Optimal. Leaf size=218 \[ \frac{4 c (d+e x)^{9/2} \left (a B e^2-2 A c d e+5 B c d^2\right )}{9 e^6}+\frac{2 (d+e x)^{5/2} \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{5 e^6}-\frac{2 (d+e x)^{3/2} \left (a e^2+c d^2\right )^2 (B d-A e)}{3 e^6}-\frac{4 c (d+e x)^{7/2} \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{7 e^6}-\frac{2 c^2 (d+e x)^{11/2} (5 B d-A e)}{11 e^6}+\frac{2 B c^2 (d+e x)^{13/2}}{13 e^6} \]
[Out]
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Rubi [A] time = 0.307187, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{4 c (d+e x)^{9/2} \left (a B e^2-2 A c d e+5 B c d^2\right )}{9 e^6}+\frac{2 (d+e x)^{5/2} \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{5 e^6}-\frac{2 (d+e x)^{3/2} \left (a e^2+c d^2\right )^2 (B d-A e)}{3 e^6}-\frac{4 c (d+e x)^{7/2} \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{7 e^6}-\frac{2 c^2 (d+e x)^{11/2} (5 B d-A e)}{11 e^6}+\frac{2 B c^2 (d+e x)^{13/2}}{13 e^6} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)*Sqrt[d + e*x]*(a + c*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 56.9648, size = 223, normalized size = 1.02 \[ \frac{2 B c^{2} \left (d + e x\right )^{\frac{13}{2}}}{13 e^{6}} + \frac{2 c^{2} \left (d + e x\right )^{\frac{11}{2}} \left (A e - 5 B d\right )}{11 e^{6}} + \frac{4 c \left (d + e x\right )^{\frac{9}{2}} \left (- 2 A c d e + B a e^{2} + 5 B c d^{2}\right )}{9 e^{6}} + \frac{4 c \left (d + e x\right )^{\frac{7}{2}} \left (A a e^{3} + 3 A c d^{2} e - 3 B a d e^{2} - 5 B c d^{3}\right )}{7 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (a e^{2} + c d^{2}\right ) \left (- 4 A c d e + B a e^{2} + 5 B c d^{2}\right )}{5 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (A e - B d\right ) \left (a e^{2} + c d^{2}\right )^{2}}{3 e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+a)**2*(e*x+d)**(1/2),x)
[Out]
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Mathematica [A] time = 0.292539, size = 214, normalized size = 0.98 \[ \frac{2 (d+e x)^{3/2} \left (13 A e \left (1155 a^2 e^4+66 a c e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+c^2 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )+B \left (3003 a^2 e^4 (3 e x-2 d)+286 a c e^2 \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )-5 c^2 \left (256 d^5-384 d^4 e x+480 d^3 e^2 x^2-560 d^2 e^3 x^3+630 d e^4 x^4-693 e^5 x^5\right )\right )\right )}{45045 e^6} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)*Sqrt[d + e*x]*(a + c*x^2)^2,x]
[Out]
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Maple [A] time = 0.012, size = 259, normalized size = 1.2 \[{\frac{6930\,B{c}^{2}{x}^{5}{e}^{5}+8190\,A{c}^{2}{e}^{5}{x}^{4}-6300\,B{c}^{2}d{e}^{4}{x}^{4}-7280\,A{c}^{2}d{e}^{4}{x}^{3}+20020\,Bac{e}^{5}{x}^{3}+5600\,B{c}^{2}{d}^{2}{e}^{3}{x}^{3}+25740\,Aac{e}^{5}{x}^{2}+6240\,A{c}^{2}{d}^{2}{e}^{3}{x}^{2}-17160\,Bacd{e}^{4}{x}^{2}-4800\,B{c}^{2}{d}^{3}{e}^{2}{x}^{2}-20592\,Aacd{e}^{4}x-4992\,A{c}^{2}{d}^{3}{e}^{2}x+18018\,B{a}^{2}{e}^{5}x+13728\,Bac{d}^{2}{e}^{3}x+3840\,B{c}^{2}{d}^{4}ex+30030\,A{a}^{2}{e}^{5}+13728\,A{d}^{2}ac{e}^{3}+3328\,A{d}^{4}{c}^{2}e-12012\,Bd{a}^{2}{e}^{4}-9152\,aBc{d}^{3}{e}^{2}-2560\,B{c}^{2}{d}^{5}}{45045\,{e}^{6}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+a)^2*(e*x+d)^(1/2),x)
[Out]
