Optimal. Leaf size=208 \[ -\frac{14 b^6 (d+e x)^{9/2} (b d-a e)}{9 e^8}+\frac{6 b^5 (d+e x)^{7/2} (b d-a e)^2}{e^8}-\frac{14 b^4 (d+e x)^{5/2} (b d-a e)^3}{e^8}+\frac{70 b^3 (d+e x)^{3/2} (b d-a e)^4}{3 e^8}-\frac{42 b^2 \sqrt{d+e x} (b d-a e)^5}{e^8}-\frac{14 b (b d-a e)^6}{e^8 \sqrt{d+e x}}+\frac{2 (b d-a e)^7}{3 e^8 (d+e x)^{3/2}}+\frac{2 b^7 (d+e x)^{11/2}}{11 e^8} \]
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Rubi [A] time = 0.0755252, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {27, 43} \[ -\frac{14 b^6 (d+e x)^{9/2} (b d-a e)}{9 e^8}+\frac{6 b^5 (d+e x)^{7/2} (b d-a e)^2}{e^8}-\frac{14 b^4 (d+e x)^{5/2} (b d-a e)^3}{e^8}+\frac{70 b^3 (d+e x)^{3/2} (b d-a e)^4}{3 e^8}-\frac{42 b^2 \sqrt{d+e x} (b d-a e)^5}{e^8}-\frac{14 b (b d-a e)^6}{e^8 \sqrt{d+e x}}+\frac{2 (b d-a e)^7}{3 e^8 (d+e x)^{3/2}}+\frac{2 b^7 (d+e x)^{11/2}}{11 e^8} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{5/2}} \, dx &=\int \frac{(a+b x)^7}{(d+e x)^{5/2}} \, dx\\ &=\int \left (\frac{(-b d+a e)^7}{e^7 (d+e x)^{5/2}}+\frac{7 b (b d-a e)^6}{e^7 (d+e x)^{3/2}}-\frac{21 b^2 (b d-a e)^5}{e^7 \sqrt{d+e x}}+\frac{35 b^3 (b d-a e)^4 \sqrt{d+e x}}{e^7}-\frac{35 b^4 (b d-a e)^3 (d+e x)^{3/2}}{e^7}+\frac{21 b^5 (b d-a e)^2 (d+e x)^{5/2}}{e^7}-\frac{7 b^6 (b d-a e) (d+e x)^{7/2}}{e^7}+\frac{b^7 (d+e x)^{9/2}}{e^7}\right ) \, dx\\ &=\frac{2 (b d-a e)^7}{3 e^8 (d+e x)^{3/2}}-\frac{14 b (b d-a e)^6}{e^8 \sqrt{d+e x}}-\frac{42 b^2 (b d-a e)^5 \sqrt{d+e x}}{e^8}+\frac{70 b^3 (b d-a e)^4 (d+e x)^{3/2}}{3 e^8}-\frac{14 b^4 (b d-a e)^3 (d+e x)^{5/2}}{e^8}+\frac{6 b^5 (b d-a e)^2 (d+e x)^{7/2}}{e^8}-\frac{14 b^6 (b d-a e) (d+e x)^{9/2}}{9 e^8}+\frac{2 b^7 (d+e x)^{11/2}}{11 e^8}\\ \end{align*}
Mathematica [A] time = 0.100922, size = 167, normalized size = 0.8 \[ \frac{2 \left (-2079 b^2 (d+e x)^2 (b d-a e)^5+1155 b^3 (d+e x)^3 (b d-a e)^4-693 b^4 (d+e x)^4 (b d-a e)^3+297 b^5 (d+e x)^5 (b d-a e)^2-77 b^6 (d+e x)^6 (b d-a e)-693 b (d+e x) (b d-a e)^6+33 (b d-a e)^7+9 b^7 (d+e x)^7\right )}{99 e^8 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 498, normalized size = 2.4 \begin{align*} -{\frac{-18\,{b}^{7}{x}^{7}{e}^{7}-154\,a{b}^{6}{e}^{7}{x}^{6}+28\,{b}^{7}d{e}^{6}{x}^{6}-594\,{a}^{2}{b}^{5}{e}^{7}{x}^{5}+264\,a{b}^{6}d{e}^{6}{x}^{5}-48\,{b}^{7}{d}^{2}{e}^{5}{x}^{5}-1386\,{a}^{3}{b}^{4}{e}^{7}{x}^{4}+1188\,{a}^{2}{b}^{5}d{e}^{6}{x}^{4}-528\,a{b}^{6}{d}^{2}{e}^{5}{x}^{4}+96\,{b}^{7}{d}^{3}{e}^{4}{x}^{4}-2310\,{a}^{4}{b}^{3}{e}^{7}{x}^{3}+3696\,{a}^{3}{b}^{4}d{e}^{6}{x}^{3}-3168\,{a}^{2}{b}^{5}{d}^{2}{e}^{5}{x}^{3}+1408\,a{b}^{6}{d}^{3}{e}^{4}{x}^{3}-256\,{b}^{7}{d}^{4}{e}^{3}{x}^{3}-4158\,{a}^{5}{b}^{2}{e}^{7}{x}^{2}+13860\,{a}^{4}{b}^{3}d{e}^{6}{x}^{2}-22176\,{a}^{3}{b}^{4}{d}^{2}{e}^{5}{x}^{2}+19008\,{a}^{2}{b}^{5}{d}^{3}{e}^{4}{x}^{2}-8448\,a{b}^{6}{d}^{4}{e}^{3}{x}^{2}+1536\,{b}^{7}{d}^{5}{e}^{2}{x}^{2}+1386\,{a}^{6}b{e}^{7}x-16632\,{a}^{5}{b}^{2}d{e}^{6}x+55440\,{a}^{4}{b}^{3}{d}^{2}{e}^{5}x-88704\,{a}^{3}{b}^{4}{d}^{3}{e}^{4}x+76032\,{a}^{2}{b}^{5}{d}^{4}{e}^{3}x-33792\,a{b}^{6}{d}^{5}{e}^{2}x+6144\,{b}^{7}{d}^{6}ex+66\,{a}^{7}{e}^{7}+924\,{a}^{6}bd{e}^{6}-11088\,{a}^{5}{b}^{2}{d}^{2}{e}^{5}+36960\,{a}^{4}{b}^{3}{d}^{3}{e}^{4}-59136\,{a}^{3}{b}^{4}{d}^{4}{e}^{3}+50688\,{a}^{2}{b}^{5}{d}^{5}{e}^{2}-22528\,a{b}^{6}{d}^{6}e+4096\,{b}^{7}{d}^{7}}{99\,{e}^{8}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.01083, size = 624, normalized size = 3. \begin{align*} \frac{2 \,{\left (\frac{9 \,{\left (e x + d\right )}^{\frac{11}{2}} b^{7} - 77 \,{\left (b^{7} d - a b^{6} e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 297 \,{\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 693 \,{\left (b^{7} d^{3} - 3 \, a b^{6} d^{2} e + 3 \, a^{2} b^{5} d e^{2} - a^{3} b^{4} e^{3}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 1155 \,{\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 2079 \,{\left (b^{7} d^{5} - 5 \, a b^{6} d^{4} e + 10 \, a^{2} b^{5} d^{3} e^{2} - 10 \, a^{3} b^{4} d^{2} e^{3} + 5 \, a^{4} b^{3} d e^{4} - a^{5} b^{2} e^{5}\right )} \sqrt{e x + d}}{e^{7}} + \frac{33 \,{\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7} - 21 \,{\left (b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}\right )}{\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{7}}\right )}}{99 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.33416, size = 1077, normalized size = 5.18 \begin{align*} \frac{2 \,{\left (9 \, b^{7} e^{7} x^{7} - 2048 \, b^{7} d^{7} + 11264 \, a b^{6} d^{6} e - 25344 \, a^{2} b^{5} d^{5} e^{2} + 29568 \, a^{3} b^{4} d^{4} e^{3} - 18480 \, a^{4} b^{3} d^{3} e^{4} + 5544 \, a^{5} b^{2} d^{2} e^{5} - 462 \, a^{6} b d e^{6} - 33 \, a^{7} e^{7} - 7 \,{\left (2 \, b^{7} d e^{6} - 11 \, a b^{6} e^{7}\right )} x^{6} + 3 \,{\left (8 \, b^{7} d^{2} e^{5} - 44 \, a b^{6} d e^{6} + 99 \, a^{2} b^{5} e^{7}\right )} x^{5} - 3 \,{\left (16 \, b^{7} d^{3} e^{4} - 88 \, a b^{6} d^{2} e^{5} + 198 \, a^{2} b^{5} d e^{6} - 231 \, a^{3} b^{4} e^{7}\right )} x^{4} +{\left (128 \, b^{7} d^{4} e^{3} - 704 \, a b^{6} d^{3} e^{4} + 1584 \, a^{2} b^{5} d^{2} e^{5} - 1848 \, a^{3} b^{4} d e^{6} + 1155 \, a^{4} b^{3} e^{7}\right )} x^{3} - 3 \,{\left (256 \, b^{7} d^{5} e^{2} - 1408 \, a b^{6} d^{4} e^{3} + 3168 \, a^{2} b^{5} d^{3} e^{4} - 3696 \, a^{3} b^{4} d^{2} e^{5} + 2310 \, a^{4} b^{3} d e^{6} - 693 \, a^{5} b^{2} e^{7}\right )} x^{2} - 3 \,{\left (1024 \, b^{7} d^{6} e - 5632 \, a b^{6} d^{5} e^{2} + 12672 \, a^{2} b^{5} d^{4} e^{3} - 14784 \, a^{3} b^{4} d^{3} e^{4} + 9240 \, a^{4} b^{3} d^{2} e^{5} - 2772 \, a^{5} b^{2} d e^{6} + 231 \, a^{6} b e^{7}\right )} x\right )} \sqrt{e x + d}}{99 \,{\left (e^{10} x^{2} + 2 \, d e^{9} x + d^{2} e^{8}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 85.1283, size = 360, normalized size = 1.73 \begin{align*} \frac{2 b^{7} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{8}} - \frac{14 b \left (a e - b d\right )^{6}}{e^{8} \sqrt{d + e x}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (14 