Optimal. Leaf size=179 \[ -\frac{12 b^5 (d+e x)^{5/2} (b d-a e)}{5 e^7}+\frac{10 b^4 (d+e x)^{3/2} (b d-a e)^2}{e^7}-\frac{40 b^3 \sqrt{d+e x} (b d-a e)^3}{e^7}-\frac{30 b^2 (b d-a e)^4}{e^7 \sqrt{d+e x}}+\frac{4 b (b d-a e)^5}{e^7 (d+e x)^{3/2}}-\frac{2 (b d-a e)^6}{5 e^7 (d+e x)^{5/2}}+\frac{2 b^6 (d+e x)^{7/2}}{7 e^7} \]
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Rubi [A] time = 0.0591365, antiderivative size = 179, 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} \[ -\frac{12 b^5 (d+e x)^{5/2} (b d-a e)}{5 e^7}+\frac{10 b^4 (d+e x)^{3/2} (b d-a e)^2}{e^7}-\frac{40 b^3 \sqrt{d+e x} (b d-a e)^3}{e^7}-\frac{30 b^2 (b d-a e)^4}{e^7 \sqrt{d+e x}}+\frac{4 b (b d-a e)^5}{e^7 (d+e x)^{3/2}}-\frac{2 (b d-a e)^6}{5 e^7 (d+e x)^{5/2}}+\frac{2 b^6 (d+e x)^{7/2}}{7 e^7} \]
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 )^3}{(d+e x)^{7/2}} \, dx &=\int \frac{(a+b x)^6}{(d+e x)^{7/2}} \, dx\\ &=\int \left (\frac{(-b d+a e)^6}{e^6 (d+e x)^{7/2}}-\frac{6 b (b d-a e)^5}{e^6 (d+e x)^{5/2}}+\frac{15 b^2 (b d-a e)^4}{e^6 (d+e x)^{3/2}}-\frac{20 b^3 (b d-a e)^3}{e^6 \sqrt{d+e x}}+\frac{15 b^4 (b d-a e)^2 \sqrt{d+e x}}{e^6}-\frac{6 b^5 (b d-a e) (d+e x)^{3/2}}{e^6}+\frac{b^6 (d+e x)^{5/2}}{e^6}\right ) \, dx\\ &=-\frac{2 (b d-a e)^6}{5 e^7 (d+e x)^{5/2}}+\frac{4 b (b d-a e)^5}{e^7 (d+e x)^{3/2}}-\frac{30 b^2 (b d-a e)^4}{e^7 \sqrt{d+e x}}-\frac{40 b^3 (b d-a e)^3 \sqrt{d+e x}}{e^7}+\frac{10 b^4 (b d-a e)^2 (d+e x)^{3/2}}{e^7}-\frac{12 b^5 (b d-a e) (d+e x)^{5/2}}{5 e^7}+\frac{2 b^6 (d+e x)^{7/2}}{7 e^7}\\ \end{align*}
Mathematica [A] time = 0.0814156, size = 145, normalized size = 0.81 \[ \frac{2 \left (-525 b^2 (d+e x)^2 (b d-a e)^4-700 b^3 (d+e x)^3 (b d-a e)^3+175 b^4 (d+e x)^4 (b d-a e)^2-42 b^5 (d+e x)^5 (b d-a e)+70 b (d+e x) (b d-a e)^5-7 (b d-a e)^6+5 b^6 (d+e x)^6\right )}{35 e^7 (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.048, size = 377, normalized size = 2.1 \begin{align*} -{\frac{-10\,{b}^{6}{x}^{6}{e}^{6}-84\,{x}^{5}a{b}^{5}{e}^{6}+24\,{x}^{5}{b}^{6}d{e}^{5}-350\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+280\,{x}^{4}a{b}^{5}d{e}^{5}-80\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}-1400\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+2800\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}-2240\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+640\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+1050\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}-8400\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+16800\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-13440\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+3840\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+140\,x{a}^{5}b{e}^{6}+1400\,x{a}^{4}{b}^{2}d{e}^{5}-11200\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+22400\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}-17920\,xa{b}^{5}{d}^{4}{e}^{2}+5120\,x{b}^{6}{d}^{5}e+14\,{a}^{6}{e}^{6}+56\,{a}^{5}bd{e}^{5}+560\,{d}^{2}{e}^{4}{a}^{4}{b}^{2}-4480\,{b}^{3}{a}^{3}{d}^{3}{e}^{3}+8960\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}-7168\,a{b}^{5}{d}^{5}e+2048\,{d}^{6}{b}^{6}}{35\,{e}^{7}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.17603, size = 481, normalized size = 2.69 \begin{align*} \frac{2 \,{\left (\frac{5 \,{\left (e x + d\right )}^{\frac{7}{2}} b^{6} - 42 \,{\left (b^{6} d - a b^{5} e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 175 \,{\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 700 \,{\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} \sqrt{e x + d}}{e^{6}} - \frac{7 \,{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6} + 75 \,{\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )}{\left (e x + d\right )}^{2} - 10 \,{\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )}{\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac{5}{2}} e^{6}}\right )}}{35 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.84991, size = 834, normalized size = 4.66 \begin{align*} \frac{2 \,{\left (5 \, b^{6} e^{6} x^{6} - 1024 \, b^{6} d^{6} + 3584 \, a b^{5} d^{5} e - 4480 \, a^{2} b^{4} d^{4} e^{2} + 2240 \, a^{3} b^{3} d^{3} e^{3} - 280 \, a^{4} b^{2} d^{2} e^{4} - 28 \, a^{5} b d e^{5} - 7 \, a^{6} e^{6} - 6 \,{\left (2 \, b^{6} d e^{5} - 7 \, a b^{5} e^{6}\right )} x^{5} + 5 \,{\left (8 \, b^{6} d^{2} e^{4} - 28 \, a b^{5} d e^{5} + 35 \, a^{2} b^{4} e^{6}\right )} x^{4} - 20 \,{\left (16 \, b^{6} d^{3} e^{3} - 56 \, a b^{5} d^{2} e^{4} + 70 \, a^{2} b^{4} d e^{5} - 35 \, a^{3} b^{3} e^{6}\right )} x^{3} - 15 \,{\left (128 \, b^{6} d^{4} e^{2} - 448 \, a b^{5} d^{3} e^{3} + 560 \, a^{2} b^{4} d^{2} e^{4} - 280 \, a^{3} b^{3} d e^{5} + 35 \, a^{4} b^{2} e^{6}\right )} x^{2} - 10 \,{\left (256 \, b^{6} d^{5} e - 896 \, a b^{5} d^{4} e^{2} + 1120 \, a^{2} b^{4} d^{3} e^{3} - 560 \, a^{3} b^{3} d^{2} e^{4} + 70 \, a^{4} b^{2} d e^{5} + 7 \, a^{5} b e^{6}\right )} x\right )} \sqrt{e x + d}}{35 \,{\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 123.569, size = 221, normalized size = 1.23 \begin{align*} \frac{2 b^{6} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{7}} - \frac{30 b^{2} \left (a e - b d\right )^{4}}{e^{7} \sqrt{d + e x}} - \frac{4 b \left (a e - b d\right )^{5}}{e^{7} \left (d + e x\right )^{\frac{3}{2}}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (12 a b^{5} e - 12 b^{6} d\right )}{5 e^{7}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (30 a^{2} b^{4} e^{2} - 60 a b^{5} d e + 30 b^{6} d^{2}\right )}{3 e^{7}} + \frac{\sqrt{d + e x} \left (40 a^{3} b^{3} e^{3} - 120 a^{2} b^{4} d e^{2} + 120 a b^{5} d^{2} e - 40 b^{6} d^{3}\right )}{e^{7}} - \frac{2 \left (a e - b d\right )^{6}}{5 e^{7} \left (d + e x\right )^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23942, size = 618, normalized size = 3.45 \begin{align*} \frac{2}{35} \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{6} e^{42} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{6} d e^{42} + 175 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{6} d^{2} e^{42} - 700 \, \sqrt{x e + d} b^{6} d^{3} e^{42} + 42 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{5} e^{43} - 350 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{5} d e^{43} + 2100 \, \sqrt{x e + d} a b^{5} d^{2} e^{43} + 175 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{4} e^{44} - 2100 \, \sqrt{x e + d} a^{2} b^{4} d e^{44} + 700 \, \sqrt{x e + d} a^{3} b^{3} e^{45}\right )} e^{\left (-49\right )} - \frac{2 \,{\left (75 \,{\left (x e + d\right )}^{2} b^{6} d^{4} - 10 \,{\left (x e + d\right )} b^{6} d^{5} + b^{6} d^{6} - 300 \,{\left (x e + d\right )}^{2} a b^{5} d^{3} e + 50 \,{\left (x e + d\right )} a b^{5} d^{4} e - 6 \, a b^{5} d^{5} e + 450 \,{\left (x e + d\right )}^{2} a^{2} b^{4} d^{2} e^{2} - 100 \,{\left (x e + d\right )} a^{2} b^{4} d^{3} e^{2} + 15 \, a^{2} b^{4} d^{4} e^{2} - 300 \,{\left (x e + d\right )}^{2} a^{3} b^{3} d e^{3} + 100 \,{\left (x e + d\right )} a^{3} b^{3} d^{2} e^{3} - 20 \, a^{3} b^{3} d^{3} e^{3} + 75 \,{\left (x e + d\right )}^{2} a^{4} b^{2} e^{4} - 50 \,{\left (x e + d\right )} a^{4} b^{2} d e^{4} + 15 \, a^{4} b^{2} d^{2} e^{4} + 10 \,{\left (x e + d\right )} a^{5} b e^{5} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} e^{\left (-7\right )}}{5 \,{\left (x e + d\right )}^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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