Optimal. Leaf size=73 \[ -\frac {\left (b^2-4 a c\right )^2}{224 c^3 d^8 (b+2 c x)^7}+\frac {b^2-4 a c}{80 c^3 d^8 (b+2 c x)^5}-\frac {1}{96 c^3 d^8 (b+2 c x)^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {697}
\begin {gather*} -\frac {\left (b^2-4 a c\right )^2}{224 c^3 d^8 (b+2 c x)^7}+\frac {b^2-4 a c}{80 c^3 d^8 (b+2 c x)^5}-\frac {1}{96 c^3 d^8 (b+2 c x)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 697
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^8} \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^2}{16 c^2 d^8 (b+2 c x)^8}+\frac {-b^2+4 a c}{8 c^2 d^8 (b+2 c x)^6}+\frac {1}{16 c^2 d^8 (b+2 c x)^4}\right ) \, dx\\ &=-\frac {\left (b^2-4 a c\right )^2}{224 c^3 d^8 (b+2 c x)^7}+\frac {b^2-4 a c}{80 c^3 d^8 (b+2 c x)^5}-\frac {1}{96 c^3 d^8 (b+2 c x)^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 59, normalized size = 0.81 \begin {gather*} \frac {-15 \left (b^2-4 a c\right )^2+42 \left (b^2-4 a c\right ) (b+2 c x)^2-35 (b+2 c x)^4}{3360 c^3 d^8 (b+2 c x)^7} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.74, size = 74, normalized size = 1.01
method | result | size |
default | \(\frac {-\frac {4 a c -b^{2}}{80 c^{3} \left (2 c x +b \right )^{5}}-\frac {16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}}{224 c^{3} \left (2 c x +b \right )^{7}}-\frac {1}{96 c^{3} \left (2 c x +b \right )^{3}}}{d^{8}}\) | \(74\) |
risch | \(\frac {-\frac {x^{4} c}{6}-\frac {b \,x^{3}}{3}-\frac {\left (a c +b^{2}\right ) x^{2}}{5 c}-\frac {b \left (6 a c +b^{2}\right ) x}{30 c^{2}}-\frac {30 a^{2} c^{2}+6 a c \,b^{2}+b^{4}}{420 c^{3}}}{d^{8} \left (2 c x +b \right )^{7}}\) | \(80\) |
gosper | \(-\frac {70 c^{4} x^{4}+140 b \,x^{3} c^{3}+84 a \,c^{3} x^{2}+84 b^{2} c^{2} x^{2}+84 a b \,c^{2} x +14 b^{3} c x +30 a^{2} c^{2}+6 a c \,b^{2}+b^{4}}{420 \left (2 c x +b \right )^{7} d^{8} c^{3}}\) | \(88\) |
norman | \(\frac {\frac {a^{2} x}{b d}+\frac {\left (6 a^{2} c +a \,b^{2}\right ) x^{2}}{b^{2} d}+\frac {\left (60 a^{2} c^{2}+12 a c \,b^{2}+b^{4}\right ) x^{3}}{3 b^{3} d}+\frac {16 c^{3} \left (30 a^{2} c^{2}+6 a c \,b^{2}+b^{4}\right ) x^{6}}{15 b^{6} d}+\frac {8 c^{2} \left (30 a^{2} c^{2}+6 a c \,b^{2}+b^{4}\right ) x^{5}}{5 b^{5} d}+\frac {c \left (240 a^{2} c^{2}+48 a c \,b^{2}+7 b^{4}\right ) x^{4}}{6 b^{4} d}+\frac {32 c^{4} \left (30 a^{2} c^{2}+6 a c \,b^{2}+b^{4}\right ) x^{7}}{105 b^{7} d}}{d^{7} \left (2 c x +b \right )^{7}}\) | \(209\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 176 vs.
