Optimal. Leaf size=134 \[ \frac {x^5 \left (a (3 a C+4 b B)+24 A b^2\right )}{15 a^3 \left (a+b x^2\right )^{7/2}}+\frac {x^3 (a B+6 A b)}{3 a^2 \left (a+b x^2\right )^{7/2}}+\frac {x^7 \left (a \left (15 a^2 D+6 a b C+8 b^2 B\right )+48 A b^3\right )}{105 a^4 \left (a+b x^2\right )^{7/2}}+\frac {A x}{a \left (a+b x^2\right )^{7/2}} \]
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Rubi [A] time = 0.21, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1813, 1803, 12, 264} \begin {gather*} \frac {x^7 \left (a \left (15 a^2 D+6 a b C+8 b^2 B\right )+48 A b^3\right )}{105 a^4 \left (a+b x^2\right )^{7/2}}+\frac {x^5 \left (a (3 a C+4 b B)+24 A b^2\right )}{15 a^3 \left (a+b x^2\right )^{7/2}}+\frac {x^3 (a B+6 A b)}{3 a^2 \left (a+b x^2\right )^{7/2}}+\frac {A x}{a \left (a+b x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 264
Rule 1803
Rule 1813
Rubi steps
\begin {align*} \int \frac {A+B x^2+C x^4+D x^6}{\left (a+b x^2\right )^{9/2}} \, dx &=\frac {A x}{a \left (a+b x^2\right )^{7/2}}+\frac {\int \frac {x^2 \left (6 A b+a \left (B+C x^2+D x^4\right )\right )}{\left (a+b x^2\right )^{9/2}} \, dx}{a}\\ &=\frac {A x}{a \left (a+b x^2\right )^{7/2}}+\frac {(6 A b+a B) x^3}{3 a^2 \left (a+b x^2\right )^{7/2}}+\frac {\int \frac {x^4 \left (4 b (6 A b+a B)+3 a \left (a C+a D x^2\right )\right )}{\left (a+b x^2\right )^{9/2}} \, dx}{3 a^2}\\ &=\frac {A x}{a \left (a+b x^2\right )^{7/2}}+\frac {(6 A b+a B) x^3}{3 a^2 \left (a+b x^2\right )^{7/2}}+\frac {\left (24 A b^2+a (4 b B+3 a C)\right ) x^5}{15 a^3 \left (a+b x^2\right )^{7/2}}+\frac {\int \frac {\left (2 b \left (24 A b^2+4 a b B+3 a^2 C\right )+15 a^3 D\right ) x^6}{\left (a+b x^2\right )^{9/2}} \, dx}{15 a^3}\\ &=\frac {A x}{a \left (a+b x^2\right )^{7/2}}+\frac {(6 A b+a B) x^3}{3 a^2 \left (a+b x^2\right )^{7/2}}+\frac {\left (24 A b^2+a (4 b B+3 a C)\right ) x^5}{15 a^3 \left (a+b x^2\right )^{7/2}}+\frac {\left (48 A b^3+a \left (8 b^2 B+6 a b C+15 a^2 D\right )\right ) \int \frac {x^6}{\left (a+b x^2\right )^{9/2}} \, dx}{15 a^3}\\ &=\frac {A x}{a \left (a+b x^2\right )^{7/2}}+\frac {(6 A b+a B) x^3}{3 a^2 \left (a+b x^2\right )^{7/2}}+\frac {\left (24 A b^2+a (4 b B+3 a C)\right ) x^5}{15 a^3 \left (a+b x^2\right )^{7/2}}+\frac {\left (48 A b^3+a \left (8 b^2 B+6 a b C+15 a^2 D\right )\right ) x^7}{105 a^4 \left (a+b x^2\right )^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 98, normalized size = 0.73 \begin {gather*} \frac {a^3 \left (105 A x+35 B x^3+21 C x^5+15 D x^7\right )+2 a^2 b x^3 \left (105 A+14 B x^2+3 C x^4\right )+8 a b^2 x^5 \left (21 A+B x^2\right )+48 A b^3 x^7}{105 a^4 \left (a+b x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.46, size = 112, normalized size = 0.84 \begin {gather*} \frac {105 a^3 A x+35 a^3 B x^3+21 a^3 C x^5+15 a^3 D x^7+210 a^2 A b x^3+28 a^2 b B x^5+6 a^2 b C x^7+168 a A b^2 x^5+8 a b^2 B x^7+48 A b^3 x^7}{105 a^4 \left (a+b x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 141, normalized size = 1.05 \begin {gather*} \frac {{\left ({\left (15 \, D a^{3} + 6 \, C a^{2} b + 8 \, B a b^{2} + 48 \, A b^{3}\right )} x^{7} + 7 \, {\left (3 \, C a^{3} + 4 \, B a^{2} b + 24 \, A a b^{2}\right )} x^{5} + 105 \, A a^{3} x + 35 \, {\left (B a^{3} + 6 \, A a^{2} b\right )} x^{3}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{4} b^{4} x^{8} + 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.53, size = 131, normalized size = 0.98 \begin {gather*} \frac {{\left ({\left (x^{2} {\left (\frac {{\left (15 \, D a^{3} b^{3} + 6 \, C a^{2} b^{4} + 8 \, B a b^{5} + 48 \, A b^{6}\right )} x^{2}}{a^{4} b^{3}} + \frac {7 \, {\left (3 \, C a^{3} b^{3} + 4 \, B a^{2} b^{4} + 24 \, A a b^{5}\right )}}{a^{4} b^{3}}\right )} + \frac {35 \, {\left (B a^{3} b^{3} + 6 \, A a^{2} b^{4}\right )}}{a^{4} b^{3}}\right )} x^{2} + \frac {105 \, A}{a}\right )} x}{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 = 109, normalized size = 0.81 \begin {gather*} \frac {\left (48 A \,b^{3} x^{6}+8 B a \,b^{2} x^{6}+6 a^{2} b C \,x^{6}+15 D a^{3} x^{6}+168 A a \,b^{2} x^{4}+28 B \,a^{2} b \,x^{4}+21 a^{3} C \,x^{4}+210 A \,a^{2} b \,x^{2}+35 B \,a^{3} x^{2}+105 A \,a^{3}\right ) x}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.42, size = 335, normalized size = 2.50 \begin {gather*} -\frac {D x^{5}}{2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {5 \, D a x^{3}}{8 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {C x^{3}}{4 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {16 \, A x}{35 \, \sqrt {b x^{2} + a} a^{4}} + \frac {8 \, A x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3}} + \frac {6 \, A x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2}} + \frac {A x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a} + \frac {D x}{14 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3}} + \frac {D x}{7 \, \sqrt {b x^{2} + a} a b^{3}} + \frac {3 \, D a x}{56 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}} - \frac {15 \, D a^{2} x}{56 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} + \frac {3 \, C x}{140 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2}} + \frac {2 \, C x}{35 \, \sqrt {b x^{2} + a} a^{2} b^{2}} + \frac {C x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a b^{2}} - \frac {3 \, C a x}{28 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {B x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {8 \, B x}{105 \, \sqrt {b x^{2} + a} a^{3} b} + \frac {4 \, B x}{105 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} b} + \frac {B x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {A+B\,x^2+C\,x^4+x^6\,D}{{\left (b\,x^2+a\right )}^{9/2}} \,d x \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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