a b^{6} e - 14 b^{7} d\right )}{9 e^{8}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (42 a^{2} b^{5} e^{2} - 84 a b^{6} d e + 42 b^{7} d^{2}\right )}{7 e^{8}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (70 a^{3} b^{4} e^{3} - 210 a^{2} b^{5} d e^{2} + 210 a b^{6} d^{2} e - 70 b^{7} d^{3}\right )}{5 e^{8}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (70 a^{4} b^{3} e^{4} - 280 a^{3} b^{4} d e^{3} + 420 a^{2} b^{5} d^{2} e^{2} - 280 a b^{6} d^{3} e + 70 b^{7} d^{4}\right )}{3 e^{8}} + \frac{\sqrt{d + e x} \left (42 a^{5} b^{2} e^{5} - 210 a^{4} b^{3} d e^{4} + 420 a^{3} b^{4} d^{2} e^{3} - 420 a^{2} b^{5} d^{3} e^{2} + 210 a b^{6} d^{4} e - 42 b^{7} d^{5}\right )}{e^{8}} - \frac{2 \left (a e - b d\right )^{7}}{3 e^{8} \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 = 1.21075, size = 822, normalized size = 3.95 \begin{align*} \frac{2}{99} \,{\left (9 \,{\left (x e + d\right )}^{\frac{11}{2}} b^{7} e^{80} - 77 \,{\left (x e + d\right )}^{\frac{9}{2}} b^{7} d e^{80} + 297 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{7} d^{2} e^{80} - 693 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{7} d^{3} e^{80} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{7} d^{4} e^{80} - 2079 \, \sqrt{x e + d} b^{7} d^{5} e^{80} + 77 \,{\left (x e + d\right )}^{\frac{9}{2}} a b^{6} e^{81} - 594 \,{\left (x e + d\right )}^{\frac{7}{2}} a b^{6} d e^{81} + 2079 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{6} d^{2} e^{81} - 4620 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{6} d^{3} e^{81} + 10395 \, \sqrt{x e + d} a b^{6} d^{4} e^{81} + 297 \,{\left (x e + d\right )}^{\frac{7}{2}} a^{2} b^{5} e^{82} - 2079 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} b^{5} d e^{82} + 6930 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{5} d^{2} e^{82} - 20790 \, \sqrt{x e + d} a^{2} b^{5} d^{3} e^{82} + 693 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{3} b^{4} e^{83} - 4620 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} b^{4} d e^{83} + 20790 \, \sqrt{x e + d} a^{3} b^{4} d^{2} e^{83} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{4} b^{3} e^{84} - 10395 \, \sqrt{x e + d} a^{4} b^{3} d e^{84} + 2079 \, \sqrt{x e + d} a^{5} b^{2} e^{85}\right )} e^{\left (-88\right )} - \frac{2 \,{\left (21 \,{\left (x e + d\right )} b^{7} d^{6} - b^{7} d^{7} - 126 \,{\left (x e + d\right )} a b^{6} d^{5} e + 7 \, a b^{6} d^{6} e + 315 \,{\left (x e + d\right )} a^{2} b^{5} d^{4} e^{2} - 21 \, a^{2} b^{5} d^{5} e^{2} - 420 \,{\left (x e + d\right )} a^{3} b^{4} d^{3} e^{3} + 35 \, a^{3} b^{4} d^{4} e^{3} + 315 \,{\left (x e + d\right )} a^{4} b^{3} d^{2} e^{4} - 35 \, a^{4} b^{3} d^{3} e^{4} - 126 \,{\left (x e + d\right )} a^{5} b^{2} d e^{5} + 21 \, a^{5} b^{2} d^{2} e^{5} + 21 \,{\left (x e + d\right )} a^{6} b e^{6} - 7 \, a^{6} b d e^{6} + a^{7} e^{7}\right )} e^{\left (-8\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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