Optimal. Leaf size=209 \[ -\frac{28 b^6 (b c-a d)^2}{d^9 (c+d x)}+\frac{28 b^5 (b c-a d)^3}{d^9 (c+d x)^2}-\frac{70 b^4 (b c-a d)^4}{3 d^9 (c+d x)^3}+\frac{14 b^3 (b c-a d)^5}{d^9 (c+d x)^4}-\frac{28 b^2 (b c-a d)^6}{5 d^9 (c+d x)^5}-\frac{8 b^7 (b c-a d) \log (c+d x)}{d^9}+\frac{4 b (b c-a d)^7}{3 d^9 (c+d x)^6}-\frac{(b c-a d)^8}{7 d^9 (c+d x)^7}+\frac{b^8 x}{d^8} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.277555, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ -\frac{28 b^6 (b c-a d)^2}{d^9 (c+d x)}+\frac{28 b^5 (b c-a d)^3}{d^9 (c+d x)^2}-\frac{70 b^4 (b c-a d)^4}{3 d^9 (c+d x)^3}+\frac{14 b^3 (b c-a d)^5}{d^9 (c+d x)^4}-\frac{28 b^2 (b c-a d)^6}{5 d^9 (c+d x)^5}-\frac{8 b^7 (b c-a d) \log (c+d x)}{d^9}+\frac{4 b (b c-a d)^7}{3 d^9 (c+d x)^6}-\frac{(b c-a d)^8}{7 d^9 (c+d x)^7}+\frac{b^8 x}{d^8} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^8}{(c+d x)^8} \, dx &=\int \left (\frac{b^8}{d^8}+\frac{(-b c+a d)^8}{d^8 (c+d x)^8}-\frac{8 b (b c-a d)^7}{d^8 (c+d x)^7}+\frac{28 b^2 (b c-a d)^6}{d^8 (c+d x)^6}-\frac{56 b^3 (b c-a d)^5}{d^8 (c+d x)^5}+\frac{70 b^4 (b c-a d)^4}{d^8 (c+d x)^4}-\frac{56 b^5 (b c-a d)^3}{d^8 (c+d x)^3}+\frac{28 b^6 (b c-a d)^2}{d^8 (c+d x)^2}-\frac{8 b^7 (b c-a d)}{d^8 (c+d x)}\right ) \, dx\\ &=\frac{b^8 x}{d^8}-\frac{(b c-a d)^8}{7 d^9 (c+d x)^7}+\frac{4 b (b c-a d)^7}{3 d^9 (c+d x)^6}-\frac{28 b^2 (b c-a d)^6}{5 d^9 (c+d x)^5}+\frac{14 b^3 (b c-a d)^5}{d^9 (c+d x)^4}-\frac{70 b^4 (b c-a d)^4}{3 d^9 (c+d x)^3}+\frac{28 b^5 (b c-a d)^3}{d^9 (c+d x)^2}-\frac{28 b^6 (b c-a d)^2}{d^9 (c+d x)}-\frac{8 b^7 (b c-a d) \log (c+d x)}{d^9}\\ \end{align*}
Mathematica [B] time = 0.194985, size = 474, normalized size = 2.27 \[ -\frac{420 a^2 b^6 d^2 \left (21 c^4 d^2 x^2+35 c^3 d^3 x^3+35 c^2 d^4 x^4+7 c^5 d x+c^6+21 c d^5 x^5+7 d^6 x^6\right )+140 a^3 b^5 d^3 \left (21 c^3 d^2 x^2+35 c^2 d^3 x^3+7 c^4 d x+c^5+35 c d^4 x^4+21 d^5 x^5\right )+70 a^4 b^4 d^4 \left (21 c^2 d^2 x^2+7 c^3 d x+c^4+35 c d^3 x^3+35 d^4 x^4\right )+42 a^5 b^3 d^5 \left (7 c^2 d x+c^3+21 c d^2 x^2+35 d^3 x^3\right )+28 a^6 b^2 d^6 \left (c^2+7 c d x+21 d^2 x^2\right )+20 a^7 b d^7 (c+7 d x)+15 a^8 d^8-2 a b^7 c d \left (20139 c^4 d^2 x^2+30625 c^3 d^3 x^3+26950 c^2 d^4 x^4+7203 c^5 d x+1089 c^6+13230 c d^5 x^5+2940 d^6 x^6\right )+840 b^7 (c+d x)^7 (b c-a d) \log (c+d x)+b^8 \left (24843 c^6 d^2 x^2+35525 c^5 d^3 x^3+28175 c^4 d^4 x^4+11025 c^3 d^5 x^5+735 c^2 d^6 x^6+9261 c^7 d x+1443 c^8-735 c d^7 x^7-105 d^8 x^8\right )}{105 d^9 (c+d x)^7} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.013, size = 845, normalized size = 4. