Optimal. Leaf size=70 \[ a^3 c x+\frac {1}{3} a^2 x^3 (a d+3 b c)+\frac {1}{7} b^2 x^7 (3 a d+b c)+\frac {3}{5} a b x^5 (a d+b c)+\frac {1}{9} b^3 d x^9 \]
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Rubi [A] time = 0.04, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {373} \begin {gather*} \frac {1}{3} a^2 x^3 (a d+3 b c)+a^3 c x+\frac {1}{7} b^2 x^7 (3 a d+b c)+\frac {3}{5} a b x^5 (a d+b c)+\frac {1}{9} b^3 d x^9 \end {gather*}
Antiderivative was successfully verified.
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Rule 373
Rubi steps
\begin {align*} \int \left (a+b x^2\right )^3 \left (c+d x^2\right ) \, dx &=\int \left (a^3 c+a^2 (3 b c+a d) x^2+3 a b (b c+a d) x^4+b^2 (b c+3 a d) x^6+b^3 d x^8\right ) \, dx\\ &=a^3 c x+\frac {1}{3} a^2 (3 b c+a d) x^3+\frac {3}{5} a b (b c+a d) x^5+\frac {1}{7} b^2 (b c+3 a d) x^7+\frac {1}{9} b^3 d x^9\\ \end {align*}
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Mathematica [A] time = 0.01, size = 70, normalized size = 1.00 \begin {gather*} a^3 c x+\frac {1}{3} a^2 x^3 (a d+3 b c)+\frac {1}{7} b^2 x^7 (3 a d+b c)+\frac {3}{5} a b x^5 (a d+b c)+\frac {1}{9} b^3 d x^9 \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a+b x^2\right )^3 \left (c+d x^2\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.49, size = 73, normalized size = 1.04 \begin {gather*} \frac {1}{9} x^{9} d b^{3} + \frac {1}{7} x^{7} c b^{3} + \frac {3}{7} x^{7} d b^{2} a + \frac {3}{5} x^{5} c b^{2} a + \frac {3}{5} x^{5} d b a^{2} + x^{3} c b a^{2} + \frac {1}{3} x^{3} d a^{3} + x c a^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.56, size = 73, normalized size = 1.04 \begin {gather*} \frac {1}{9} \, b^{3} d x^{9} + \frac {1}{7} \, b^{3} c x^{7} + \frac {3}{7} \, a b^{2} d x^{7} + \frac {3}{5} \, a b^{2} c x^{5} + \frac {3}{5} \, a^{2} b d x^{5} + a^{2} b c x^{3} + \frac {1}{3} \, a^{3} d x^{3} + a^{3} c x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 73, normalized size = 1.04 \begin {gather*} \frac {b^{3} d \,x^{9}}{9}+\frac {\left (3 a \,b^{2} d +b^{3} c \right ) x^{7}}{7}+a^{3} c x +\frac {\left (3 a^{2} b d +3 a \,b^{2} c \right ) x^{5}}{5}+\frac {\left (a^{3} d +3 a^{2} b c \right ) x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 70, normalized size = 1.00 \begin {gather*} \frac {1}{9} \, b^{3} d x^{9} + \frac {1}{7} \, {\left (b^{3} c + 3 \, a b^{2} d\right )} x^{7} + \frac {3}{5} \, {\left (a b^{2} c + a^{2} b d\right )} x^{5} + a^{3} c x + \frac {1}{3} \, {\left (3 \, a^{2} b c + a^{3} d\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 65, normalized size = 0.93 \begin {gather*} x^7\,\left (\frac {c\,b^3}{7}+\frac {3\,a\,d\,b^2}{7}\right )+x^3\,\left (\frac {d\,a^3}{3}+b\,c\,a^2\right )+\frac {b^3\,d\,x^9}{9}+a^3\,c\,x+\frac {3\,a\,b\,x^5\,\left (a\,d+b\,c\right )}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 76, normalized size = 1.09 \begin {gather*} a^{3} c x + \frac {b^{3} d x^{9}}{9} + x^{7} \left (\frac {3 a b^{2} d}{7} + \frac {b^{3} c}{7}\right ) + x^{5} \left (\frac {3 a^{2} b d}{5} + \frac {3 a b^{2} c}{5}\right ) + x^{3} \left (\frac {a^{3} d}{3} + a^{2} b c\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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