Optimal. Leaf size=230 \[ \frac {c^4 (3 c d-5 b e)}{b^4 (b+c x) (c d-b e)^3}+\frac {2 b e+3 c d}{b^4 d^3 x}+\frac {c^4}{2 b^3 (b+c x)^2 (c d-b e)^2}-\frac {1}{2 b^3 d^2 x^2}+\frac {3 \log (x) \left (b^2 e^2+2 b c d e+2 c^2 d^2\right )}{b^5 d^4}-\frac {3 c^4 \left (5 b^2 e^2-6 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 (c d-b e)^4}+\frac {3 e^5 (2 c d-b e) \log (d+e x)}{d^4 (c d-b e)^4}-\frac {e^5}{d^3 (d+e x) (c d-b e)^3} \]
________________________________________________________________________________________
Rubi [A] time = 0.34, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {698} \begin {gather*} \frac {3 \log (x) \left (b^2 e^2+2 b c d e+2 c^2 d^2\right )}{b^5 d^4}-\frac {3 c^4 \left (5 b^2 e^2-6 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 (c d-b e)^4}+\frac {c^4 (3 c d-5 b e)}{b^4 (b+c x) (c d-b e)^3}+\frac {c^4}{2 b^3 (b+c x)^2 (c d-b e)^2}+\frac {2 b e+3 c d}{b^4 d^3 x}-\frac {1}{2 b^3 d^2 x^2}-\frac {e^5}{d^3 (d+e x) (c d-b e)^3}+\frac {3 e^5 (2 c d-b e) \log (d+e x)}{d^4 (c d-b e)^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 698
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^3} \, dx &=\int \left (\frac {1}{b^3 d^2 x^3}+\frac {-3 c d-2 b e}{b^4 d^3 x^2}+\frac {3 \left (2 c^2 d^2+2 b c d e+b^2 e^2\right )}{b^5 d^4 x}-\frac {c^5}{b^3 (-c d+b e)^2 (b+c x)^3}-\frac {c^5 (-3 c d+5 b e)}{b^4 (-c d+b e)^3 (b+c x)^2}-\frac {3 c^5 \left (2 c^2 d^2-6 b c d e+5 b^2 e^2\right )}{b^5 (-c d+b e)^4 (b+c x)}+\frac {e^6}{d^3 (c d-b e)^3 (d+e x)^2}+\frac {3 e^6 (2 c d-b e)}{d^4 (c d-b e)^4 (d+e x)}\right ) \, dx\\ &=-\frac {1}{2 b^3 d^2 x^2}+\frac {3 c d+2 b e}{b^4 d^3 x}+\frac {c^4}{2 b^3 (c d-b e)^2 (b+c x)^2}+\frac {c^4 (3 c d-5 b e)}{b^4 (c d-b e)^3 (b+c x)}-\frac {e^5}{d^3 (c d-b e)^3 (d+e x)}+\frac {3 \left (2 c^2 d^2+2 b c d e+b^2 e^2\right ) \log (x)}{b^5 d^4}-\frac {3 c^4 \left (2 c^2 d^2-6 b c d e+5 b^2 e^2\right ) \log (b+c x)}{b^5 (c d-b e)^4}+\frac {3 e^5 (2 c d-b e) \log (d+e x)}{d^4 (c d-b e)^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.25, size = 230, normalized size = 1.00 \begin {gather*} \frac {c^4 (5 b e-3 c d)}{b^4 (b+c x) (b e-c d)^3}+\frac {2 b e+3 c d}{b^4 d^3 x}+\frac {c^4}{2 b^3 (b+c x)^2 (c d-b e)^2}-\frac {1}{2 b^3 d^2 x^2}+\frac {3 \log (x) \left (b^2 e^2+2 b c d e+2 c^2 d^2\right )}{b^5 d^4}-\frac {3 c^4 \left (5 b^2 e^2-6 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 (c d-b e)^4}+\frac {3 e^5 (2 c d-b e) \log (d+e x)}{d^4 (c d-b e)^4}-\frac {e^5}{d^3 (d+e x) (c d-b e)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 56.14, size = 1305, normalized size = 5.67
