Optimal. Leaf size=179 \[ -\frac{4 b^5 (d+e x)^{9/2} (b d-a e)}{3 e^7}+\frac{30 b^4 (d+e x)^{7/2} (b d-a e)^2}{7 e^7}-\frac{8 b^3 (d+e x)^{5/2} (b d-a e)^3}{e^7}+\frac{10 b^2 (d+e x)^{3/2} (b d-a e)^4}{e^7}-\frac{12 b \sqrt{d+e x} (b d-a e)^5}{e^7}-\frac{2 (b d-a e)^6}{e^7 \sqrt{d+e x}}+\frac{2 b^6 (d+e x)^{11/2}}{11 e^7} \]
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Rubi [A] time = 0.0590916, 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} \[ -\frac{4 b^5 (d+e x)^{9/2} (b d-a e)}{3 e^7}+\frac{30 b^4 (d+e x)^{7/2} (b d-a e)^2}{7 e^7}-\frac{8 b^3 (d+e x)^{5/2} (b d-a e)^3}{e^7}+\frac{10 b^2 (d+e x)^{3/2} (b d-a e)^4}{e^7}-\frac{12 b \sqrt{d+e x} (b d-a e)^5}{e^7}-\frac{2 (b d-a e)^6}{e^7 \sqrt{d+e x}}+\frac{2 b^6 (d+e x)^{11/2}}{11 e^7} \]
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)^{3/2}} \, dx &=\int \frac{(a+b x)^6}{(d+e x)^{3/2}} \, dx\\ &=\int \left (\frac{(-b d+a e)^6}{e^6 (d+e x)^{3/2}}-\frac{6 b (b d-a e)^5}{e^6 \sqrt{d+e x}}+\frac{15 b^2 (b d-a e)^4 \sqrt{d+e x}}{e^6}-\frac{20 b^3 (b d-a e)^3 (d+e x)^{3/2}}{e^6}+\frac{15 b^4 (b d-a e)^2 (d+e x)^{5/2}}{e^6}-\frac{6 b^5 (b d-a e) (d+e x)^{7/2}}{e^6}+\frac{b^6 (d+e x)^{9/2}}{e^6}\right ) \, dx\\ &=-\frac{2 (b d-a e)^6}{e^7 \sqrt{d+e x}}-\frac{12 b (b d-a e)^5 \sqrt{d+e x}}{e^7}+\frac{10 b^2 (b d-a e)^4 (d+e x)^{3/2}}{e^7}-\frac{8 b^3 (b d-a e)^3 (d+e x)^{5/2}}{e^7}+\frac{30 b^4 (b d-a e)^2 (d+e x)^{7/2}}{7 e^7}-\frac{4 b^5 (b d-a e) (d+e x)^{9/2}}{3 e^7}+\frac{2 b^6 (d+e x)^{11/2}}{11 e^7}\\ \end{align*}
Mathematica [A] time = 0.0804628, size = 145, normalized size = 0.81 \[ \frac{2 \left (1155 b^2 (d+e x)^2 (b d-a e)^4-924 b^3 (d+e x)^3 (b d-a e)^3+495 b^4 (d+e x)^4 (b d-a e)^2-154 b^5 (d+e x)^5 (b d-a e)-1386 b (d+e x) (b d-a e)^5-231 (b d-a e)^6+21 b^6 (d+e x)^6\right )}{231 e^7 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.047, size = 377, normalized size = 2.1 \begin{align*} -{\frac{-42\,{b}^{6}{x}^{6}{e}^{6}-308\,{x}^{5}a{b}^{5}{e}^{6}+56\,{x}^{5}{b}^{6}d{e}^{5}-990\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+440\,{x}^{4}a{b}^{5}d{e}^{5}-80\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}-1848\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+1584\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}-704\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+128\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}-2310\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+3696\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}-3168\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+1408\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}-256\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}-2772\,x{a}^{5}b{e}^{6}+9240\,x{a}^{4}{b}^{2}d{e}^{5}-14784\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+12672\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}-5632\,xa{b}^{5}{d}^{4}{e}^{2}+1024\,x{b}^{6}{d}^{5}e+462\,{a}^{6}{e}^{6}-5544\,{a}^{5}bd{e}^{5}+18480\,{d}^{2}{e}^{4}{a}^{4}{b}^{2}-29568\,{b}^{3}{a}^{3}{d}^{3}{e}^{3}+25344\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}-11264\,a{b}^{5}{d}^{5}e+2048\,{d}^{6}{b}^{6}}{231\,{e}^{7}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.01675, size = 483, normalized size = 2.7 \begin{align*} \frac{2 \,{\left (\frac{21 \,{\left (e x + d\right )}^{\frac{11}{2}} b^{6} - 154 \,{\left (b^{6} d - a b^{5} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 495 \,{\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 924 \,{\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 )}{\left (e x + d\right )}^{\frac{5}{2}} + 1155 \,{\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 )}^{\frac{3}{2}} - 1386 \,{\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 )} \sqrt{e x + d}}{e^{6}} - \frac{231 \,{\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}\right )}}{\sqrt{e x + d} e^{6}}\right )}}{231 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.86338, size = 814, normalized size = 4.55 \begin{align*} \frac{2 \,{\left (21 \, b^{6} e^{6} x^{6} - 1024 \, b^{6} d^{6} + 5632 \, a b^{5} d^{5} e - 12672 \, a^{2} b^{4} d^{4} e^{2} + 14784 \, a^{3} b^{3} d^{3} e^{3} - 9240 \, a^{4} b^{2} d^{2} e^{4} + 2772 \, a^{5} b d e^{5} - 231 \, a^{6} e^{6} - 14 \,{\left (2 \, b^{6} d e^{5} - 11 \, a b^{5} e^{6}\right )} x^{5} + 5 \,{\left (8 \, b^{6} d^{2} e^{4} - 44 \, a b^{5} d e^{5} + 99 \, a^{2} b^{4} e^{6}\right )} x^{4} - 4 \,{\left (16 \, b^{6} d^{3} e^{3} - 88 \, a b^{5} d^{2} e^{4} + 198 \, a^{2} b^{4} d e^{5} - 231 \, a^{3} b^{3} e^{6}\right )} x^{3} +{\left (128 \, b^{6} d^{4} e^{2} - 704 \, a b^{5} d^{3} e^{3} + 1584 \, a^{2} b^{4} d^{2} e^{4} - 1848 \, a^{3} b^{3} d e^{5} + 1155 \, a^{4} b^{2} e^{6}\right )} x^{2} - 2 \,{\left (256 \, b^{6} d^{5} e - 1408 \, a b^{5} d^{4} e^{2} + 3168 \, a^{2} b^{4} d^{3} e^{3} - 3696 \, a^{3} b^{3} d^{2} e^{4} + 2310 \, a^{4} b^{2} d e^{5} - 693 \, a^{5} b e^{6}\right )} x\right )} \sqrt{e x + d}}{231 \,{\left (e^{8} x + d e^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 55.1659, size = 333, normalized size = 1.86 \begin{align*} \frac{2 b^{6} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{7}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (12 a b^{5} e - 12 b^{6} d\right )}{9 e^{7}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (30 a^{2} b^{4} e^{2} - 60 a b^{5} d e + 30 b^{6} d^{2}\right )}{7 e^{7}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \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 )}{5 e^{7}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (30 a^{4} b^{2} e^{4} - 120 a^{3} b^{3} d e^{3} + 180 a^{2} b^{4} d^{2} e^{2} - 120 a b^{5} d^{3} e + 30 b^{6} d^{4}\right )}{3 e^{7}} + \frac{\sqrt{d + e x} \left (12 a^{5} b e^{5} - 60 a^{4} b^{2} d e^{4} + 120 a^{3} b^{3} d^{2} e^{3} - 120 a^{2} b^{4} d^{3} e^{2} + 60 a b^{5} d^{4} e - 12 b^{6} d^{5}\right )}{e^{7}} - \frac{2 \left (a e - b d\right )^{6}}{e^{7} \sqrt{d + e x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.30341, size = 640, normalized size = 3.58 \begin{align*} \frac{2}{231} \,{\left (21 \,{\left (x e + d\right )}^{\frac{11}{2}} b^{6} e^{70} - 154 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{6} d e^{70} + 495 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{6} d^{2} e^{70} - 924 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{6} d^{3} e^{70} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{6} d^{4} e^{70} - 1386 \, \sqrt{x e + d} b^{6} d^{5} e^{70} + 154 \,{\left (x e + d\right )}^{\frac{9}{2}} a b^{5} e^{71} - 990 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{5} d e^{71} + 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{5} d^{2} e^{71} - 4620 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{5} d^{3} e^{71} + 6930 \, \sqrt{x e + d} a b^{5} d^{4} e^{71} + 495 \,{\left (x e + d\right )}^{\frac{7}{2}} a^{2} b^{4} e^{72} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{4} d e^{72} + 6930 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{4} d^{2} e^{72} - 13860 \, \sqrt{x e + d} a^{2} b^{4} d^{3} e^{72} + 924 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{3} b^{3} e^{73} - 4620 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b^{3} d e^{73} + 13860 \, \sqrt{x e + d} a^{3} b^{3} d^{2} e^{73} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{4} b^{2} e^{74} - 6930 \, \sqrt{x e + d} a^{4} b^{2} d e^{74} + 1386 \, \sqrt{x e + d} a^{5} b e^{75}\right )} e^{\left (-77\right )} - \frac{2 \,{\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}\right )} e^{\left (-7\right )}}{\sqrt{x e + d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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