Optimal. Leaf size=115 \[ \frac{2 c (d+e x)^5 \left (a e^2+3 c d^2\right )}{5 e^5}-\frac{c d (d+e x)^4 \left (a e^2+c d^2\right )}{e^5}+\frac{(d+e x)^3 \left (a e^2+c d^2\right )^2}{3 e^5}+\frac{c^2 (d+e x)^7}{7 e^5}-\frac{2 c^2 d (d+e x)^6}{3 e^5} \]
[Out]
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Rubi [A] time = 0.195079, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{2 c (d+e x)^5 \left (a e^2+3 c d^2\right )}{5 e^5}-\frac{c d (d+e x)^4 \left (a e^2+c d^2\right )}{e^5}+\frac{(d+e x)^3 \left (a e^2+c d^2\right )^2}{3 e^5}+\frac{c^2 (d+e x)^7}{7 e^5}-\frac{2 c^2 d (d+e x)^6}{3 e^5} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^2*(a + c*x^2)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ 2 a^{2} d e \int x\, dx + a c d e x^{4} + \frac{a x^{3} \left (a e^{2} + 2 c d^{2}\right )}{3} + \frac{c^{2} d e x^{6}}{3} + \frac{c^{2} e^{2} x^{7}}{7} + \frac{c x^{5} \left (2 a e^{2} + c d^{2}\right )}{5} + d^{2} \int a^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**2*(c*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.0259199, size = 91, normalized size = 0.79 \[ a^2 d^2 x+a^2 d e x^2+\frac{1}{5} c x^5 \left (2 a e^2+c d^2\right )+\frac{1}{3} a x^3 \left (a e^2+2 c d^2\right )+a c d e x^4+\frac{1}{3} c^2 d e x^6+\frac{1}{7} c^2 e^2 x^7 \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^2*(a + c*x^2)^2,x]
[Out]
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Maple [A] time = 0.001, size = 88, normalized size = 0.8 \[{\frac{{c}^{2}{e}^{2}{x}^{7}}{7}}+{\frac{de{c}^{2}{x}^{6}}{3}}+{\frac{ \left ( 2\,{e}^{2}ac+{c}^{2}{d}^{2} \right ){x}^{5}}{5}}+acde{x}^{4}+{\frac{ \left ({a}^{2}{e}^{2}+2\,{d}^{2}ac \right ){x}^{3}}{3}}+de{a}^{2}{x}^{2}+{a}^{2}{d}^{2}x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^2*(c*x^2+a)^2,x)
[Out]
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Maxima [A] time = 0.693889, size = 117, normalized size = 1.02 \[ \frac{1}{7} \, c^{2} e^{2} x^{7} + \frac{1}{3} \, c^{2} d e x^{6} + a c d e x^{4} + a^{2} d e x^{2} + \frac{1}{5} \,{\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} x^{5} + a^{2} d^{2} x + \frac{1}{3} \,{\left (2 \, a c d^{2} + a^{2} e^{2}\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(e*x + d)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.180352, size = 1, normalized size = 0.01 \[ \frac{1}{7} x^{7} e^{2} c^{2} + \frac{1}{3} x^{6} e d c^{2} + \frac{1}{5} x^{5} d^{2} c^{2} + \frac{2}{5} x^{5} e^{2} c a + x^{4} e d c a + \frac{2}{3} x^{3} d^{2} c a + \frac{1}{3} x^{3} e^{2} a^{2} + x^{2} e d a^{2} + x d^{2} a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(e*x + d)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.148985, size = 95, normalized size = 0.83 \[ a^{2} d^{2} x + a^{2} d e x^{2} + a c d e x^{4} + \frac{c^{2} d e x^{6}}{3} + \frac{c^{2} e^{2} x^{7}}{7} + x^{5} \left (\frac{2 a c e^{2}}{5} + \frac{c^{2} d^{2}}{5}\right ) + x^{3} \left (\frac{a^{2} e^{2}}{3} + \frac{2 a c d^{2}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**2*(c*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.210736, size = 120, normalized size = 1.04 \[ \frac{1}{7} \, c^{2} x^{7} e^{2} + \frac{1}{3} \, c^{2} d x^{6} e + \frac{1}{5} \, c^{2} d^{2} x^{5} + \frac{2}{5} \, a c x^{5} e^{2} + a c d x^{4} e + \frac{2}{3} \, a c d^{2} x^{3} + \frac{1}{3} \, a^{2} x^{3} e^{2} + a^{2} d x^{2} e + a^{2} d^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(e*x + d)^2,x, algorithm="giac")
[Out]