^{3} + x^{2} \left (a^{2} b e^{3} - 2 a b^{2} d e^{2} + b^{3} d^{2} e\right ) + x \left (a^{3} e^{3} - a^{2} b d e^{2} - a b^{2} d^{2} e + b^{3} d^{3}\right )} + \frac {\left (- 2 A b e + B a e + B b d\right ) \log {\left (x + \frac {- 2 A a b e^{2} - 2 A b^{2} d e + B a^{2} e^{2} + 2 B a b d e + B b^{2} d^{2} - \frac {a^{4} e^{4} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} + \frac {4 a^{3} b d e^{3} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} - \frac {6 a^{2} b^{2} d^{2} e^{2} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} + \frac {4 a b^{3} d^{3} e \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} - \frac {b^{4} d^{4} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}}}{- 4 A b^{2} e^{2} + 2 B a b e^{2} + 2 B b^{2} d e} \right )}}{\left (a e - b d\right )^{3}} - \frac {\left (- 2 A b e + B a e + B b d\right ) \log {\left (x + \frac {- 2 A a b e^{2} - 2 A b^{2} d e + B a^{2} e^{2} + 2 B a b d e + B b^{2} d^{2} + \frac {a^{4} e^{4} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} - \frac {4 a^{3} b d e^{3} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} + \frac {6 a^{2} b^{2} d^{2} e^{2} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} - \frac {4 a b^{3} d^{3} e \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} + \frac {b^{4} d^{4} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}}}{- 4 A b^{2} e^{2} + 2 B a b e^{2} + 2 B b^{2} d e} \right )}}{\left (a e - b d\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________