Optimal. Leaf size=153 \[ -\frac {a^3 A}{2 x^2}-\frac {a^2 (a B+3 A b)}{x}+c x^3 \left (a B c+A b c+b^2 B\right )+3 a \log (x) \left (A \left (a c+b^2\right )+a b B\right )+\frac {1}{2} x^2 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+x \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac {1}{4} c^2 x^4 (A c+3 b B)+\frac {1}{5} B c^3 x^5 \]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {765} \begin {gather*} -\frac {a^2 (a B+3 A b)}{x}-\frac {a^3 A}{2 x^2}+\frac {1}{2} x^2 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+c x^3 \left (a B c+A b c+b^2 B\right )+x \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+3 a \log (x) \left (A \left (a c+b^2\right )+a b B\right )+\frac {1}{4} c^2 x^4 (A c+3 b B)+\frac {1}{5} B c^3 x^5 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 765
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^3} \, dx &=\int \left (A b^3 \left (1+\frac {3 a \left (b^2 B+2 A b c+a B c\right )}{A b^3}\right )+\frac {a^3 A}{x^3}+\frac {a^2 (3 A b+a B)}{x^2}+\frac {3 a \left (a b B+A \left (b^2+a c\right )\right )}{x}+\left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x+3 c \left (b^2 B+A b c+a B c\right ) x^2+c^2 (3 b B+A c) x^3+B c^3 x^4\right ) \, dx\\ &=-\frac {a^3 A}{2 x^2}-\frac {a^2 (3 A b+a B)}{x}+\left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x+\frac {1}{2} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^2+c \left (b^2 B+A b c+a B c\right ) x^3+\frac {1}{4} c^2 (3 b B+A c) x^4+\frac {1}{5} B c^3 x^5+3 a \left (a b B+A \left (b^2+a c\right )\right ) \log (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 153, normalized size = 1.00 \begin {gather*} -\frac {a^3 A}{2 x^2}-\frac {a^2 (a B+3 A b)}{x}+c x^3 \left (a B c+A b c+b^2 B\right )+3 a \log (x) \left (A \left (a c+b^2\right )+a b B\right )+\frac {1}{2} x^2 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+x \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac {1}{4} c^2 x^4 (A c+3 b B)+\frac {1}{5} B c^3 x^5 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{x^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 168, normalized size = 1.10 \begin {gather*} \frac {4 \, B c^{3} x^{7} + 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 20 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{5} + 10 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} - 10 \, A a^{3} + 20 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 60 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} \log \relax (x) - 20 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{20 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 174, normalized size = 1.14 \begin {gather*} \frac {1}{5} \, B c^{3} x^{5} + \frac {3}{4} \, B b c^{2} x^{4} + \frac {1}{4} \, A c^{3} x^{4} + B b^{2} c x^{3} + B a c^{2} x^{3} + A b c^{2} x^{3} + \frac {1}{2} \, B b^{3} x^{2} + 3 \, B a b c x^{2} + \frac {3}{2} \, A b^{2} c x^{2} + \frac {3}{2} \, A a c^{2} x^{2} + 3 \, B a b^{2} x + A b^{3} x + 3 \, B a^{2} c x + 6 \, A a b c x + 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} \log \left ({\left | x \right |}\right ) - \frac {A a^{3} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 179, normalized size = 1.17 \begin {gather*} \frac {B \,c^{3} x^{5}}{5}+\frac {A \,c^{3} x^{4}}{4}+\frac {3 B b \,c^{2} x^{4}}{4}+A b \,c^{2} x^{3}+B a \,c^{2} x^{3}+B \,b^{2} c \,x^{3}+\frac {3 A a \,c^{2} x^{2}}{2}+\frac {3 A \,b^{2} c \,x^{2}}{2}+3 B a b c \,x^{2}+\frac {B \,b^{3} x^{2}}{2}+3 A \,a^{2} c \ln \relax (x )+3 A a \,b^{2} \ln \relax (x )+6 A a b c x +A \,b^{3} x +3 B \,a^{2} b \ln \relax (x )+3 B \,a^{2} c x +3 B a \,b^{2} x -\frac {3 A \,a^{2} b}{x}-\frac {B \,a^{3}}{x}-\frac {A \,a^{3}}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.58, size = 161, normalized size = 1.05 \begin {gather*} \frac {1}{5} \, B c^{3} x^{5} + \frac {1}{4} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{4} + {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{2} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x + 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} \log \relax (x) - \frac {A a^{3} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.06, size = 162, normalized size = 1.06 \begin {gather*} x^2\,\left (\frac {B\,b^3}{2}+\frac {3\,A\,b^2\,c}{2}+3\,B\,a\,b\,c+\frac {3\,A\,a\,c^2}{2}\right )-\frac {x\,\left (B\,a^3+3\,A\,b\,a^2\right )+\frac {A\,a^3}{2}}{x^2}+x^4\,\left (\frac {A\,c^3}{4}+\frac {3\,B\,b\,c^2}{4}\right )+x^3\,\left (B\,b^2\,c+A\,b\,c^2+B\,a\,c^2\right )+x\,\left (3\,B\,c\,a^2+3\,B\,a\,b^2+6\,A\,c\,a\,b+A\,b^3\right )+\ln \relax (x)\,\left (3\,B\,a^2\,b+3\,A\,c\,a^2+3\,A\,a\,b^2\right )+\frac {B\,c^3\,x^5}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.60, size = 175, normalized size = 1.14 \begin {gather*} \frac {B c^{3} x^{5}}{5} + 3 a \left (A a c + A b^{2} + B a b\right ) \log {\relax (x )} + x^{4} \left (\frac {A c^{3}}{4} + \frac {3 B b c^{2}}{4}\right ) + x^{3} \left (A b c^{2} + B a c^{2} + B b^{2} c\right ) + x^{2} \left (\frac {3 A a c^{2}}{2} + \frac {3 A b^{2} c}{2} + 3 B a b c + \frac {B b^{3}}{2}\right ) + x \left (6 A a b c + A b^{3} + 3 B a^{2} c + 3 B a b^{2}\right ) + \frac {- A a^{3} + x \left (- 6 A a^{2} b - 2 B a^{3}\right )}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________