Optimal. Leaf size=129 \[ \frac{3}{2} a^2 A b x^2+a^3 A \log (x)+\frac{1}{3} a^2 x^3 (a D+3 b B)+a^3 B x+\frac{3}{4} a A b^2 x^4+\frac{1}{7} b^2 x^7 (3 a D+b B)+\frac{3}{5} a b x^5 (a D+b B)+\frac{C \left (a+b x^2\right )^4}{8 b}+\frac{1}{6} A b^3 x^6+\frac{1}{9} b^3 D x^9 \]
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Rubi [A] time = 0.0904408, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {1583, 1802} \[ \frac{3}{2} a^2 A b x^2+a^3 A \log (x)+\frac{1}{3} a^2 x^3 (a D+3 b B)+a^3 B x+\frac{3}{4} a A b^2 x^4+\frac{1}{7} b^2 x^7 (3 a D+b B)+\frac{3}{5} a b x^5 (a D+b B)+\frac{C \left (a+b x^2\right )^4}{8 b}+\frac{1}{6} A b^3 x^6+\frac{1}{9} b^3 D x^9 \]
Antiderivative was successfully verified.
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Rule 1583
Rule 1802
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^3 \left (A+B x+C x^2+D x^3\right )}{x} \, dx &=\frac{C \left (a+b x^2\right )^4}{8 b}+\int \frac{\left (a+b x^2\right )^3 \left (A+B x+D x^3\right )}{x} \, dx\\ &=\frac{C \left (a+b x^2\right )^4}{8 b}+\int \left (a^3 B+\frac{a^3 A}{x}+3 a^2 A b x+a^2 (3 b B+a D) x^2+3 a A b^2 x^3+3 a b (b B+a D) x^4+A b^3 x^5+b^2 (b B+3 a D) x^6+b^3 D x^8\right ) \, dx\\ &=a^3 B x+\frac{3}{2} a^2 A b x^2+\frac{1}{3} a^2 (3 b B+a D) x^3+\frac{3}{4} a A b^2 x^4+\frac{3}{5} a b (b B+a D) x^5+\frac{1}{6} A b^3 x^6+\frac{1}{7} b^2 (b B+3 a D) x^7+\frac{1}{9} b^3 D x^9+\frac{C \left (a+b x^2\right )^4}{8 b}+a^3 A \log (x)\\ \end{align*}
Mathematica [A] time = 0.0606095, size = 121, normalized size = 0.94 \[ \frac{x \left (126 a^2 b x (30 A+x (20 B+3 x (5 C+4 D x)))+420 a^3 (6 B+x (3 C+2 D x))+18 a b^2 x^3 (105 A+2 x (42 B+5 x (7 C+6 D x)))+5 b^3 x^5 (84 A+x (72 B+7 x (9 C+8 D x)))\right )}{2520}+a^3 A \log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 148, normalized size = 1.2 \begin{align*}{\frac{{b}^{3}D{x}^{9}}{9}}+{\frac{C{b}^{3}{x}^{8}}{8}}+{\frac{B{x}^{7}{b}^{3}}{7}}+{\frac{3\,D{x}^{7}a{b}^{2}}{7}}+{\frac{A{x}^{6}{b}^{3}}{6}}+{\frac{C{x}^{6}a{b}^{2}}{2}}+{\frac{3\,B{x}^{5}a{b}^{2}}{5}}+{\frac{3\,D{x}^{5}{a}^{2}b}{5}}+{\frac{3\,A{x}^{4}a{b}^{2}}{4}}+{\frac{3\,C{x}^{4}{a}^{2}b}{4}}+B{x}^{3}{a}^{2}b+{\frac{D{x}^{3}{a}^{3}}{3}}+{\frac{3\,A{x}^{2}{a}^{2}b}{2}}+{\frac{C{x}^{2}{a}^{3}}{2}}+{a}^{3}Bx+{a}^{3}A\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.977444, size = 189, normalized size = 1.47 \begin{align*} \frac{1}{9} \, D b^{3} x^{9} + \frac{1}{8} \, C b^{3} x^{8} + \frac{1}{7} \,{\left (3 \, D a b^{2} + B b^{3}\right )} x^{7} + \frac{1}{6} \,{\left (3 \, C a b^{2} + A b^{3}\right )} x^{6} + \frac{3}{5} \,{\left (D a^{2} b + B a b^{2}\right )} x^{5} + B a^{3} x + \frac{3}{4} \,{\left (C a^{2} b + A a b^{2}\right )} x^{4} + A a^{3} \log \left (x\right ) + \frac{1}{3} \,{\left (D a^{3} + 3 \, B a^{2} b\right )} x^{3} + \frac{1}{2} \,{\left (C a^{3} + 3 \, A a^{2} b\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.444424, size = 158, normalized size = 1.22 \begin{align*} A a^{3} \log{\left (x \right )} + B a^{3} x + \frac{C b^{3} x^{8}}{8} + \frac{D b^{3} x^{9}}{9} + x^{7} \left (\frac{B b^{3}}{7} + \frac{3 D a b^{2}}{7}\right ) + x^{6} \left (\frac{A b^{3}}{6} + \frac{C a b^{2}}{2}\right ) + x^{5} \left (\frac{3 B a b^{2}}{5} + \frac{3 D a^{2} b}{5}\right ) + x^{4} \left (\frac{3 A a b^{2}}{4} + \frac{3 C a^{2} b}{4}\right ) + x^{3} \left (B a^{2} b + \frac{D a^{3}}{3}\right ) + x^{2} \left (\frac{3 A a^{2} b}{2} + \frac{C a^{3}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17059, size = 200, normalized size = 1.55 \begin{align*} \frac{1}{9} \, D b^{3} x^{9} + \frac{1}{8} \, C b^{3} x^{8} + \frac{3}{7} \, D a b^{2} x^{7} + \frac{1}{7} \, B b^{3} x^{7} + \frac{1}{2} \, C a b^{2} x^{6} + \frac{1}{6} \, A b^{3} x^{6} + \frac{3}{5} \, D a^{2} b x^{5} + \frac{3}{5} \, B a b^{2} x^{5} + \frac{3}{4} \, C a^{2} b x^{4} + \frac{3}{4} \, A a b^{2} x^{4} + \frac{1}{3} \, D a^{3} x^{3} + B a^{2} b x^{3} + \frac{1}{2} \, C a^{3} x^{2} + \frac{3}{2} \, A a^{2} b x^{2} + B a^{3} x + A a^{3} \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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