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 = 0.09, antiderivative size = 133, 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} \[ \frac {1}{3} a^2 x^3 (a C+3 A b)+a^3 A x+\frac {1}{2} a^2 b D x^6+\frac {1}{4} a^3 D x^4+\frac {1}{7} b^2 x^7 (3 a C+A b)+\frac {3}{5} a b x^5 (a C+A b)+\frac {3}{8} a b^2 D x^8+\frac {B \left (a+b x^2\right )^4}{8 b}+\frac {1}{9} b^3 C x^9+\frac {1}{10} b^3 D x^{10} \]
Antiderivative was successfully verified.
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Rule 1582
Rule 1810
Rubi steps
\begin {align*} \int \left (a+b x^2\right )^3 \left (A+B x+C x^2+D x^3\right ) \, dx &=\frac {B \left (a+b x^2\right )^4}{8 b}+\int \left (a+b x^2\right )^3 \left (A+C x^2+D x^3\right ) \, dx\\ &=\frac {B \left (a+b x^2\right )^4}{8 b}+\int \left (a^3 A+a^2 (3 A b+a C) x^2+a^3 D x^3+3 a b (A b+a C) x^4+3 a^2 b D x^5+b^2 (A b+3 a C) x^6+3 a b^2 D x^7+b^3 C x^8+b^3 D x^9\right ) \, dx\\ &=a^3 A x+\frac {1}{3} a^2 (3 A b+a C) x^3+\frac {1}{4} a^3 D x^4+\frac {3}{5} a b (A b+a C) x^5+\frac {1}{2} a^2 b D x^6+\frac {1}{7} b^2 (A b+3 a C) x^7+\frac {3}{8} a b^2 D x^8+\frac {1}{9} b^3 C x^9+\frac {1}{10} b^3 D x^{10}+\frac {B \left (a+b x^2\right )^4}{8 b}\\ \end {align*}
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Mathematica [A] time = 0.05, 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.
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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 = 0.40, size = 149, normalized size = 1.12 \[ \frac {1}{10} \, D b^{3} x^{10} + \frac {1}{9} \, C b^{3} x^{9} + \frac {3}{8} \, D a b^{2} x^{8} + \frac {1}{8} \, B b^{3} x^{8} + \frac {3}{7} \, C a b^{2} x^{7} + \frac {1}{7} \, A b^{3} x^{7} + \frac {1}{2} \, D a^{2} b x^{6} + \frac {1}{2} \, B a b^{2} x^{6} + \frac {3}{5} \, C a^{2} b x^{5} + \frac {3}{5} \, A a b^{2} x^{5} + \frac {1}{4} \, D a^{3} x^{4} + \frac {3}{4} \, B a^{2} b x^{4} + \frac {1}{3} \, C a^{3} x^{3} + A a^{2} b x^{3} + \frac {1}{2} \, B a^{3} x^{2} + A a^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 147, normalized size = 1.11 \[ \frac {D b^{3} x^{10}}{10}+\frac {C \,b^{3} x^{9}}{9}+\frac {\left (b^{3} B +3 a \,b^{2} D\right ) x^{8}}{8}+\frac {\left (A \,b^{3}+3 a \,b^{2} C \right ) x^{7}}{7}+\frac {B \,a^{3} x^{2}}{2}+\frac {\left (3 a \,b^{2} B +3 a^{2} b D\right ) x^{6}}{6}+A \,a^{3} x +\frac {\left (3 a \,b^{2} A +3 a^{2} b C \right ) x^{5}}{5}+\frac {\left (3 a^{2} b B +a^{3} D\right ) x^{4}}{4}+\frac {\left (3 A \,a^{2} b +a^{3} C \right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 142, normalized size = 1.07 \[ \frac {1}{10} \, D b^{3} x^{10} + \frac {1}{9} \, C b^{3} x^{9} + \frac {1}{8} \, {\left (3 \, D a b^{2} + B b^{3}\right )} x^{8} + \frac {1}{7} \, {\left (3 \, C a b^{2} + A b^{3}\right )} x^{7} + \frac {1}{2} \, {\left (D a^{2} b + B a b^{2}\right )} x^{6} + \frac {1}{2} \, B a^{3} x^{2} + \frac {3}{5} \, {\left (C a^{2} b + A a b^{2}\right )} x^{5} + A a^{3} x + \frac {1}{4} \, {\left (D a^{3} + 3 \, B a^{2} b\right )} x^{4} + \frac {1}{3} \, {\left (C a^{3} + 3 \, A a^{2} b\right )} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 149, normalized size = 1.12 \[ \frac {B\,a^3\,x^2}{2}+\frac {A\,b^3\,x^7}{7}+\frac {C\,a^3\,x^3}{3}+\frac {B\,b^3\,x^8}{8}+\frac {C\,b^3\,x^9}{9}+\frac {a^3\,x^4\,D}{4}+\frac {b^3\,x^{10}\,D}{10}+A\,a^3\,x+\frac {a^2\,b\,x^6\,D}{2}+\frac {3\,a\,b^2\,x^8\,D}{8}+A\,a^2\,b\,x^3+\frac {3\,A\,a\,b^2\,x^5}{5}+\frac {3\,B\,a^2\,b\,x^4}{4}+\frac {B\,a\,b^2\,x^6}{2}+\frac {3\,C\,a^2\,b\,x^5}{5}+\frac {3\,C\,a\,b^2\,x^7}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, 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.
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