Optimal. Leaf size=218 \[ -\frac{2 b^3 (d+e x)^{13/2} (-4 a B e-A b e+5 b B d)}{13 e^6}+\frac{4 b^2 (d+e x)^{11/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{11 e^6}-\frac{4 b (d+e x)^{9/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{9 e^6}+\frac{2 (d+e x)^{7/2} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{7 e^6}-\frac{2 (d+e x)^{5/2} (b d-a e)^4 (B d-A e)}{5 e^6}+\frac{2 b^4 B (d+e x)^{15/2}}{15 e^6} \]
[Out]
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Rubi [A] time = 0.263694, antiderivative size = 218, 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 \[ -\frac{2 b^3 (d+e x)^{13/2} (-4 a B e-A b e+5 b B d)}{13 e^6}+\frac{4 b^2 (d+e x)^{11/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{11 e^6}-\frac{4 b (d+e x)^{9/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{9 e^6}+\frac{2 (d+e x)^{7/2} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{7 e^6}-\frac{2 (d+e x)^{5/2} (b d-a e)^4 (B d-A e)}{5 e^6}+\frac{2 b^4 B (d+e x)^{15/2}}{15 e^6} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)*(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 105.898, size = 221, normalized size = 1.01 \[ \frac{2 B b^{4} \left (d + e x\right )^{\frac{15}{2}}}{15 e^{6}} + \frac{2 b^{3} \left (d + e x\right )^{\frac{13}{2}} \left (A b e + 4 B a e - 5 B b d\right )}{13 e^{6}} + \frac{4 b^{2} \left (d + e x\right )^{\frac{11}{2}} \left (a e - b d\right ) \left (2 A b e + 3 B a e - 5 B b d\right )}{11 e^{6}} + \frac{4 b \left (d + e x\right )^{\frac{9}{2}} \left (a e - b d\right )^{2} \left (3 A b e + 2 B a e - 5 B b d\right )}{9 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (a e - b d\right )^{3} \left (4 A b e + B a e - 5 B b d\right )}{7 e^{6}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (A e - B d\right ) \left (a e - b d\right )^{4}}{5 e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**(3/2)*(b**2*x**2+2*a*b*x+a**2)**2,x)
[Out]
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Mathematica [A] time = 0.565823, size = 339, normalized size = 1.56 \[ \frac{2 (d+e x)^{5/2} \left (1287 a^4 e^4 (7 A e-2 B d+5 B e x)+572 a^3 b e^3 \left (9 A e (5 e x-2 d)+B \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )-78 a^2 b^2 e^2 \left (3 B \left (16 d^3-40 d^2 e x+70 d e^2 x^2-105 e^3 x^3\right )-11 A e \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )+12 a b^3 e \left (13 A e \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+B \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )+b^4 \left (3 A e \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )+B \left (-256 d^5+640 d^4 e x-1120 d^3 e^2 x^2+1680 d^2 e^3 x^3-2310 d e^4 x^4+3003 e^5 x^5\right )\right )\right )}{45045 e^6} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)*(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
[Out]
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Maple [B] time = 0.013, size = 469, normalized size = 2.2 \[{\frac{6006\,{b}^{4}B{x}^{5}{e}^{5}+6930\,A{b}^{4}{e}^{5}{x}^{4}+27720\,Ba{b}^{3}{e}^{5}{x}^{4}-4620\,B{b}^{4}d{e}^{4}{x}^{4}+32760\,Aa{b}^{3}{e}^{5}{x}^{3}-5040\,A{b}^{4}d{e}^{4}{x}^{3}+49140\,B{a}^{2}{b}^{2}{e}^{5}{x}^{3}-20160\,Ba{b}^{3}d{e}^{4}{x}^{3}+3360\,B{b}^{4}{d}^{2}{e}^{3}{x}^{3}+60060\,A{a}^{2}{b}^{2}{e}^{5}{x}^{2}-21840\,Aa{b}^{3}d{e}^{4}{x}^{2}+3360\,A{b}^{4}{d}^{2}{e}^{3}{x}^{2}+40040\,B{a}^{3}b{e}^{5}{x}^{2}-32760\,B{a}^{2}{b}^{2}d{e}^{4}{x}^{2}+13440\,Ba{b}^{3}{d}^{2}{e}^{3}{x}^{2}-2240\,B{b}^{4}{d}^{3}{e}^{2}{x}^{2}+51480\,A{a}^{3}b{e}^{5}x-34320\,A{a}^{2}{b}^{2}d{e}^{4}x+12480\,Aa{b}^{3}{d}^{2}{e}^{3}x-1920\,A{b}^{4}{d}^{3}{e}^{2}x+12870\,B{a}^{4}{e}^{5}x-22880\,B{a}^{3}bd{e}^{4}x+18720\,B{a}^{2}{b}^{2}{d}^{2}{e}^{3}x-7680\,Ba{b}^{3}{d}^{3}{e}^{2}x+1280\,B{b}^{4}{d}^{4}ex+18018\,A{a}^{4}{e}^{5}-20592\,Ad{a}^{3}b{e}^{4}+13728\,A{a}^{2}{b}^{2}{d}^{2}{e}^{3}-4992\,Aa{b}^{3}{d}^{3}{e}^{2}+768\,A{d}^{4}{b}^{4}e-5148\,B{a}^{4}d{e}^{4}+9152\,B{d}^{2}{a}^{3}b{e}^{3}-7488\,B{d}^{3}{a}^{2}{b}^{2}{e}^{2}+3072\,B{d}^{4}a{b}^{3}e-512\,{b}^{4}B{d}^{5}}{45045\,{e}^{6}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^2,x)
