Optimal. Leaf size=117 \[ -\frac {2 e (b d-a e)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^2}{2 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^2 (a+b x) \log (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.07, antiderivative size = 117, 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} \[ -\frac {2 e (b d-a e)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^2}{2 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^2 (a+b x) \log (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 646
Rubi steps
\begin {align*} \int \frac {(d+e x)^2}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^2}{\left (a b+b^2 x\right )^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \left (\frac {(b d-a e)^2}{b^5 (a+b x)^3}+\frac {2 e (b d-a e)}{b^5 (a+b x)^2}+\frac {e^2}{b^5 (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {2 e (b d-a e)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^2}{2 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {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 = 67, normalized size = 0.57 \[ \frac {2 e^2 (a+b x)^2 \log (a+b x)-(b d-a e) (3 a e+b (d+4 e x))}{2 b^3 (a+b x) \sqrt {(a+b x)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 99, normalized size = 0.85 \[ -\frac {b^{2} d^{2} + 2 \, a b d e - 3 \, a^{2} e^{2} + 4 \, {\left (b^{2} d e - a b e^{2}\right )} x - 2 \, {\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \log \left (b x + a\right )}{2 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 104, normalized size = 0.89 \[ \frac {\left (2 b^{2} e^{2} x^{2} \ln \left (b x +a \right )+4 a b \,e^{2} x \ln \left (b x +a \right )+2 a^{2} e^{2} \ln \left (b x +a \right )+4 a b \,e^{2} x -4 b^{2} d e x +3 a^{2} e^{2}-2 a b d e -b^{2} d^{2}\right ) \left (b x +a \right )}{2 \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 113, normalized size = 0.97 \[ \frac {e^{2} \log \left (x + \frac {a}{b}\right )}{b^{3}} - \frac {2 \, d e}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} + \frac {2 \, a e^{2} x}{b^{4} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {d^{2}}{2 \, b^{3} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {a d e}{b^{4} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {3 \, a^{2} e^{2}}{2 \, b^{5} {\left (x + \frac {a}{b}\right )}^{2}} \]
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 \[ \int \frac {{\left (d+e\,x\right )}^2}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d + e x\right )^{2}}{\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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