Optimal. Leaf size=124 \[ \frac {e x (a+b x) (b d-a e)}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (d+e x)^2}{2 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(a+b x) (b d-a e)^2 \log (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.05, 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 = {646, 43} \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 43
Rule 646
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 = 0.03, size = 59, normalized size = 0.48 \begin {gather*} \frac {(a+b x) \left (b e x (-2 a e+4 b d+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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IntegrateAlgebraic [B] time = 0.60, size = 309, normalized size = 2.49 \begin {gather*} \frac {\left (-a^2 \sqrt {b^2} e^2-a^2 b e^2+2 a b^2 d e+2 a \sqrt {b^2} b d e+b^3 \left (-d^2\right )-\left (b^2\right )^{3/2} d^2\right ) \log \left (\sqrt {a^2+2 a b x+b^2 x^2}-a-\sqrt {b^2} x\right )}{2 b^3 \sqrt {b^2}}+\frac {\left (a^2 \sqrt {b^2} e^2-a^2 b e^2+2 a b^2 d e-2 a \sqrt {b^2} b d e+b^3 \left (-d^2\right )+\left (b^2\right )^{3/2} d^2\right ) \log \left (\sqrt {a^2+2 a b x+b^2 x^2}+a-\sqrt {b^2} x\right )}{2 b^3 \sqrt {b^2}}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-3 a e^2+4 b d e+b e^2 x\right )}{4 b^3}+\frac {2 a e^2 x-4 b d e x-b e^2 x^2}{4 b \sqrt {b^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 63, normalized size = 0.51 \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 = 0.17, 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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maple [A] time = 0.05, size = 87, normalized size = 0.70 \begin {gather*} \frac {\left (b x +a \right ) \left (b^{2} e^{2} x^{2}+2 a^{2} e^{2} \ln \left (b x +a \right )-4 a b d e \ln \left (b x +a \right )-2 a b \,e^{2} x +2 b^{2} d^{2} \ln \left (b x +a \right )+4 b^{2} d e x \right )}{2 \sqrt {\left (b x +a \right )^{2}}\, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.07, size = 96, normalized size = 0.77 \begin {gather*} \frac {e^{2} x^{2}}{2 \, b} - \frac {a e^{2} x}{b^{2}} + \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^{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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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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sympy [A] time = 0.25, 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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