Optimal. Leaf size=217 \[ \frac{e^2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^3}+\frac{3 b e^2 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{3 b e^2 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac{2 b e}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{b}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2} \]
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Rubi [A] time = 0.126441, antiderivative size = 217, 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} \[ \frac{e^2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^3}+\frac{3 b e^2 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{3 b e^2 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac{2 b e}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{b}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2} \]
Antiderivative was successfully verified.
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Rule 646
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{(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{1}{\left (a b+b^2 x\right )^3 (d+e x)^2} \, 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 (b d-a e)^2 (a+b x)^3}-\frac{2 e}{b (b d-a e)^3 (a+b x)^2}+\frac{3 e^2}{b (b d-a e)^4 (a+b x)}-\frac{e^3}{b^3 (b d-a e)^3 (d+e x)^2}-\frac{3 e^3}{b^2 (b d-a e)^4 (d+e x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{2 b e}{(b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b}{2 (b d-a e)^2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^2 (a+b x)}{(b d-a e)^3 (d+e x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 b e^2 (a+b x) \log (a+b x)}{(b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 b e^2 (a+b x) \log (d+e x)}{(b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0992049, size = 141, normalized size = 0.65 \[ \frac{-(b d-a e) \left (-2 a^2 e^2-a b e (5 d+9 e x)+b^2 \left (d^2-3 d e x-6 e^2 x^2\right )\right )+6 b e^2 (a+b x)^2 (d+e x) \log (a+b x)-6 b e^2 (a+b x)^2 (d+e x) \log (d+e x)}{2 (a+b x) \sqrt{(a+b x)^2} (d+e x) (b d-a e)^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.204, size = 329, normalized size = 1.5 \begin{align*} -{\frac{ \left ( 6\,\ln \left ( ex+d \right ){x}^{3}{b}^{3}{e}^{3}-6\,\ln \left ( bx+a \right ){x}^{3}{b}^{3}{e}^{3}+12\,\ln \left ( ex+d \right ){x}^{2}a{b}^{2}{e}^{3}+6\,\ln \left ( ex+d \right ){x}^{2}{b}^{3}d{e}^{2}-12\,\ln \left ( bx+a \right ){x}^{2}a{b}^{2}{e}^{3}-6\,\ln \left ( bx+a \right ){x}^{2}{b}^{3}d{e}^{2}+6\,\ln \left ( ex+d \right ) x{a}^{2}b{e}^{3}+12\,\ln \left ( ex+d \right ) xa{b}^{2}d{e}^{2}-6\,\ln \left ( bx+a \right ) x{a}^{2}b{e}^{3}-12\,\ln \left ( bx+a \right ) xa{b}^{2}d{e}^{2}+6\,{x}^{2}a{b}^{2}{e}^{3}-6\,{x}^{2}{b}^{3}d{e}^{2}+6\,\ln \left ( ex+d \right ){a}^{2}bd{e}^{2}-6\,\ln \left ( bx+a \right ){a}^{2}bd{e}^{2}+9\,x{a}^{2}b{e}^{3}-6\,xa{b}^{2}d{e}^{2}-3\,x{b}^{3}{d}^{2}e+2\,{a}^{3}{e}^{3}+3\,d{e}^{2}{a}^{2}b-6\,a{b}^{2}{d}^{2}e+{b}^{3}{d}^{3} \right ) \left ( bx+a \right ) }{ \left ( 2\,ex+2\,d \right ) \left ( ae-bd \right ) ^{4}} \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 = 1.70269, size = 991, normalized size = 4.57 \begin{align*} -\frac{b^{3} d^{3} - 6 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} + 2 \, a^{3} e^{3} - 6 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} - 3 \,{\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2} - 3 \, a^{2} b e^{3}\right )} x - 6 \,{\left (b^{3} e^{3} x^{3} + a^{2} b d e^{2} +{\left (b^{3} d e^{2} + 2 \, a b^{2} e^{3}\right )} x^{2} +{\left (2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \log \left (b x + a\right ) + 6 \,{\left (b^{3} e^{3} x^{3} + a^{2} b d e^{2} +{\left (b^{3} d e^{2} + 2 \, a b^{2} e^{3}\right )} x^{2} +{\left (2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (a^{2} b^{4} d^{5} - 4 \, a^{3} b^{3} d^{4} e + 6 \, a^{4} b^{2} d^{3} e^{2} - 4 \, a^{5} b d^{2} e^{3} + a^{6} d e^{4} +{\left (b^{6} d^{4} e - 4 \, a b^{5} d^{3} e^{2} + 6 \, a^{2} b^{4} d^{2} e^{3} - 4 \, a^{3} b^{3} d e^{4} + a^{4} b^{2} e^{5}\right )} x^{3} +{\left (b^{6} d^{5} - 2 \, a b^{5} d^{4} e - 2 \, a^{2} b^{4} d^{3} e^{2} + 8 \, a^{3} b^{3} d^{2} e^{3} - 7 \, a^{4} b^{2} d e^{4} + 2 \, a^{5} b e^{5}\right )} x^{2} +{\left (2 \, a b^{5} d^{5} - 7 \, a^{2} b^{4} d^{4} e + 8 \, a^{3} b^{3} d^{3} e^{2} - 2 \, a^{4} b^{2} d^{2} e^{3} - 2 \, a^{5} b d e^{4} + a^{6} e^{5}\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 )^{2} \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*} \int \frac{1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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