Optimal. Leaf size=149 \[ \frac{x (5 a C+2 A b)}{35 a^2 b^3 \sqrt{a+b x^2}}-\frac{x^2 (x (5 a C+2 A b)+4 a B)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac{3 x (5 a C+2 A b)+8 a B}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac{x^4 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}} \]
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Rubi [A] time = 0.181302, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {1804, 819, 778, 191} \[ \frac{x (5 a C+2 A b)}{35 a^2 b^3 \sqrt{a+b x^2}}-\frac{x^2 (x (5 a C+2 A b)+4 a B)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac{3 x (5 a C+2 A b)+8 a B}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac{x^4 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 1804
Rule 819
Rule 778
Rule 191
Rubi steps
\begin{align*} \int \frac{x^4 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx &=-\frac{x^4 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{\int \frac{x^3 (-4 a B-(2 A b+5 a C) x)}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a b}\\ &=-\frac{x^4 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{x^2 (4 a B+(2 A b+5 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac{\int \frac{x \left (-8 a^2 B-3 a (2 A b+5 a C) x\right )}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^2 b^2}\\ &=-\frac{x^4 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{x^2 (4 a B+(2 A b+5 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac{8 a B+3 (2 A b+5 a C) x}{105 a b^3 \left (a+b x^2\right )^{3/2}}+\frac{(2 A b+5 a C) \int \frac{1}{\left (a+b x^2\right )^{3/2}} \, dx}{35 a b^3}\\ &=-\frac{x^4 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac{x^2 (4 a B+(2 A b+5 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac{8 a B+3 (2 A b+5 a C) x}{105 a b^3 \left (a+b x^2\right )^{3/2}}+\frac{(2 A b+5 a C) x}{35 a^2 b^3 \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [A] time = 0.0819922, size = 78, normalized size = 0.52 \[ \frac{-35 a^2 b^2 B x^4-28 a^3 b B x^2-8 a^4 B+3 a b^3 x^5 \left (7 A+5 C x^2\right )+6 A b^4 x^7}{105 a^2 b^3 \left (a+b x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 76, normalized size = 0.5 \begin{align*}{\frac{6\,A{b}^{4}{x}^{7}+15\,Ca{b}^{3}{x}^{7}+21\,A{x}^{5}a{b}^{3}-35\,B{x}^{4}{a}^{2}{b}^{2}-28\,B{a}^{3}{x}^{2}b-8\,{a}^{4}B}{105\,{a}^{2}{b}^{3}} \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 [A] time = 1.02215, size = 342, normalized size = 2.3 \begin{align*} -\frac{C x^{5}}{2 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} - \frac{B x^{4}}{3 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} - \frac{5 \, C a x^{3}}{8 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2}} - \frac{A x^{3}}{4 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} - \frac{4 \, B a x^{2}}{15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2}} + \frac{C x}{14 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} b^{3}} + \frac{C x}{7 \, \sqrt{b x^{2} + a} a b^{3}} + \frac{3 \, C a x}{56 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} b^{3}} - \frac{15 \, C a^{2} x}{56 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{3}} + \frac{3 \, A x}{140 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} b^{2}} + \frac{2 \, A x}{35 \, \sqrt{b x^{2} + a} a^{2} b^{2}} + \frac{A x}{35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a b^{2}} - \frac{3 \, A a x}{28 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2}} - \frac{8 \, B 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 = 1.63217, size = 254, normalized size = 1.7 \begin{align*} \frac{{\left (21 \, A a b^{3} x^{5} - 35 \, B a^{2} b^{2} x^{4} + 3 \,{\left (5 \, C a b^{3} + 2 \, A b^{4}\right )} x^{7} - 28 \, B a^{3} b x^{2} - 8 \, B a^{4}\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{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 = 1.19, size = 109, normalized size = 0.73 \begin{align*} \frac{{\left ({\left (3 \, x{\left (\frac{7 \, A}{a} + \frac{{\left (5 \, C a^{2} b^{3} + 2 \, A a b^{4}\right )} x^{2}}{a^{3} b^{3}}\right )} - \frac{35 \, B}{b}\right )} x^{2} - \frac{28 \, B a}{b^{2}}\right )} x^{2} - \frac{8 \, B a^{2}}{b^{3}}}{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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