Optimal. Leaf size=92 \[ a^2 A \log (x)+a^2 B x+a A b x^2+\frac {1}{5} b x^5 (2 a D+b B)+\frac {1}{3} a x^3 (a D+2 b B)+\frac {C \left (a+b x^2\right )^3}{6 b}+\frac {1}{4} A b^2 x^4+\frac {1}{7} b^2 D x^7 \]
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Rubi [A] time = 0.07, antiderivative size = 92, normalized size of antiderivative = 1.00, 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} \[ a^2 A \log (x)+a^2 B x+a A b x^2+\frac {1}{5} b x^5 (2 a D+b B)+\frac {1}{3} a x^3 (a D+2 b B)+\frac {C \left (a+b x^2\right )^3}{6 b}+\frac {1}{4} A b^2 x^4+\frac {1}{7} b^2 D x^7 \]
Antiderivative was successfully verified.
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Rule 1583
Rule 1802
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \left (A+B x+C x^2+D x^3\right )}{x} \, dx &=\frac {C \left (a+b x^2\right )^3}{6 b}+\int \frac {\left (a+b x^2\right )^2 \left (A+B x+D x^3\right )}{x} \, dx\\ &=\frac {C \left (a+b x^2\right )^3}{6 b}+\int \left (a^2 B+\frac {a^2 A}{x}+2 a A b x+a (2 b B+a D) x^2+A b^2 x^3+b (b B+2 a D) x^4+b^2 D x^6\right ) \, dx\\ &=a^2 B x+a A b x^2+\frac {1}{3} a (2 b B+a D) x^3+\frac {1}{4} A b^2 x^4+\frac {1}{5} b (b B+2 a D) x^5+\frac {1}{7} b^2 D x^7+\frac {C \left (a+b x^2\right )^3}{6 b}+a^2 A \log (x)\\ \end {align*}
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Mathematica [A] time = 0.08, size = 88, normalized size = 0.96 \[ \frac {1}{420} x \left (70 a^2 (6 B+x (3 C+2 D x))+14 a b x (30 A+x (20 B+3 x (5 C+4 D x)))+b^2 x^3 (105 A+2 x (42 B+5 x (7 C+6 D x)))\right )+a^2 A \log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 96, normalized size = 1.04 \[ \frac {1}{7} \, D b^{2} x^{7} + \frac {1}{6} \, C b^{2} x^{6} + \frac {1}{5} \, {\left (2 \, D a b + B b^{2}\right )} x^{5} + \frac {1}{4} \, {\left (2 \, C a b + A b^{2}\right )} x^{4} + B a^{2} x + \frac {1}{3} \, {\left (D a^{2} + 2 \, B a b\right )} x^{3} + A a^{2} \log \relax (x) + \frac {1}{2} \, {\left (C a^{2} + 2 \, A a b\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 100, normalized size = 1.09 \[ \frac {1}{7} \, D b^{2} x^{7} + \frac {1}{6} \, C b^{2} x^{6} + \frac {2}{5} \, D a b x^{5} + \frac {1}{5} \, B b^{2} x^{5} + \frac {1}{2} \, C a b x^{4} + \frac {1}{4} \, A b^{2} x^{4} + \frac {1}{3} \, D a^{2} x^{3} + \frac {2}{3} \, B a b x^{3} + \frac {1}{2} \, C a^{2} x^{2} + A a b x^{2} + B a^{2} x + A a^{2} \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 100, normalized size = 1.09 \[ \frac {D b^{2} x^{7}}{7}+\frac {C \,b^{2} x^{6}}{6}+\frac {B \,b^{2} x^{5}}{5}+\frac {2 D a b \,x^{5}}{5}+\frac {A \,b^{2} x^{4}}{4}+\frac {C a b \,x^{4}}{2}+\frac {2 B a b \,x^{3}}{3}+\frac {D a^{2} x^{3}}{3}+A a b \,x^{2}+\frac {C \,a^{2} x^{2}}{2}+A \,a^{2} \ln \relax (x )+B \,a^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 96, normalized size = 1.04 \[ \frac {1}{7} \, D b^{2} x^{7} + \frac {1}{6} \, C b^{2} x^{6} + \frac {1}{5} \, {\left (2 \, D a b + B b^{2}\right )} x^{5} + \frac {1}{4} \, {\left (2 \, C a b + A b^{2}\right )} x^{4} + B a^{2} x + \frac {1}{3} \, {\left (D a^{2} + 2 \, B a b\right )} x^{3} + A a^{2} \log \relax (x) + \frac {1}{2} \, {\left (C a^{2} + 2 \, A a b\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.11, size = 103, normalized size = 1.12 \[ \frac {A\,\left (4\,a^2\,\ln \relax (x)+b^2\,x^4+4\,a\,b\,x^2\right )}{4}+\frac {B\,x\,\left (15\,a^2+10\,a\,b\,x^2+3\,b^2\,x^4\right )}{15}+\frac {a^2\,x^3\,D}{3}+\frac {b^2\,x^7\,D}{7}+\frac {C\,x^2\,\left (3\,a^2+3\,a\,b\,x^2+b^2\,x^4\right )}{6}+\frac {2\,a\,b\,x^5\,D}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.32, size = 104, normalized size = 1.13 \[ A a^{2} \log {\relax (x )} + B a^{2} x + \frac {C b^{2} x^{6}}{6} + \frac {D b^{2} x^{7}}{7} + x^{5} \left (\frac {B b^{2}}{5} + \frac {2 D a b}{5}\right ) + x^{4} \left (\frac {A b^{2}}{4} + \frac {C a b}{2}\right ) + x^{3} \left (\frac {2 B a b}{3} + \frac {D a^{2}}{3}\right ) + x^{2} \left (A a b + \frac {C a^{2}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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