Optimal. Leaf size=206 \[ -\frac{14 b^6 (d+e x)^{11/2} (b d-a e)}{11 e^8}+\frac{14 b^5 (d+e x)^{9/2} (b d-a e)^2}{3 e^8}-\frac{10 b^4 (d+e x)^{7/2} (b d-a e)^3}{e^8}+\frac{14 b^3 (d+e x)^{5/2} (b d-a e)^4}{e^8}-\frac{14 b^2 (d+e x)^{3/2} (b d-a e)^5}{e^8}+\frac{14 b \sqrt{d+e x} (b d-a e)^6}{e^8}+\frac{2 (b d-a e)^7}{e^8 \sqrt{d+e x}}+\frac{2 b^7 (d+e x)^{13/2}}{13 e^8} \]
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Rubi [A] time = 0.0788397, antiderivative size = 206, 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} \[ -\frac{14 b^6 (d+e x)^{11/2} (b d-a e)}{11 e^8}+\frac{14 b^5 (d+e x)^{9/2} (b d-a e)^2}{3 e^8}-\frac{10 b^4 (d+e x)^{7/2} (b d-a e)^3}{e^8}+\frac{14 b^3 (d+e x)^{5/2} (b d-a e)^4}{e^8}-\frac{14 b^2 (d+e x)^{3/2} (b d-a e)^5}{e^8}+\frac{14 b \sqrt{d+e x} (b d-a e)^6}{e^8}+\frac{2 (b d-a e)^7}{e^8 \sqrt{d+e x}}+\frac{2 b^7 (d+e x)^{13/2}}{13 e^8} \]
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)^{3/2}} \, dx &=\int \frac{(a+b x)^7}{(d+e x)^{3/2}} \, dx\\ &=\int \left (\frac{(-b d+a e)^7}{e^7 (d+e x)^{3/2}}+\frac{7 b (b d-a e)^6}{e^7 \sqrt{d+e x}}-\frac{21 b^2 (b d-a e)^5 \sqrt{d+e x}}{e^7}+\frac{35 b^3 (b d-a e)^4 (d+e x)^{3/2}}{e^7}-\frac{35 b^4 (b d-a e)^3 (d+e x)^{5/2}}{e^7}+\frac{21 b^5 (b d-a e)^2 (d+e x)^{7/2}}{e^7}-\frac{7 b^6 (b d-a e) (d+e x)^{9/2}}{e^7}+\frac{b^7 (d+e x)^{11/2}}{e^7}\right ) \, dx\\ &=\frac{2 (b d-a e)^7}{e^8 \sqrt{d+e x}}+\frac{14 b (b d-a e)^6 \sqrt{d+e x}}{e^8}-\frac{14 b^2 (b d-a e)^5 (d+e x)^{3/2}}{e^8}+\frac{14 b^3 (b d-a e)^4 (d+e x)^{5/2}}{e^8}-\frac{10 b^4 (b d-a e)^3 (d+e x)^{7/2}}{e^8}+\frac{14 b^5 (b d-a e)^2 (d+e x)^{9/2}}{3 e^8}-\frac{14 b^6 (b d-a e) (d+e x)^{11/2}}{11 e^8}+\frac{2 b^7 (d+e x)^{13/2}}{13 e^8}\\ \end{align*}
Mathematica [A] time = 0.0986877, size = 167, normalized size = 0.81 \[ \frac{2 \left (-3003 b^2 (d+e x)^2 (b d-a e)^5+3003 b^3 (d+e x)^3 (b d-a e)^4-2145 b^4 (d+e x)^4 (b d-a e)^3+1001 b^5 (d+e x)^5 (b d-a e)^2-273 b^6 (d+e x)^6 (b d-a e)+3003 b (d+e x) (b d-a e)^6+429 (b d-a e)^7+33 b^7 (d+e x)^7\right )}{429 e^8 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 498, normalized size = 2.4 \begin{align*} -{\frac{-66\,{b}^{7}{x}^{7}{e}^{7}-546\,a{b}^{6}{e}^{7}{x}^{6}+84\,{b}^{7}d{e}^{6}{x}^{6}-2002\,{a}^{2}{b}^{5}{e}^{7}{x}^{5}+728\,a{b}^{6}d{e}^{6}{x}^{5}-112\,{b}^{7}{d}^{2}{e}^{5}{x}^{5}-4290\,{a}^{3}{b}^{4}{e}^{7}{x}^{4}+2860\,{a}^{2}{b}^{5}d{e}^{6}{x}^{4}-1040\,a{b}^{6}{d}^{2}{e}^{5}{x}^{4}+160\,{b}^{7}{d}^{3}{e}^{4}{x}^{4}-6006\,{a}^{4}{b}^{3}{e}^{7}{x}^{3}+6864\,{a}^{3}{b}^{4}d{e}^{6}{x}^{3}-4576\,{a}^{2}{b}^{5}{d}^{2}{e}^{5}{x}^{3}+1664\,a{b}^{6}{d}^{3}{e}^{4}{x}^{3}-256\,{b}^{7}{d}^{4}{e}^{3}{x}^{3}-6006\,{a}^{5}{b}^{2}{e}^{7}{x}^{2}+12012\,{a}^{4}{b}^{3}d{e}^{6}{x}^{2}-13728\,{a}^{3}{b}^{4}{d}^{2}{e}^{5}{x}^{2}+9152\,{a}^{2}{b}^{5}{d}^{3}{e}^{4}{x}^{2}-3328\,a{b}^{6}{d}^{4}{e}^{3}{x}^{2}+512\,{b}^{7}{d}^{5}{e}^{2}{x}^{2}-6006\,{a}^{6}b{e}^{7}x+24024\,{a}^{5}{b}^{2}d{e}^{6}x-48048\,{a}^{4}{b}^{3}{d}^{2}{e}^{5}x+54912\,{a}^{3}{b}^{4}{d}^{3}{e}^{4}x-36608\,{a}^{2}{b}^{5}{d}^{4}{e}^{3}x+13312\,a{b}^{6}{d}^{5}{e}^{2}x-2048\,{b}^{7}{d}^{6}ex+858\,{a}^{7}{e}^{7}-12012\,{a}^{6}bd{e}^{6}+48048\,{a}^{5}{b}^{2}{d}^{2}{e}^{5}-96096\,{a}^{4}{b}^{3}{d}^{3}{e}^{4}+109824\,{a}^{3}{b}^{4}{d}^{4}{e}^{3}-73216\,{a}^{2}{b}^{5}{d}^{5}{e}^{2}+26624\,a{b}^{6}{d}^{6}e-4096\,{b}^{7}{d}^{7}}{429\,{e}^{8}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.993249, size = 626, normalized size = 3.04 \begin{align*} \frac{2 \,{\left (\frac{33 \,{\left (e x + d\right )}^{\frac{13}{2}} b^{7} - 273 \,{\left (b^{7} d - a b^{6} e\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 1001 \,{\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 2145 \,{\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{7}{2}} + 3003 \,{\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{5}{2}} - 3003 \,{\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 )}{\left (e x + d\right )}^{\frac{3}{2}} + 3003 \,{\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 )} \sqrt{e x + d}}{e^{7}} + \frac{429 \,{\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}\right )}}{\sqrt{e x + d} e^{7}}\right )}}{429 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.66599, size = 1065, normalized size = 5.17 \begin{align*} \frac{2 \,{\left (33 \, b^{7} e^{7} x^{7} + 2048 \, b^{7} d^{7} - 13312 \, a b^{6} d^{6} e + 36608 \, a^{2} b^{5} d^{5} e^{2} - 54912 \, a^{3} b^{4} d^{4} e^{3} + 48048 \, a^{4} b^{3} d^{3} e^{4} - 24024 \, a^{5} b^{2} d^{2} e^{5} + 6006 \, a^{6} b d e^{6} - 429 \, a^{7} e^{7} - 21 \,{\left (2 \, b^{7} d e^{6} - 13 \, a b^{6} e^{7}\right )} x^{6} + 7 \,{\left (8 \, b^{7} d^{2} e^{5} - 52 \, a b^{6} d e^{6} + 143 \, a^{2} b^{5} e^{7}\right )} x^{5} - 5 \,{\left (16 \, b^{7} d^{3} e^{4} - 104 \, a b^{6} d^{2} e^{5} + 286 \, a^{2} b^{5} d e^{6} - 429 \, a^{3} b^{4} e^{7}\right )} x^{4} +{\left (128 \, b^{7} d^{4} e^{3} - 832 \, a b^{6} d^{3} e^{4} + 2288 \, a^{2} b^{5} d^{2} e^{5} - 3432 \, a^{3} b^{4} d e^{6} + 3003 \, a^{4} b^{3} e^{7}\right )} x^{3} -{\left (256 \, b^{7} d^{5} e^{2} - 1664 \, a b^{6} d^{4} e^{3} + 4576 \, a^{2} b^{5} d^{3} e^{4} - 6864 \, a^{3} b^{4} d^{2} e^{5} + 6006 \, a^{4} b^{3} d e^{6} - 3003 \, a^{5} b^{2} e^{7}\right )} x^{2} +{\left (1024 \, b^{7} d^{6} e - 6656 \, a b^{6} d^{5} e^{2} + 18304 \, a^{2} b^{5} d^{4} e^{3} - 27456 \, a^{3} b^{4} d^{3} e^{4} + 24024 \, a^{4} b^{3} d^{2} e^{5} - 12012 \, a^{5} b^{2} d e^{6} + 3003 \, a^{6} b e^{7}\right )} x\right )} \sqrt{e x + d}}{429 \,{\left (e^{9} x + d e^{8}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 75.588, size = 439, normalized size = 2.13 \begin{align*} \frac{2 b^{7} \left (d + e x\right )^{\frac{13}{2}}}{13 e^{8}} + \frac{\left (d + e x\right )^{\frac{11}{2}} \left (14 a b^{6} e - 14 b^{7} d\right )}{11 e^{8}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (42 a^{2} b^{5} e^{2} - 84 a b^{6} d e + 42 b^{7} d^{2}\right )}{9 e^{8}} + \frac{\left (d + e x\right )^{\frac{7}{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 )}{7 e^{8}} + \frac{\left (d + e x\right )^{\frac{5}{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 )}{5 e^{8}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \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 )}{3 e^{8}} + \frac{\sqrt{d + e x} \left (14 a^{6} b e^{6} - 84 a^{5} b^{2} d e^{5} + 210 a^{4} b^{3} d^{2} e^{4} - 280 a^{3} b^{4} d^{3} e^{3} + 210 a^{2} b^{5} d^{4} e^{2} - 84 a b^{6} d^{5} e + 14 b^{7} d^{6}\right )}{e^{8}} - \frac{2 \left (a e - b d\right )^{7}}{e^{8} \sqrt{d + e x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15032, size = 844, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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