Optimal. Leaf size=127 \[ \frac{1}{11} d x^{11} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{9} c x^9 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{1}{5} a^2 c^3 x^5+\frac{1}{7} a c^2 x^7 (3 a d+2 b c)+\frac{1}{13} b d^2 x^{13} (2 a d+3 b c)+\frac{1}{15} b^2 d^3 x^{15} \]
[Out]
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Rubi [A] time = 0.251225, antiderivative size = 127, 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 \[ \frac{1}{11} d x^{11} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{9} c x^9 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{1}{5} a^2 c^3 x^5+\frac{1}{7} a c^2 x^7 (3 a d+2 b c)+\frac{1}{13} b d^2 x^{13} (2 a d+3 b c)+\frac{1}{15} b^2 d^3 x^{15} \]
Antiderivative was successfully verified.
[In] Int[x^4*(a + b*x^2)^2*(c + d*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 34.2225, size = 124, normalized size = 0.98 \[ \frac{a^{2} c^{3} x^{5}}{5} + \frac{a c^{2} x^{7} \left (3 a d + 2 b c\right )}{7} + \frac{b^{2} d^{3} x^{15}}{15} + \frac{b d^{2} x^{13} \left (2 a d + 3 b c\right )}{13} + \frac{c x^{9} \left (3 a^{2} d^{2} + 6 a b c d + b^{2} c^{2}\right )}{9} + \frac{d x^{11} \left (a^{2} d^{2} + 6 a b c d + 3 b^{2} c^{2}\right )}{11} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4*(b*x**2+a)**2*(d*x**2+c)**3,x)
[Out]
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Mathematica [A] time = 0.0451218, size = 127, normalized size = 1. \[ \frac{1}{11} d x^{11} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{9} c x^9 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{1}{5} a^2 c^3 x^5+\frac{1}{7} a c^2 x^7 (3 a d+2 b c)+\frac{1}{13} b d^2 x^{13} (2 a d+3 b c)+\frac{1}{15} b^2 d^3 x^{15} \]
Antiderivative was successfully verified.
[In] Integrate[x^4*(a + b*x^2)^2*(c + d*x^2)^3,x]
[Out]
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Maple [A] time = 0.001, size = 128, normalized size = 1. \[{\frac{{b}^{2}{d}^{3}{x}^{15}}{15}}+{\frac{ \left ( 2\,ab{d}^{3}+3\,{b}^{2}c{d}^{2} \right ){x}^{13}}{13}}+{\frac{ \left ({a}^{2}{d}^{3}+6\,abc{d}^{2}+3\,{b}^{2}{c}^{2}d \right ){x}^{11}}{11}}+{\frac{ \left ( 3\,{a}^{2}c{d}^{2}+6\,ab{c}^{2}d+{b}^{2}{c}^{3} \right ){x}^{9}}{9}}+{\frac{ \left ( 3\,{a}^{2}{c}^{2}d+2\,ab{c}^{3} \right ){x}^{7}}{7}}+{\frac{{a}^{2}{c}^{3}{x}^{5}}{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(b*x^2+a)^2*(d*x^2+c)^3,x)
[Out]
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Maxima [A] time = 1.3555, size = 171, normalized size = 1.35 \[ \frac{1}{15} \, b^{2} d^{3} x^{15} + \frac{1}{13} \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{13} + \frac{1}{11} \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{11} + \frac{1}{5} \, a^{2} c^{3} x^{5} + \frac{1}{9} \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{9} + \frac{1}{7} \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)^3*x^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.205901, size = 1, normalized size = 0.01 \[ \frac{1}{15} x^{15} d^{3} b^{2} + \frac{3}{13} x^{13} d^{2} c b^{2} + \frac{2}{13} x^{13} d^{3} b a + \frac{3}{11} x^{11} d c^{2} b^{2} + \frac{6}{11} x^{11} d^{2} c b a + \frac{1}{11} x^{11} d^{3} a^{2} + \frac{1}{9} x^{9} c^{3} b^{2} + \frac{2}{3} x^{9} d c^{2} b a + \frac{1}{3} x^{9} d^{2} c a^{2} + \frac{2}{7} x^{7} c^{3} b a + \frac{3}{7} x^{7} d c^{2} a^{2} + \frac{1}{5} x^{5} c^{3} a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)^3*x^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.17853, size = 143, normalized size = 1.13 \[ \frac{a^{2} c^{3} x^{5}}{5} + \frac{b^{2} d^{3} x^{15}}{15} + x^{13} \left (\frac{2 a b d^{3}}{13} + \frac{3 b^{2} c d^{2}}{13}\right ) + x^{11} \left (\frac{a^{2} d^{3}}{11} + \frac{6 a b c d^{2}}{11} + \frac{3 b^{2} c^{2} d}{11}\right ) + x^{9} \left (\frac{a^{2} c d^{2}}{3} + \frac{2 a b c^{2} d}{3} + \frac{b^{2} c^{3}}{9}\right ) + x^{7} \left (\frac{3 a^{2} c^{2} d}{7} + \frac{2 a b c^{3}}{7}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4*(b*x**2+a)**2*(d*x**2+c)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.222654, size = 182, normalized size = 1.43 \[ \frac{1}{15} \, b^{2} d^{3} x^{15} + \frac{3}{13} \, b^{2} c d^{2} x^{13} + \frac{2}{13} \, a b d^{3} x^{13} + \frac{3}{11} \, b^{2} c^{2} d x^{11} + \frac{6}{11} \, a b c d^{2} x^{11} + \frac{1}{11} \, a^{2} d^{3} x^{11} + \frac{1}{9} \, b^{2} c^{3} x^{9} + \frac{2}{3} \, a b c^{2} d x^{9} + \frac{1}{3} \, a^{2} c d^{2} x^{9} + \frac{2}{7} \, a b c^{3} x^{7} + \frac{3}{7} \, a^{2} c^{2} d x^{7} + \frac{1}{5} \, a^{2} c^{3} x^{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^2*(d*x^2 + c)^3*x^4,x, algorithm="giac")
[Out]