\(2 (67) = 134\).
time = 0.30, size = 176, normalized size = 2.41 \begin {gather*} -\frac {70 \, c^{4} x^{4} + 140 \, b c^{3} x^{3} + b^{4} + 6 \, a b^{2} c + 30 \, a^{2} c^{2} + 84 \, {\left (b^{2} c^{2} + a c^{3}\right )} x^{2} + 14 \, {\left (b^{3} c + 6 \, a b c^{2}\right )} x}{420 \, {\left (128 \, c^{10} d^{8} x^{7} + 448 \, b c^{9} d^{8} x^{6} + 672 \, b^{2} c^{8} d^{8} x^{5} + 560 \, b^{3} c^{7} d^{8} x^{4} + 280 \, b^{4} c^{6} d^{8} x^{3} + 84 \, b^{5} c^{5} d^{8} x^{2} + 14 \, b^{6} c^{4} d^{8} x + b^{7} c^{3} d^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 176 vs.
\(2 (67) = 134\).
time = 2.31, size = 176, normalized size = 2.41 \begin {gather*} -\frac {70 \, c^{4} x^{4} + 140 \, b c^{3} x^{3} + b^{4} + 6 \, a b^{2} c + 30 \, a^{2} c^{2} + 84 \, {\left (b^{2} c^{2} + a c^{3}\right )} x^{2} + 14 \, {\left (b^{3} c + 6 \, a b c^{2}\right )} x}{420 \, {\left (128 \, c^{10} d^{8} x^{7} + 448 \, b c^{9} d^{8} x^{6} + 672 \, b^{2} c^{8} d^{8} x^{5} + 560 \, b^{3} c^{7} d^{8} x^{4} + 280 \, b^{4} c^{6} d^{8} x^{3} + 84 \, b^{5} c^{5} d^{8} x^{2} + 14 \, b^{6} c^{4} d^{8} x + b^{7} c^{3} d^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 189 vs.
\(2 (70) = 140\).
time = 1.06, size = 189, normalized size = 2.59 \begin {gather*} \frac {- 30 a^{2} c^{2} - 6 a b^{2} c - b^{4} - 140 b c^{3} x^{3} - 70 c^{4} x^{4} + x^{2} \left (- 84 a c^{3} - 84 b^{2} c^{2}\right ) + x \left (- 84 a b c^{2} - 14 b^{3} c\right )}{420 b^{7} c^{3} d^{8} + 5880 b^{6} c^{4} d^{8} x + 35280 b^{5} c^{5} d^{8} x^{2} + 117600 b^{4} c^{6} d^{8} x^{3} + 235200 b^{3} c^{7} d^{8} x^{4} + 282240 b^{2} c^{8} d^{8} x^{5} + 188160 b c^{9} d^{8} x^{6} + 53760 c^{10} d^{8} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.14, size = 87, normalized size = 1.19 \begin {gather*} -\frac {70 \, c^{4} x^{4} + 140 \, b c^{3} x^{3} + 84 \, b^{2} c^{2} x^{2} + 84 \, a c^{3} x^{2} + 14 \, b^{3} c x + 84 \, a b c^{2} x + b^{4} + 6 \, a b^{2} c + 30 \, a^{2} c^{2}}{420 \, {\left (2 \, c x + b\right )}^{7} c^{3} d^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.12, size = 168, normalized size = 2.30 \begin {gather*} -\frac {\frac {30\,a^2\,c^2+6\,a\,b^2\,c+b^4}{420\,c^3}+\frac {b\,x^3}{3}+\frac {c\,x^4}{6}+\frac {x^2\,\left (b^2+a\,c\right )}{5\,c}+\frac {b\,x\,\left (b^2+6\,a\,c\right )}{30\,c^2}}{b^7\,d^8+14\,b^6\,c\,d^8\,x+84\,b^5\,c^2\,d^8\,x^2+280\,b^4\,c^3\,d^8\,x^3+560\,b^3\,c^4\,d^8\,x^4+672\,b^2\,c^5\,d^8\,x^5+448\,b\,c^6\,d^8\,x^6+128\,c^7\,d^8\,x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________