Optimal. Leaf size=60 \[ -\frac {b (B d-A e) x}{e^2}+\frac {B (a+b x)^2}{2 b e}+\frac {(b d-a e) (B d-A e) \log (d+e x)}{e^3} \]
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Rubi [A]
time = 0.03, 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 {(b d-a e) (B d-A e) \log (d+e x)}{e^3}+\frac {B (a+b x)^2}{2 b e}-\frac {b x (B d-A e)}{e^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rubi steps
\begin {align*} \int \frac {(a+b x) (A+B x)}{d+e x} \, dx &=\int \left (\frac {b (-B d+A e)}{e^2}+\frac {B (a+b x)}{e}+\frac {(-b d+a e) (-B d+A e)}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac {b (B d-A e) x}{e^2}+\frac {B (a+b x)^2}{2 b e}+\frac {(b d-a e) (B d-A e) \log (d+e x)}{e^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 56, normalized size = 0.93 \begin {gather*} \frac {e x (2 a B e+b (-2 B d+2 A e+B e x))+2 (b d-a e) (B d-A e) \log (d+e x)}{2 e^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 66, normalized size = 1.10
method | result | size |
default | \(\frac {\frac {1}{2} B b e \,x^{2}+A b e x +B a e x -B b d x}{e^{2}}+\frac {\left (A a \,e^{2}-A b d e -B a d e +B b \,d^{2}\right ) \ln \left (e x +d \right )}{e^{3}}\) | \(66\) |
norman | \(\frac {\left (A b e +B a e -B b d \right ) x}{e^{2}}+\frac {B b \,x^{2}}{2 e}+\frac {\left (A a \,e^{2}-A b d e -B a d e +B b \,d^{2}\right ) \ln \left (e x +d \right )}{e^{3}}\) | \(66\) |
risch | \(\frac {B b \,x^{2}}{2 e}+\frac {A b x}{e}+\frac {B a x}{e}-\frac {B b d x}{e^{2}}+\frac {\ln \left (e x +d \right ) A a}{e}-\frac {\ln \left (e x +d \right ) A b d}{e^{2}}-\frac {\ln \left (e x +d \right ) B a d}{e^{2}}+\frac {\ln \left (e x +d \right ) B b \,d^{2}}{e^{3}}\) | \(90\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 70, normalized size = 1.17 \begin {gather*} {\left (B b d^{2} + A a e^{2} - {\left (B a e + A b e\right )} d\right )} e^{\left (-3\right )} \log \left (x e + d\right ) + \frac {1}{2} \, {\left (B b x^{2} e - 2 \, {\left (B b d - B a e - A b e\right )} x\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 67, normalized size = 1.12 \begin {gather*} -\frac {1}{2} \, {\left (2 \, B b d x e - {\left (B b x^{2} + 2 \, {\left (B a + A b\right )} x\right )} e^{2} - 2 \, {\left (B b d^{2} + A a e^{2} - {\left (B a + A b\right )} d e\right )} \log \left (x e + d\right )\right )} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.14, size = 53, normalized size = 0.88 \begin {gather*} \frac {B b x^{2}}{2 e} + x \left (\frac {A b}{e} + \frac {B a}{e} - \frac {B b d}{e^{2}}\right ) - \frac {\left (- A e + B d\right ) \left (a e - b d\right ) \log {\left (d + e x \right )}}{e^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.72, size = 71, normalized size = 1.18 \begin {gather*} {\left (B b d^{2} - B a d e - A b d e + A a e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{2} \, {\left (B b x^{2} e - 2 \, B b d x + 2 \, B a x e + 2 \, A b x e\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 68, normalized size = 1.13 \begin {gather*} x\,\left (\frac {A\,b+B\,a}{e}-\frac {B\,b\,d}{e^2}\right )+\frac {\ln \left (d+e\,x\right )\,\left (A\,a\,e^2+B\,b\,d^2-A\,b\,d\,e-B\,a\,d\,e\right )}{e^3}+\frac {B\,b\,x^2}{2\,e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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