Optimal. Leaf size=124 \[ \frac {e (b d-a e) x (a+b x)}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (d+e x)^2}{2 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(b d-a e)^2 (a+b x) \log (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {660, 45}
\begin {gather*} \frac {e x (a+b x) (b d-a e)}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (d+e x)^2}{2 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (b d-a e)^2 \log (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 660
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {(d+e x)^2}{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 \left (\frac {e (b d-a e)}{b^3}+\frac {(b d-a e)^2}{b^2 \left (a b+b^2 x\right )}+\frac {e (d+e x)}{b^2}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {e (b d-a e) x (a+b x)}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (d+e x)^2}{2 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(b d-a e)^2 (a+b x) \log (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 59, normalized size = 0.48 \begin {gather*} \frac {(a+b x) \left (b e x (4 b d-2 a e+b e x)+2 (b d-a e)^2 \log (a+b x)\right )}{2 b^3 \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.52, size = 87, normalized size = 0.70
method | result | size |
default | \(\frac {\left (b x +a \right ) \left (b^{2} x^{2} e^{2}+2 \ln \left (b x +a \right ) a^{2} e^{2}-4 \ln \left (b x +a \right ) a b d e +2 \ln \left (b x +a \right ) b^{2} d^{2}-2 a b \,e^{2} x +4 b^{2} d e x \right )}{2 \sqrt {\left (b x +a \right )^{2}}\, b^{3}}\) | \(87\) |
risch | \(-\frac {\sqrt {\left (b x +a \right )^{2}}\, e \left (-\frac {1}{2} b e \,x^{2}+a e x -2 x b d \right )}{\left (b x +a \right ) b^{2}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \ln \left (b x +a \right )}{\left (b x +a \right ) b^{3}}\) | \(88\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 95, normalized size = 0.77 \begin {gather*} \frac {x^{2} e^{2}}{2 \, b} + \frac {d^{2} \log \left (x + \frac {a}{b}\right )}{b} - \frac {2 \, a d e \log \left (x + \frac {a}{b}\right )}{b^{2}} - \frac {a x e^{2}}{b^{2}} + \frac {a^{2} e^{2} \log \left (x + \frac {a}{b}\right )}{b^{3}} + \frac {2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} d e}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.25, size = 60, normalized size = 0.48 \begin {gather*} \frac {4 \, b^{2} d x e + {\left (b^{2} x^{2} - 2 \, a b x\right )} e^{2} + 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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Sympy [A]
time = 0.11, size = 44, normalized size = 0.35 \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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Giac [A]
time = 1.78, size = 95, normalized size = 0.77 \begin {gather*} \frac {b x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, b d x e \mathrm {sgn}\left (b x + a\right ) - 2 \, a x e^{2} \mathrm {sgn}\left (b x + a\right )}{2 \, b^{2}} + \frac {{\left (b^{2} d^{2} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b d e \mathrm {sgn}\left (b x + a\right ) + a^{2} e^{2} \mathrm {sgn}\left (b x + a\right )\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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^2}{\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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