Optimal. Leaf size=77 \[ \frac{2}{5} a^2 A x^{5/2}+\frac{2}{7} a^2 B x^{7/2}+\frac{4}{9} a A c x^{9/2}+\frac{4}{11} a B c x^{11/2}+\frac{2}{13} A c^2 x^{13/2}+\frac{2}{15} B c^2 x^{15/2} \]
[Out]
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Rubi [A] time = 0.0717559, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{2}{5} a^2 A x^{5/2}+\frac{2}{7} a^2 B x^{7/2}+\frac{4}{9} a A c x^{9/2}+\frac{4}{11} a B c x^{11/2}+\frac{2}{13} A c^2 x^{13/2}+\frac{2}{15} B c^2 x^{15/2} \]
Antiderivative was successfully verified.
[In] Int[x^(3/2)*(A + B*x)*(a + c*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 8.7153, size = 80, normalized size = 1.04 \[ \frac{2 A a^{2} x^{\frac{5}{2}}}{5} + \frac{4 A a c x^{\frac{9}{2}}}{9} + \frac{2 A c^{2} x^{\frac{13}{2}}}{13} + \frac{2 B a^{2} x^{\frac{7}{2}}}{7} + \frac{4 B a c x^{\frac{11}{2}}}{11} + \frac{2 B c^{2} x^{\frac{15}{2}}}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3/2)*(B*x+A)*(c*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.0299843, size = 54, normalized size = 0.7 \[ \frac{2 x^{5/2} \left (1287 a^2 (7 A+5 B x)+910 a c x^2 (11 A+9 B x)+231 c^2 x^4 (15 A+13 B x)\right )}{45045} \]
Antiderivative was successfully verified.
[In] Integrate[x^(3/2)*(A + B*x)*(a + c*x^2)^2,x]
[Out]
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Maple [A] time = 0.009, size = 54, normalized size = 0.7 \[{\frac{6006\,B{c}^{2}{x}^{5}+6930\,A{c}^{2}{x}^{4}+16380\,aBc{x}^{3}+20020\,aAc{x}^{2}+12870\,{a}^{2}Bx+18018\,A{a}^{2}}{45045}{x}^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(3/2)*(B*x+A)*(c*x^2+a)^2,x)
[Out]
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Maxima [A] time = 0.675944, size = 72, normalized size = 0.94 \[ \frac{2}{15} \, B c^{2} x^{\frac{15}{2}} + \frac{2}{13} \, A c^{2} x^{\frac{13}{2}} + \frac{4}{11} \, B a c x^{\frac{11}{2}} + \frac{4}{9} \, A a c x^{\frac{9}{2}} + \frac{2}{7} \, B a^{2} x^{\frac{7}{2}} + \frac{2}{5} \, A a^{2} x^{\frac{5}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(B*x + A)*x^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.270694, size = 78, normalized size = 1.01 \[ \frac{2}{45045} \,{\left (3003 \, B c^{2} x^{7} + 3465 \, A c^{2} x^{6} + 8190 \, B a c x^{5} + 10010 \, A a c x^{4} + 6435 \, B a^{2} x^{3} + 9009 \, A a^{2} x^{2}\right )} \sqrt{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(B*x + A)*x^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 9.29356, size = 80, normalized size = 1.04 \[ \frac{2 A a^{2} x^{\frac{5}{2}}}{5} + \frac{4 A a c x^{\frac{9}{2}}}{9} + \frac{2 A c^{2} x^{\frac{13}{2}}}{13} + \frac{2 B a^{2} x^{\frac{7}{2}}}{7} + \frac{4 B a c x^{\frac{11}{2}}}{11} + \frac{2 B c^{2} x^{\frac{15}{2}}}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(3/2)*(B*x+A)*(c*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.268328, size = 72, normalized size = 0.94 \[ \frac{2}{15} \, B c^{2} x^{\frac{15}{2}} + \frac{2}{13} \, A c^{2} x^{\frac{13}{2}} + \frac{4}{11} \, B a c x^{\frac{11}{2}} + \frac{4}{9} \, A a c x^{\frac{9}{2}} + \frac{2}{7} \, B a^{2} x^{\frac{7}{2}} + \frac{2}{5} \, A a^{2} x^{\frac{5}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(B*x + A)*x^(3/2),x, algorithm="giac")
[Out]