Optimal. Leaf size=73 \[ \frac {(b c-a d)^3 \log (a+b x)}{b^4}+\frac {d x (b c-a d)^2}{b^3}+\frac {(c+d x)^2 (b c-a d)}{2 b^2}+\frac {(c+d x)^3}{3 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {43} \[ \frac {d x (b c-a d)^2}{b^3}+\frac {(c+d x)^2 (b c-a d)}{2 b^2}+\frac {(b c-a d)^3 \log (a+b x)}{b^4}+\frac {(c+d x)^3}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rubi steps
\begin {align*} \int \frac {(c+d x)^3}{a+b x} \, dx &=\int \left (\frac {d (b c-a d)^2}{b^3}+\frac {(b c-a d)^3}{b^3 (a+b x)}+\frac {d (b c-a d) (c+d x)}{b^2}+\frac {d (c+d x)^2}{b}\right ) \, dx\\ &=\frac {d (b c-a d)^2 x}{b^3}+\frac {(b c-a d) (c+d x)^2}{2 b^2}+\frac {(c+d x)^3}{3 b}+\frac {(b c-a d)^3 \log (a+b x)}{b^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 74, normalized size = 1.01 \[ \frac {b d x \left (6 a^2 d^2-3 a b d (6 c+d x)+b^2 \left (18 c^2+9 c d x+2 d^2 x^2\right )\right )+6 (b c-a d)^3 \log (a+b x)}{6 b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.71, size = 116, normalized size = 1.59 \[ \frac {2 \, b^{3} d^{3} x^{3} + 3 \, {\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 6 \, {\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x + 6 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{6 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.07, size = 115, normalized size = 1.58 \[ \frac {2 \, b^{2} d^{3} x^{3} + 9 \, b^{2} c d^{2} x^{2} - 3 \, a b d^{3} x^{2} + 18 \, b^{2} c^{2} d x - 18 \, a b c d^{2} x + 6 \, a^{2} d^{3} x}{6 \, b^{3}} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 133, normalized size = 1.82 \[ \frac {d^{3} x^{3}}{3 b}-\frac {a \,d^{3} x^{2}}{2 b^{2}}+\frac {3 c \,d^{2} x^{2}}{2 b}-\frac {a^{3} d^{3} \ln \left (b x +a \right )}{b^{4}}+\frac {3 a^{2} c \,d^{2} \ln \left (b x +a \right )}{b^{3}}+\frac {a^{2} d^{3} x}{b^{3}}-\frac {3 a \,c^{2} d \ln \left (b x +a \right )}{b^{2}}-\frac {3 a c \,d^{2} x}{b^{2}}+\frac {c^{3} \ln \left (b x +a \right )}{b}+\frac {3 c^{2} d x}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.98, size = 114, normalized size = 1.56 \[ \frac {2 \, b^{2} d^{3} x^{3} + 3 \, {\left (3 \, b^{2} c d^{2} - a b d^{3}\right )} x^{2} + 6 \, {\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} x}{6 \, b^{3}} + \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.08, size = 118, normalized size = 1.62 \[ x\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )-x^2\,\left (\frac {a\,d^3}{2\,b^2}-\frac {3\,c\,d^2}{2\,b}\right )+\frac {d^3\,x^3}{3\,b}-\frac {\ln \left (a+b\,x\right )\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{b^4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.36, size = 83, normalized size = 1.14 \[ x^{2} \left (- \frac {a d^{3}}{2 b^{2}} + \frac {3 c d^{2}}{2 b}\right ) + x \left (\frac {a^{2} d^{3}}{b^{3}} - \frac {3 a c d^{2}}{b^{2}} + \frac {3 c^{2} d}{b}\right ) + \frac {d^{3} x^{3}}{3 b} - \frac {\left (a d - b c\right )^{3} \log {\left (a + b x \right )}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________