Optimal. Leaf size=91 \[ -\frac {a^3 \log (a+b x)}{b^2 (b c-a d)^2}-\frac {c^3}{d^3 (c+d x) (b c-a d)}-\frac {c^2 (2 b c-3 a d) \log (c+d x)}{d^3 (b c-a d)^2}+\frac {x}{b d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 91, 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} \[ -\frac {a^3 \log (a+b x)}{b^2 (b c-a d)^2}-\frac {c^3}{d^3 (c+d x) (b c-a d)}-\frac {c^2 (2 b c-3 a d) \log (c+d x)}{d^3 (b c-a d)^2}+\frac {x}{b d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 88
Rubi steps
\begin {align*} \int \frac {x^3}{(a+b x) (c+d x)^2} \, dx &=\int \left (\frac {1}{b d^2}-\frac {a^3}{b (b c-a d)^2 (a+b x)}-\frac {c^3}{d^2 (-b c+a d) (c+d x)^2}-\frac {c^2 (2 b c-3 a d)}{d^2 (-b c+a d)^2 (c+d x)}\right ) \, dx\\ &=\frac {x}{b d^2}-\frac {c^3}{d^3 (b c-a d) (c+d x)}-\frac {a^3 \log (a+b x)}{b^2 (b c-a d)^2}-\frac {c^2 (2 b c-3 a d) \log (c+d x)}{d^3 (b c-a d)^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.14, size = 87, normalized size = 0.96 \[ \frac {\frac {c^3}{(c+d x) (a d-b c)}-\frac {c^2 (2 b c-3 a d) \log (c+d x)}{(b c-a d)^2}+\frac {d x}{b}}{d^3}-\frac {a^3 \log (a+b x)}{b^2 (b c-a d)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.95, size = 230, normalized size = 2.53 \[ -\frac {b^{3} c^{4} - a b^{2} c^{3} d - {\left (b^{3} c^{2} d^{2} - 2 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{2} - {\left (b^{3} c^{3} d - 2 \, a b^{2} c^{2} d^{2} + a^{2} b c d^{3}\right )} x + {\left (a^{3} d^{4} x + a^{3} c d^{3}\right )} \log \left (b x + a\right ) + {\left (2 \, b^{3} c^{4} - 3 \, a b^{2} c^{3} d + {\left (2 \, b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2}\right )} x\right )} \log \left (d x + c\right )}{b^{4} c^{3} d^{3} - 2 \, a b^{3} c^{2} d^{4} + a^{2} b^{2} c d^{5} + {\left (b^{4} c^{2} d^{4} - 2 \, a b^{3} c d^{5} + a^{2} b^{2} d^{6}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.81, size = 139, normalized size = 1.53 \[ -\frac {a^{3} d \log \left ({\left | b - \frac {b c}{d x + c} + \frac {a d}{d x + c} \right |}\right )}{b^{4} c^{2} d - 2 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}} - \frac {c^{3} d^{2}}{{\left (b c d^{5} - a d^{6}\right )} {\left (d x + c\right )}} + \frac {d x + c}{b d^{3}} + \frac {{\left (2 \, b c + a d\right )} \log \left (\frac {{\left | d x + c \right |}}{{\left (d x + c\right )}^{2} {\left | d \right |}}\right )}{b^{2} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 108, normalized size = 1.19 \[ -\frac {a^{3} \ln \left (b x +a \right )}{\left (a d -b c \right )^{2} b^{2}}+\frac {3 a \,c^{2} \ln \left (d x +c \right )}{\left (a d -b c \right )^{2} d^{2}}-\frac {2 b \,c^{3} \ln \left (d x +c \right )}{\left (a d -b c \right )^{2} d^{3}}+\frac {c^{3}}{\left (a d -b c \right ) \left (d x +c \right ) d^{3}}+\frac {x}{b \,d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.20, size = 136, normalized size = 1.49 \[ -\frac {a^{3} \log \left (b x + a\right )}{b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}} - \frac {c^{3}}{b c^{2} d^{3} - a c d^{4} + {\left (b c d^{4} - a d^{5}\right )} x} - \frac {{\left (2 \, b c^{3} - 3 \, a c^{2} d\right )} \log \left (d x + c\right )}{b^{2} c^{2} d^{3} - 2 \, a b c d^{4} + a^{2} d^{5}} + \frac {x}{b d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.52, size = 131, normalized size = 1.44 \[ \frac {x}{b\,d^2}-\frac {\ln \left (c+d\,x\right )\,\left (2\,b\,c^3-3\,a\,c^2\,d\right )}{a^2\,d^5-2\,a\,b\,c\,d^4+b^2\,c^2\,d^3}-\frac {a^3\,\ln \left (a+b\,x\right )}{a^2\,b^2\,d^2-2\,a\,b^3\,c\,d+b^4\,c^2}+\frac {b\,c^3}{d\,\left (b\,x\,d^3+b\,c\,d^2\right )\,\left (a\,d-b\,c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 3.29, size = 400, normalized size = 4.40 \[ - \frac {a^{3} \log {\left (x + \frac {\frac {a^{6} d^{5}}{b \left (a d - b c\right )^{2}} - \frac {3 a^{5} c d^{4}}{\left (a d - b c\right )^{2}} + \frac {3 a^{4} b c^{2} d^{3}}{\left (a d - b c\right )^{2}} - \frac {a^{3} b^{2} c^{3} d^{2}}{\left (a d - b c\right )^{2}} + a^{3} c d^{2} + 3 a^{2} b c^{2} d - 2 a b^{2} c^{3}}{a^{3} d^{3} + 3 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{b^{2} \left (a d - b c\right )^{2}} + \frac {c^{3}}{a c d^{4} - b c^{2} d^{3} + x \left (a d^{5} - b c d^{4}\right )} + \frac {c^{2} \left (3 a d - 2 b c\right ) \log {\left (x + \frac {- \frac {a^{3} b c^{2} d^{2} \left (3 a d - 2 b c\right )}{\left (a d - b c\right )^{2}} + a^{3} c d^{2} + \frac {3 a^{2} b^{2} c^{3} d \left (3 a d - 2 b c\right )}{\left (a d - b c\right )^{2}} + 3 a^{2} b c^{2} d - \frac {3 a b^{3} c^{4} \left (3 a d - 2 b c\right )}{\left (a d - b c\right )^{2}} - 2 a b^{2} c^{3} + \frac {b^{4} c^{5} \left (3 a d - 2 b c\right )}{d \left (a d - b c\right )^{2}}}{a^{3} d^{3} + 3 a b^{2} c^{2} d - 2 b^{3} c^{3}} \right )}}{d^{3} \left (a d - b c\right )^{2}} + \frac {x}{b d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________