Optimal. Leaf size=139 \[ \frac {2 B x}{35 a^2 b^2 \sqrt {a+b x^2}}-\frac {2 (4 a C+3 A b)-3 b B x}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac {x (x (4 a C+3 A b)+3 a B)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {x^3 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}} \]
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Rubi [A] time = 0.15, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1804, 819, 639, 191} \begin {gather*} \frac {2 B x}{35 a^2 b^2 \sqrt {a+b x^2}}-\frac {x (x (4 a C+3 A b)+3 a B)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {2 (4 a C+3 A b)-3 b B x}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac {x^3 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 191
Rule 639
Rule 819
Rule 1804
Rubi steps
\begin {align*} \int \frac {x^3 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx &=-\frac {x^3 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {x^2 (-3 a B-(3 A b+4 a C) x)}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a b}\\ &=-\frac {x^3 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x (3 a B+(3 A b+4 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {\int \frac {-3 a^2 B-2 a (3 A b+4 a C) x}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^2 b^2}\\ &=-\frac {x^3 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x (3 a B+(3 A b+4 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {2 (3 A b+4 a C)-3 b B x}{105 a b^3 \left (a+b x^2\right )^{3/2}}+\frac {(2 B) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{35 a b^2}\\ &=-\frac {x^3 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x (3 a B+(3 A b+4 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {2 (3 A b+4 a C)-3 b B x}{105 a b^3 \left (a+b x^2\right )^{3/2}}+\frac {2 B x}{35 a^2 b^2 \sqrt {a+b x^2}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 84, normalized size = 0.60 \begin {gather*} \frac {-8 a^4 C-2 a^3 b \left (3 A+14 C x^2\right )-7 a^2 b^2 x^2 \left (3 A+5 C x^2\right )+21 a b^3 B x^5+6 b^4 B x^7}{105 a^2 b^3 \left (a+b x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.15, size = 88, normalized size = 0.63 \begin {gather*} \frac {-8 a^4 C-6 a^3 A b-28 a^3 b C x^2-21 a^2 A b^2 x^2-35 a^2 b^2 C x^4+21 a b^3 B x^5+6 b^4 B x^7}{105 a^2 b^3 \left (a+b x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 131, normalized size = 0.94 \begin {gather*} \frac {{\left (6 \, B b^{4} x^{7} + 21 \, B a b^{3} x^{5} - 35 \, C a^{2} b^{2} x^{4} - 8 \, C a^{4} - 6 \, A a^{3} b - 7 \, {\left (4 \, C a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{2} b^{7} x^{8} + 4 \, a^{3} b^{6} x^{6} + 6 \, a^{4} b^{5} x^{4} + 4 \, a^{5} b^{4} x^{2} + a^{6} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.59, size = 95, normalized size = 0.68 \begin {gather*} \frac {{\left ({\left (3 \, {\left (\frac {2 \, B b x^{2}}{a^{2}} + \frac {7 \, B}{a}\right )} x - \frac {35 \, C}{b}\right )} x^{2} - \frac {7 \, {\left (4 \, C a^{4} b + 3 \, A a^{3} b^{2}\right )}}{a^{3} b^{3}}\right )} x^{2} - \frac {2 \, {\left (4 \, C a^{5} + 3 \, A a^{4} b\right )}}{a^{3} b^{3}}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 85, normalized size = 0.61 \begin {gather*} -\frac {-6 B \,x^{7} b^{4}-21 B \,x^{5} a \,b^{3}+35 C \,a^{2} b^{2} x^{4}+21 A \,a^{2} b^{2} x^{2}+28 C \,a^{3} b \,x^{2}+6 A \,a^{3} b +8 C \,a^{4}}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{2} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 179, normalized size = 1.29 \begin {gather*} -\frac {C x^{4}}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {B x^{3}}{4 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {4 \, C a x^{2}}{15 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {A x^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {3 \, B x}{140 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2}} + \frac {2 \, B x}{35 \, \sqrt {b x^{2} + a} a^{2} b^{2}} + \frac {B x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a b^{2}} - \frac {3 \, B a x}{28 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {8 \, C a^{2}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} - \frac {2 \, A a}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 133, normalized size = 0.96 \begin {gather*} \frac {\frac {a\,\left (\frac {A}{7\,b}-\frac {C\,a}{7\,b^2}\right )}{b}+\frac {B\,a\,x}{7\,b^2}}{{\left (b\,x^2+a\right )}^{7/2}}-\frac {\frac {C}{3\,b^3}-\frac {B\,x}{35\,a\,b^2}}{{\left (b\,x^2+a\right )}^{3/2}}+\frac {\frac {a\,\left (\frac {C}{5\,b^2}-\frac {7\,A\,b-7\,C\,a}{35\,a\,b^2}\right )}{b}-\frac {8\,B\,x}{35\,b^2}}{{\left (b\,x^2+a\right )}^{5/2}}+\frac {2\,B\,x}{35\,a^2\,b^2\,\sqrt {b\,x^2+a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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