Optimal. Leaf size=125 \[ -\frac{8 b^3 (d+e x)^{3/2} (b d-a e)}{3 e^5}+\frac{12 b^2 \sqrt{d+e x} (b d-a e)^2}{e^5}+\frac{8 b (b d-a e)^3}{e^5 \sqrt{d+e x}}-\frac{2 (b d-a e)^4}{3 e^5 (d+e x)^{3/2}}+\frac{2 b^4 (d+e x)^{5/2}}{5 e^5} \]
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Rubi [A] time = 0.0415078, antiderivative size = 125, 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 = {27, 43} \[ -\frac{8 b^3 (d+e x)^{3/2} (b d-a e)}{3 e^5}+\frac{12 b^2 \sqrt{d+e x} (b d-a e)^2}{e^5}+\frac{8 b (b d-a e)^3}{e^5 \sqrt{d+e x}}-\frac{2 (b d-a e)^4}{3 e^5 (d+e x)^{3/2}}+\frac{2 b^4 (d+e x)^{5/2}}{5 e^5} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{5/2}} \, dx &=\int \frac{(a+b x)^4}{(d+e x)^{5/2}} \, dx\\ &=\int \left (\frac{(-b d+a e)^4}{e^4 (d+e x)^{5/2}}-\frac{4 b (b d-a e)^3}{e^4 (d+e x)^{3/2}}+\frac{6 b^2 (b d-a e)^2}{e^4 \sqrt{d+e x}}-\frac{4 b^3 (b d-a e) \sqrt{d+e x}}{e^4}+\frac{b^4 (d+e x)^{3/2}}{e^4}\right ) \, dx\\ &=-\frac{2 (b d-a e)^4}{3 e^5 (d+e x)^{3/2}}+\frac{8 b (b d-a e)^3}{e^5 \sqrt{d+e x}}+\frac{12 b^2 (b d-a e)^2 \sqrt{d+e x}}{e^5}-\frac{8 b^3 (b d-a e) (d+e x)^{3/2}}{3 e^5}+\frac{2 b^4 (d+e x)^{5/2}}{5 e^5}\\ \end{align*}
Mathematica [A] time = 0.0716295, size = 101, normalized size = 0.81 \[ \frac{2 \left (90 b^2 (d+e x)^2 (b d-a e)^2-20 b^3 (d+e x)^3 (b d-a e)+60 b (d+e x) (b d-a e)^3-5 (b d-a e)^4+3 b^4 (d+e x)^4\right )}{15 e^5 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 186, normalized size = 1.5 \begin{align*} -{\frac{-6\,{x}^{4}{b}^{4}{e}^{4}-40\,{x}^{3}a{b}^{3}{e}^{4}+16\,{x}^{3}{b}^{4}d{e}^{3}-180\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+240\,{x}^{2}a{b}^{3}d{e}^{3}-96\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+120\,x{a}^{3}b{e}^{4}-720\,x{a}^{2}{b}^{2}d{e}^{3}+960\,xa{b}^{3}{d}^{2}{e}^{2}-384\,x{b}^{4}{d}^{3}e+10\,{a}^{4}{e}^{4}+80\,{a}^{3}bd{e}^{3}-480\,{d}^{2}{e}^{2}{a}^{2}{b}^{2}+640\,a{b}^{3}{d}^{3}e-256\,{b}^{4}{d}^{4}}{15\,{e}^{5}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12195, size = 252, normalized size = 2.02 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (e x + d\right )}^{\frac{5}{2}} b^{4} - 20 \,{\left (b^{4} d - a b^{3} e\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 90 \,{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} \sqrt{e x + d}}{e^{4}} - \frac{5 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4} - 12 \,{\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )}{\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{4}}\right )}}{15 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61883, size = 431, normalized size = 3.45 \begin{align*} \frac{2 \,{\left (3 \, b^{4} e^{4} x^{4} + 128 \, b^{4} d^{4} - 320 \, a b^{3} d^{3} e + 240 \, a^{2} b^{2} d^{2} e^{2} - 40 \, a^{3} b d e^{3} - 5 \, a^{4} e^{4} - 4 \,{\left (2 \, b^{4} d e^{3} - 5 \, a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (8 \, b^{4} d^{2} e^{2} - 20 \, a b^{3} d e^{3} + 15 \, a^{2} b^{2} e^{4}\right )} x^{2} + 12 \,{\left (16 \, b^{4} d^{3} e - 40 \, a b^{3} d^{2} e^{2} + 30 \, a^{2} b^{2} d e^{3} - 5 \, a^{3} b e^{4}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 40.3786, size = 136, normalized size = 1.09 \begin{align*} \frac{2 b^{4} \left (d + e x\right )^{\frac{5}{2}}}{5 e^{5}} - \frac{8 b \left (a e - b d\right )^{3}}{e^{5} \sqrt{d + e x}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (8 a b^{3} e - 8 b^{4} d\right )}{3 e^{5}} + \frac{\sqrt{d + e x} \left (12 a^{2} b^{2} e^{2} - 24 a b^{3} d e + 12 b^{4} d^{2}\right )}{e^{5}} - \frac{2 \left (a e - b d\right )^{4}}{3 e^{5} \left (d + e x\right )^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15454, size = 309, normalized size = 2.47 \begin{align*} \frac{2}{15} \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} e^{20} - 20 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d e^{20} + 90 \, \sqrt{x e + d} b^{4} d^{2} e^{20} + 20 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} e^{21} - 180 \, \sqrt{x e + d} a b^{3} d e^{21} + 90 \, \sqrt{x e + d} a^{2} b^{2} e^{22}\right )} e^{\left (-25\right )} + \frac{2 \,{\left (12 \,{\left (x e + d\right )} b^{4} d^{3} - b^{4} d^{4} - 36 \,{\left (x e + d\right )} a b^{3} d^{2} e + 4 \, a b^{3} d^{3} e + 36 \,{\left (x e + d\right )} a^{2} b^{2} d e^{2} - 6 \, a^{2} b^{2} d^{2} e^{2} - 12 \,{\left (x e + d\right )} a^{3} b e^{3} + 4 \, a^{3} b d e^{3} - a^{4} e^{4}\right )} e^{\left (-5\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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