Optimal. Leaf size=55 \[ \frac {1}{2} A b^2 x^2+\frac {1}{4} c x^4 (A c+2 b B)+\frac {1}{3} b x^3 (2 A c+b B)+\frac {1}{5} B c^2 x^5 \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {765} \begin {gather*} \frac {1}{2} A b^2 x^2+\frac {1}{4} c x^4 (A c+2 b B)+\frac {1}{3} b x^3 (2 A c+b B)+\frac {1}{5} B c^2 x^5 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 765
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^2}{x} \, dx &=\int \left (A b^2 x+b (b B+2 A c) x^2+c (2 b B+A c) x^3+B c^2 x^4\right ) \, dx\\ &=\frac {1}{2} A b^2 x^2+\frac {1}{3} b (b B+2 A c) x^3+\frac {1}{4} c (2 b B+A c) x^4+\frac {1}{5} B c^2 x^5\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 49, normalized size = 0.89 \begin {gather*} \frac {1}{60} x^2 \left (30 A b^2+15 c x^2 (A c+2 b B)+20 b x (2 A c+b B)+12 B c^2 x^3\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.03, size = 65, normalized size = 1.18 \begin {gather*} \frac {1}{2} A b^2 x^2+\frac {2}{3} A b c x^3+\frac {1}{4} A c^2 x^4+\frac {1}{3} b^2 B x^3+\frac {1}{2} b B c x^4+\frac {1}{5} B c^2 x^5 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.39, size = 51, normalized size = 0.93 \begin {gather*} \frac {1}{5} \, B c^{2} x^{5} + \frac {1}{2} \, A b^{2} x^{2} + \frac {1}{4} \, {\left (2 \, B b c + A c^{2}\right )} x^{4} + \frac {1}{3} \, {\left (B b^{2} + 2 \, A b c\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 53, normalized size = 0.96 \begin {gather*} \frac {1}{5} \, B c^{2} x^{5} + \frac {1}{2} \, B b c x^{4} + \frac {1}{4} \, A c^{2} x^{4} + \frac {1}{3} \, B b^{2} x^{3} + \frac {2}{3} \, A b c x^{3} + \frac {1}{2} \, A b^{2} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 52, normalized size = 0.95 \begin {gather*} \frac {B \,c^{2} x^{5}}{5}+\frac {A \,b^{2} x^{2}}{2}+\frac {\left (A \,c^{2}+2 b B c \right ) x^{4}}{4}+\frac {\left (2 A b c +b^{2} B \right ) x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.90, size = 51, normalized size = 0.93 \begin {gather*} \frac {1}{5} \, B c^{2} x^{5} + \frac {1}{2} \, A b^{2} x^{2} + \frac {1}{4} \, {\left (2 \, B b c + A c^{2}\right )} x^{4} + \frac {1}{3} \, {\left (B b^{2} + 2 \, A b c\right )} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 51, normalized size = 0.93 \begin {gather*} x^3\,\left (\frac {B\,b^2}{3}+\frac {2\,A\,c\,b}{3}\right )+x^4\,\left (\frac {A\,c^2}{4}+\frac {B\,b\,c}{2}\right )+\frac {A\,b^2\,x^2}{2}+\frac {B\,c^2\,x^5}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.08, size = 54, normalized size = 0.98 \begin {gather*} \frac {A b^{2} x^{2}}{2} + \frac {B c^{2} x^{5}}{5} + x^{4} \left (\frac {A c^{2}}{4} + \frac {B b c}{2}\right ) + x^{3} \left (\frac {2 A b c}{3} + \frac {B b^{2}}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________