Optimal. Leaf size=59 \[ \frac {(A b-a B) (b d-a e) \log (a+b x)}{b^3}+\frac {B x (b d-a e)}{b^2}+\frac {e (A+B x)^2}{2 b B} \]
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Rubi [A] time = 0.04, antiderivative size = 59, 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 = {77} \[ \frac {(A b-a B) (b d-a e) \log (a+b x)}{b^3}+\frac {B x (b d-a e)}{b^2}+\frac {e (A+B x)^2}{2 b B} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)}{a+b x} \, dx &=\int \left (\frac {B (b d-a e)}{b^2}+\frac {(A b-a B) (b d-a e)}{b^2 (a+b x)}+\frac {e (A+B x)}{b}\right ) \, dx\\ &=\frac {B (b d-a e) x}{b^2}+\frac {e (A+B x)^2}{2 b B}+\frac {(A b-a B) (b d-a e) \log (a+b x)}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 56, normalized size = 0.95 \[ \frac {b x (b (2 A e+2 B d+B e x)-2 a B e)+2 (A b-a B) (b d-a e) \log (a+b x)}{2 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 75, normalized size = 1.27 \[ \frac {B b^{2} e x^{2} + 2 \, {\left (B b^{2} d - {\left (B a b - A b^{2}\right )} e\right )} x - 2 \, {\left ({\left (B a b - A b^{2}\right )} d - {\left (B a^{2} - A a b\right )} e\right )} \log \left (b x + a\right )}{2 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.15, size = 74, normalized size = 1.25 \[ \frac {B b x^{2} e + 2 \, B b d x - 2 \, B a x e + 2 \, A b x e}{2 \, b^{2}} - \frac {{\left (B a b d - A b^{2} d - B a^{2} e + A a b e\right )} \log \left ({\left | b x + a \right |}\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 90, normalized size = 1.53 \[ \frac {B e \,x^{2}}{2 b}-\frac {A a e \ln \left (b x +a \right )}{b^{2}}+\frac {A d \ln \left (b x +a \right )}{b}+\frac {A e x}{b}+\frac {B \,a^{2} e \ln \left (b x +a \right )}{b^{3}}-\frac {B a d \ln \left (b x +a \right )}{b^{2}}-\frac {B a e x}{b^{2}}+\frac {B d x}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 72, normalized size = 1.22 \[ \frac {B b e x^{2} + 2 \, {\left (B b d - {\left (B a - A b\right )} e\right )} x}{2 \, b^{2}} - \frac {{\left ({\left (B a b - A b^{2}\right )} d - {\left (B a^{2} - A a b\right )} e\right )} \log \left (b x + a\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.10, size = 68, normalized size = 1.15 \[ x\,\left (\frac {A\,e+B\,d}{b}-\frac {B\,a\,e}{b^2}\right )+\frac {\ln \left (a+b\,x\right )\,\left (A\,b^2\,d+B\,a^2\,e-A\,a\,b\,e-B\,a\,b\,d\right )}{b^3}+\frac {B\,e\,x^2}{2\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.28, size = 53, normalized size = 0.90 \[ \frac {B e x^{2}}{2 b} + x \left (\frac {A e}{b} - \frac {B a e}{b^{2}} + \frac {B d}{b}\right ) + \frac {\left (- A b + B a\right ) \left (a e - b d\right ) \log {\left (a + b x \right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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