Optimal. Leaf size=214 \[ -\frac{14 b^6 (d+e x)^{15/2} (b d-a e)}{15 e^8}+\frac{42 b^5 (d+e x)^{13/2} (b d-a e)^2}{13 e^8}-\frac{70 b^4 (d+e x)^{11/2} (b d-a e)^3}{11 e^8}+\frac{70 b^3 (d+e x)^{9/2} (b d-a e)^4}{9 e^8}-\frac{6 b^2 (d+e x)^{7/2} (b d-a e)^5}{e^8}+\frac{14 b (d+e x)^{5/2} (b d-a e)^6}{5 e^8}-\frac{2 (d+e x)^{3/2} (b d-a e)^7}{3 e^8}+\frac{2 b^7 (d+e x)^{17/2}}{17 e^8} \]
[Out]
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Rubi [A] time = 0.173855, antiderivative size = 214, 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 \[ -\frac{14 b^6 (d+e x)^{15/2} (b d-a e)}{15 e^8}+\frac{42 b^5 (d+e x)^{13/2} (b d-a e)^2}{13 e^8}-\frac{70 b^4 (d+e x)^{11/2} (b d-a e)^3}{11 e^8}+\frac{70 b^3 (d+e x)^{9/2} (b d-a e)^4}{9 e^8}-\frac{6 b^2 (d+e x)^{7/2} (b d-a e)^5}{e^8}+\frac{14 b (d+e x)^{5/2} (b d-a e)^6}{5 e^8}-\frac{2 (d+e x)^{3/2} (b d-a e)^7}{3 e^8}+\frac{2 b^7 (d+e x)^{17/2}}{17 e^8} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)*Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 105.565, size = 199, normalized size = 0.93 \[ \frac{2 b^{7} \left (d + e x\right )^{\frac{17}{2}}}{17 e^{8}} + \frac{14 b^{6} \left (d + e x\right )^{\frac{15}{2}} \left (a e - b d\right )}{15 e^{8}} + \frac{42 b^{5} \left (d + e x\right )^{\frac{13}{2}} \left (a e - b d\right )^{2}}{13 e^{8}} + \frac{70 b^{4} \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right )^{3}}{11 e^{8}} + \frac{70 b^{3} \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{4}}{9 e^{8}} + \frac{6 b^{2} \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{5}}{e^{8}} + \frac{14 b \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{6}}{5 e^{8}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{7}}{3 e^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**3*(e*x+d)**(1/2),x)
[Out]
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Mathematica [A] time = 0.430545, size = 376, normalized size = 1.76 \[ \frac{2 (d+e x)^{3/2} \left (36465 a^7 e^7+51051 a^6 b e^6 (3 e x-2 d)+21879 a^5 b^2 e^5 \left (8 d^2-12 d e x+15 e^2 x^2\right )+12155 a^4 b^3 e^4 \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+1105 a^3 b^4 e^3 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )+255 a^2 b^5 e^2 \left (-256 d^5+384 d^4 e x-480 d^3 e^2 x^2+560 d^2 e^3 x^3-630 d e^4 x^4+693 e^5 x^5\right )+17 a b^6 e \left (1024 d^6-1536 d^5 e x+1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-2772 d e^5 x^5+3003 e^6 x^6\right )+b^7 \left (-2048 d^7+3072 d^6 e x-3840 d^5 e^2 x^2+4480 d^4 e^3 x^3-5040 d^3 e^4 x^4+5544 d^2 e^5 x^5-6006 d e^6 x^6+6435 e^7 x^7\right )\right )}{109395 e^8} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)*Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
