Optimal. Leaf size=85 \[ -\frac {(b c-a d)^3 \log (a+b x)}{a^3 b}+\frac {c^2 (b c-3 a d)}{a^2 x}+\frac {c \log (x) \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{a^3}-\frac {c^3}{2 a x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {88} \[ \frac {c \log (x) \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{a^3}+\frac {c^2 (b c-3 a d)}{a^2 x}-\frac {(b c-a d)^3 \log (a+b x)}{a^3 b}-\frac {c^3}{2 a x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 88
Rubi steps
\begin {align*} \int \frac {(c+d x)^3}{x^3 (a+b x)} \, dx &=\int \left (\frac {c^3}{a x^3}+\frac {c^2 (-b c+3 a d)}{a^2 x^2}+\frac {c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right )}{a^3 x}+\frac {(-b c+a d)^3}{a^3 (a+b x)}\right ) \, dx\\ &=-\frac {c^3}{2 a x^2}+\frac {c^2 (b c-3 a d)}{a^2 x}+\frac {c \left (b^2 c^2-3 a b c d+3 a^2 d^2\right ) \log (x)}{a^3}-\frac {(b c-a d)^3 \log (a+b x)}{a^3 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 78, normalized size = 0.92 \[ -\frac {-2 c \log (x) \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )+\frac {a c^2 (a (c+6 d x)-2 b c x)}{x^2}+\frac {2 (b c-a d)^3 \log (a+b x)}{b}}{2 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.87, size = 124, normalized size = 1.46 \[ -\frac {a^{2} b c^{3} + 2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2} \log \left (b x + a\right ) - 2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2}\right )} x^{2} \log \relax (x) - 2 \, {\left (a b^{2} c^{3} - 3 \, a^{2} b c^{2} d\right )} x}{2 \, a^{3} b x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.10, size = 119, normalized size = 1.40 \[ \frac {{\left (b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{3}} - \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{3} b} - \frac {a^{2} c^{3} - 2 \, {\left (a b c^{3} - 3 \, a^{2} c^{2} d\right )} x}{2 \, a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 132, normalized size = 1.55 \[ \frac {3 c \,d^{2} \ln \relax (x )}{a}-\frac {3 c \,d^{2} \ln \left (b x +a \right )}{a}-\frac {3 b \,c^{2} d \ln \relax (x )}{a^{2}}+\frac {3 b \,c^{2} d \ln \left (b x +a \right )}{a^{2}}+\frac {b^{2} c^{3} \ln \relax (x )}{a^{3}}-\frac {b^{2} c^{3} \ln \left (b x +a \right )}{a^{3}}+\frac {d^{3} \ln \left (b x +a \right )}{b}-\frac {3 c^{2} d}{a x}+\frac {b \,c^{3}}{a^{2} x}-\frac {c^{3}}{2 a \,x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.05, size = 112, normalized size = 1.32 \[ \frac {{\left (b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} \log \relax (x)}{a^{3}} - \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{a^{3} b} - \frac {a c^{3} - 2 \, {\left (b c^{3} - 3 \, a c^{2} d\right )} x}{2 \, a^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.46, size = 84, normalized size = 0.99 \[ \frac {\ln \left (a+b\,x\right )\,{\left (a\,d-b\,c\right )}^3}{a^3\,b}-\frac {\frac {c^3}{2\,a}+\frac {c^2\,x\,\left (3\,a\,d-b\,c\right )}{a^2}}{x^2}+\frac {c\,\ln \relax (x)\,\left (3\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2\right )}{a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 2.22, size = 257, normalized size = 3.02 \[ \frac {- a c^{3} + x \left (- 6 a c^{2} d + 2 b c^{3}\right )}{2 a^{2} x^{2}} + \frac {c \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right ) \log {\left (x + \frac {- 3 a^{3} c d^{2} + 3 a^{2} b c^{2} d - a b^{2} c^{3} + a c \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{a^{3} d^{3} - 6 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{a^{3}} + \frac {\left (a d - b c\right )^{3} \log {\left (x + \frac {- 3 a^{3} c d^{2} + 3 a^{2} b c^{2} d - a b^{2} c^{3} + \frac {a \left (a d - b c\right )^{3}}{b}}{a^{3} d^{3} - 6 a^{2} b c d^{2} + 6 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{a^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________