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.28, size = 839, normalized size = 3.65 \begin {gather*} -\frac {3 \, {\left (4 \, c^{6} d^{6} e^{2} - 12 \, b c^{5} d^{5} e^{3} + 10 \, b^{2} c^{4} d^{4} e^{4} - 2 \, b^{5} c d e^{7} + b^{6} e^{8}\right )} e^{\left (-2\right )} \log \left (\frac {{\left | -2 \, c d e + \frac {2 \, c d^{2} e}{x e + d} + b e^{2} - \frac {2 \, b d e^{2}}{x e + d} - {\left | b \right |} e^{2} \right |}}{{\left | -2 \, c d e + \frac {2 \, c d^{2} e}{x e + d} + b e^{2} - \frac {2 \, b d e^{2}}{x e + d} + {\left | b \right |} e^{2} \right |}}\right )}{2 \, {\left (b^{4} c^{4} d^{8} - 4 \, b^{5} c^{3} d^{7} e + 6 \, b^{6} c^{2} d^{6} e^{2} - 4 \, b^{7} c d^{5} e^{3} + b^{8} d^{4} e^{4}\right )} {\left | b \right |}} - \frac {3 \, {\left (2 \, c d e^{5} - b e^{6}\right )} \log \left ({\left | -c + \frac {2 \, c d}{x e + d} - \frac {c d^{2}}{{\left (x e + d\right )}^{2}} - \frac {b e}{x e + d} + \frac {b d e}{{\left (x e + d\right )}^{2}} \right |}\right )}{2 \, {\left (c^{4} d^{8} - 4 \, b c^{3} d^{7} e + 6 \, b^{2} c^{2} d^{6} e^{2} - 4 \, b^{3} c d^{5} e^{3} + b^{4} d^{4} e^{4}\right )}} - \frac {e^{11}}{{\left (c^{3} d^{6} e^{6} - 3 \, b c^{2} d^{5} e^{7} + 3 \, b^{2} c d^{4} e^{8} - b^{3} d^{3} e^{9}\right )} {\left (x e + d\right )}} + \frac {12 \, c^{7} d^{5} e - 30 \, b c^{6} d^{4} e^{2} + 16 \, b^{2} c^{5} d^{3} e^{3} + 6 \, b^{3} c^{4} d^{2} e^{4} - 14 \, b^{4} c^{3} d e^{5} + 5 \, b^{5} c^{2} e^{6} - \frac {2 \, {\left (18 \, c^{7} d^{6} e^{2} - 54 \, b c^{6} d^{5} e^{3} + 47 \, b^{2} c^{5} d^{4} e^{4} - 4 \, b^{3} c^{4} d^{3} e^{5} - 29 \, b^{4} c^{3} d^{2} e^{6} + 22 \, b^{5} c^{2} d e^{7} - 5 \, b^{6} c e^{8}\right )} e^{\left (-1\right )}}{x e + d} + \frac {{\left (36 \, c^{7} d^{7} e^{3} - 126 \, b c^{6} d^{6} e^{4} + 144 \, b^{2} c^{5} d^{5} e^{5} - 45 \, b^{3} c^{4} d^{4} e^{6} - 70 \, b^{4} c^{3} d^{3} e^{7} + 87 \, b^{5} c^{2} d^{2} e^{8} - 36 \, b^{6} c d e^{9} + 5 \, b^{7} e^{10}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {6 \, {\left (2 \, c^{7} d^{8} e^{4} - 8 \, b c^{6} d^{7} e^{5} + 11 \, b^{2} c^{5} d^{6} e^{6} - 5 \, b^{3} c^{4} d^{5} e^{7} - 