Optimal. Leaf size=276 \[ \frac{3 b e^2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^4}+\frac{e^2 (a+b x)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^3}+\frac{6 b^2 e^2 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{6 b^2 e^2 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{b^2}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac{3 b^2 e}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4} \]
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Rubi [A] time = 0.151705, antiderivative size = 276, 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} \[ \frac{3 b e^2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^4}+\frac{e^2 (a+b x)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^3}+\frac{6 b^2 e^2 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{6 b^2 e^2 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{b^2}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac{3 b^2 e}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4} \]
Antiderivative was successfully verified.
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Rule 646
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^3 \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)^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{1}{(b d-a e)^3 (a+b x)^3}-\frac{3 e}{(b d-a e)^4 (a+b x)^2}+\frac{6 e^2}{(b d-a e)^5 (a+b x)}-\frac{e^3}{b^3 (b d-a e)^3 (d+e x)^3}-\frac{3 e^3}{b^2 (b d-a e)^4 (d+e x)^2}-\frac{6 e^3}{b (b d-a e)^5 (d+e x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{3 b^2 e}{(b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b^2}{2 (b d-a e)^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^2 (a+b x)}{2 (b d-a e)^3 (d+e x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 b e^2 (a+b x)}{(b d-a e)^4 (d+e x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{6 b^2 e^2 (a+b x) \log (a+b x)}{(b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{6 b^2 e^2 (a+b x) \log (d+e x)}{(b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0980738, size = 163, normalized size = 0.59 \[ \frac{(a+b x) \left (-12 b^2 e^2 (a+b x)^2 \log (d+e x)+6 b^2 e (a+b x) (b d-a e)+b^2 \left (-(b d-a e)^2\right )+12 b^2 e^2 (a+b x)^2 \log (a+b x)+\frac{6 b e^2 (a+b x)^2 (b d-a e)}{d+e x}+\frac{e^2 (a+b x)^2 (b d-a e)^2}{(d+e x)^2}\right )}{2 \left ((a+b x)^2\right )^{3/2} (b d-a e)^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.212, size = 508, normalized size = 1.8 \begin{align*}{\frac{ \left ( -24\,\ln \left ( bx+a \right ) xa{b}^{3}{d}^{2}{e}^{2}+24\,\ln \left ( ex+d \right ){x}^{3}a{b}^{3}{e}^{4}-8\,a{b}^{3}{d}^{3}e-48\,\ln \left ( bx+a \right ){x}^{2}a{b}^{3}d{e}^{3}-{a}^{4}{e}^{4}+{b}^{4}{d}^{4}-24\,\ln \left ( bx+a \right ){x}^{3}a{b}^{3}{e}^{4}-24\,\ln \left ( bx+a \right ){x}^{3}{b}^{4}d{e}^{3}+12\,\ln \left ( ex+d \right ){x}^{2}{a}^{2}{b}^{2}{e}^{4}+12\,\ln \left ( ex+d \right ){x}^{2}{b}^{4}{d}^{2}{e}^{2}+12\,\ln \left ( ex+d \right ){a}^{2}{b}^{2}{d}^{2}{e}^{2}+12\,{x}^{3}a{b}^{3}{e}^{4}+48\,\ln \left ( ex+d \right ){x}^{2}a{b}^{3}d{e}^{3}+24\,\ln \left ( ex+d \right ) x{a}^{2}{b}^{2}d{e}^{3}+24\,\ln \left ( ex+d \right ) xa{b}^{3}{d}^{2}{e}^{2}+12\,\ln \left ( ex+d \right ){x}^{4}{b}^{4}{e}^{4}-12\,\ln \left ( bx+a \right ){x}^{4}{b}^{4}{e}^{4}-12\,{x}^{3}{b}^{4}d{e}^{3}+18\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}-18\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+4\,x{a}^{3}b{e}^{4}-4\,x{b}^{4}{d}^{3}e-12\,\ln \left ( bx+a \right ){a}^{2}{b}^{2}{d}^{2}{e}^{2}+24\,x{a}^{2}{b}^{2}d{e}^{3}-24\,xa{b}^{3}{d}^{2}{e}^{2}-24\,\ln \left ( bx+a \right ) x{a}^{2}{b}^{2}d{e}^{3}-12\,\ln \left ( bx+a \right ){x}^{2}{b}^{4}{d}^{2}{e}^{2}-12\,\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{2}{e}^{4}+24\,\ln \left ( ex+d \right ){x}^{3}{b}^{4}d{e}^{3}+8\,{a}^{3}bd{e}^{3} \right ) \left ( bx+a \right ) }{2\, \left ( ex+d \right ) ^{2} \left ( ae-bd \right ) ^{5}} \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 = 1.75502, size = 1488, normalized size = 5.39 \begin{align*} -\frac{b^{4} d^{4} - 8 \, a b^{3} d^{3} e + 8 \, a^{3} b d e^{3} - a^{4} e^{4} - 12 \,{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} - 18 \,{\left (b^{4} d^{2} e^{2} - a^{2} b^{2} e^{4}\right )} x^{2} - 4 \,{\left (b^{4} d^{3} e + 6 \, a b^{3} d^{2} e^{2} - 6 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x - 12 \,{\left (b^{4} e^{4} x^{4} + a^{2} b^{2} d^{2} e^{2} + 2 \,{\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} +{\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3}\right )} x\right )} \log \left (b x + a\right ) + 12 \,{\left (b^{4} e^{4} x^{4} + a^{2} b^{2} d^{2} e^{2} + 2 \,{\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} +{\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (a^{2} b^{5} d^{7} - 5 \, a^{3} b^{4} d^{6} e + 10 \, a^{4} b^{3} d^{5} e^{2} - 10 \, a^{5} b^{2} d^{4} e^{3} + 5 \, a^{6} b d^{3} e^{4} - a^{7} d^{2} e^{5} +{\left (b^{7} d^{5} e^{2} - 5 \, a b^{6} d^{4} e^{3} + 10 \, a^{2} b^{5} d^{3} e^{4} - 10 \, a^{3} b^{4} d^{2} e^{5} + 5 \, a^{4} b^{3} d e^{6} - a^{5} b^{2} e^{7}\right )} x^{4} + 2 \,{\left (b^{7} d^{6} e - 4 \, a b^{6} d^{5} e^{2} + 5 \, a^{2} b^{5} d^{4} e^{3} - 5 \, a^{4} b^{3} d^{2} e^{5} + 4 \, a^{5} b^{2} d e^{6} - a^{6} b e^{7}\right )} x^{3} +{\left (b^{7} d^{7} - a b^{6} d^{6} e - 9 \, a^{2} b^{5} d^{5} e^{2} + 25 \, a^{3} b^{4} d^{4} e^{3} - 25 \, a^{4} b^{3} d^{3} e^{4} + 9 \, a^{5} b^{2} d^{2} e^{5} + a^{6} b d e^{6} - a^{7} e^{7}\right )} x^{2} + 2 \,{\left (a b^{6} d^{7} - 4 \, a^{2} b^{5} d^{6} e + 5 \, a^{3} b^{4} d^{5} e^{2} - 5 \, a^{5} b^{2} d^{3} e^{4} + 4 \, a^{6} b d^{2} e^{5} - a^{7} d e^{6}\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 )^{3} \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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