Optimal. Leaf size=165 \[ \frac{e^2 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{e^2 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac{e}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{1}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]
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Rubi [A] time = 0.0969946, antiderivative size = 165, normalized size of antiderivative = 1., 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} \[ \frac{e^2 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{e^2 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac{e}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac{1}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 646
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{(d+e x) \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{1}{\left (a b+b^2 x\right )^3 (d+e x)} \, 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{1}{b^2 (b d-a e) (a+b x)^3}-\frac{e}{b^2 (b d-a e)^2 (a+b x)^2}+\frac{e^2}{b^2 (b d-a e)^3 (a+b x)}-\frac{e^3}{b^3 (b d-a e)^3 (d+e x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{e}{(b d-a e)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{1}{2 (b d-a e) (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 d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e^2 (a+b x) \log (d+e x)}{(b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0550638, size = 92, normalized size = 0.56 \[ \frac{-2 e^2 (a+b x)^2 \log (d+e x)-(b d-a e) (b (d-2 e x)-3 a e)+2 e^2 (a+b x)^2 \log (a+b x)}{2 (a+b x) \sqrt{(a+b x)^2} (b d-a e)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.203, size = 155, normalized size = 0.9 \begin{align*}{\frac{ \left ( 2\,\ln \left ( ex+d \right ){x}^{2}{b}^{2}{e}^{2}-2\,\ln \left ( bx+a \right ){x}^{2}{b}^{2}{e}^{2}+4\,\ln \left ( ex+d \right ) xab{e}^{2}-4\,\ln \left ( bx+a \right ) xab{e}^{2}+2\,\ln \left ( ex+d \right ){a}^{2}{e}^{2}-2\,\ln \left ( bx+a \right ){a}^{2}{e}^{2}+2\,xab{e}^{2}-2\,x{b}^{2}de+3\,{a}^{2}{e}^{2}-4\,abde+{b}^{2}{d}^{2} \right ) \left ( bx+a \right ) }{2\, \left ( ae-bd \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.49254, size = 491, normalized size = 2.98 \begin{align*} -\frac{b^{2} d^{2} - 4 \, a b d e + 3 \, a^{2} e^{2} - 2 \,{\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^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \log \left (e x + d\right )}{2 \,{\left (a^{2} b^{3} d^{3} - 3 \, a^{3} b^{2} d^{2} e + 3 \, a^{4} b d e^{2} - a^{5} e^{3} +{\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} x^{2} + 2 \,{\left (a b^{4} d^{3} - 3 \, a^{2} b^{3} d^{2} e + 3 \, a^{3} b^{2} d e^{2} - a^{4} b e^{3}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d + e x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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