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Maple [B] time = 0.012, size = 498, normalized size = 2.3 \[{\frac{12870\,{b}^{7}{x}^{7}{e}^{7}+102102\,a{b}^{6}{e}^{7}{x}^{6}-12012\,{b}^{7}d{e}^{6}{x}^{6}+353430\,{a}^{2}{b}^{5}{e}^{7}{x}^{5}-94248\,a{b}^{6}d{e}^{6}{x}^{5}+11088\,{b}^{7}{d}^{2}{e}^{5}{x}^{5}+696150\,{a}^{3}{b}^{4}{e}^{7}{x}^{4}-321300\,{a}^{2}{b}^{5}d{e}^{6}{x}^{4}+85680\,a{b}^{6}{d}^{2}{e}^{5}{x}^{4}-10080\,{b}^{7}{d}^{3}{e}^{4}{x}^{4}+850850\,{a}^{4}{b}^{3}{e}^{7}{x}^{3}-618800\,{a}^{3}{b}^{4}d{e}^{6}{x}^{3}+285600\,{a}^{2}{b}^{5}{d}^{2}{e}^{5}{x}^{3}-76160\,a{b}^{6}{d}^{3}{e}^{4}{x}^{3}+8960\,{b}^{7}{d}^{4}{e}^{3}{x}^{3}+656370\,{a}^{5}{b}^{2}{e}^{7}{x}^{2}-729300\,{a}^{4}{b}^{3}d{e}^{6}{x}^{2}+530400\,{a}^{3}{b}^{4}{d}^{2}{e}^{5}{x}^{2}-244800\,{a}^{2}{b}^{5}{d}^{3}{e}^{4}{x}^{2}+65280\,a{b}^{6}{d}^{4}{e}^{3}{x}^{2}-7680\,{b}^{7}{d}^{5}{e}^{2}{x}^{2}+306306\,{a}^{6}b{e}^{7}x-525096\,{a}^{5}{b}^{2}d{e}^{6}x+583440\,{a}^{4}{b}^{3}{d}^{2}{e}^{5}x-424320\,{a}^{3}{b}^{4}{d}^{3}{e}^{4}x+195840\,{a}^{2}{b}^{5}{d}^{4}{e}^{3}x-52224\,a{b}^{6}{d}^{5}{e}^{2}x+6144\,{b}^{7}{d}^{6}ex+72930\,{a}^{7}{e}^{7}-204204\,{a}^{6}bd{e}^{6}+350064\,{a}^{5}{b}^{2}{d}^{2}{e}^{5}-388960\,{a}^{4}{b}^{3}{d}^{3}{e}^{4}+282880\,{a}^{3}{b}^{4}{d}^{4}{e}^{3}-130560\,{a}^{2}{b}^{5}{d}^{5}{e}^{2}+34816\,a{b}^{6}{d}^{6}e-4096\,{b}^{7}{d}^{7}}{109395\,{e}^{8}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^3*(e*x+d)^(1/2),x)
[Out]
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Maxima [A] time = 0.718152, size = 616, normalized size = 2.88 \[ \frac{2 \,{\left (6435 \,{\left (e x + d\right )}^{\frac{17}{2}} b^{7} - 51051 \,{\left (b^{7} d - a b^{6} e\right )}{\left (e x + d\right )}^{\frac{15}{2}} + 176715 \,{\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )}{\left (e x + d\right )}^{\frac{13}{2}} - 348075 \,{\left (b^{7} d^{3} - 3 \, a b^{6} d^{2} e + 3 \, a^{2} b^{5} d e^{2} - a^{3} b^{4} e^{3}\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 425425 \,{\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 328185 \,{\left (b^{7} d^{5} - 5 \, a b^{6} d^{4} e + 10 \, a^{2} b^{5} d^{3} e^{2} - 10 \, a^{3} b^{4} d^{2} e^{3} + 5 \, a^{4} b^{3} d e^{4} - a^{5} b^{2} e^{5}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 153153 \,{\left (b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 36465 \,{\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{109395 \, e^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(b*x + a)*sqrt(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.283107, size = 767, normalized size = 3.58 \[ \frac{2 \,{\left (6435 \, b^{7} e^{8} x^{8} - 2048 \, b^{7} d^{8} + 17408 \, a b^{6} d^{7} e - 65280 \, a^{2} b^{5} d^{6} e^{2} + 141440 \, a^{3} b^{4} d^{5} e^{3} - 194480 \, a^{4} b^{3} d^{4} e^{4} + 175032 \, a^{5} b^{2} d^{3} e^{5} - 102102 \, a^{6} b d^{2} e^{6} + 36465 \, a^{7} d e^{7} + 429 \,{\left (b^{7} d e^{7} + 119 \, a b^{6} e^{8}\right )} x^{7} - 231 \,{\left (2 \, b^{7} d^{2} e^{6} - 17 \, a b^{6} d