Optimal. Leaf size=71 \[ -\frac {b \left (c+d x^2\right )^5 (b c-a d)}{5 d^3}+\frac {\left (c+d x^2\right )^4 (b c-a d)^2}{8 d^3}+\frac {b^2 \left (c+d x^2\right )^6}{12 d^3} \]
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Rubi [A] time = 0.12, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {444, 43} \[ -\frac {b \left (c+d x^2\right )^5 (b c-a d)}{5 d^3}+\frac {\left (c+d x^2\right )^4 (b c-a d)^2}{8 d^3}+\frac {b^2 \left (c+d x^2\right )^6}{12 d^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 444
Rubi steps
\begin {align*} \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int (a+b x)^2 (c+d x)^3 \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {(-b c+a d)^2 (c+d x)^3}{d^2}-\frac {2 b (b c-a d) (c+d x)^4}{d^2}+\frac {b^2 (c+d x)^5}{d^2}\right ) \, dx,x,x^2\right )\\ &=\frac {(b c-a d)^2 \left (c+d x^2\right )^4}{8 d^3}-\frac {b (b c-a d) \left (c+d x^2\right )^5}{5 d^3}+\frac {b^2 \left (c+d x^2\right )^6}{12 d^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 119, normalized size = 1.68 \[ \frac {1}{120} x^2 \left (15 d x^6 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+20 c x^4 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+60 a^2 c^3+30 a c^2 x^2 (3 a d+2 b c)+12 b d^2 x^8 (2 a d+3 b c)+10 b^2 d^3 x^{10}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.38, size = 134, normalized size = 1.89 \[ \frac {1}{12} x^{12} d^{3} b^{2} + \frac {3}{10} x^{10} d^{2} c b^{2} + \frac {1}{5} x^{10} d^{3} b a + \frac {3}{8} x^{8} d c^{2} b^{2} + \frac {3}{4} x^{8} d^{2} c b a + \frac {1}{8} x^{8} d^{3} a^{2} + \frac {1}{6} x^{6} c^{3} b^{2} + x^{6} d c^{2} b a + \frac {1}{2} x^{6} d^{2} c a^{2} + \frac {1}{2} x^{4} c^{3} b a + \frac {3}{4} x^{4} d c^{2} a^{2} + \frac {1}{2} x^{2} c^{3} a^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 134, normalized size = 1.89 \[ \frac {1}{12} \, b^{2} d^{3} x^{12} + \frac {3}{10} \, b^{2} c d^{2} x^{10} + \frac {1}{5} \, a b d^{3} x^{10} + \frac {3}{8} \, b^{2} c^{2} d x^{8} + \frac {3}{4} \, a b c d^{2} x^{8} + \frac {1}{8} \, a^{2} d^{3} x^{8} + \frac {1}{6} \, b^{2} c^{3} x^{6} + a b c^{2} d x^{6} + \frac {1}{2} \, a^{2} c d^{2} x^{6} + \frac {1}{2} \, a b c^{3} x^{4} + \frac {3}{4} \, a^{2} c^{2} d x^{4} + \frac {1}{2} \, a^{2} c^{3} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 128, normalized size = 1.80 \[ \frac {b^{2} d^{3} x^{12}}{12}+\frac {\left (2 a b \,d^{3}+3 b^{2} c \,d^{2}\right ) x^{10}}{10}+\frac {\left (a^{2} d^{3}+6 a b c \,d^{2}+3 b^{2} c^{2} d \right ) x^{8}}{8}+\frac {a^{2} c^{3} x^{2}}{2}+\frac {\left (3 a^{2} c \,d^{2}+6 a b \,c^{2} d +b^{2} c^{3}\right ) x^{6}}{6}+\frac {\left (3 a^{2} c^{2} d +2 a b \,c^{3}\right ) x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 127, normalized size = 1.79 \[ \frac {1}{12} \, b^{2} d^{3} x^{12} + \frac {1}{10} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{10} + \frac {1}{8} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{8} + \frac {1}{2} \, a^{2} c^{3} x^{2} + \frac {1}{6} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{6} + \frac {1}{4} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 118, normalized size = 1.66 \[ x^6\,\left (\frac {a^2\,c\,d^2}{2}+a\,b\,c^2\,d+\frac {b^2\,c^3}{6}\right )+x^8\,\left (\frac {a^2\,d^3}{8}+\frac {3\,a\,b\,c\,d^2}{4}+\frac {3\,b^2\,c^2\,d}{8}\right )+\frac {a^2\,c^3\,x^2}{2}+\frac {b^2\,d^3\,x^{12}}{12}+\frac {a\,c^2\,x^4\,\left (3\,a\,d+2\,b\,c\right )}{4}+\frac {b\,d^2\,x^{10}\,\left (2\,a\,d+3\,b\,c\right )}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.09, size = 136, normalized size = 1.92 \[ \frac {a^{2} c^{3} x^{2}}{2} + \frac {b^{2} d^{3} x^{12}}{12} + x^{10} \left (\frac {a b d^{3}}{5} + \frac {3 b^{2} c d^{2}}{10}\right ) + x^{8} \left (\frac {a^{2} d^{3}}{8} + \frac {3 a b c d^{2}}{4} + \frac {3 b^{2} c^{2} d}{8}\right ) + x^{6} \left (\frac {a^{2} c d^{2}}{2} + a b c^{2} d + \frac {b^{2} c^{3}}{6}\right ) + x^{4} \left (\frac {3 a^{2} c^{2} d}{4} + \frac {a b c^{3}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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