Optimal. Leaf size=139 \[ \frac{2}{15} d x^{15/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{11} c x^{11/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{2}{3} a^2 c^3 x^{3/2}+\frac{2}{7} a c^2 x^{7/2} (3 a d+2 b c)+\frac{2}{19} b d^2 x^{19/2} (2 a d+3 b c)+\frac{2}{23} b^2 d^3 x^{23/2} \]
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Rubi [A] time = 0.171956, antiderivative size = 139, 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 \[ \frac{2}{15} d x^{15/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{11} c x^{11/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{2}{3} a^2 c^3 x^{3/2}+\frac{2}{7} a c^2 x^{7/2} (3 a d+2 b c)+\frac{2}{19} b d^2 x^{19/2} (2 a d+3 b c)+\frac{2}{23} b^2 d^3 x^{23/2} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]*(a + b*x^2)^2*(c + d*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 30.5208, size = 144, normalized size = 1.04 \[ \frac{2 a^{2} c^{3} x^{\frac{3}{2}}}{3} + \frac{2 a c^{2} x^{\frac{7}{2}} \left (3 a d + 2 b c\right )}{7} + \frac{2 b^{2} d^{3} x^{\frac{23}{2}}}{23} + \frac{2 b d^{2} x^{\frac{19}{2}} \left (2 a d + 3 b c\right )}{19} + \frac{2 c x^{\frac{11}{2}} \left (3 a^{2} d^{2} + 6 a b c d + b^{2} c^{2}\right )}{11} + \frac{2 d x^{\frac{15}{2}} \left (a^{2} d^{2} + 6 a b c d + 3 b^{2} c^{2}\right )}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**2*(d*x**2+c)**3*x**(1/2),x)
[Out]
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Mathematica [A] time = 0.0628594, size = 139, normalized size = 1. \[ \frac{2}{15} d x^{15/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{11} c x^{11/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{2}{3} a^2 c^3 x^{3/2}+\frac{2}{7} a c^2 x^{7/2} (3 a d+2 b c)+\frac{2}{19} b d^2 x^{19/2} (2 a d+3 b c)+\frac{2}{23} b^2 d^3 x^{23/2} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x]*(a + b*x^2)^2*(c + d*x^2)^3,x]
[Out]
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Maple [A] time = 0.01, size = 138, normalized size = 1. \[{\frac{43890\,{b}^{2}{d}^{3}{x}^{10}+106260\,{x}^{8}ab{d}^{3}+159390\,{x}^{8}{b}^{2}c{d}^{2}+67298\,{x}^{6}{a}^{2}{d}^{3}+403788\,{x}^{6}abc{d}^{2}+201894\,{x}^{6}{b}^{2}{c}^{2}d+275310\,{x}^{4}{a}^{2}c{d}^{2}+550620\,{x}^{4}ab{c}^{2}d+91770\,{x}^{4}{b}^{2}{c}^{3}+432630\,{x}^{2}{a}^{2}{c}^{2}d+288420\,{x}^{2}ab{c}^{3}+336490\,{a}^{2}{c}^{3}}{504735}{x}^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^2*(d*x^2+c)^3*x^(1/2),x)
[Out]
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Maxima [A] time = 1.34414, size = 171, normalized size = 1.23 \[ \frac{2}{23} \, b^{2} d^{3} x^{\frac{23}{2}} + \frac{2}{19} \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{\frac{19}{2}} + \frac{2}{15} \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac{15}{2}} + \frac{2}{3} \, a^{2} c^{3} x^{\frac{3}{2}} + \frac{2}{11} \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{\frac{11}{2}} + \frac{2}{7} \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{\frac{7}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)^3*sqrt(x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.214311, size = 176, normalized size = 1.27 \[ \frac{2}{504735} \,{\left (21945 \, b^{2} d^{3} x^{11} + 26565 \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{9} + 33649 \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{7} + 168245 \, a^{2} c^{3} x + 45885 \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{5} + 72105 \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{3}\right )} \sqrt{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)^3*sqrt(x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 16.2215, size = 155, normalized size = 1.12 \[ \frac{2 a^{2} c^{3} x^{\frac{3}{2}}}{3} + \frac{2 b^{2} d^{3} x^{\frac{23}{2}}}{23} + \frac{2 x^{\frac{19}{2}} \left (2 a b d^{3} + 3 b^{2} c d^{2}\right )}{19} + \frac{2 x^{\frac{15}{2}} \left (a^{2} d^{3} + 6 a b c d^{2} + 3 b^{2} c^{2} d\right )}{15} + \frac{2 x^{\frac{11}{2}} \left (3 a^{2} c d^{2} + 6 a b c^{2} d + b^{2} c^{3}\right )}{11} + \frac{2 x^{\frac{7}{2}} \left (3 a^{2} c^{2} d + 2 a b c^{3}\right )}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**2*(d*x**2+c)**3*x**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.233863, size = 182, normalized size = 1.31 \[ \frac{2}{23} \, b^{2} d^{3} x^{\frac{23}{2}} + \frac{6}{19} \, b^{2} c d^{2} x^{\frac{19}{2}} + \frac{4}{19} \, a b d^{3} x^{\frac{19}{2}} + \frac{2}{5} \, b^{2} c^{2} d x^{\frac{15}{2}} + \frac{4}{5} \, a b c d^{2} x^{\frac{15}{2}} + \frac{2}{15} \, a^{2} d^{3} x^{\frac{15}{2}} + \frac{2}{11} \, b^{2} c^{3} x^{\frac{11}{2}} + \frac{12}{11} \, a b c^{2} d x^{\frac{11}{2}} + \frac{6}{11} \, a^{2} c d^{2} x^{\frac{11}{2}} + \frac{4}{7} \, a b c^{3} x^{\frac{7}{2}} + \frac{6}{7} \, a^{2} c^{2} d x^{\frac{7}{2}} + \frac{2}{3} \, a^{2} c^{3} x^{\frac{3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)^3*sqrt(x),x, algorithm="giac")
[Out]