Optimal. Leaf size=133 \[ a^3 A x+\frac{1}{4} a^3 D x^4+\frac{1}{3} a^2 x^3 (a C+3 A b)+\frac{1}{2} a^2 b D x^6+\frac{1}{7} b^2 x^7 (3 a C+A b)+\frac{3}{5} a b x^5 (a C+A b)+\frac{3}{8} a b^2 D x^8+\frac{B \left (a+b x^2\right )^4}{8 b}+\frac{1}{9} b^3 C x^9+\frac{1}{10} b^3 D x^{10} \]
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Rubi [A] time = 0.249969, antiderivative size = 133, 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 \[ a^3 A x+\frac{1}{4} a^3 D x^4+\frac{1}{3} a^2 x^3 (a C+3 A b)+\frac{1}{2} a^2 b D x^6+\frac{1}{7} b^2 x^7 (3 a C+A b)+\frac{3}{5} a b x^5 (a C+A b)+\frac{3}{8} a b^2 D x^8+\frac{B \left (a+b x^2\right )^4}{8 b}+\frac{1}{9} b^3 C x^9+\frac{1}{10} b^3 D x^{10} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^3*(A + B*x + C*x^2 + D*x^3),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{B \left (a + b x^{2}\right )^{4}}{8 b} + \frac{C b^{3} x^{9}}{9} + \frac{D a^{3} x^{4}}{4} + \frac{D a^{2} b x^{6}}{2} + \frac{3 D a b^{2} x^{8}}{8} + \frac{D b^{3} x^{10}}{10} + a^{3} \int A\, dx + \frac{a^{2} x^{3} \left (3 A b + C a\right )}{3} + \frac{3 a b x^{5} \left (A b + C a\right )}{5} + \frac{b^{2} x^{7} \left (A b + 3 C a\right )}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**3*(D*x**3+C*x**2+B*x+A),x)
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Mathematica [A] time = 0.137867, size = 121, normalized size = 0.91 \[ \frac{210 a^3 x (12 A+x (6 B+x (4 C+3 D x)))+126 a^2 b x^3 (20 A+x (15 B+2 x (6 C+5 D x)))+9 a b^2 x^5 (168 A+5 x (28 B+3 x (8 C+7 D x)))+b^3 x^7 (360 A+7 x (45 B+4 x (10 C+9 D x)))}{2520} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^3*(A + B*x + C*x^2 + D*x^3),x]
[Out]
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Maple [A] time = 0.003, size = 147, normalized size = 1.1 \[{\frac{{b}^{3}D{x}^{10}}{10}}+{\frac{{b}^{3}C{x}^{9}}{9}}+{\frac{ \left ({b}^{3}B+3\,a{b}^{2}D \right ){x}^{8}}{8}}+{\frac{ \left ( A{b}^{3}+3\,a{b}^{2}C \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,a{b}^{2}B+3\,{a}^{2}bD \right ){x}^{6}}{6}}+{\frac{ \left ( 3\,a{b}^{2}A+3\,{a}^{2}bC \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,{a}^{2}bB+{a}^{3}D \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,A{a}^{2}b+{a}^{3}C \right ){x}^{3}}{3}}+{\frac{{a}^{3}B{x}^{2}}{2}}+{a}^{3}Ax \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^2+a)^3*(D*x^3+C*x^2+B*x+A),x)
[Out]
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Maxima [A] time = 1.42694, size = 192, normalized size = 1.44 \[ \frac{1}{10} \, D b^{3} x^{10} + \frac{1}{9} \, C b^{3} x^{9} + \frac{1}{8} \,{\left (3 \, D a b^{2} + B b^{3}\right )} x^{8} + \frac{1}{7} \,{\left (3 \, C a b^{2} + A b^{3}\right )} x^{7} + \frac{1}{2} \,{\left (D a^{2} b + B a b^{2}\right )} x^{6} + \frac{1}{2} \, B a^{3} x^{2} + \frac{3}{5} \,{\left (C a^{2} b + A a b^{2}\right )} x^{5} + A a^{3} x + \frac{1}{4} \,{\left (D a^{3} + 3 \, B a^{2} b\right )} x^{4} + \frac{1}{3} \,{\left (C a^{3} + 3 \, A a^{2} b\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((D*x^3 + C*x^2 + B*x + A)*(b*x^2 + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.196794, size = 1, normalized size = 0.01 \[ \frac{1}{10} x^{10} b^{3} D + \frac{1}{9} x^{9} b^{3} C + \frac{3}{8} x^{8} b^{2} a D + \frac{1}{8} x^{8} b^{3} B + \frac{3}{7} x^{7} b^{2} a C + \frac{1}{7} x^{7} b^{3} A + \frac{1}{2} x^{6} b a^{2} D + \frac{1}{2} x^{6} b^{2} a B + \frac{3}{5} x^{5} b a^{2} C + \frac{3}{5} x^{5} b^{2} a A + \frac{1}{4} x^{4} a^{3} D + \frac{3}{4} x^{4} b a^{2} B + \frac{1}{3} x^{3} a^{3} C + x^{3} b a^{2} A + \frac{1}{2} x^{2} a^{3} B + x a^{3} A \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((D*x^3 + C*x^2 + B*x + A)*(b*x^2 + a)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.08537, size = 158, normalized size = 1.19 \[ A a^{3} x + \frac{B a^{3} x^{2}}{2} + \frac{C b^{3} x^{9}}{9} + \frac{D b^{3} x^{10}}{10} + x^{8} \left (\frac{B b^{3}}{8} + \frac{3 D a b^{2}}{8}\right ) + x^{7} \left (\frac{A b^{3}}{7} + \frac{3 C a b^{2}}{7}\right ) + x^{6} \left (\frac{B a b^{2}}{2} + \frac{D a^{2} b}{2}\right ) + x^{5} \left (\frac{3 A a b^{2}}{5} + \frac{3 C a^{2} b}{5}\right ) + x^{4} \left (\frac{3 B a^{2} b}{4} + \frac{D a^{3}}{4}\right ) + x^{3} \left (A a^{2} b + \frac{C a^{3}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**3*(D*x**3+C*x**2+B*x+A),x)
[Out]
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GIAC/XCAS [A] time = 0.222359, size = 201, normalized size = 1.51 \[ \frac{1}{10} \, D b^{3} x^{10} + \frac{1}{9} \, C b^{3} x^{9} + \frac{3}{8} \, D a b^{2} x^{8} + \frac{1}{8} \, B b^{3} x^{8} + \frac{3}{7} \, C a b^{2} x^{7} + \frac{1}{7} \, A b^{3} x^{7} + \frac{1}{2} \, D a^{2} b x^{6} + \frac{1}{2} \, B a b^{2} x^{6} + \frac{3}{5} \, C a^{2} b x^{5} + \frac{3}{5} \, A a b^{2} x^{5} + \frac{1}{4} \, D a^{3} x^{4} + \frac{3}{4} \, B a^{2} b x^{4} + \frac{1}{3} \, C a^{3} x^{3} + A a^{2} b x^{3} + \frac{1}{2} \, B a^{3} x^{2} + A a^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((D*x^3 + C*x^2 + B*x + A)*(b*x^2 + a)^3,x, algorithm="giac")
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