Optimal. Leaf size=109 \[ a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{3} a^2 e x^3+\frac{2}{5} a b c x^5+\frac{1}{3} a b d x^6+\frac{2}{7} a b e x^7+\frac{f \left (a+b x^4\right )^3}{12 b}+\frac{1}{9} b^2 c x^9+\frac{1}{10} b^2 d x^{10}+\frac{1}{11} b^2 e x^{11} \]
[Out]
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Rubi [A] time = 0.163476, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{3} a^2 e x^3+\frac{2}{5} a b c x^5+\frac{1}{3} a b d x^6+\frac{2}{7} a b e x^7+\frac{f \left (a+b x^4\right )^3}{12 b}+\frac{1}{9} b^2 c x^9+\frac{1}{10} b^2 d x^{10}+\frac{1}{11} b^2 e x^{11} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ a^{2} d \int x\, dx + \frac{a^{2} e x^{3}}{3} + a^{2} \int c\, dx + \frac{2 a b c x^{5}}{5} + \frac{a b d x^{6}}{3} + \frac{2 a b e x^{7}}{7} + \frac{b^{2} c x^{9}}{9} + \frac{b^{2} d x^{10}}{10} + \frac{b^{2} e x^{11}}{11} + \frac{f \left (a + b x^{4}\right )^{3}}{12 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x**3+e*x**2+d*x+c)*(b*x**4+a)**2,x)
[Out]
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Mathematica [A] time = 0.0119555, size = 124, normalized size = 1.14 \[ a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{3} a^2 e x^3+\frac{1}{4} a^2 f x^4+\frac{2}{5} a b c x^5+\frac{1}{3} a b d x^6+\frac{2}{7} a b e x^7+\frac{1}{4} a b f x^8+\frac{1}{9} b^2 c x^9+\frac{1}{10} b^2 d x^{10}+\frac{1}{11} b^2 e x^{11}+\frac{1}{12} b^2 f x^{12} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^2,x]
[Out]
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Maple [A] time = 0.001, size = 103, normalized size = 0.9 \[{\frac{{b}^{2}f{x}^{12}}{12}}+{\frac{{b}^{2}e{x}^{11}}{11}}+{\frac{{b}^{2}d{x}^{10}}{10}}+{\frac{{b}^{2}c{x}^{9}}{9}}+{\frac{fab{x}^{8}}{4}}+{\frac{2\,abe{x}^{7}}{7}}+{\frac{abd{x}^{6}}{3}}+{\frac{2\,abc{x}^{5}}{5}}+{\frac{f{a}^{2}{x}^{4}}{4}}+{\frac{{a}^{2}e{x}^{3}}{3}}+{\frac{{a}^{2}d{x}^{2}}{2}}+{a}^{2}cx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x^3+e*x^2+d*x+c)*(b*x^4+a)^2,x)
[Out]
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Maxima [A] time = 5.93498, size = 138, normalized size = 1.27 \[ \frac{1}{12} \, b^{2} f x^{12} + \frac{1}{11} \, b^{2} e x^{11} + \frac{1}{10} \, b^{2} d x^{10} + \frac{1}{9} \, b^{2} c x^{9} + \frac{1}{4} \, a b f x^{8} + \frac{2}{7} \, a b e x^{7} + \frac{1}{3} \, a b d x^{6} + \frac{2}{5} \, a b c x^{5} + \frac{1}{4} \, a^{2} f x^{4} + \frac{1}{3} \, a^{2} e x^{3} + \frac{1}{2} \, a^{2} d x^{2} + a^{2} c x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^2*(f*x^3 + e*x^2 + d*x + c),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.198317, size = 1, normalized size = 0.01 \[ \frac{1}{12} x^{12} f b^{2} + \frac{1}{11} x^{11} e b^{2} + \frac{1}{10} x^{10} d b^{2} + \frac{1}{9} x^{9} c b^{2} + \frac{1}{4} x^{8} f b a + \frac{2}{7} x^{7} e b a + \frac{1}{3} x^{6} d b a + \frac{2}{5} x^{5} c b a + \frac{1}{4} x^{4} f a^{2} + \frac{1}{3} x^{3} e a^{2} + \frac{1}{2} x^{2} d a^{2} + x c a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^2*(f*x^3 + e*x^2 + d*x + c),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.074912, size = 121, normalized size = 1.11 \[ a^{2} c x + \frac{a^{2} d x^{2}}{2} + \frac{a^{2} e x^{3}}{3} + \frac{a^{2} f x^{4}}{4} + \frac{2 a b c x^{5}}{5} + \frac{a b d x^{6}}{3} + \frac{2 a b e x^{7}}{7} + \frac{a b f x^{8}}{4} + \frac{b^{2} c x^{9}}{9} + \frac{b^{2} d x^{10}}{10} + \frac{b^{2} e x^{11}}{11} + \frac{b^{2} f x^{12}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x**3+e*x**2+d*x+c)*(b*x**4+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.215479, size = 142, normalized size = 1.3 \[ \frac{1}{12} \, b^{2} f x^{12} + \frac{1}{11} \, b^{2} x^{11} e + \frac{1}{10} \, b^{2} d x^{10} + \frac{1}{9} \, b^{2} c x^{9} + \frac{1}{4} \, a b f x^{8} + \frac{2}{7} \, a b x^{7} e + \frac{1}{3} \, a b d x^{6} + \frac{2}{5} \, a b c x^{5} + \frac{1}{4} \, a^{2} f x^{4} + \frac{1}{3} \, a^{2} x^{3} e + \frac{1}{2} \, a^{2} d x^{2} + a^{2} c x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^2*(f*x^3 + e*x^2 + d*x + c),x, algorithm="giac")
[Out]