Optimal. Leaf size=43 \[ 4 c d^3 \log \left (a+b x+c x^2\right )-\frac {d^3 (b+2 c x)^2}{a+b x+c x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {686, 628} \[ 4 c d^3 \log \left (a+b x+c x^2\right )-\frac {d^3 (b+2 c x)^2}{a+b x+c x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 628
Rule 686
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac {d^3 (b+2 c x)^2}{a+b x+c x^2}+\left (4 c d^2\right ) \int \frac {b d+2 c d x}{a+b x+c x^2} \, dx\\ &=-\frac {d^3 (b+2 c x)^2}{a+b x+c x^2}+4 c d^3 \log \left (a+b x+c x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 42, normalized size = 0.98 \[ d^3 \left (\frac {4 a c-b^2}{a+b x+c x^2}+4 c \log \left (a+b x+c x^2\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.96, size = 64, normalized size = 1.49 \[ -\frac {{\left (b^{2} - 4 \, a c\right )} d^{3} - 4 \, {\left (c^{2} d^{3} x^{2} + b c d^{3} x + a c d^{3}\right )} \log \left (c x^{2} + b x + a\right )}{c x^{2} + b x + a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.18, size = 101, normalized size = 2.35 \[ -4 \, {\left (d \log \left (\frac {{\left | a + \frac {c d x^{2} + b d x}{d} \right |} {\left | d \right |}}{{\left (a + \frac {c d x^{2} + b d x}{d}\right )}^{2}}\right ) - \frac {a d}{a + \frac {c d x^{2} + b d x}{d}}\right )} c d^{2} - \frac {b^{2} d^{3}}{a + \frac {c d x^{2} + b d x}{d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 58, normalized size = 1.35 \[ \frac {4 a c \,d^{3}}{c \,x^{2}+b x +a}-\frac {b^{2} d^{3}}{c \,x^{2}+b x +a}+4 c \,d^{3} \ln \left (c \,x^{2}+b x +a \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.38, size = 43, normalized size = 1.00 \[ 4 \, c d^{3} \log \left (c x^{2} + b x + a\right ) - \frac {{\left (b^{2} - 4 \, a c\right )} d^{3}}{c x^{2} + b x + a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.07, size = 47, normalized size = 1.09 \[ 4\,c\,d^3\,\ln \left (c\,x^2+b\,x+a\right )-\frac {b^2\,d^3-4\,a\,c\,d^3}{c\,x^2+b\,x+a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.89, size = 42, normalized size = 0.98 \[ 4 c d^{3} \log {\left (a + b x + c x^{2} \right )} + \frac {4 a c d^{3} - b^{2} d^{3}}{a + b x + c x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________