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Maxima [A] time = 0.686161, size = 335, normalized size = 1.54 \[ \frac{2 \,{\left (3465 \,{\left (e x + d\right )}^{\frac{13}{2}} B c^{2} - 4095 \,{\left (5 \, B c^{2} d - A c^{2} e\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 10010 \,{\left (5 \, B c^{2} d^{2} - 2 \, A c^{2} d e + B a c e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 12870 \,{\left (5 \, B c^{2} d^{3} - 3 \, A c^{2} d^{2} e + 3 \, B a c d e^{2} - A a c e^{3}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 9009 \,{\left (5 \, B c^{2} d^{4} - 4 \, A c^{2} d^{3} e + 6 \, B a c d^{2} e^{2} - 4 \, A a c d e^{3} + B a^{2} e^{4}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 15015 \,{\left (B c^{2} d^{5} - A c^{2} d^{4} e + 2 \, B a c d^{3} e^{2} - 2 \, A a c d^{2} e^{3} + B a^{2} d e^{4} - A a^{2} e^{5}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{45045 \, e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(B*x + A)*sqrt(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.26666, size = 432, normalized size = 1.98 \[ \frac{2 \,{\left (3465 \, B c^{2} e^{6} x^{6} - 1280 \, B c^{2} d^{6} + 1664 \, A c^{2} d^{5} e - 4576 \, B a c d^{4} e^{2} + 6864 \, A a c d^{3} e^{3} - 6006 \, B a^{2} d^{2} e^{4} + 15015 \, A a^{2} d e^{5} + 315 \,{\left (B c^{2} d e^{5} + 13 \, A c^{2} e^{6}\right )} x^{5} - 35 \,{\left (10 \, B c^{2} d^{2} e^{4} - 13 \, A c^{2} d e^{5} - 286 \, B a c e^{6}\right )} x^{4} + 10 \,{\left (40 \, B c^{2} d^{3} e^{3} - 52 \, A c^{2} d^{2} e^{4} + 143 \, B a c d e^{5} + 1287 \, A a c e^{6}\right )} x^{3} - 3 \,{\left (160 \, B c^{2} d^{4} e^{2} - 208 \, A c^{2} d^{3} e^{3} + 572 \, B a c d^{2} e^{4} - 858 \, A a c d e^{5} - 3003 \, B a^{2} e^{6}\right )} x^{2} +{\left (640 \, B c^{2} d^{5} e - 832 \, A c^{2} d^{4} e^{2} + 2288 \, B a c d^{3} e^{3} - 3432 \, A a c d^{2} e^{4} + 3003 \, B a^{2} d e^{5} + 15015 \, A a^{2} e^{6}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(B*x + A)*sqrt(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.88147, size = 308, normalized size = 1.41 \[ \frac{2 \left (\frac{B c^{2} \left (d + e x\right )^{\frac{13}{2}}}{13 e^{5}} + \frac{\left (d + e x\right )^{\frac{11}{2}} \left (A c^{2} e - 5 B c^{2} d\right )}{11 e^{5}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (- 4 A c^{2} d e + 2 B a c e^{2} + 10 B c^{2} d^{2}\right )}{9 e^{5}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (2 A a c e^{3} + 6 A c^{2} d^{2} e - 6 B a c d e^{2} - 10 B c^{2} d^{3}\right )}{7 e^{5}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (- 4 A a c d e^{3} - 4 A c^{2} d^{3} e + B a^{2} e^{4} + 6 B a c d^{2} e^{2} + 5 B c^{2} d^{4}\right )}{5 e^{5}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (A a^{2} e^{5} + 2 A a c d^{2} e^{3} + A c^{2} d^{4} e - B a^{2} d e^{4} - 2 B a c d^{3} e^{2} - B c^{2} d^{5}\right )}{3 e^{5}}\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+a)**2*(e*x+d)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.293261, size = 450, normalized size = 2.06 \[ \frac{2}{45045} \,{\left (3003 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} B a^{2} e^{\left (-1\right )} + 858 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} A a c e^{\left (-14\right )} + 286 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} B a c e^{\left (-27\right )} + 13 \,{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} e^{40} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d e^{40} + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} e^{40} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} e^{40} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4} e^{40}\right )} A c^{2} e^{\left (-44\right )} + 5 \,{\left (693 \,{\left (x e + d\right )}^{\frac{13}{2}} e^{60} - 4095 \,{\left (x e + d\right )}^{\frac{11}{2}} d e^{60} + 10010 \,{\left (x e + d\right )}^{\frac{9}{2}} d^{2} e^{60} - 12870 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{3} e^{60} + 9009 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{4} e^{60} - 3003 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{5} e^{60}\right )} B c^{2} e^{\left (-65\right )} + 15015 \,{\left (x e + d\right )}^{\frac{3}{2}} A a^{2}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(B*x + A)*sqrt(e*x + d),x, algorithm="giac")
[Out]