Optimal. Leaf size=70 \[ \frac {\log \left (a+b x+c x^2\right )}{d^3 \left (b^2-4 a c\right )^2}+\frac {1}{d^3 \left (b^2-4 a c\right ) (b+2 c x)^2}-\frac {2 \log (b+2 c x)}{d^3 \left (b^2-4 a c\right )^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {693, 681, 31, 628} \begin {gather*} \frac {\log \left (a+b x+c x^2\right )}{d^3 \left (b^2-4 a c\right )^2}+\frac {1}{d^3 \left (b^2-4 a c\right ) (b+2 c x)^2}-\frac {2 \log (b+2 c x)}{d^3 \left (b^2-4 a c\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 628
Rule 681
Rule 693
Rubi steps
\begin {align*} \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )} \, dx &=\frac {1}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2}+\frac {\int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )} \, dx}{\left (b^2-4 a c\right ) d^2}\\ &=\frac {1}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2}+\frac {\int \frac {b d+2 c d x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^2 d^4}-\frac {(4 c) \int \frac {1}{b+2 c x} \, dx}{\left (b^2-4 a c\right )^2 d^3}\\ &=\frac {1}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2}-\frac {2 \log (b+2 c x)}{\left (b^2-4 a c\right )^2 d^3}+\frac {\log \left (a+b x+c x^2\right )}{\left (b^2-4 a c\right )^2 d^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 65, normalized size = 0.93 \begin {gather*} \frac {\frac {\log \left (a+b x+c x^2\right )}{\left (b^2-4 a c\right )^2}+\frac {1}{\left (b^2-4 a c\right ) (b+2 c x)^2}-\frac {2 \log (b+2 c x)}{\left (b^2-4 a c\right )^2}}{d^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}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.40, size = 156, normalized size = 2.23 \begin {gather*} \frac {b^{2} - 4 \, a c + {\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \log \left (c x^{2} + b x + a\right ) - 2 \, {\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \log \left (2 \, c x + b\right )}{4 \, {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{3} x^{2} + 4 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{3} x + {\left (b^{6} - 8 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2}\right )} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 111, normalized size = 1.59 \begin {gather*} -\frac {2 \, c \log \left ({\left | 2 \, c x + b \right |}\right )}{b^{4} c d^{3} - 8 \, a b^{2} c^{2} d^{3} + 16 \, a^{2} c^{3} d^{3}} + \frac {\log \left (c x^{2} + b x + a\right )}{b^{4} d^{3} - 8 \, a b^{2} c d^{3} + 16 \, a^{2} c^{2} d^{3}} + \frac {1}{{\left (b^{2} - 4 \, a c\right )} {\left (2 \, c x + b\right )}^{2} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 78, normalized size = 1.11 \begin {gather*} -\frac {2 \ln \left (2 c x +b \right )}{\left (4 a c -b^{2}\right )^{2} d^{3}}+\frac {\ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right )^{2} d^{3}}-\frac {1}{\left (4 a c -b^{2}\right ) \left (2 c x +b \right )^{2} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.36, size = 129, normalized size = 1.84 \begin {gather*} \frac {1}{4 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{3} x^{2} + 4 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d^{3} x + {\left (b^{4} - 4 \, a b^{2} c\right )} d^{3}} + \frac {\log \left (c x^{2} + b x + a\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{3}} - \frac {2 \, \log \left (2 \, c x + b\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.22, size = 151, normalized size = 2.16 \begin {gather*} \frac {\ln \left (c\,x^2+b\,x+a\right )}{16\,a^2\,c^2\,d^3-8\,a\,b^2\,c\,d^3+b^4\,d^3}-\frac {2\,\ln \left (b+2\,c\,x\right )}{16\,a^2\,c^2\,d^3-8\,a\,b^2\,c\,d^3+b^4\,d^3}+\frac {1}{b^4\,d^3+4\,b^3\,c\,d^3\,x+4\,b^2\,c^2\,d^3\,x^2-4\,a\,b^2\,c\,d^3-16\,a\,b\,c^2\,d^3\,x-16\,a\,c^3\,d^3\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.67, size = 119, normalized size = 1.70 \begin {gather*} - \frac {1}{4 a b^{2} c d^{3} - b^{4} d^{3} + x^{2} \left (16 a c^{3} d^{3} - 4 b^{2} c^{2} d^{3}\right ) + x \left (16 a b c^{2} d^{3} - 4 b^{3} c d^{3}\right )} - \frac {2 \log {\left (\frac {b}{2 c} + x \right )}}{d^{3} \left (4 a c - b^{2}\right )^{2}} + \frac {\log {\left (\frac {a}{c} + \frac {b x}{c} + x^{2} \right )}}{d^{3} \left (4 a c - b^{2}\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________