Optimal. Leaf size=117 \[ \frac{c (d+e x)^6 \left (a e^2+3 c d^2\right )}{3 e^5}-\frac{4 c d (d+e x)^5 \left (a e^2+c d^2\right )}{5 e^5}+\frac{(d+e x)^4 \left (a e^2+c d^2\right )^2}{4 e^5}+\frac{c^2 (d+e x)^8}{8 e^5}-\frac{4 c^2 d (d+e x)^7}{7 e^5} \]
[Out]
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Rubi [A] time = 0.25531, antiderivative size = 117, 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 \[ \frac{c (d+e x)^6 \left (a e^2+3 c d^2\right )}{3 e^5}-\frac{4 c d (d+e x)^5 \left (a e^2+c d^2\right )}{5 e^5}+\frac{(d+e x)^4 \left (a e^2+c d^2\right )^2}{4 e^5}+\frac{c^2 (d+e x)^8}{8 e^5}-\frac{4 c^2 d (d+e x)^7}{7 e^5} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^3*(a + c*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 33.4069, size = 109, normalized size = 0.93 \[ - \frac{4 c^{2} d \left (d + e x\right )^{7}}{7 e^{5}} + \frac{c^{2} \left (d + e x\right )^{8}}{8 e^{5}} - \frac{4 c d \left (d + e x\right )^{5} \left (a e^{2} + c d^{2}\right )}{5 e^{5}} + \frac{c \left (d + e x\right )^{6} \left (a e^{2} + 3 c d^{2}\right )}{3 e^{5}} + \frac{\left (d + e x\right )^{4} \left (a e^{2} + c d^{2}\right )^{2}}{4 e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3*(c*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.0409524, size = 117, normalized size = 1. \[ \frac{1}{4} a^2 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+\frac{1}{30} a c x^3 \left (20 d^3+45 d^2 e x+36 d e^2 x^2+10 e^3 x^3\right )+\frac{1}{280} c^2 x^5 \left (56 d^3+140 d^2 e x+120 d e^2 x^2+35 e^3 x^3\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^3*(a + c*x^2)^2,x]
[Out]
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Maple [A] time = 0.001, size = 131, normalized size = 1.1 \[{\frac{{c}^{2}{e}^{3}{x}^{8}}{8}}+{\frac{3\,d{e}^{2}{c}^{2}{x}^{7}}{7}}+{\frac{ \left ( 2\,{e}^{3}ac+3\,{d}^{2}e{c}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 6\,d{e}^{2}ac+{c}^{2}{d}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ({a}^{2}{e}^{3}+6\,{d}^{2}eac \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,d{e}^{2}{a}^{2}+2\,{d}^{3}ac \right ){x}^{3}}{3}}+{\frac{3\,{d}^{2}e{a}^{2}{x}^{2}}{2}}+{a}^{2}{d}^{3}x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3*(c*x^2+a)^2,x)
[Out]
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Maxima [A] time = 0.699846, size = 176, normalized size = 1.5 \[ \frac{1}{8} \, c^{2} e^{3} x^{8} + \frac{3}{7} \, c^{2} d e^{2} x^{7} + \frac{3}{2} \, a^{2} d^{2} e x^{2} + \frac{1}{6} \,{\left (3 \, c^{2} d^{2} e + 2 \, a c e^{3}\right )} x^{6} + a^{2} d^{3} x + \frac{1}{5} \,{\left (c^{2} d^{3} + 6 \, a c d e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (6 \, a c d^{2} e + a^{2} e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (2 \, a c d^{3} + 3 \, a^{2} d e^{2}\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(e*x + d)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.184038, size = 1, normalized size = 0.01 \[ \frac{1}{8} x^{8} e^{3} c^{2} + \frac{3}{7} x^{7} e^{2} d c^{2} + \frac{1}{2} x^{6} e d^{2} c^{2} + \frac{1}{3} x^{6} e^{3} c a + \frac{1}{5} x^{5} d^{3} c^{2} + \frac{6}{5} x^{5} e^{2} d c a + \frac{3}{2} x^{4} e d^{2} c a + \frac{1}{4} x^{4} e^{3} a^{2} + \frac{2}{3} x^{3} d^{3} c a + x^{3} e^{2} d a^{2} + \frac{3}{2} x^{2} e d^{2} a^{2} + x d^{3} a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(e*x + d)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.161947, size = 141, normalized size = 1.21 \[ a^{2} d^{3} x + \frac{3 a^{2} d^{2} e x^{2}}{2} + \frac{3 c^{2} d e^{2} x^{7}}{7} + \frac{c^{2} e^{3} x^{8}}{8} + x^{6} \left (\frac{a c e^{3}}{3} + \frac{c^{2} d^{2} e}{2}\right ) + x^{5} \left (\frac{6 a c d e^{2}}{5} + \frac{c^{2} d^{3}}{5}\right ) + x^{4} \left (\frac{a^{2} e^{3}}{4} + \frac{3 a c d^{2} e}{2}\right ) + x^{3} \left (a^{2} d e^{2} + \frac{2 a c d^{3}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**3*(c*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.210343, size = 173, normalized size = 1.48 \[ \frac{1}{8} \, c^{2} x^{8} e^{3} + \frac{3}{7} \, c^{2} d x^{7} e^{2} + \frac{1}{2} \, c^{2} d^{2} x^{6} e + \frac{1}{5} \, c^{2} d^{3} x^{5} + \frac{1}{3} \, a c x^{6} e^{3} + \frac{6}{5} \, a c d x^{5} e^{2} + \frac{3}{2} \, a c d^{2} x^{4} e + \frac{2}{3} \, a c d^{3} x^{3} + \frac{1}{4} \, a^{2} x^{4} e^{3} + a^{2} d x^{3} e^{2} + \frac{3}{2} \, a^{2} d^{2} x^{2} e + a^{2} d^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2*(e*x + d)^3,x, algorithm="giac")
[Out]