Optimal. Leaf size=75 \[ \frac {1}{3} A b^3 x^3+\frac {1}{4} b^2 (b B+3 A c) x^4+\frac {3}{5} b c (b B+A c) x^5+\frac {1}{6} c^2 (3 b B+A c) x^6+\frac {1}{7} B c^3 x^7 \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 75, 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 = {779}
\begin {gather*} \frac {1}{3} A b^3 x^3+\frac {1}{4} b^2 x^4 (3 A c+b B)+\frac {1}{6} c^2 x^6 (A c+3 b B)+\frac {3}{5} b c x^5 (A c+b B)+\frac {1}{7} B c^3 x^7 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 779
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^3}{x} \, dx &=\int \left (A b^3 x^2+b^2 (b B+3 A c) x^3+3 b c (b B+A c) x^4+c^2 (3 b B+A c) x^5+B c^3 x^6\right ) \, dx\\ &=\frac {1}{3} A b^3 x^3+\frac {1}{4} b^2 (b B+3 A c) x^4+\frac {3}{5} b c (b B+A c) x^5+\frac {1}{6} c^2 (3 b B+A c) x^6+\frac {1}{7} B c^3 x^7\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 75, normalized size = 1.00 \begin {gather*} \frac {1}{3} A b^3 x^3+\frac {1}{4} b^2 (b B+3 A c) x^4+\frac {3}{5} b c (b B+A c) x^5+\frac {1}{6} c^2 (3 b B+A c) x^6+\frac {1}{7} B c^3 x^7 \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.49, size = 76, normalized size = 1.01
method | result | size |
norman | \(\frac {B \,c^{3} x^{7}}{7}+\left (\frac {1}{6} A \,c^{3}+\frac {1}{2} B b \,c^{2}\right ) x^{6}+\left (\frac {3}{5} A b \,c^{2}+\frac {3}{5} B \,b^{2} c \right ) x^{5}+\left (\frac {3}{4} A \,b^{2} c +\frac {1}{4} B \,b^{3}\right ) x^{4}+\frac {A \,b^{3} x^{3}}{3}\) | \(75\) |
gosper | \(\frac {x^{3} \left (60 B \,c^{3} x^{4}+70 A \,c^{3} x^{3}+210 B b \,c^{2} x^{3}+252 A b \,c^{2} x^{2}+252 B \,b^{2} c \,x^{2}+315 A \,b^{2} c x +105 B \,b^{3} x +140 A \,b^{3}\right )}{420}\) | \(76\) |
default | \(\frac {B \,c^{3} x^{7}}{7}+\frac {\left (A \,c^{3}+3 B b \,c^{2}\right ) x^{6}}{6}+\frac {\left (3 A b \,c^{2}+3 B \,b^{2} c \right ) x^{5}}{5}+\frac {\left (3 A \,b^{2} c +B \,b^{3}\right ) x^{4}}{4}+\frac {A \,b^{3} x^{3}}{3}\) | \(76\) |
risch | \(\frac {1}{7} B \,c^{3} x^{7}+\frac {1}{6} A \,c^{3} x^{6}+\frac {1}{2} x^{6} B b \,c^{2}+\frac {3}{5} x^{5} A b \,c^{2}+\frac {3}{5} x^{5} B \,b^{2} c +\frac {3}{4} x^{4} A \,b^{2} c +\frac {1}{4} x^{4} B \,b^{3}+\frac {1}{3} A \,b^{3} x^{3}\) | \(78\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 73, normalized size = 0.97 \begin {gather*} \frac {1}{7} \, B c^{3} x^{7} + \frac {1}{3} \, A b^{3} x^{3} + \frac {1}{6} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + \frac {3}{5} \, {\left (B b^{2} c + A b c^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.64, size = 73, normalized size = 0.97 \begin {gather*} \frac {1}{7} \, B c^{3} x^{7} + \frac {1}{3} \, A b^{3} x^{3} + \frac {1}{6} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + \frac {3}{5} \, {\left (B b^{2} c + A b c^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.01, size = 82, normalized size = 1.09 \begin {gather*} \frac {A b^{3} x^{3}}{3} + \frac {B c^{3} x^{7}}{7} + x^{6} \left (\frac {A c^{3}}{6} + \frac {B b c^{2}}{2}\right ) + x^{5} \cdot \left (\frac {3 A b c^{2}}{5} + \frac {3 B b^{2} c}{5}\right ) + x^{4} \cdot \left (\frac {3 A b^{2} c}{4} + \frac {B b^{3}}{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 2.81, size = 77, normalized size = 1.03 \begin {gather*} \frac {1}{7} \, B c^{3} x^{7} + \frac {1}{2} \, B b c^{2} x^{6} + \frac {1}{6} \, A c^{3} x^{6} + \frac {3}{5} \, B b^{2} c x^{5} + \frac {3}{5} \, A b c^{2} x^{5} + \frac {1}{4} \, B b^{3} x^{4} + \frac {3}{4} \, A b^{2} c x^{4} + \frac {1}{3} \, A b^{3} x^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.03, size = 69, normalized size = 0.92 \begin {gather*} x^4\,\left (\frac {B\,b^3}{4}+\frac {3\,A\,c\,b^2}{4}\right )+x^6\,\left (\frac {A\,c^3}{6}+\frac {B\,b\,c^2}{2}\right )+\frac {A\,b^3\,x^3}{3}+\frac {B\,c^3\,x^7}{7}+\frac {3\,b\,c\,x^5\,\left (A\,c+B\,b\right )}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________