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.14791, size = 876, normalized size = 4.19 \begin{align*} \frac{b^{8} x}{d^{8}} - \frac{1443 \, b^{8} c^{8} - 2178 \, a b^{7} c^{7} d + 420 \, a^{2} b^{6} c^{6} d^{2} + 140 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} + 42 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} + 20 \, a^{7} b c d^{7} + 15 \, a^{8} d^{8} + 2940 \,{\left (b^{8} c^{2} d^{6} - 2 \, a b^{7} c d^{7} + a^{2} b^{6} d^{8}\right )} x^{6} + 2940 \,{\left (5 \, b^{8} c^{3} d^{5} - 9 \, a b^{7} c^{2} d^{6} + 3 \, a^{2} b^{6} c d^{7} + a^{3} b^{5} d^{8}\right )} x^{5} + 2450 \,{\left (13 \, b^{8} c^{4} d^{4} - 22 \, a b^{7} c^{3} d^{5} + 6 \, a^{2} b^{6} c^{2} d^{6} + 2 \, a^{3} b^{5} c d^{7} + a^{4} b^{4} d^{8}\right )} x^{4} + 490 \,{\left (77 \, b^{8} c^{5} d^{3} - 125 \, a b^{7} c^{4} d^{4} + 30 \, a^{2} b^{6} c^{3} d^{5} + 10 \, a^{3} b^{5} c^{2} d^{6} + 5 \, a^{4} b^{4} c d^{7} + 3 \, a^{5} b^{3} d^{8}\right )} x^{3} + 294 \,{\left (87 \, b^{8} c^{6} d^{2} - 137 \, a b^{7} c^{5} d^{3} + 30 \, a^{2} b^{6} c^{4} d^{4} + 10 \, a^{3} b^{5} c^{3} d^{5} + 5 \, a^{4} b^{4} c^{2} d^{6} + 3 \, a^{5} b^{3} c d^{7} + 2 \, a^{6} b^{2} d^{8}\right )} x^{2} + 14 \,{\left (669 \, b^{8} c^{7} d - 1029 \, a b^{7} c^{6} d^{2} + 210 \, a^{2} b^{6} c^{5} d^{3} + 70 \, a^{3} b^{5} c^{4} d^{4} + 35 \, a^{4} b^{4} c^{3} d^{5} + 21 \, a^{5} b^{3} c^{2} d^{6} + 14 \, a^{6} b^{2} c d^{7} + 10 \, a^{7} b d^{8}\right )} x}{105 \,{\left (d^{16} x^{7} + 7 \, c d^{15} x^{6} + 21 \, c^{2} d^{14} x^{5} + 35 \, c^{3} d^{13} x^{4} + 35 \, c^{4} d^{12} x^{3} + 21 \, c^{5} d^{11} x^{2} + 7 \, c^{6} d^{10} x + c^{7} d^{9}\right )}} - \frac{8 \,{\left (b^{8} c - a b^{7} d\right )} \log \left (d x + c\right )}{d^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.0194, size = 1770, normalized size = 8.47 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.06049, size = 784, normalized size = 3.75 \begin{align*} \frac{b^{8} x}{d^{8}} - \frac{8 \,{\left (b^{8} c - a b^{7} d\right )} \log \left ({\left | d x + c \right |}\right )}{d^{9}} - \frac{1443 \, b^{8} c^{8} - 2178 \, a b^{7} c^{7} d + 420 \, a^{2} b^{6} c^{6} d^{2} + 140 \, a^{3} b^{5} c^{5} d^{3} + 70 \, a^{4} b^{4} c^{4} d^{4} + 42 \, a^{5} b^{3} c^{3} d^{5} + 28 \, a^{6} b^{2} c^{2} d^{6} + 20 \, a^{7} b c d^{7} + 15 \, a^{8} d^{8} + 2940 \,{\left (b^{8} c^{2} d^{6} - 2 \, a b^{7} c d^{7} + a^{2} b^{6} d^{8}\right )} x^{6} + 2940 \,{\left (5 \, b^{8} c^{3} d^{5} - 9 \, a b^{7} c^{2} d^{6} + 3 \, a^{2} b^{6} c d^{7} + a^{3} b^{5} d^{8}\right )} x^{5} + 2450 \,{\left (13 \, b^{8} c^{4} d^{4} - 22 \, a b^{7} c^{3} d^{5} + 6 \, a^{2} b^{6} c^{2} d^{6} + 2 \, a^{3} b^{5} c d^{7} + a^{4} b^{4} d^{8}\right )} x^{4} + 490 \,{\left (77 \, b^{8} c^{5} d^{3} - 125 \, a b^{7} c^{4} d^{4} + 30 \, a^{2} b^{6} c^{3} d^{5} + 10 \, a^{3} b^{5} c^{2} d^{6} + 5 \, a^{4} b^{4} c d^{7} + 3 \, a^{5} b^{3} d^{8}\right )} x^{3} + 294 \,{\left (87 \, b^{8} c^{6} d^{2} - 137 \, a b^{7} c^{5} d^{3} + 30 \, a^{2} b^{6} c^{4} d^{4} + 10 \, a^{3} b^{5} c^{3} d^{5} + 5 \, a^{4} b^{4} c^{2} d^{6} + 3 \, a^{5} b^{3} c d^{7} + 2 \, a^{6} b^{2} d^{8}\right )} x^{2} + 14 \,{\left (669 \, b^{8} c^{7} d - 1029 \, a b^{7} c^{6} d^{2} + 210 \, a^{2} b^{6} c^{5} d^{3} + 70 \, a^{3} b^{5} c^{4} d^{4} + 35 \, a^{4} b^{4} c^{3} d^{5} + 21 \, a^{5} b^{3} c^{2} d^{6} + 14 \, a^{6} b^{2} c d^{7} + 10 \, a^{7} b d^{8}\right )} x}{105 \,{\left (d x + c\right )}^{7} d^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]