Optimal. Leaf size=125 \[ -\frac{8 b^3 \sqrt{d+e x} (b d-a e)}{e^5}-\frac{12 b^2 (b d-a e)^2}{e^5 \sqrt{d+e x}}+\frac{8 b (b d-a e)^3}{3 e^5 (d+e x)^{3/2}}-\frac{2 (b d-a e)^4}{5 e^5 (d+e x)^{5/2}}+\frac{2 b^4 (d+e x)^{3/2}}{3 e^5} \]
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Rubi [A] time = 0.0430044, 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} \[ -\frac{8 b^3 \sqrt{d+e x} (b d-a e)}{e^5}-\frac{12 b^2 (b d-a e)^2}{e^5 \sqrt{d+e x}}+\frac{8 b (b d-a e)^3}{3 e^5 (d+e x)^{3/2}}-\frac{2 (b d-a e)^4}{5 e^5 (d+e x)^{5/2}}+\frac{2 b^4 (d+e x)^{3/2}}{3 e^5} \]
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)^{7/2}} \, dx &=\int \frac{(a+b x)^4}{(d+e x)^{7/2}} \, dx\\ &=\int \left (\frac{(-b d+a e)^4}{e^4 (d+e x)^{7/2}}-\frac{4 b (b d-a e)^3}{e^4 (d+e x)^{5/2}}+\frac{6 b^2 (b d-a e)^2}{e^4 (d+e x)^{3/2}}-\frac{4 b^3 (b d-a e)}{e^4 \sqrt{d+e x}}+\frac{b^4 \sqrt{d+e x}}{e^4}\right ) \, dx\\ &=-\frac{2 (b d-a e)^4}{5 e^5 (d+e x)^{5/2}}+\frac{8 b (b d-a e)^3}{3 e^5 (d+e x)^{3/2}}-\frac{12 b^2 (b d-a e)^2}{e^5 \sqrt{d+e x}}-\frac{8 b^3 (b d-a e) \sqrt{d+e x}}{e^5}+\frac{2 b^4 (d+e x)^{3/2}}{3 e^5}\\ \end{align*}
Mathematica [A] time = 0.0715284, size = 101, normalized size = 0.81 \[ \frac{2 \left (-90 b^2 (d+e x)^2 (b d-a e)^2-60 b^3 (d+e x)^3 (b d-a e)+20 b (d+e x) (b d-a e)^3-3 (b d-a e)^4+5 b^4 (d+e x)^4\right )}{15 e^5 (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 186, normalized size = 1.5 \begin{align*} -{\frac{-10\,{x}^{4}{b}^{4}{e}^{4}-120\,{x}^{3}a{b}^{3}{e}^{4}+80\,{x}^{3}{b}^{4}d{e}^{3}+180\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}-720\,{x}^{2}a{b}^{3}d{e}^{3}+480\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+40\,x{a}^{3}b{e}^{4}+240\,x{a}^{2}{b}^{2}d{e}^{3}-960\,xa{b}^{3}{d}^{2}{e}^{2}+640\,x{b}^{4}{d}^{3}e+6\,{a}^{4}{e}^{4}+16\,{a}^{3}bd{e}^{3}+96\,{d}^{2}{e}^{2}{a}^{2}{b}^{2}-384\,a{b}^{3}{d}^{3}e+256\,{b}^{4}{d}^{4}}{15\,{e}^{5}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.21617, size = 255, normalized size = 2.04 \begin{align*} \frac{2 \,{\left (\frac{5 \,{\left ({\left (e x + d\right )}^{\frac{3}{2}} b^{4} - 12 \,{\left (b^{4} d - a b^{3} e\right )} \sqrt{e x + d}\right )}}{e^{4}} - \frac{3 \, b^{4} d^{4} - 12 \, a b^{3} d^{3} e + 18 \, a^{2} b^{2} d^{2} e^{2} - 12 \, a^{3} b d e^{3} + 3 \, a^{4} e^{4} + 90 \,{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )}{\left (e x + d\right )}^{2} - 20 \,{\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 )}}{{\left (e x + d\right )}^{\frac{5}{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 = 1.49603, size = 447, normalized size = 3.58 \begin{align*} \frac{2 \,{\left (5 \, b^{4} e^{4} x^{4} - 128 \, b^{4} d^{4} + 192 \, a b^{3} d^{3} e - 48 \, a^{2} b^{2} d^{2} e^{2} - 8 \, a^{3} b d e^{3} - 3 \, a^{4} e^{4} - 20 \,{\left (2 \, b^{4} d e^{3} - 3 \, a b^{3} e^{4}\right )} x^{3} - 30 \,{\left (8 \, b^{4} d^{2} e^{2} - 12 \, a b^{3} d e^{3} + 3 \, a^{2} b^{2} e^{4}\right )} x^{2} - 20 \,{\left (16 \, b^{4} d^{3} e - 24 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.30085, size = 1008, normalized size = 8.06 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.24122, size = 305, normalized size = 2.44 \begin{align*} \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} b^{4} e^{10} - 12 \, \sqrt{x e + d} b^{4} d e^{10} + 12 \, \sqrt{x e + d} a b^{3} e^{11}\right )} e^{\left (-15\right )} - \frac{2 \,{\left (90 \,{\left (x e + d\right )}^{2} b^{4} d^{2} - 20 \,{\left (x e + d\right )} b^{4} d^{3} + 3 \, b^{4} d^{4} - 180 \,{\left (x e + d\right )}^{2} a b^{3} d e + 60 \,{\left (x e + d\right )} a b^{3} d^{2} e - 12 \, a b^{3} d^{3} e + 90 \,{\left (x e + d\right )}^{2} a^{2} b^{2} e^{2} - 60 \,{\left (x e + d\right )} a^{2} b^{2} d e^{2} + 18 \, a^{2} b^{2} d^{2} e^{2} + 20 \,{\left (x e + d\right )} a^{3} b e^{3} - 12 \, a^{3} b d e^{3} + 3 \, a^{4} e^{4}\right )} e^{\left (-5\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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