5 \, b^{4} c^{3} d^{4} e^{8} + 9 \, b^{5} c^{2} d^{3} e^{9} - 5 \, b^{6} c d^{2} e^{10} + b^{7} d e^{11}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}}}{2 \, {\left (c d - b e\right )}^{4} b^{4} {\left (c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {b e}{x e + d} - \frac {b d e}{{\left (x e + d\right )}^{2}}\right )}^{2} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 306, normalized size = 1.33 \begin {gather*} -\frac {3 b \,e^{6} \ln \left (e x +d \right )}{\left (b e -c d \right )^{4} d^{4}}-\frac {15 c^{4} e^{2} \ln \left (c x +b \right )}{\left (b e -c d \right )^{4} b^{3}}+\frac {18 c^{5} d e \ln \left (c x +b \right )}{\left (b e -c d \right )^{4} b^{4}}-\frac {6 c^{6} d^{2} \ln \left (c x +b \right )}{\left (b e -c d \right )^{4} b^{5}}+\frac {6 c \,e^{5} \ln \left (e x +d \right )}{\left (b e -c d \right )^{4} d^{3}}+\frac {5 c^{4} e}{\left (b e -c d \right )^{3} \left (c x +b \right ) b^{3}}-\frac {3 c^{5} d}{\left (b e -c d \right )^{3} \left (c x +b \right ) b^{4}}+\frac {e^{5}}{\left (b e -c d \right )^{3} \left (e x +d \right ) d^{3}}+\frac {c^{4}}{2 \left (b e -c d \right )^{2} \left (c x +b \right )^{2} b^{3}}+\frac {3 e^{2} \ln \relax (x )}{b^{3} d^{4}}+\frac {6 c e \ln \relax (x )}{b^{4} d^{3}}+\frac {6 c^{2} \ln \relax (x )}{b^{5} d^{2}}+\frac {2 e}{b^{3} d^{3} x}+\frac {3 c}{b^{4} d^{2} x}-\frac {1}{2 b^{3} d^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 2.02, size = 752, normalized size = 3.27 \begin {gather*} -\frac {3 \, {\left (2 \, c^{6} d^{2} - 6 \, b c^{5} d e + 5 \, b^{2} c^{4} e^{2}\right )} \log \left (c x + b\right )}{b^{5} c^{4} d^{4} - 4 \, b^{6} c^{3} d^{3} e + 6 \, b^{7} c^{2} d^{2} e^{2} - 4 \, b^{8} c d e^{3} + b^{9} e^{4}} + \frac {3 \, {\left (2 \, c d e^{5} - b e^{6}\right )} \log \left (e x + d\right )}{c^{4} d^{8} - 4 \, b c^{3} d^{7} e + 6 \, b^{2} c^{2} d^{6} e^{2} - 4 \, b^{3} c d^{5} e^{3} + b^{4} d^{4} e^{4}} - \frac {b^{3} c^{3} d^{5} - 3 \, b^{4} c^{2} d^{4} e + 3 \, b^{5} c d^{3} e^{2} - b^{6} d^{2} e^{3} - 6 \, {\left (2 \, c^{6} d^{4} e - 4 \, b c^{5} d^{3} e^{2} + b^{2} c^{4} d^{2} e^{3} + b^{3} c^{3} d e^{4} - b^{4} c^{2} e^{5}\right )} x^{4} - 3 \, {\left (4 \, c^{6} d^{5} - 2 \, b c^{5} d^{4} e - 10 \, b^{2} c^{4} d^{3} e^{2} + 5 \, b^{3} c^{3} d^{2} e^{3} + 3 \, b^{4} c^{2} d e^{4} - 4 \, b^{5} c e^{5}\right )} x^{3} - {\left (18 \, b c^{5} d^{5} - 32 \, b^{2} c^{4} d^{4} e + b^{3} c^{3} d^{3} e^{2} + 13 \, b^{4} c^{2} d^{2} e^{3} - 6 \, b^{6} e^{5}\right )} x^{2} - {\left (4 \, b^{2} c^{4} d^{5} - 9 \, b^{3} c^{3} d^{4} e + 3 \, b^{4} c^{2} d^{3} e^{2} + 5 \, b^{5} c d^{2} e^{3} - 3 \, b^{6} d e^{4}\right )} x}{2 \, {\left ({\left (b^{4} c^{5} d^{6} e - 3 \, b^{5} c^{4} d^{5} e^{2} + 3 \, b^{6} c^{3} d^{4} e^{3} - b^{7} c^{2} d^{3} e^{4}\right )} x^{5} + {\left (b^{4} c^{5} d^{7} - b^{5} c^{4} d^{6} e - 3 \, b^{6} c^{3} d^{5} e^{2} + 5 \, b^{7} c^{2} d^{4} e^{3} - 2 \, b^{8} c d^{3} e^{4}\right )} x^{4} + {\left (2 \, b^{5} c^{4} d^{7} - 5 \, b^{6} c^{3} d^{6} e + 3 \, b^{7} c^{2} d^{5} e^{2} + b^{8} c d^{4} e^{3} - b^{9} d^{3} e^{4}\right )} x^{3} + {\left (b^{6} c^{3} d^{7} - 3 \, b^{7} c^{2} d^{6} e + 3 \, b^{8} c d^{5} e^{2} - b^{9} d^{4} e^{3}\right )} x^{2}\right )}} + \frac {3 \, {\left (2 \, c^{2} d^{2} + 2 \, b c d e + b^{2} e^{2}\right )} \log \relax (x)}{b^{5} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.07, size = 603, normalized size = 2.62 \begin {gather*} \frac {\ln \relax (x)\,\left (3\,b^2\,e^2+6\,b\,c\,d\,e+6\,c^2\,d^2\right )}{b^5\,d^4}-\frac {\ln \left (d+e\,x\right )\,\left (3\,b\,e^6-6\,c\,d\,e^5\right )}{b^4\,d^4\,e^4-4\,b^3\,c\,d^5\,e^3+6\,b^2\,c^2\,d^6\,e^2-4\,b\,c^3\,d^7\,e+c^4\,d^8}-\frac {\ln \left (b+c\,x\right )\,\left (15\,b^2\,c^4\,e^2-18\,b\,c^5\,d\,e+6\,c^6\,d^2\right )}{b^9\,e^4-4\,b^8\,c\,d\,e^3+6\,b^7\,c^2\,d^2\,e^2-4\,b^6\,c^3\,d^3\,e+b^5\,c^4\,d^4}-\frac {\frac {1}{2\,b\,d}-\frac {x\,\left (3\,b\,e+4\,c\,d\right )}{2\,b^2\,d^2}+\frac {x^2\,\left (-6\,b^5\,e^5+13\,b^3\,c^2\,d^2\,e^3+b^2\,c^3\,d^3\,e^2-32\,b\,c^4\,d^4\,e+18\,c^5\,d^5\right )}{2\,b^3\,d^3\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-c^3\,d^3\right )}+\frac {3\,x^3\,\left (-4\,b^5\,c\,e^5+3\,b^4\,c^2\,d\,e^4+5\,b^3\,c^3\,d^2\,e^3-10\,b^2\,c^4\,d^3\,e^2-2\,b\,c^5\,d^4\,e+4\,c^6\,d^5\right )}{2\,b^4\,d^3\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-c^3\,d^3\right )}+\frac {3\,c^2\,e\,x^4\,\left (-b^4\,e^4+b^3\,c\,d\,e^3+b^2\,c^2\,d^2\,e^2-4\,b\,c^3\,d^3\,e+2\,c^4\,d^4\right )}{b^4\,d^3\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-c^3\,d^3\right )}}{x^3\,\left (e\,b^2+2\,c\,d\,b\right )+x^4\,\left (d\,c^2+2\,b\,e\,c\right )+b^2\,d\,x^2+c^2\,e\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________