Optimal. Leaf size=231 \[ \frac{b^2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^3}+\frac{b (a+b x)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^2}+\frac{a+b x}{3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)}+\frac{b^3 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{b^3 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4} \]
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Rubi [A] time = 0.102261, antiderivative size = 231, 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} \[ \frac{b^2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^3}+\frac{b (a+b x)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^2}+\frac{a+b x}{3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^3 (b d-a e)}+\frac{b^3 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{b^3 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4} \]
Antiderivative was successfully verified.
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Rule 646
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^4 \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)^4} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (a b+b^2 x\right ) \int \left (\frac{b^3}{(b d-a e)^4 (a+b x)}-\frac{e}{b (b d-a e) (d+e x)^4}-\frac{e}{(b d-a e)^2 (d+e x)^3}-\frac{b e}{(b d-a e)^3 (d+e x)^2}-\frac{b^2 e}{(b d-a e)^4 (d+e x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{a+b x}{3 (b d-a e) (d+e x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b (a+b x)}{2 (b d-a e)^2 (d+e x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2 (a+b x)}{(b d-a e)^3 (d+e x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^3 (a+b x) \log (a+b x)}{(b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b^3 (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 = 0.0749858, size = 124, normalized size = 0.54 \[ \frac{(a+b x) \left (6 b^2 (d+e x)^2 (b d-a e)+6 b^3 (d+e x)^3 \log (a+b x)+3 b (d+e x) (b d-a e)^2+2 (b d-a e)^3-6 b^3 (d+e x)^3 \log (d+e x)\right )}{6 \sqrt{(a+b x)^2} (d+e x)^3 (b d-a e)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.165, size = 256, normalized size = 1.1 \begin{align*} -{\frac{ \left ( bx+a \right ) \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}+18\,\ln \left ( ex+d \right ){x}^{2}{b}^{3}d{e}^{2}-18\,\ln \left ( bx+a \right ){x}^{2}{b}^{3}d{e}^{2}+18\,\ln \left ( ex+d \right ) x{b}^{3}{d}^{2}e-18\,\ln \left ( bx+a \right ) x{b}^{3}{d}^{2}e+6\,{x}^{2}a{b}^{2}{e}^{3}-6\,{x}^{2}{b}^{3}d{e}^{2}+6\,\ln \left ( ex+d \right ){b}^{3}{d}^{3}-6\,\ln \left ( bx+a \right ){b}^{3}{d}^{3}-3\,x{a}^{2}b{e}^{3}+18\,xa{b}^{2}d{e}^{2}-15\,x{b}^{3}{d}^{2}e+2\,{a}^{3}{e}^{3}-9\,d{e}^{2}{a}^{2}b+18\,a{b}^{2}{d}^{2}e-11\,{b}^{3}{d}^{3} \right ) }{6\, \left ( ae-bd \right ) ^{4} \left ( ex+d \right ) ^{3}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{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 = 1.62372, size = 855, normalized size = 3.7 \begin{align*} \frac{11 \, b^{3} d^{3} - 18 \, a b^{2} d^{2} e + 9 \, 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 (5 \, b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x + 6 \,{\left (b^{3} e^{3} x^{3} + 3 \, b^{3} d e^{2} x^{2} + 3 \, b^{3} d^{2} e x + b^{3} d^{3}\right )} \log \left (b x + a\right ) - 6 \,{\left (b^{3} e^{3} x^{3} + 3 \, b^{3} d e^{2} x^{2} + 3 \, b^{3} d^{2} e x + b^{3} d^{3}\right )} \log \left (e x + d\right )}{6 \,{\left (b^{4} d^{7} - 4 \, a b^{3} d^{6} e + 6 \, a^{2} b^{2} d^{5} e^{2} - 4 \, a^{3} b d^{4} e^{3} + a^{4} d^{3} e^{4} +{\left (b^{4} d^{4} e^{3} - 4 \, a b^{3} d^{3} e^{4} + 6 \, a^{2} b^{2} d^{2} e^{5} - 4 \, a^{3} b d e^{6} + a^{4} e^{7}\right )} x^{3} + 3 \,{\left (b^{4} d^{5} e^{2} - 4 \, a b^{3} d^{4} e^{3} + 6 \, a^{2} b^{2} d^{3} e^{4} - 4 \, a^{3} b d^{2} e^{5} + a^{4} d e^{6}\right )} x^{2} + 3 \,{\left (b^{4} d^{6} e - 4 \, a b^{3} d^{5} e^{2} + 6 \, a^{2} b^{2} d^{4} e^{3} - 4 \, a^{3} b d^{3} e^{4} + a^{4} d^{2} e^{5}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.15661, size = 570, normalized size = 2.47 \begin{align*} - \frac{b^{3} \log{\left (x + \frac{- \frac{a^{5} b^{3} e^{5}}{\left (a e - b d\right )^{4}} + \frac{5 a^{4} b^{4} d e^{4}}{\left (a e - b d\right )^{4}} - \frac{10 a^{3} b^{5} d^{2} e^{3}}{\left (a e - b d\right )^{4}} + \frac{10 a^{2} b^{6} d^{3} e^{2}}{\left (a e - b d\right )^{4}} - \frac{5 a b^{7} d^{4} e}{\left (a e - b d\right )^{4}} + a b^{3} e + \frac{b^{8} d^{5}}{\left (a e - b d\right )^{4}} + b^{4} d}{2 b^{4} e} \right )}}{\left (a e - b d\right )^{4}} + \frac{b^{3} \log{\left (x + \frac{\frac{a^{5} b^{3} e^{5}}{\left (a e - b d\right )^{4}} - \frac{5 a^{4} b^{4} d e^{4}}{\left (a e - b d\right )^{4}} + \frac{10 a^{3} b^{5} d^{2} e^{3}}{\left (a e - b d\right )^{4}} - \frac{10 a^{2} b^{6} d^{3} e^{2}}{\left (a e - b d\right )^{4}} + \frac{5 a b^{7} d^{4} e}{\left (a e - b d\right )^{4}} + a b^{3} e - \frac{b^{8} d^{5}}{\left (a e - b d\right )^{4}} + b^{4} d}{2 b^{4} e} \right )}}{\left (a e - b d\right )^{4}} - \frac{2 a^{2} e^{2} - 7 a b d e + 11 b^{2} d^{2} + 6 b^{2} e^{2} x^{2} + x \left (- 3 a b e^{2} + 15 b^{2} d e\right )}{6 a^{3} d^{3} e^{3} - 18 a^{2} b d^{4} e^{2} + 18 a b^{2} d^{5} e - 6 b^{3} d^{6} + x^{3} \left (6 a^{3} e^{6} - 18 a^{2} b d e^{5} + 18 a b^{2} d^{2} e^{4} - 6 b^{3} d^{3} e^{3}\right ) + x^{2} \left (18 a^{3} d e^{5} - 54 a^{2} b d^{2} e^{4} + 54 a b^{2} d^{3} e^{3} - 18 b^{3} d^{4} e^{2}\right ) + x \left (18 a^{3} d^{2} e^{4} - 54 a^{2} b d^{3} e^{3} + 54 a b^{2} d^{4} e^{2} - 18 b^{3} d^{5} e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1684, size = 332, normalized size = 1.44 \begin{align*} \frac{1}{6} \,{\left (\frac{6 \, b^{4} \log \left ({\left | b x + a \right |}\right )}{b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}} - \frac{6 \, b^{3} e \log \left ({\left | x e + d \right |}\right )}{b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}} + \frac{11 \, b^{3} d^{3} - 18 \, a b^{2} d^{2} e + 9 \, 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 (5 \, b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{{\left (b d - a e\right )}^{4}{\left (x e + d\right )}^{3}}\right )} \mathrm{sgn}\left (b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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