Optimal. Leaf size=73 \[ a^4 c x+\frac{4}{3} a^3 b c x^3+\frac{6}{5} a^2 b^2 c x^5+\frac{4}{7} a b^3 c x^7+\frac{d \left (a+b x^2\right )^5}{10 b}+\frac{1}{9} b^4 c x^9 \]
[Out]
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Rubi [A] time = 0.0894868, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ a^4 c x+\frac{4}{3} a^3 b c x^3+\frac{6}{5} a^2 b^2 c x^5+\frac{4}{7} a b^3 c x^7+\frac{d \left (a+b x^2\right )^5}{10 b}+\frac{1}{9} b^4 c x^9 \]
Antiderivative was successfully verified.
[In] Int[(c + d*x)*(a + b*x^2)^4,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{4 a^{3} b c x^{3}}{3} + \frac{6 a^{2} b^{2} c x^{5}}{5} + \frac{4 a b^{3} c x^{7}}{7} + \frac{b^{4} c x^{9}}{9} + c \int a^{4}\, dx + \frac{d \left (a + b x^{2}\right )^{5}}{10 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x+c)*(b*x**2+a)**4,x)
[Out]
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Mathematica [A] time = 0.00627615, size = 110, normalized size = 1.51 \[ a^4 c x+\frac{1}{2} a^4 d x^2+\frac{4}{3} a^3 b c x^3+a^3 b d x^4+\frac{6}{5} a^2 b^2 c x^5+a^2 b^2 d x^6+\frac{4}{7} a b^3 c x^7+\frac{1}{2} a b^3 d x^8+\frac{1}{9} b^4 c x^9+\frac{1}{10} b^4 d x^{10} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x)*(a + b*x^2)^4,x]
[Out]
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Maple [A] time = 0.002, size = 97, normalized size = 1.3 \[{\frac{d{b}^{4}{x}^{10}}{10}}+{\frac{{b}^{4}c{x}^{9}}{9}}+{\frac{ad{b}^{3}{x}^{8}}{2}}+{\frac{4\,a{b}^{3}c{x}^{7}}{7}}+{a}^{2}d{b}^{2}{x}^{6}+{\frac{6\,{a}^{2}{b}^{2}c{x}^{5}}{5}}+{a}^{3}bd{x}^{4}+{\frac{4\,{a}^{3}bc{x}^{3}}{3}}+{\frac{{a}^{4}d{x}^{2}}{2}}+{a}^{4}cx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x+c)*(b*x^2+a)^4,x)
[Out]
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Maxima [A] time = 0.698636, size = 130, normalized size = 1.78 \[ \frac{1}{10} \, b^{4} d x^{10} + \frac{1}{9} \, b^{4} c x^{9} + \frac{1}{2} \, a b^{3} d x^{8} + \frac{4}{7} \, a b^{3} c x^{7} + a^{2} b^{2} d x^{6} + \frac{6}{5} \, a^{2} b^{2} c x^{5} + a^{3} b d x^{4} + \frac{4}{3} \, a^{3} b c x^{3} + \frac{1}{2} \, a^{4} d x^{2} + a^{4} c x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^4*(d*x + c),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.189352, size = 1, normalized size = 0.01 \[ \frac{1}{10} x^{10} d b^{4} + \frac{1}{9} x^{9} c b^{4} + \frac{1}{2} x^{8} d b^{3} a + \frac{4}{7} x^{7} c b^{3} a + x^{6} d b^{2} a^{2} + \frac{6}{5} x^{5} c b^{2} a^{2} + x^{4} d b a^{3} + \frac{4}{3} x^{3} c b a^{3} + \frac{1}{2} x^{2} d a^{4} + x c a^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^4*(d*x + c),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.144977, size = 112, normalized size = 1.53 \[ a^{4} c x + \frac{a^{4} d x^{2}}{2} + \frac{4 a^{3} b c x^{3}}{3} + a^{3} b d x^{4} + \frac{6 a^{2} b^{2} c x^{5}}{5} + a^{2} b^{2} d x^{6} + \frac{4 a b^{3} c x^{7}}{7} + \frac{a b^{3} d x^{8}}{2} + \frac{b^{4} c x^{9}}{9} + \frac{b^{4} d x^{10}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x+c)*(b*x**2+a)**4,x)
[Out]
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GIAC/XCAS [A] time = 0.211148, size = 130, normalized size = 1.78 \[ \frac{1}{10} \, b^{4} d x^{10} + \frac{1}{9} \, b^{4} c x^{9} + \frac{1}{2} \, a b^{3} d x^{8} + \frac{4}{7} \, a b^{3} c x^{7} + a^{2} b^{2} d x^{6} + \frac{6}{5} \, a^{2} b^{2} c x^{5} + a^{3} b d x^{4} + \frac{4}{3} \, a^{3} b c x^{3} + \frac{1}{2} \, a^{4} d x^{2} + a^{4} c x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^4*(d*x + c),x, algorithm="giac")
[Out]