Optimal. Leaf size=85 \[ \frac{2}{5} a^3 A x^{5/2}+\frac{2}{9} a^2 x^{9/2} (a B+3 A b)+\frac{2}{17} b^2 x^{17/2} (3 a B+A b)+\frac{6}{13} a b x^{13/2} (a B+A b)+\frac{2}{21} b^3 B x^{21/2} \]
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Rubi [A] time = 0.116648, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{2}{5} a^3 A x^{5/2}+\frac{2}{9} a^2 x^{9/2} (a B+3 A b)+\frac{2}{17} b^2 x^{17/2} (3 a B+A b)+\frac{6}{13} a b x^{13/2} (a B+A b)+\frac{2}{21} b^3 B x^{21/2} \]
Antiderivative was successfully verified.
[In] Int[x^(3/2)*(a + b*x^2)^3*(A + B*x^2),x]
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Rubi in Sympy [A] time = 16.6313, size = 85, normalized size = 1. \[ \frac{2 A a^{3} x^{\frac{5}{2}}}{5} + \frac{2 B b^{3} x^{\frac{21}{2}}}{21} + \frac{2 a^{2} x^{\frac{9}{2}} \left (3 A b + B a\right )}{9} + \frac{6 a b x^{\frac{13}{2}} \left (A b + B a\right )}{13} + \frac{2 b^{2} x^{\frac{17}{2}} \left (A b + 3 B a\right )}{17} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3/2)*(b*x**2+a)**3*(B*x**2+A),x)
[Out]
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Mathematica [A] time = 0.0381439, size = 85, normalized size = 1. \[ \frac{2}{5} a^3 A x^{5/2}+\frac{2}{9} a^2 x^{9/2} (a B+3 A b)+\frac{2}{17} b^2 x^{17/2} (3 a B+A b)+\frac{6}{13} a b x^{13/2} (a B+A b)+\frac{2}{21} b^3 B x^{21/2} \]
Antiderivative was successfully verified.
[In] Integrate[x^(3/2)*(a + b*x^2)^3*(A + B*x^2),x]
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Maple [A] time = 0.008, size = 80, normalized size = 0.9 \[{\frac{6630\,B{b}^{3}{x}^{8}+8190\,{x}^{6}A{b}^{3}+24570\,{x}^{6}Ba{b}^{2}+32130\,{x}^{4}Aa{b}^{2}+32130\,{x}^{4}B{a}^{2}b+46410\,{x}^{2}A{a}^{2}b+15470\,{x}^{2}B{a}^{3}+27846\,A{a}^{3}}{69615}{x}^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(3/2)*(b*x^2+a)^3*(B*x^2+A),x)
[Out]
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Maxima [A] time = 1.36807, size = 99, normalized size = 1.16 \[ \frac{2}{21} \, B b^{3} x^{\frac{21}{2}} + \frac{2}{17} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{17}{2}} + \frac{6}{13} \,{\left (B a^{2} b + A a b^{2}\right )} x^{\frac{13}{2}} + \frac{2}{5} \, A a^{3} x^{\frac{5}{2}} + \frac{2}{9} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac{9}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)^3*x^(3/2),x, algorithm="maxima")
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Fricas [A] time = 0.216124, size = 105, normalized size = 1.24 \[ \frac{2}{69615} \,{\left (3315 \, B b^{3} x^{10} + 4095 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{8} + 16065 \,{\left (B a^{2} b + A a b^{2}\right )} x^{6} + 13923 \, A a^{3} x^{2} + 7735 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{4}\right )} \sqrt{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)^3*x^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 38.1527, size = 114, normalized size = 1.34 \[ \frac{2 A a^{3} x^{\frac{5}{2}}}{5} + \frac{2 A a^{2} b x^{\frac{9}{2}}}{3} + \frac{6 A a b^{2} x^{\frac{13}{2}}}{13} + \frac{2 A b^{3} x^{\frac{17}{2}}}{17} + \frac{2 B a^{3} x^{\frac{9}{2}}}{9} + \frac{6 B a^{2} b x^{\frac{13}{2}}}{13} + \frac{6 B a b^{2} x^{\frac{17}{2}}}{17} + \frac{2 B b^{3} x^{\frac{21}{2}}}{21} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(3/2)*(b*x**2+a)**3*(B*x**2+A),x)
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GIAC/XCAS [A] time = 0.21342, size = 104, normalized size = 1.22 \[ \frac{2}{21} \, B b^{3} x^{\frac{21}{2}} + \frac{6}{17} \, B a b^{2} x^{\frac{17}{2}} + \frac{2}{17} \, A b^{3} x^{\frac{17}{2}} + \frac{6}{13} \, B a^{2} b x^{\frac{13}{2}} + \frac{6}{13} \, A a b^{2} x^{\frac{13}{2}} + \frac{2}{9} \, B a^{3} x^{\frac{9}{2}} + \frac{2}{3} \, A a^{2} b x^{\frac{9}{2}} + \frac{2}{5} \, A a^{3} x^{\frac{5}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)^3*x^(3/2),x, algorithm="giac")
[Out]