Optimal. Leaf size=104 \[ \frac {1}{3} a^2 B x^3+\frac {1}{4} a^2 C x^4+\frac {A \left (a+b x^2\right )^3}{6 b}+\frac {1}{7} b x^7 (2 a D+b B)+\frac {1}{5} a x^5 (a D+2 b B)+\frac {1}{3} a b C x^6+\frac {1}{8} b^2 C x^8+\frac {1}{9} b^2 D x^9 \]
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Rubi [A] time = 0.07, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1582, 1810} \[ \frac {1}{3} a^2 B x^3+\frac {1}{4} a^2 C x^4+\frac {A \left (a+b x^2\right )^3}{6 b}+\frac {1}{7} b x^7 (2 a D+b B)+\frac {1}{5} a x^5 (a D+2 b B)+\frac {1}{3} a b C x^6+\frac {1}{8} b^2 C x^8+\frac {1}{9} b^2 D x^9 \]
Antiderivative was successfully verified.
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Rule 1582
Rule 1810
Rubi steps
\begin {align*} \int x \left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right ) \, dx &=\frac {A \left (a+b x^2\right )^3}{6 b}+\int \left (a+b x^2\right )^2 \left (-A x+x \left (A+B x+C x^2+D x^3\right )\right ) \, dx\\ &=\frac {A \left (a+b x^2\right )^3}{6 b}+\int \left (a^2 B x^2+a^2 C x^3+a (2 b B+a D) x^4+2 a b C x^5+b (b B+2 a D) x^6+b^2 C x^7+b^2 D x^8\right ) \, dx\\ &=\frac {1}{3} a^2 B x^3+\frac {1}{4} a^2 C x^4+\frac {1}{5} a (2 b B+a D) x^5+\frac {1}{3} a b C x^6+\frac {1}{7} b (b B+2 a D) x^7+\frac {1}{8} b^2 C x^8+\frac {1}{9} b^2 D x^9+\frac {A \left (a+b x^2\right )^3}{6 b}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 92, normalized size = 0.88 \[ \frac {42 a^2 x^2 (30 A+x (20 B+3 x (5 C+4 D x)))+12 a b x^4 (105 A+2 x (42 B+5 x (7 C+6 D x)))+5 b^2 x^6 (84 A+x (72 B+7 x (9 C+8 D x)))}{2520} \]
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 = 0.36, size = 105, normalized size = 1.01 \[ \frac {1}{9} \, D b^{2} x^{9} + \frac {1}{8} \, C b^{2} x^{8} + \frac {2}{7} \, D a b x^{7} + \frac {1}{7} \, B b^{2} x^{7} + \frac {1}{3} \, C a b x^{6} + \frac {1}{6} \, A b^{2} x^{6} + \frac {1}{5} \, D a^{2} x^{5} + \frac {2}{5} \, B a b x^{5} + \frac {1}{4} \, C a^{2} x^{4} + \frac {1}{2} \, A a b x^{4} + \frac {1}{3} \, B a^{2} x^{3} + \frac {1}{2} \, A a^{2} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 102, normalized size = 0.98 \[ \frac {D b^{2} x^{9}}{9}+\frac {C \,b^{2} x^{8}}{8}+\frac {\left (b^{2} B +2 a b D\right ) x^{7}}{7}+\frac {B \,a^{2} x^{3}}{3}+\frac {\left (b^{2} A +2 a b C \right ) x^{6}}{6}+\frac {A \,a^{2} x^{2}}{2}+\frac {\left (2 a b B +a^{2} D\right ) x^{5}}{5}+\frac {\left (2 A a b +a^{2} C \right ) x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 101, normalized size = 0.97 \[ \frac {1}{9} \, D b^{2} x^{9} + \frac {1}{8} \, C b^{2} x^{8} + \frac {1}{7} \, {\left (2 \, D a b + B b^{2}\right )} x^{7} + \frac {1}{6} \, {\left (2 \, C a b + A b^{2}\right )} x^{6} + \frac {1}{3} \, B a^{2} x^{3} + \frac {1}{5} \, {\left (D a^{2} + 2 \, B a b\right )} x^{5} + \frac {1}{2} \, A a^{2} x^{2} + \frac {1}{4} \, {\left (C a^{2} + 2 \, A a b\right )} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.11, size = 107, normalized size = 1.03 \[ \frac {a^2\,x^5\,D}{5}+\frac {b^2\,x^9\,D}{9}+\frac {A\,x^2\,\left (3\,a^2+3\,a\,b\,x^2+b^2\,x^4\right )}{6}+\frac {B\,x^3\,\left (35\,a^2+42\,a\,b\,x^2+15\,b^2\,x^4\right )}{105}+\frac {C\,x^4\,\left (6\,a^2+8\,a\,b\,x^2+3\,b^2\,x^4\right )}{24}+\frac {2\,a\,b\,x^7\,D}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 110, normalized size = 1.06 \[ \frac {A a^{2} x^{2}}{2} + \frac {B a^{2} x^{3}}{3} + \frac {C b^{2} x^{8}}{8} + \frac {D b^{2} x^{9}}{9} + x^{7} \left (\frac {B b^{2}}{7} + \frac {2 D a b}{7}\right ) + x^{6} \left (\frac {A b^{2}}{6} + \frac {C a b}{3}\right ) + x^{5} \left (\frac {2 B a b}{5} + \frac {D a^{2}}{5}\right ) + x^{4} \left (\frac {A a b}{2} + \frac {C a^{2}}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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