Optimal. Leaf size=132 \[ -\frac{x^4 (6 a C+A b-5 b B x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac{4 (6 a C+A b)}{35 a b^4 \sqrt{a+b x^2}}+\frac{4 (6 a C+A b)}{105 b^4 \left (a+b x^2\right )^{3/2}}-\frac{x^5 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}} \]
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Rubi [A] time = 0.166223, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {1804, 805, 266, 43} \[ -\frac{x^4 (6 a C+A b-5 b B x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac{4 (6 a C+A b)}{35 a b^4 \sqrt{a+b x^2}}+\frac{4 (6 a C+A b)}{105 b^4 \left (a+b x^2\right )^{3/2}}-\frac{x^5 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 1804
Rule 805
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^5 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx &=-\frac{x^5 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{\int \frac{x^4 (-5 a B-(A b+6 a C) x)}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a b}\\ &=-\frac{x^5 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{x^4 (A b+6 a C-5 b B x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}+\frac{(4 (A b+6 a C)) \int \frac{x^3}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a b^2}\\ &=-\frac{x^5 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{x^4 (A b+6 a C-5 b B x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}+\frac{(2 (A b+6 a C)) \operatorname{Subst}\left (\int \frac{x}{(a+b x)^{5/2}} \, dx,x,x^2\right )}{35 a b^2}\\ &=-\frac{x^5 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{x^4 (A b+6 a C-5 b B x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}+\frac{(2 (A b+6 a C)) \operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^{5/2}}+\frac{1}{b (a+b x)^{3/2}}\right ) \, dx,x,x^2\right )}{35 a b^2}\\ &=-\frac{x^5 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{x^4 (A b+6 a C-5 b B x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}+\frac{4 (A b+6 a C)}{105 b^4 \left (a+b x^2\right )^{3/2}}-\frac{4 (A b+6 a C)}{35 a b^4 \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [A] time = 0.0792782, size = 89, normalized size = 0.67 \[ \frac{-14 a^2 b^2 x^2 \left (2 A+15 C x^2\right )-8 a^3 b \left (A+21 C x^2\right )-48 a^4 C-35 a b^3 x^4 \left (A+3 C x^2\right )+15 b^4 B x^7}{105 a b^4 \left (a+b x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 95, normalized size = 0.7 \begin{align*} -{\frac{-15\,B{x}^{7}{b}^{4}+105\,C{x}^{6}a{b}^{3}+35\,Aa{b}^{3}{x}^{4}+210\,C{a}^{2}{b}^{2}{x}^{4}+28\,A{a}^{2}{b}^{2}{x}^{2}+168\,C{a}^{3}b{x}^{2}+8\,A{a}^{3}b+48\,C{a}^{4}}{105\,a{b}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.08059, size = 324, normalized size = 2.45 \begin{align*} -\frac{C x^{6}}{{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} - \frac{B x^{5}}{2 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} - \frac{2 \, C a x^{4}}{{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2}} - \frac{A x^{4}}{3 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} - \frac{5 \, B a x^{3}}{8 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2}} - \frac{8 \, C a^{2} x^{2}}{5 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{3}} - \frac{4 \, A a x^{2}}{15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2}} + \frac{B x}{14 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} b^{3}} + \frac{B x}{7 \, \sqrt{b x^{2} + a} a b^{3}} + \frac{3 \, B a x}{56 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} b^{3}} - \frac{15 \, B a^{2} x}{56 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{3}} - \frac{16 \, C a^{3}}{35 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{4}} - \frac{8 \, A a^{2}}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6037, size = 290, normalized size = 2.2 \begin{align*} \frac{{\left (15 \, B b^{4} x^{7} - 105 \, C a b^{3} x^{6} - 48 \, C a^{4} - 8 \, A a^{3} b - 35 \,{\left (6 \, C a^{2} b^{2} + A a b^{3}\right )} x^{4} - 28 \,{\left (6 \, C a^{3} b + A a^{2} b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{105 \,{\left (a b^{8} x^{8} + 4 \, a^{2} b^{7} x^{6} + 6 \, a^{3} b^{6} x^{4} + 4 \, a^{4} b^{5} x^{2} + a^{5} b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20047, size = 151, normalized size = 1.14 \begin{align*} \frac{{\left (5 \,{\left (3 \,{\left (\frac{B x}{a} - \frac{7 \, C}{b}\right )} x^{2} - \frac{7 \,{\left (6 \, C a^{4} b^{2} + A a^{3} b^{3}\right )}}{a^{3} b^{4}}\right )} x^{2} - \frac{28 \,{\left (6 \, C a^{5} b + A a^{4} b^{2}\right )}}{a^{3} b^{4}}\right )} x^{2} - \frac{8 \,{\left (6 \, C a^{6} + A a^{5} b\right )}}{a^{3} b^{4}}}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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