Optimal. Leaf size=131 \[ \frac {a+b x}{\sqrt {a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)}+\frac {b (a+b x) \log (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac {b (a+b x) \log (d+e x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2} \]
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Rubi [A] time = 0.06, antiderivative size = 131, 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, 44} \begin {gather*} \frac {a+b x}{\sqrt {a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)}+\frac {b (a+b x) \log (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac {b (a+b x) \log (d+e x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 646
Rubi steps
\begin {align*} \int \frac {1}{(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 {1}{\left (a b+b^2 x\right ) (d+e x)^2} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int \left (\frac {b}{(b d-a e)^2 (a+b x)}-\frac {e}{b (b d-a e) (d+e x)^2}-\frac {e}{(b d-a e)^2 (d+e x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {a+b x}{(b d-a e) (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b (a+b x) \log (a+b x)}{(b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b (a+b x) \log (d+e x)}{(b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 69, normalized size = 0.53 \begin {gather*} \frac {(a+b x) (b (d+e x) \log (a+b x)-a e-b (d+e x) \log (d+e x)+b d)}{\sqrt {(a+b x)^2} (d+e x) (b d-a e)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.99, size = 328, normalized size = 2.50 \begin {gather*} -\frac {e \sqrt {a^2+2 a b x+b^2 x^2}}{2 (d+e x) (b d-a e)^2}+\frac {\left (-\sqrt {b^2}-b\right ) \log \left (\sqrt {a^2+2 a b x+b^2 x^2}-a-\sqrt {b^2} x\right )}{2 (b d-a e)^2}+\frac {\left (b-\sqrt {b^2}\right ) \log \left (\sqrt {a^2+2 a b x+b^2 x^2}+a-\sqrt {b^2} x\right )}{2 (b d-a e)^2}+\frac {\left (\sqrt {b^2}+b\right ) \log \left (-e \sqrt {a^2+2 a b x+b^2 x^2}-a e+\sqrt {b^2} e x+2 b d\right )}{2 (b d-a e)^2}+\frac {\left (\sqrt {b^2}-b\right ) \log \left (e \sqrt {a^2+2 a b x+b^2 x^2}-a e-\sqrt {b^2} e x+2 b d\right )}{2 (b d-a e)^2}-\frac {\sqrt {b^2}}{2 b (d+e x) (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 92, normalized size = 0.70 \begin {gather*} \frac {b d - a e + {\left (b e x + b d\right )} \log \left (b x + a\right ) - {\left (b e x + b d\right )} \log \left (e x + d\right )}{b^{2} d^{3} - 2 \, a b d^{2} e + a^{2} d e^{2} + {\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 103, normalized size = 0.79 \begin {gather*} {\left (\frac {b^{2} \log \left ({\left | b x + a \right |}\right )}{b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}} - \frac {b e \log \left ({\left | x e + d \right |}\right )}{b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}} + \frac {1}{{\left (b d - a e\right )} {\left (x e + d\right )}}\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 = 0.06, size = 81, normalized size = 0.62 \begin {gather*} \frac {\left (b x +a \right ) \left (b e x \ln \left (b x +a \right )-b e x \ln \left (e x +d \right )+b d \ln \left (b x +a \right )-b d \ln \left (e x +d \right )-a e +b d \right )}{\sqrt {\left (b x +a \right )^{2}}\, \left (a e -b d \right )^{2} \left (e x +d \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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 {1}{\sqrt {{\left (a+b\,x\right )}^2}\,{\left (d+e\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.71, size = 233, normalized size = 1.78 \begin {gather*} - \frac {b \log {\left (x + \frac {- \frac {a^{3} b e^{3}}{\left (a e - b d\right )^{2}} + \frac {3 a^{2} b^{2} d e^{2}}{\left (a e - b d\right )^{2}} - \frac {3 a b^{3} d^{2} e}{\left (a e - b d\right )^{2}} + a b e + \frac {b^{4} d^{3}}{\left (a e - b d\right )^{2}} + b^{2} d}{2 b^{2} e} \right )}}{\left (a e - b d\right )^{2}} + \frac {b \log {\left (x + \frac {\frac {a^{3} b e^{3}}{\left (a e - b d\right )^{2}} - \frac {3 a^{2} b^{2} d e^{2}}{\left (a e - b d\right )^{2}} + \frac {3 a b^{3} d^{2} e}{\left (a e - b d\right )^{2}} + a b e - \frac {b^{4} d^{3}}{\left (a e - b d\right )^{2}} + b^{2} d}{2 b^{2} e} \right )}}{\left (a e - b d\right )^{2}} - \frac {1}{a d e - b d^{2} + x \left (a e^{2} - b d e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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