Optimal. Leaf size=99 \[ a^2 A x+\frac {1}{4} a^2 D x^4+\frac {1}{5} b x^5 (2 a C+A b)+\frac {1}{3} a x^3 (a C+2 A b)+\frac {B \left (a+b x^2\right )^3}{6 b}+\frac {1}{3} a b D x^6+\frac {1}{7} b^2 C x^7+\frac {1}{8} b^2 D x^8 \]
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Rubi [A] time = 0.07, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1582, 1810} \[ a^2 A x+\frac {1}{4} a^2 D x^4+\frac {1}{5} b x^5 (2 a C+A b)+\frac {1}{3} a x^3 (a C+2 A b)+\frac {B \left (a+b x^2\right )^3}{6 b}+\frac {1}{3} a b D x^6+\frac {1}{7} b^2 C x^7+\frac {1}{8} b^2 D x^8 \]
Antiderivative was successfully verified.
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Rule 1582
Rule 1810
Rubi steps
\begin {align*} \int \left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right ) \, dx &=\frac {B \left (a+b x^2\right )^3}{6 b}+\int \left (a+b x^2\right )^2 \left (A+C x^2+D x^3\right ) \, dx\\ &=\frac {B \left (a+b x^2\right )^3}{6 b}+\int \left (a^2 A+a (2 A b+a C) x^2+a^2 D x^3+b (A b+2 a C) x^4+2 a b D x^5+b^2 C x^6+b^2 D x^7\right ) \, dx\\ &=a^2 A x+\frac {1}{3} a (2 A b+a C) x^3+\frac {1}{4} a^2 D x^4+\frac {1}{5} b (A b+2 a C) x^5+\frac {1}{3} a b D x^6+\frac {1}{7} b^2 C x^7+\frac {1}{8} b^2 D x^8+\frac {B \left (a+b x^2\right )^3}{6 b}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 88, normalized size = 0.89 \[ \frac {1}{840} \left (70 a^2 x (12 A+x (6 B+x (4 C+3 D x)))+28 a b x^3 (20 A+x (15 B+2 x (6 C+5 D x)))+b^2 x^5 (168 A+5 x (28 B+3 x (8 C+7 D x)))\right ) \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 102, normalized size = 1.03 \[ \frac {1}{8} \, D b^{2} x^{8} + \frac {1}{7} \, C b^{2} x^{7} + \frac {1}{3} \, D a b x^{6} + \frac {1}{6} \, B b^{2} x^{6} + \frac {2}{5} \, C a b x^{5} + \frac {1}{5} \, A b^{2} x^{5} + \frac {1}{4} \, D a^{2} x^{4} + \frac {1}{2} \, B a b x^{4} + \frac {1}{3} \, C a^{2} x^{3} + \frac {2}{3} \, A a b x^{3} + \frac {1}{2} \, B a^{2} x^{2} + A a^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 99, normalized size = 1.00 \[ \frac {D b^{2} x^{8}}{8}+\frac {C \,b^{2} x^{7}}{7}+\frac {\left (b^{2} B +2 a b D\right ) x^{6}}{6}+\frac {B \,a^{2} x^{2}}{2}+\frac {\left (b^{2} A +2 a b C \right ) x^{5}}{5}+A \,a^{2} x +\frac {\left (2 a b B +a^{2} D\right ) x^{4}}{4}+\frac {\left (2 A a b +a^{2} C \right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 98, normalized size = 0.99 \[ \frac {1}{8} \, D b^{2} x^{8} + \frac {1}{7} \, C b^{2} x^{7} + \frac {1}{6} \, {\left (2 \, D a b + B b^{2}\right )} x^{6} + \frac {1}{5} \, {\left (2 \, C a b + A b^{2}\right )} x^{5} + \frac {1}{2} \, B a^{2} x^{2} + \frac {1}{4} \, {\left (D a^{2} + 2 \, B a b\right )} x^{4} + A a^{2} x + \frac {1}{3} \, {\left (C a^{2} + 2 \, A a b\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.11, size = 105, normalized size = 1.06 \[ \frac {A\,x\,\left (15\,a^2+10\,a\,b\,x^2+3\,b^2\,x^4\right )}{15}+\frac {a^2\,x^4\,D}{4}+\frac {b^2\,x^8\,D}{8}+\frac {B\,x^2\,\left (3\,a^2+3\,a\,b\,x^2+b^2\,x^4\right )}{6}+\frac {C\,x^3\,\left (35\,a^2+42\,a\,b\,x^2+15\,b^2\,x^4\right )}{105}+\frac {a\,b\,x^6\,D}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 107, normalized size = 1.08 \[ A a^{2} x + \frac {B a^{2} x^{2}}{2} + \frac {C b^{2} x^{7}}{7} + \frac {D b^{2} x^{8}}{8} + x^{6} \left (\frac {B b^{2}}{6} + \frac {D a b}{3}\right ) + x^{5} \left (\frac {A b^{2}}{5} + \frac {2 C a b}{5}\right ) + x^{4} \left (\frac {B a b}{2} + \frac {D a^{2}}{4}\right ) + x^{3} \left (\frac {2 A a b}{3} + \frac {C a^{2}}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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