Optimal. Leaf size=210 \[ \frac{e^3 x (a+b x) (4 b d-3 a e)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{6 e^2 (a+b x) (b d-a e)^2 \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{4 e (b d-a e)^3}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^4}{2 b^5 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^4 x^2 (a+b x)}{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.13473, antiderivative size = 210, 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, 43} \[ \frac{e^3 x (a+b x) (4 b d-3 a e)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{6 e^2 (a+b x) (b d-a e)^2 \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{4 e (b d-a e)^3}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^4}{2 b^5 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^4 x^2 (a+b x)}{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 646
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x)^4}{\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{(d+e x)^4}{\left (a b+b^2 x\right )^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{e^3 (4 b d-3 a e)}{b^7}+\frac{e^4 x}{b^6}+\frac{(b d-a e)^4}{b^7 (a+b x)^3}+\frac{4 e (b d-a e)^3}{b^7 (a+b x)^2}+\frac{6 e^2 (b d-a e)^2}{b^7 (a+b x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{4 e (b d-a e)^3}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^4}{2 b^5 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^3 (4 b d-3 a e) x (a+b x)}{b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^4 x^2 (a+b x)}{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{6 e^2 (b d-a e)^2 (a+b x) \log (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0843384, size = 174, normalized size = 0.83 \[ \frac{a^2 b^2 e^2 \left (18 d^2-16 d e x-11 e^2 x^2\right )+2 a^3 b e^3 (e x-10 d)+7 a^4 e^4-4 a b^3 e \left (-6 d^2 e x+d^3-4 d e^2 x^2+e^3 x^3\right )+12 e^2 (a+b x)^2 (b d-a e)^2 \log (a+b x)+b^4 \left (-8 d^3 e x-d^4+8 d e^3 x^3+e^4 x^4\right )}{2 b^5 (a+b x) \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.204, size = 341, normalized size = 1.6 \begin{align*}{\frac{ \left ({x}^{4}{b}^{4}{e}^{4}+12\,\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{2}{e}^{4}-24\,\ln \left ( bx+a \right ){x}^{2}a{b}^{3}d{e}^{3}+12\,\ln \left ( bx+a \right ){x}^{2}{b}^{4}{d}^{2}{e}^{2}-4\,{x}^{3}a{b}^{3}{e}^{4}+8\,{x}^{3}{b}^{4}d{e}^{3}+24\,\ln \left ( bx+a \right ) x{a}^{3}b{e}^{4}-48\,\ln \left ( bx+a \right ) x{a}^{2}{b}^{2}d{e}^{3}+24\,\ln \left ( bx+a \right ) xa{b}^{3}{d}^{2}{e}^{2}-11\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+16\,{x}^{2}a{b}^{3}d{e}^{3}+12\,\ln \left ( bx+a \right ){a}^{4}{e}^{4}-24\,\ln \left ( bx+a \right ){a}^{3}bd{e}^{3}+12\,\ln \left ( bx+a \right ){a}^{2}{b}^{2}{d}^{2}{e}^{2}+2\,x{a}^{3}b{e}^{4}-16\,x{a}^{2}{b}^{2}d{e}^{3}+24\,xa{b}^{3}{d}^{2}{e}^{2}-8\,x{b}^{4}{d}^{3}e+7\,{a}^{4}{e}^{4}-20\,{a}^{3}bd{e}^{3}+18\,{d}^{2}{e}^{2}{a}^{2}{b}^{2}-4\,a{b}^{3}{d}^{3}e-{b}^{4}{d}^{4} \right ) \left ( bx+a \right ) }{2\,{b}^{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 [B] time = 1.08357, size = 657, normalized size = 3.13 \begin{align*} \frac{e^{4} x^{3}}{2 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} + \frac{4 \, d e^{3} x^{2}}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac{5 \, a e^{4} x^{2}}{2 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{3}} + \frac{6 \, d^{2} e^{2} \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{3}{2}}} - \frac{12 \, a d e^{3} \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{3}{2}} b} + \frac{6 \, a^{2} e^{4} \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{3}{2}} b^{2}} + \frac{9 \, a^{2} b^{2} d^{2} e^{2}}{{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{2}} - \frac{18 \, a^{3} b d e^{3}}{{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{9 \, a^{4} e^{4}}{{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{12 \, a b d^{2} e^{2} x}{{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{2}} - \frac{24 \, a^{2} d e^{3} x}{{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{12 \, a^{3} e^{4} x}{{\left (b^{2}\right )}^{\frac{5}{2}} b{\left (x + \frac{a}{b}\right )}^{2}} - \frac{4 \, d^{3} e}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} + \frac{8 \, a^{2} d e^{3}}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{4}} - \frac{5 \, a^{3} e^{4}}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{5}} - \frac{d^{4}}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{2 \, a d^{3} e}{{\left (b^{2}\right )}^{\frac{3}{2}} b{\left (x + \frac{a}{b}\right )}^{2}} - \frac{4 \, a^{3} d e^{3}}{{\left (b^{2}\right )}^{\frac{3}{2}} b^{3}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{5 \, a^{4} e^{4}}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}} b^{4}{\left (x + \frac{a}{b}\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58009, size = 586, normalized size = 2.79 \begin{align*} \frac{b^{4} e^{4} x^{4} - b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 18 \, a^{2} b^{2} d^{2} e^{2} - 20 \, a^{3} b d e^{3} + 7 \, a^{4} e^{4} + 4 \,{\left (2 \, b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} +{\left (16 \, a b^{3} d e^{3} - 11 \, a^{2} b^{2} e^{4}\right )} x^{2} - 2 \,{\left (4 \, b^{4} d^{3} e - 12 \, a b^{3} d^{2} e^{2} + 8 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x + 12 \,{\left (a^{2} b^{2} d^{2} e^{2} - 2 \, a^{3} b d e^{3} + a^{4} e^{4} +{\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (a b^{3} d^{2} e^{2} - 2 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} \log \left (b x + a\right )}{2 \,{\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\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{\left (d + e x\right )^{4}}{\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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