Optimal. Leaf size=60 \[ \frac {B e x}{b^2}-\frac {(A b-a B) (b d-a e)}{b^3 (a+b x)}+\frac {(b B d+A b e-2 a B e) \log (a+b x)}{b^3} \]
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Rubi [A]
time = 0.04, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {78}
\begin {gather*} -\frac {(A b-a B) (b d-a e)}{b^3 (a+b x)}+\frac {\log (a+b x) (-2 a B e+A b e+b B d)}{b^3}+\frac {B e x}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)}{(a+b x)^2} \, dx &=\int \left (\frac {B e}{b^2}+\frac {(A b-a B) (b d-a e)}{b^2 (a+b x)^2}+\frac {b B d+A b e-2 a B e}{b^2 (a+b x)}\right ) \, dx\\ &=\frac {B e x}{b^2}-\frac {(A b-a B) (b d-a e)}{b^3 (a+b x)}+\frac {(b B d+A b e-2 a B e) \log (a+b x)}{b^3}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 56, normalized size = 0.93 \begin {gather*} \frac {b B e x-\frac {(A b-a B) (b d-a e)}{a+b x}+(b B d+A b e-2 a B e) \log (a+b x)}{b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 70, normalized size = 1.17
method | result | size |
default | \(\frac {B e x}{b^{2}}-\frac {-A a b e +A \,b^{2} d +B \,a^{2} e -B a b d}{b^{3} \left (b x +a \right )}+\frac {\left (A b e -2 B a e +B b d \right ) \ln \left (b x +a \right )}{b^{3}}\) | \(70\) |
norman | \(\frac {\frac {A a b e -A \,b^{2} d -2 B \,a^{2} e +B a b d}{b^{3}}+\frac {B e \,x^{2}}{b}}{b x +a}+\frac {\left (A b e -2 B a e +B b d \right ) \ln \left (b x +a \right )}{b^{3}}\) | \(73\) |
risch | \(\frac {B e x}{b^{2}}+\frac {A a e}{b^{2} \left (b x +a \right )}-\frac {A d}{b \left (b x +a \right )}-\frac {B \,a^{2} e}{b^{3} \left (b x +a \right )}+\frac {B a d}{b^{2} \left (b x +a \right )}+\frac {\ln \left (b x +a \right ) A e}{b^{2}}-\frac {2 \ln \left (b x +a \right ) B a e}{b^{3}}+\frac {\ln \left (b x +a \right ) B d}{b^{2}}\) | \(106\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 79, normalized size = 1.32 \begin {gather*} \frac {B x e}{b^{2}} - \frac {B a^{2} e - A a b e - {\left (B a b - A b^{2}\right )} d}{b^{4} x + a b^{3}} + \frac {{\left (B b d - 2 \, B a e + A b e\right )} \log \left (b x + a\right )}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.80, size = 105, normalized size = 1.75 \begin {gather*} \frac {{\left (B a b - A b^{2}\right )} d + {\left (B b^{2} x^{2} + B a b x - B a^{2} + A a b\right )} e + {\left (B b^{2} d x + B a b d - {\left (2 \, B a^{2} - A a b + {\left (2 \, B a b - A b^{2}\right )} x\right )} e\right )} \log \left (b x + a\right )}{b^{4} x + a b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.25, size = 71, normalized size = 1.18 \begin {gather*} \frac {B e x}{b^{2}} + \frac {A a b e - A b^{2} d - B a^{2} e + B a b d}{a b^{3} + b^{4} x} - \frac {\left (- A b e + 2 B a e - B b d\right ) \log {\left (a + b x \right )}}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.30, size = 117, normalized size = 1.95 \begin {gather*} \frac {{\left (b x + a\right )} B e}{b^{3}} - \frac {{\left (B b d - 2 \, B a e + A b e\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{3}} + \frac {\frac {B a b^{2} d}{b x + a} - \frac {A b^{3} d}{b x + a} - \frac {B a^{2} b e}{b x + a} + \frac {A a b^{2} e}{b x + a}}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.12, size = 75, normalized size = 1.25 \begin {gather*} \frac {\ln \left (a+b\,x\right )\,\left (A\,b\,e-2\,B\,a\,e+B\,b\,d\right )}{b^3}-\frac {A\,b^2\,d+B\,a^2\,e-A\,a\,b\,e-B\,a\,b\,d}{b\,\left (x\,b^3+a\,b^2\right )}+\frac {B\,e\,x}{b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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