Optimal. Leaf size=139 \[ -\frac {a^3 A}{3 x^3}-\frac {a^3 B}{2 x^2}-\frac {a^2 (a C+3 A b)}{x}+a^2 \log (x) (a D+3 b B)+\frac {1}{3} b^2 x^3 (3 a C+A b)+3 a b x (a C+A b)+\frac {1}{4} b^2 x^4 (3 a D+b B)+\frac {3}{2} a b x^2 (a D+b B)+\frac {1}{5} b^3 C x^5+\frac {1}{6} b^3 D x^6 \]
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Rubi [A] time = 0.11, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {1802} \[ -\frac {a^2 (a C+3 A b)}{x}-\frac {a^3 A}{3 x^3}+a^2 \log (x) (a D+3 b B)-\frac {a^3 B}{2 x^2}+\frac {1}{3} b^2 x^3 (3 a C+A b)+3 a b x (a C+A b)+\frac {1}{4} b^2 x^4 (3 a D+b B)+\frac {3}{2} a b x^2 (a D+b B)+\frac {1}{5} b^3 C x^5+\frac {1}{6} b^3 D x^6 \]
Antiderivative was successfully verified.
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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^4} \, dx &=\int \left (3 a b (A b+a C)+\frac {a^3 A}{x^4}+\frac {a^3 B}{x^3}+\frac {a^2 (3 A b+a C)}{x^2}+\frac {a^2 (3 b B+a D)}{x}+3 a b (b B+a D) x+b^2 (A b+3 a C) x^2+b^2 (b B+3 a D) x^3+b^3 C x^4+b^3 D x^5\right ) \, dx\\ &=-\frac {a^3 A}{3 x^3}-\frac {a^3 B}{2 x^2}-\frac {a^2 (3 A b+a C)}{x}+3 a b (A b+a C) x+\frac {3}{2} a b (b B+a D) x^2+\frac {1}{3} b^2 (A b+3 a C) x^3+\frac {1}{4} b^2 (b B+3 a D) x^4+\frac {1}{5} b^3 C x^5+\frac {1}{6} b^3 D x^6+a^2 (3 b B+a D) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.05, size = 124, normalized size = 0.89 \[ -\frac {a^3 (2 A+3 x (B+2 C x))}{6 x^3}+\frac {3 a^2 b \left (x^2 (2 C+D x)-2 A\right )}{2 x}+a^2 \log (x) (a D+3 b B)+\frac {1}{4} a b^2 x (12 A+x (6 B+x (4 C+3 D x)))+\frac {1}{60} b^3 x^3 (20 A+x (15 B+2 x (6 C+5 D x))) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 147, normalized size = 1.06 \[ \frac {10 \, D b^{3} x^{9} + 12 \, C b^{3} x^{8} + 15 \, {\left (3 \, D a b^{2} + B b^{3}\right )} x^{7} + 20 \, {\left (3 \, C a b^{2} + A b^{3}\right )} x^{6} + 90 \, {\left (D a^{2} b + B a b^{2}\right )} x^{5} - 30 \, B a^{3} x + 180 \, {\left (C a^{2} b + A a b^{2}\right )} x^{4} + 60 \, {\left (D a^{3} + 3 \, B a^{2} b\right )} x^{3} \log \relax (x) - 20 \, A a^{3} - 60 \, {\left (C a^{3} + 3 \, A a^{2} b\right )} x^{2}}{60 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 146, normalized size = 1.05 \[ \frac {1}{6} \, D b^{3} x^{6} + \frac {1}{5} \, C b^{3} x^{5} + \frac {3}{4} \, D a b^{2} x^{4} + \frac {1}{4} \, B b^{3} x^{4} + C a b^{2} x^{3} + \frac {1}{3} \, A b^{3} x^{3} + \frac {3}{2} \, D a^{2} b x^{2} + \frac {3}{2} \, B a b^{2} x^{2} + 3 \, C a^{2} b x + 3 \, A a b^{2} x + {\left (D a^{3} + 3 \, B a^{2} b\right )} \log \left ({\left | x \right |}\right ) - \frac {3 \, B a^{3} x + 2 \, A a^{3} + 6 \, {\left (C a^{3} + 3 \, A a^{2} b\right )} x^{2}}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 146, normalized size = 1.05 \[ \frac {D b^{3} x^{6}}{6}+\frac {C \,b^{3} x^{5}}{5}+\frac {B \,b^{3} x^{4}}{4}+\frac {3 D a \,b^{2} x^{4}}{4}+\frac {A \,b^{3} x^{3}}{3}+C a \,b^{2} x^{3}+\frac {3 B a \,b^{2} x^{2}}{2}+\frac {3 D a^{2} b \,x^{2}}{2}+3 A a \,b^{2} x +3 B \,a^{2} b \ln \relax (x )+3 C \,a^{2} b x +D a^{3} \ln \relax (x )-\frac {3 A \,a^{2} b}{x}-\frac {C \,a^{3}}{x}-\frac {B \,a^{3}}{2 x^{2}}-\frac {A \,a^{3}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 142, normalized size = 1.02 \[ \frac {1}{6} \, D b^{3} x^{6} + \frac {1}{5} \, C b^{3} x^{5} + \frac {1}{4} \, {\left (3 \, D a b^{2} + B b^{3}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, C a b^{2} + A b^{3}\right )} x^{3} + \frac {3}{2} \, {\left (D a^{2} b + B a b^{2}\right )} x^{2} + 3 \, {\left (C a^{2} b + A a b^{2}\right )} x + {\left (D a^{3} + 3 \, B a^{2} b\right )} \log \relax (x) - \frac {3 \, B a^{3} x + 2 \, A a^{3} + 6 \, {\left (C a^{3} + 3 \, A a^{2} b\right )} x^{2}}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.37, size = 148, normalized size = 1.06 \[ \frac {B\,b^3\,x^4}{4}-\frac {C\,a^3}{x}-\frac {B\,a^3}{2\,x^2}+\frac {C\,b^3\,x^5}{5}+\frac {b^3\,x^6\,D}{6}-\frac {A\,\left (a^3+9\,a^2\,b\,x^2-9\,a\,b^2\,x^4-b^3\,x^6\right )}{3\,x^3}+\frac {a^3\,\ln \left (x^2\right )\,D}{2}+\frac {3\,a^2\,b\,x^2\,D}{2}+3\,C\,a^2\,b\,x+\frac {3\,a\,b^2\,x^4\,D}{4}+\frac {3\,B\,a\,b^2\,x^2}{2}+C\,a\,b^2\,x^3+3\,B\,a^2\,b\,\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.09, size = 155, normalized size = 1.12 \[ \frac {C b^{3} x^{5}}{5} + \frac {D b^{3} x^{6}}{6} + a^{2} \left (3 B b + D a\right ) \log {\relax (x )} + x^{4} \left (\frac {B b^{3}}{4} + \frac {3 D a b^{2}}{4}\right ) + x^{3} \left (\frac {A b^{3}}{3} + C a b^{2}\right ) + x^{2} \left (\frac {3 B a b^{2}}{2} + \frac {3 D a^{2} b}{2}\right ) + x \left (3 A a b^{2} + 3 C a^{2} b\right ) + \frac {- 2 A a^{3} - 3 B a^{3} x + x^{2} \left (- 18 A a^{2} b - 6 C a^{3}\right )}{6 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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