Optimal. Leaf size=62 \[ \frac {d x (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e x^2 (a+b x)}{2 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 62, 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 = {770, 21} \begin {gather*} \frac {d x (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e x^2 (a+b x)}{2 \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 770
Rubi steps
\begin {align*} \int \frac {(a+b x) (d+e x)}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {(a+b x) (d+e x)}{a b+b^2 x} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int (d+e x) \, dx}{b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {d x (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e x^2 (a+b x)}{2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.45 \begin {gather*} \frac {x (a+b x) (2 d+e x)}{2 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.31, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x) (d+e x)}{\sqrt {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 = 0.42, size = 10, normalized size = 0.16 \begin {gather*} \frac {1}{2} \, e x^{2} + d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 19, normalized size = 0.31 \begin {gather*} \frac {1}{2} \, {\left (x^{2} e + 2 \, d x\right )} \mathrm {sgn}\left (b x + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 25, normalized size = 0.40 \begin {gather*} \frac {\left (e x +2 d \right ) \left (b x +a \right ) x}{2 \sqrt {\left (b x +a \right )^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.76, size = 95, normalized size = 1.53 \begin {gather*} \frac {1}{2} \, e x^{2} - \frac {a e x}{b} + \frac {a d \log \left (x + \frac {a}{b}\right )}{b} + \frac {a^{2} e \log \left (x + \frac {a}{b}\right )}{b^{2}} - \frac {{\left (b d + a e\right )} a \log \left (x + \frac {a}{b}\right )}{b^{2}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} {\left (b d + a e\right )}}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\left (a+b\,x\right )\,\left (d+e\,x\right )}{\sqrt {{\left (a+b\,x\right )}^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 8, normalized size = 0.13 \begin {gather*} d x + \frac {e x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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