Optimal. Leaf size=83 \[ -\frac{2 a^3 A}{3 x^{3/2}}+2 a^2 \sqrt{x} (a B+3 A b)+\frac{2}{9} b^2 x^{9/2} (3 a B+A b)+\frac{6}{5} a b x^{5/2} (a B+A b)+\frac{2}{13} b^3 B x^{13/2} \]
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Rubi [A] time = 0.112388, antiderivative size = 83, 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 \[ -\frac{2 a^3 A}{3 x^{3/2}}+2 a^2 \sqrt{x} (a B+3 A b)+\frac{2}{9} b^2 x^{9/2} (3 a B+A b)+\frac{6}{5} a b x^{5/2} (a B+A b)+\frac{2}{13} b^3 B x^{13/2} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^2)^3*(A + B*x^2))/x^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 16.6911, size = 83, normalized size = 1. \[ - \frac{2 A a^{3}}{3 x^{\frac{3}{2}}} + \frac{2 B b^{3} x^{\frac{13}{2}}}{13} + 2 a^{2} \sqrt{x} \left (3 A b + B a\right ) + \frac{6 a b x^{\frac{5}{2}} \left (A b + B a\right )}{5} + \frac{2 b^{2} x^{\frac{9}{2}} \left (A b + 3 B a\right )}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**3*(B*x**2+A)/x**(5/2),x)
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Mathematica [A] time = 0.0444066, size = 71, normalized size = 0.86 \[ \frac{2 \left (-195 a^3 A+585 a^2 x^2 (a B+3 A b)+65 b^2 x^6 (3 a B+A b)+351 a b x^4 (a B+A b)+45 b^3 B x^8\right )}{585 x^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^2)^3*(A + B*x^2))/x^(5/2),x]
[Out]
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Maple [A] time = 0.008, size = 80, normalized size = 1. \[ -{\frac{-90\,{b}^{3}B{x}^{8}-130\,{x}^{6}{b}^{3}A-390\,{x}^{6}a{b}^{2}B-702\,{x}^{4}a{b}^{2}A-702\,{x}^{4}{a}^{2}bB-3510\,A{a}^{2}b{x}^{2}-1170\,B{a}^{3}{x}^{2}+390\,{a}^{3}A}{585}{x}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^3*(B*x^2+A)/x^(5/2),x)
[Out]
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Maxima [A] time = 1.33559, size = 99, normalized size = 1.19 \[ \frac{2}{13} \, B b^{3} x^{\frac{13}{2}} + \frac{2}{9} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{9}{2}} + \frac{6}{5} \,{\left (B a^{2} b + A a b^{2}\right )} x^{\frac{5}{2}} - \frac{2 \, A a^{3}}{3 \, x^{\frac{3}{2}}} + 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} \sqrt{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)^3/x^(5/2),x, algorithm="maxima")
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Fricas [A] time = 0.220424, size = 101, normalized size = 1.22 \[ \frac{2 \,{\left (45 \, B b^{3} x^{8} + 65 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + 351 \,{\left (B a^{2} b + A a b^{2}\right )} x^{4} - 195 \, A a^{3} + 585 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2}\right )}}{585 \, x^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)^3/x^(5/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 24.298, size = 110, normalized size = 1.33 \[ - \frac{2 A a^{3}}{3 x^{\frac{3}{2}}} + 6 A a^{2} b \sqrt{x} + \frac{6 A a b^{2} x^{\frac{5}{2}}}{5} + \frac{2 A b^{3} x^{\frac{9}{2}}}{9} + 2 B a^{3} \sqrt{x} + \frac{6 B a^{2} b x^{\frac{5}{2}}}{5} + \frac{2 B a b^{2} x^{\frac{9}{2}}}{3} + \frac{2 B b^{3} x^{\frac{13}{2}}}{13} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**3*(B*x**2+A)/x**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.21813, size = 104, normalized size = 1.25 \[ \frac{2}{13} \, B b^{3} x^{\frac{13}{2}} + \frac{2}{3} \, B a b^{2} x^{\frac{9}{2}} + \frac{2}{9} \, A b^{3} x^{\frac{9}{2}} + \frac{6}{5} \, B a^{2} b x^{\frac{5}{2}} + \frac{6}{5} \, A a b^{2} x^{\frac{5}{2}} + 2 \, B a^{3} \sqrt{x} + 6 \, A a^{2} b \sqrt{x} - \frac{2 \, A a^{3}}{3 \, x^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)^3/x^(5/2),x, algorithm="giac")
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