[Out]
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Maxima [A] time = 0.73054, size = 552, normalized size = 2.53 \[ \frac{2 \,{\left (3003 \,{\left (e x + d\right )}^{\frac{15}{2}} B b^{4} - 3465 \,{\left (5 \, B b^{4} d -{\left (4 \, B a b^{3} + A b^{4}\right )} e\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 8190 \,{\left (5 \, B b^{4} d^{2} - 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e +{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 10010 \,{\left (5 \, B b^{4} d^{3} - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{2} -{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{3}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 6435 \,{\left (5 \, B b^{4} d^{4} - 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{2} - 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{4}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 9009 \,{\left (B b^{4} d^{5} - A a^{4} e^{5} -{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{45045 \, e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)*(e*x + d)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.294882, size = 876, normalized size = 4.02 \[ \frac{2 \,{\left (3003 \, B b^{4} e^{7} x^{7} - 256 \, B b^{4} d^{7} + 9009 \, A a^{4} d^{2} e^{5} + 384 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{6} e - 1248 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{5} e^{2} + 2288 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{4} e^{3} - 2574 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d^{3} e^{4} + 231 \,{\left (16 \, B b^{4} d e^{6} + 15 \,{\left (4 \, B a b^{3} + A b^{4}\right )} e^{7}\right )} x^{6} + 63 \,{\left (B b^{4} d^{2} e^{5} + 70 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{6} + 130 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{7}\right )} x^{5} - 35 \,{\left (2 \, B b^{4} d^{3} e^{4} - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{5} - 312 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{6} - 286 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{7}\right )} x^{4} + 5 \,{\left (16 \, B b^{4} d^{4} e^{3} - 24 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{4} + 78 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{5} + 2860 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{6} + 1287 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{7}\right )} x^{3} - 3 \,{\left (32 \, B b^{4} d^{5} e^{2} - 3003 \, A a^{4} e^{7} - 48 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e^{3} + 156 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{4} - 286 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{5} - 3432 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{6}\right )} x^{2} +{\left (128 \, B b^{4} d^{6} e + 18018 \, A a^{4} d e^{6} - 192 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{5} e^{2} + 624 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{4} e^{3} - 1144 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{3} e^{4} + 1287 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d^{2} e^{5}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)*(e*x + d)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 16.5572, size = 1297, normalized size = 5.95 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**(3/2)*(b**2*x**2+2*a*b*x+a**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.301396, 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)^2*(B*x + A)*(e*x + d)^(3/2),x, algorithm="giac")
[Out]