Optimal. Leaf size=49 \[ \frac {(b d-a e)^2 \log (a+b x)}{b^3}+\frac {e x (b d-a e)}{b^2}+\frac {(d+e x)^2}{2 b} \]
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Rubi [A] time = 0.02, antiderivative size = 49, 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, 43} \begin {gather*} \frac {e x (b d-a e)}{b^2}+\frac {(b d-a e)^2 \log (a+b x)}{b^3}+\frac {(d+e x)^2}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin {align*} \int \frac {(a+b x) (d+e x)^2}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac {(d+e x)^2}{a+b x} \, dx\\ &=\int \left (\frac {e (b d-a e)}{b^2}+\frac {(b d-a e)^2}{b^2 (a+b x)}+\frac {e (d+e x)}{b}\right ) \, dx\\ &=\frac {e (b d-a e) x}{b^2}+\frac {(d+e x)^2}{2 b}+\frac {(b d-a e)^2 \log (a+b x)}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 43, normalized size = 0.88 \begin {gather*} \frac {b e x (-2 a e+4 b d+b e x)+2 (b d-a e)^2 \log (a+b x)}{2 b^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x) (d+e x)^2}{a^2+2 a b x+b^2 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 63, normalized size = 1.29 \begin {gather*} \frac {b^{2} e^{2} x^{2} + 2 \, {\left (2 \, b^{2} d e - a b e^{2}\right )} x + 2 \, {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \log \left (b x + a\right )}{2 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 59, normalized size = 1.20 \begin {gather*} \frac {b x^{2} e^{2} + 4 \, b d x e - 2 \, a x e^{2}}{2 \, b^{2}} + \frac {{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 74, normalized size = 1.51 \begin {gather*} \frac {e^{2} x^{2}}{2 b}+\frac {a^{2} e^{2} \ln \left (b x +a \right )}{b^{3}}-\frac {2 a d e \ln \left (b x +a \right )}{b^{2}}-\frac {a \,e^{2} x}{b^{2}}+\frac {d^{2} \ln \left (b x +a \right )}{b}+\frac {2 d e x}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 61, normalized size = 1.24 \begin {gather*} \frac {b e^{2} x^{2} + 2 \, {\left (2 \, b d e - a e^{2}\right )} x}{2 \, b^{2}} + \frac {{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\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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mupad [B] time = 0.07, size = 62, normalized size = 1.27 \begin {gather*} \frac {\ln \left (a+b\,x\right )\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}{b^3}-x\,\left (\frac {a\,e^2}{b^2}-\frac {2\,d\,e}{b}\right )+\frac {e^2\,x^2}{2\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 44, normalized size = 0.90 \begin {gather*} x \left (- \frac {a e^{2}}{b^{2}} + \frac {2 d e}{b}\right ) + \frac {e^{2} x^{2}}{2 b} + \frac {\left (a e - b d\right )^{2} \log {\left (a + b x \right )}}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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