Optimal. Leaf size=109 \[ \frac{2}{3} a^3 A x^{3/2}+\frac{2}{5} a^3 B x^{5/2}+\frac{6}{7} a^2 A c x^{7/2}+\frac{2}{3} a^2 B c x^{9/2}+\frac{6}{11} a A c^2 x^{11/2}+\frac{6}{13} a B c^2 x^{13/2}+\frac{2}{15} A c^3 x^{15/2}+\frac{2}{17} B c^3 x^{17/2} \]
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Rubi [A] time = 0.0977503, antiderivative size = 109, 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 \[ \frac{2}{3} a^3 A x^{3/2}+\frac{2}{5} a^3 B x^{5/2}+\frac{6}{7} a^2 A c x^{7/2}+\frac{2}{3} a^2 B c x^{9/2}+\frac{6}{11} a A c^2 x^{11/2}+\frac{6}{13} a B c^2 x^{13/2}+\frac{2}{15} A c^3 x^{15/2}+\frac{2}{17} B c^3 x^{17/2} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]*(A + B*x)*(a + c*x^2)^3,x]
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Rubi in Sympy [A] time = 11.4166, size = 114, normalized size = 1.05 \[ \frac{2 A a^{3} x^{\frac{3}{2}}}{3} + \frac{6 A a^{2} c x^{\frac{7}{2}}}{7} + \frac{6 A a c^{2} x^{\frac{11}{2}}}{11} + \frac{2 A c^{3} x^{\frac{15}{2}}}{15} + \frac{2 B a^{3} x^{\frac{5}{2}}}{5} + \frac{2 B a^{2} c x^{\frac{9}{2}}}{3} + \frac{6 B a c^{2} x^{\frac{13}{2}}}{13} + \frac{2 B c^{3} x^{\frac{17}{2}}}{17} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+a)**3*x**(1/2),x)
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Mathematica [A] time = 0.0425104, size = 83, normalized size = 0.76 \[ \frac{2}{15} a^3 x^{3/2} (5 A+3 B x)+\frac{2}{21} a^2 c x^{7/2} (9 A+7 B x)+\frac{6}{143} a c^2 x^{11/2} (13 A+11 B x)+\frac{2}{255} c^3 x^{15/2} (17 A+15 B x) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x]*(A + B*x)*(a + c*x^2)^3,x]
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Maple [A] time = 0.009, size = 78, normalized size = 0.7 \[{\frac{30030\,B{c}^{3}{x}^{7}+34034\,A{c}^{3}{x}^{6}+117810\,aB{c}^{2}{x}^{5}+139230\,aA{c}^{2}{x}^{4}+170170\,{a}^{2}Bc{x}^{3}+218790\,{a}^{2}Ac{x}^{2}+102102\,{a}^{3}Bx+170170\,A{a}^{3}}{255255}{x}^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+a)^3*x^(1/2),x)
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Maxima [A] time = 0.678036, size = 104, normalized size = 0.95 \[ \frac{2}{17} \, B c^{3} x^{\frac{17}{2}} + \frac{2}{15} \, A c^{3} x^{\frac{15}{2}} + \frac{6}{13} \, B a c^{2} x^{\frac{13}{2}} + \frac{6}{11} \, A a c^{2} x^{\frac{11}{2}} + \frac{2}{3} \, B a^{2} c x^{\frac{9}{2}} + \frac{6}{7} \, A a^{2} c x^{\frac{7}{2}} + \frac{2}{5} \, B a^{3} x^{\frac{5}{2}} + \frac{2}{3} \, A a^{3} x^{\frac{3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3*(B*x + A)*sqrt(x),x, algorithm="maxima")
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Fricas [A] time = 0.264453, size = 108, normalized size = 0.99 \[ \frac{2}{255255} \,{\left (15015 \, B c^{3} x^{8} + 17017 \, A c^{3} x^{7} + 58905 \, B a c^{2} x^{6} + 69615 \, A a c^{2} x^{5} + 85085 \, B a^{2} c x^{4} + 109395 \, A a^{2} c x^{3} + 51051 \, B a^{3} x^{2} + 85085 \, A a^{3} x\right )} \sqrt{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3*(B*x + A)*sqrt(x),x, algorithm="fricas")
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Sympy [A] time = 5.27318, size = 114, normalized size = 1.05 \[ \frac{2 A a^{3} x^{\frac{3}{2}}}{3} + \frac{6 A a^{2} c x^{\frac{7}{2}}}{7} + \frac{6 A a c^{2} x^{\frac{11}{2}}}{11} + \frac{2 A c^{3} x^{\frac{15}{2}}}{15} + \frac{2 B a^{3} x^{\frac{5}{2}}}{5} + \frac{2 B a^{2} c x^{\frac{9}{2}}}{3} + \frac{6 B a c^{2} x^{\frac{13}{2}}}{13} + \frac{2 B c^{3} x^{\frac{17}{2}}}{17} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+a)**3*x**(1/2),x)
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GIAC/XCAS [A] time = 0.269533, size = 104, normalized size = 0.95 \[ \frac{2}{17} \, B c^{3} x^{\frac{17}{2}} + \frac{2}{15} \, A c^{3} x^{\frac{15}{2}} + \frac{6}{13} \, B a c^{2} x^{\frac{13}{2}} + \frac{6}{11} \, A a c^{2} x^{\frac{11}{2}} + \frac{2}{3} \, B a^{2} c x^{\frac{9}{2}} + \frac{6}{7} \, A a^{2} c x^{\frac{7}{2}} + \frac{2}{5} \, B a^{3} x^{\frac{5}{2}} + \frac{2}{3} \, A a^{3} x^{\frac{3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^3*(B*x + A)*sqrt(x),x, algorithm="giac")
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