e^{7} - 765 \, a^{2} b^{5} e^{8}\right )} x^{6} + 63 \,{\left (8 \, b^{7} d^{3} e^{5} - 68 \, a b^{6} d^{2} e^{6} + 255 \, a^{2} b^{5} d e^{7} + 5525 \, a^{3} b^{4} e^{8}\right )} x^{5} - 35 \,{\left (16 \, b^{7} d^{4} e^{4} - 136 \, a b^{6} d^{3} e^{5} + 510 \, a^{2} b^{5} d^{2} e^{6} - 1105 \, a^{3} b^{4} d e^{7} - 12155 \, a^{4} b^{3} e^{8}\right )} x^{4} + 5 \,{\left (128 \, b^{7} d^{5} e^{3} - 1088 \, a b^{6} d^{4} e^{4} + 4080 \, a^{2} b^{5} d^{3} e^{5} - 8840 \, a^{3} b^{4} d^{2} e^{6} + 12155 \, a^{4} b^{3} d e^{7} + 65637 \, a^{5} b^{2} e^{8}\right )} x^{3} - 3 \,{\left (256 \, b^{7} d^{6} e^{2} - 2176 \, a b^{6} d^{5} e^{3} + 8160 \, a^{2} b^{5} d^{4} e^{4} - 17680 \, a^{3} b^{4} d^{3} e^{5} + 24310 \, a^{4} b^{3} d^{2} e^{6} - 21879 \, a^{5} b^{2} d e^{7} - 51051 \, a^{6} b e^{8}\right )} x^{2} +{\left (1024 \, b^{7} d^{7} e - 8704 \, a b^{6} d^{6} e^{2} + 32640 \, a^{2} b^{5} d^{5} e^{3} - 70720 \, a^{3} b^{4} d^{4} e^{4} + 97240 \, a^{4} b^{3} d^{3} e^{5} - 87516 \, a^{5} b^{2} d^{2} e^{6} + 51051 \, a^{6} b d e^{7} + 36465 \, a^{7} e^{8}\right )} x\right )} \sqrt{e x + d}}{109395 \, e^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(b*x + a)*sqrt(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 11.5366, size = 544, normalized size = 2.54 \[ \frac{2 \left (\frac{b^{7} \left (d + e x\right )^{\frac{17}{2}}}{17 e^{7}} + \frac{\left (d + e x\right )^{\frac{15}{2}} \left (7 a b^{6} e - 7 b^{7} d\right )}{15 e^{7}} + \frac{\left (d + e x\right )^{\frac{13}{2}} \left (21 a^{2} b^{5} e^{2} - 42 a b^{6} d e + 21 b^{7} d^{2}\right )}{13 e^{7}} + \frac{\left (d + e x\right )^{\frac{11}{2}} \left (35 a^{3} b^{4} e^{3} - 105 a^{2} b^{5} d e^{2} + 105 a b^{6} d^{2} e - 35 b^{7} d^{3}\right )}{11 e^{7}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (35 a^{4} b^{3} e^{4} - 140 a^{3} b^{4} d e^{3} + 210 a^{2} b^{5} d^{2} e^{2} - 140 a b^{6} d^{3} e + 35 b^{7} d^{4}\right )}{9 e^{7}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (21 a^{5} b^{2} e^{5} - 105 a^{4} b^{3} d e^{4} + 210 a^{3} b^{4} d^{2} e^{3} - 210 a^{2} b^{5} d^{3} e^{2} + 105 a b^{6} d^{4} e - 21 b^{7} d^{5}\right )}{7 e^{7}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (7 a^{6} b e^{6} - 42 a^{5} b^{2} d e^{5} + 105 a^{4} b^{3} d^{2} e^{4} - 140 a^{3} b^{4} d^{3} e^{3} + 105 a^{2} b^{5} d^{4} e^{2} - 42 a b^{6} d^{5} e + 7 b^{7} d^{6}\right )}{5 e^{7}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (a^{7} e^{7} - 7 a^{6} b d e^{6} + 21 a^{5} b^{2} d^{2} e^{5} - 35 a^{4} b^{3} d^{3} e^{4} + 35 a^{3} b^{4} d^{4} e^{3} - 21 a^{2} b^{5} d^{5} e^{2} + 7 a b^{6} d^{6} e - b^{7} d^{7}\right )}{3 e^{7}}\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**3*(e*x+d)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.291891, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(b*x + a)*sqrt(e*x + d),x, algorithm="giac")
[Out]