Optimal. Leaf size=135 \[ -\frac {a^3 A}{2 x^2}-\frac {a^3 B}{x}+a^2 \log (x) (a C+3 A b)+a^2 x (a D+3 b B)+\frac {1}{4} b^2 x^4 (3 a C+A b)+\frac {3}{2} a b x^2 (a C+A b)+\frac {1}{5} b^2 x^5 (3 a D+b B)+a b x^3 (a D+b B)+\frac {1}{6} b^3 C x^6+\frac {1}{7} b^3 D x^7 \]
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Rubi [A] time = 0.11, antiderivative size = 135, 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} \[ a^2 \log (x) (a C+3 A b)-\frac {a^3 A}{2 x^2}+a^2 x (a D+3 b B)-\frac {a^3 B}{x}+\frac {1}{4} b^2 x^4 (3 a C+A b)+\frac {3}{2} a b x^2 (a C+A b)+\frac {1}{5} b^2 x^5 (3 a D+b B)+a b x^3 (a D+b B)+\frac {1}{6} b^3 C x^6+\frac {1}{7} b^3 D x^7 \]
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^3} \, dx &=\int \left (a^2 (3 b B+a D)+\frac {a^3 A}{x^3}+\frac {a^3 B}{x^2}+\frac {a^2 (3 A b+a C)}{x}+3 a b (A b+a C) x+3 a b (b B+a D) x^2+b^2 (A b+3 a C) x^3+b^2 (b B+3 a D) x^4+b^3 C x^5+b^3 D x^6\right ) \, dx\\ &=-\frac {a^3 A}{2 x^2}-\frac {a^3 B}{x}+a^2 (3 b B+a D) x+\frac {3}{2} a b (A b+a C) x^2+a b (b B+a D) x^3+\frac {1}{4} b^2 (A b+3 a C) x^4+\frac {1}{5} b^2 (b B+3 a D) x^5+\frac {1}{6} b^3 C x^6+\frac {1}{7} b^3 D x^7+a^2 (3 A b+a C) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.06, size = 124, normalized size = 0.92 \[ -\frac {a^3 \left (A+2 B x-2 D x^3\right )}{2 x^2}+a^2 \log (x) (a C+3 A b)+\frac {1}{2} a^2 b x (6 B+x (3 C+2 D x))+\frac {1}{20} a b^2 x^2 (30 A+x (20 B+3 x (5 C+4 D x)))+\frac {1}{420} b^3 x^4 (105 A+2 x (42 B+5 x (7 C+6 D x))) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 147, normalized size = 1.09 \[ \frac {60 \, D b^{3} x^{9} + 70 \, C b^{3} x^{8} + 84 \, {\left (3 \, D a b^{2} + B b^{3}\right )} x^{7} + 105 \, {\left (3 \, C a b^{2} + A b^{3}\right )} x^{6} + 420 \, {\left (D a^{2} b + B a b^{2}\right )} x^{5} - 420 \, B a^{3} x + 630 \, {\left (C a^{2} b + A a b^{2}\right )} x^{4} - 210 \, A a^{3} + 420 \, {\left (D a^{3} + 3 \, B a^{2} b\right )} x^{3} + 420 \, {\left (C a^{3} + 3 \, A a^{2} b\right )} x^{2} \log \relax (x)}{420 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 144, normalized size = 1.07 \[ \frac {1}{7} \, D b^{3} x^{7} + \frac {1}{6} \, C b^{3} x^{6} + \frac {3}{5} \, D a b^{2} x^{5} + \frac {1}{5} \, B b^{3} x^{5} + \frac {3}{4} \, C a b^{2} x^{4} + \frac {1}{4} \, A b^{3} x^{4} + D a^{2} b x^{3} + B a b^{2} x^{3} + \frac {3}{2} \, C a^{2} b x^{2} + \frac {3}{2} \, A a b^{2} x^{2} + D a^{3} x + 3 \, B a^{2} b x + {\left (C a^{3} + 3 \, A a^{2} b\right )} \log \left ({\left | x \right |}\right ) - \frac {2 \, B a^{3} x + A a^{3}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 144, normalized size = 1.07 \[ \frac {D b^{3} x^{7}}{7}+\frac {C \,b^{3} x^{6}}{6}+\frac {B \,b^{3} x^{5}}{5}+\frac {3 D a \,b^{2} x^{5}}{5}+\frac {A \,b^{3} x^{4}}{4}+\frac {3 C a \,b^{2} x^{4}}{4}+B a \,b^{2} x^{3}+D a^{2} b \,x^{3}+\frac {3 A a \,b^{2} x^{2}}{2}+\frac {3 C \,a^{2} b \,x^{2}}{2}+3 A \,a^{2} b \ln \relax (x )+3 B \,a^{2} b x +C \,a^{3} \ln \relax (x )+D a^{3} x -\frac {B \,a^{3}}{x}-\frac {A \,a^{3}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 139, normalized size = 1.03 \[ \frac {1}{7} \, D b^{3} x^{7} + \frac {1}{6} \, C b^{3} x^{6} + \frac {1}{5} \, {\left (3 \, D a b^{2} + B b^{3}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, C a b^{2} + A b^{3}\right )} x^{4} + {\left (D a^{2} b + B a b^{2}\right )} x^{3} + \frac {3}{2} \, {\left (C a^{2} b + A a b^{2}\right )} x^{2} + {\left (D a^{3} + 3 \, B a^{2} b\right )} x + {\left (C a^{3} + 3 \, A a^{2} b\right )} \log \relax (x) - \frac {2 \, B a^{3} x + A a^{3}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 143, normalized size = 1.06 \[ \frac {A\,b^3\,x^4}{4}-\frac {B\,a^3}{x}-\frac {A\,a^3}{2\,x^2}+\frac {B\,b^3\,x^5}{5}+\frac {C\,b^3\,x^6}{6}+C\,a^3\,\ln \relax (x)+a^3\,x\,D+\frac {b^3\,x^7\,D}{7}+a^2\,b\,x^3\,D+\frac {3\,a\,b^2\,x^5\,D}{5}+3\,B\,a^2\,b\,x+\frac {3\,A\,a\,b^2\,x^2}{2}+B\,a\,b^2\,x^3+\frac {3\,C\,a^2\,b\,x^2}{2}+\frac {3\,C\,a\,b^2\,x^4}{4}+3\,A\,a^2\,b\,\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.67, size = 151, normalized size = 1.12 \[ \frac {C b^{3} x^{6}}{6} + \frac {D b^{3} x^{7}}{7} + a^{2} \left (3 A b + C a\right ) \log {\relax (x )} + x^{5} \left (\frac {B b^{3}}{5} + \frac {3 D a b^{2}}{5}\right ) + x^{4} \left (\frac {A b^{3}}{4} + \frac {3 C a b^{2}}{4}\right ) + x^{3} \left (B a b^{2} + D a^{2} b\right ) + x^{2} \left (\frac {3 A a b^{2}}{2} + \frac {3 C a^{2} b}{2}\right ) + x \left (3 B a^{2} b + D a^{3}\right ) + \frac {- A a^{3} - 2 B a^{3} x}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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