Optimal. Leaf size=139 \[ \frac{2 B x}{35 a^2 b^2 \sqrt{a+b x^2}}-\frac{x (x (4 a C+3 A b)+3 a B)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac{2 (4 a C+3 A b)-3 b B x}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac{x^3 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}} \]
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Rubi [A] time = 0.151264, antiderivative size = 139, 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, 639, 191} \[ \frac{2 B x}{35 a^2 b^2 \sqrt{a+b x^2}}-\frac{x (x (4 a C+3 A b)+3 a B)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac{2 (4 a C+3 A b)-3 b B x}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac{x^3 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 1804
Rule 819
Rule 639
Rule 191
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*}
Mathematica [A] time = 0.089708, size = 84, normalized size = 0.6 \[ \frac{-7 a^2 b^2 x^2 \left (3 A+5 C x^2\right )-2 a^3 b \left (3 A+14 C x^2\right )-8 a^4 C+21 a b^3 B x^5+6 b^4 B x^7}{105 a^2 b^3 \left (a+b x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 85, normalized size = 0.6 \begin{align*} -{\frac{-6\,B{x}^{7}{b}^{4}-21\,B{x}^{5}a{b}^{3}+35\,C{x}^{4}{a}^{2}{b}^{2}+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\,{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 = 0.988154, size = 242, normalized size = 1.74 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6596, size = 271, normalized size = 1.95 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 142.489, size = 660, normalized size = 4.75 \begin{align*} A \left (\begin{cases} - \frac{2 a}{35 a^{3} b^{2} \sqrt{a + b x^{2}} + 105 a^{2} b^{3} x^{2} \sqrt{a + b x^{2}} + 105 a b^{4} x^{4} \sqrt{a + b x^{2}} + 35 b^{5} x^{6} \sqrt{a + b x^{2}}} - \frac{7 b x^{2}}{35 a^{3} b^{2} \sqrt{a + b x^{2}} + 105 a^{2} b^{3} x^{2} \sqrt{a + b x^{2}} + 105 a b^{4} x^{4} \sqrt{a + b x^{2}} + 35 b^{5} x^{6} \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 a^{\frac{9}{2}}} & \text{otherwise} \end{cases}\right ) + B \left (\frac{7 a x^{5}}{35 a^{\frac{11}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 105 a^{\frac{9}{2}} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}} + 105 a^{\frac{7}{2}} b^{2} x^{4} \sqrt{1 + \frac{b x^{2}}{a}} + 35 a^{\frac{5}{2}} b^{3} x^{6} \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{2 b x^{7}}{35 a^{\frac{11}{2}} \sqrt{1 + \frac{b x^{2}}{a}} + 105 a^{\frac{9}{2}} b x^{2} \sqrt{1 + \frac{b x^{2}}{a}} + 105 a^{\frac{7}{2}} b^{2} x^{4} \sqrt{1 + \frac{b x^{2}}{a}} + 35 a^{\frac{5}{2}} b^{3} x^{6} \sqrt{1 + \frac{b x^{2}}{a}}}\right ) + C \left (\begin{cases} - \frac{8 a^{2}}{105 a^{3} b^{3} \sqrt{a + b x^{2}} + 315 a^{2} b^{4} x^{2} \sqrt{a + b x^{2}} + 315 a b^{5} x^{4} \sqrt{a + b x^{2}} + 105 b^{6} x^{6} \sqrt{a + b x^{2}}} - \frac{28 a b x^{2}}{105 a^{3} b^{3} \sqrt{a + b x^{2}} + 315 a^{2} b^{4} x^{2} \sqrt{a + b x^{2}} + 315 a b^{5} x^{4} \sqrt{a + b x^{2}} + 105 b^{6} x^{6} \sqrt{a + b x^{2}}} - \frac{35 b^{2} x^{4}}{105 a^{3} b^{3} \sqrt{a + b x^{2}} + 315 a^{2} b^{4} x^{2} \sqrt{a + b x^{2}} + 315 a b^{5} x^{4} \sqrt{a + b x^{2}} + 105 b^{6} x^{6} \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{6}}{6 a^{\frac{9}{2}}} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21818, size = 128, normalized size = 0.92 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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