Optimal. Leaf size=137 \[ \frac{2}{11} d x^{11/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{7} c x^{7/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac{2 a^2 c^3}{\sqrt{x}}+\frac{2}{3} a c^2 x^{3/2} (3 a d+2 b c)+\frac{2}{15} b d^2 x^{15/2} (2 a d+3 b c)+\frac{2}{19} b^2 d^3 x^{19/2} \]
[Out]
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Rubi [A] time = 0.173131, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{2}{11} d x^{11/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{7} c x^{7/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac{2 a^2 c^3}{\sqrt{x}}+\frac{2}{3} a c^2 x^{3/2} (3 a d+2 b c)+\frac{2}{15} b d^2 x^{15/2} (2 a d+3 b c)+\frac{2}{19} b^2 d^3 x^{19/2} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x^2)^2*(c + d*x^2)^3)/x^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 29.9344, size = 143, normalized size = 1.04 \[ - \frac{2 a^{2} c^{3}}{\sqrt{x}} + \frac{2 a c^{2} x^{\frac{3}{2}} \left (3 a d + 2 b c\right )}{3} + \frac{2 b^{2} d^{3} x^{\frac{19}{2}}}{19} + \frac{2 b d^{2} x^{\frac{15}{2}} \left (2 a d + 3 b c\right )}{15} + \frac{2 c x^{\frac{7}{2}} \left (3 a^{2} d^{2} + 6 a b c d + b^{2} c^{2}\right )}{7} + \frac{2 d x^{\frac{11}{2}} \left (a^{2} d^{2} + 6 a b c d + 3 b^{2} c^{2}\right )}{11} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2*(d*x**2+c)**3/x**(3/2),x)
[Out]
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Mathematica [A] time = 0.0882251, size = 137, normalized size = 1. \[ \frac{2}{11} d x^{11/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{7} c x^{7/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )-\frac{2 a^2 c^3}{\sqrt{x}}+\frac{2}{3} a c^2 x^{3/2} (3 a d+2 b c)+\frac{2}{15} b d^2 x^{15/2} (2 a d+3 b c)+\frac{2}{19} b^2 d^3 x^{19/2} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x^2)^2*(c + d*x^2)^3)/x^(3/2),x]
[Out]
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Maple [A] time = 0.01, size = 138, normalized size = 1. \[ -{\frac{-2310\,{b}^{2}{d}^{3}{x}^{10}-5852\,{x}^{8}ab{d}^{3}-8778\,{x}^{8}{b}^{2}c{d}^{2}-3990\,{x}^{6}{a}^{2}{d}^{3}-23940\,{x}^{6}abc{d}^{2}-11970\,{x}^{6}{b}^{2}{c}^{2}d-18810\,{x}^{4}{a}^{2}c{d}^{2}-37620\,{x}^{4}ab{c}^{2}d-6270\,{x}^{4}{b}^{2}{c}^{3}-43890\,{x}^{2}{a}^{2}{c}^{2}d-29260\,{x}^{2}ab{c}^{3}+43890\,{a}^{2}{c}^{3}}{21945}{\frac{1}{\sqrt{x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2*(d*x^2+c)^3/x^(3/2),x)
[Out]
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Maxima [A] time = 1.36984, size = 171, normalized size = 1.25 \[ \frac{2}{19} \, b^{2} d^{3} x^{\frac{19}{2}} + \frac{2}{15} \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{\frac{15}{2}} + \frac{2}{11} \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac{11}{2}} - \frac{2 \, a^{2} c^{3}}{\sqrt{x}} + \frac{2}{7} \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{\frac{7}{2}} + \frac{2}{3} \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{\frac{3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)^3/x^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.218475, size = 174, normalized size = 1.27 \[ \frac{2 \,{\left (1155 \, b^{2} d^{3} x^{10} + 1463 \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{8} + 1995 \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} - 21945 \, a^{2} c^{3} + 3135 \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} + 7315 \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}\right )}}{21945 \, \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)^3/x^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 41.9939, size = 189, normalized size = 1.38 \[ - \frac{2 a^{2} c^{3}}{\sqrt{x}} + 2 a^{2} c^{2} d x^{\frac{3}{2}} + \frac{6 a^{2} c d^{2} x^{\frac{7}{2}}}{7} + \frac{2 a^{2} d^{3} x^{\frac{11}{2}}}{11} + \frac{4 a b c^{3} x^{\frac{3}{2}}}{3} + \frac{12 a b c^{2} d x^{\frac{7}{2}}}{7} + \frac{12 a b c d^{2} x^{\frac{11}{2}}}{11} + \frac{4 a b d^{3} x^{\frac{15}{2}}}{15} + \frac{2 b^{2} c^{3} x^{\frac{7}{2}}}{7} + \frac{6 b^{2} c^{2} d x^{\frac{11}{2}}}{11} + \frac{2 b^{2} c d^{2} x^{\frac{15}{2}}}{5} + \frac{2 b^{2} d^{3} x^{\frac{19}{2}}}{19} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**2*(d*x**2+c)**3/x**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.238057, size = 182, normalized size = 1.33 \[ \frac{2}{19} \, b^{2} d^{3} x^{\frac{19}{2}} + \frac{2}{5} \, b^{2} c d^{2} x^{\frac{15}{2}} + \frac{4}{15} \, a b d^{3} x^{\frac{15}{2}} + \frac{6}{11} \, b^{2} c^{2} d x^{\frac{11}{2}} + \frac{12}{11} \, a b c d^{2} x^{\frac{11}{2}} + \frac{2}{11} \, a^{2} d^{3} x^{\frac{11}{2}} + \frac{2}{7} \, b^{2} c^{3} x^{\frac{7}{2}} + \frac{12}{7} \, a b c^{2} d x^{\frac{7}{2}} + \frac{6}{7} \, a^{2} c d^{2} x^{\frac{7}{2}} + \frac{4}{3} \, a b c^{3} x^{\frac{3}{2}} + 2 \, a^{2} c^{2} d x^{\frac{3}{2}} - \frac{2 \, a^{2} c^{3}}{\sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)^3/x^(3/2),x, algorithm="giac")
[Out]