Optimal. Leaf size=21 \[ -\frac{x^{-7 n}}{7 n \left (b+c x^n\right )^7} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0324075, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {1584, 446, 74} \[ -\frac{x^{-7 n}}{7 n \left (b+c x^n\right )^7} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1584
Rule 446
Rule 74
Rubi steps
\begin{align*} \int \frac{x^{-1+n} \left (b+2 c x^n\right )}{\left (b x^n+c x^{2 n}\right )^8} \, dx &=\int \frac{x^{-1-7 n} \left (b+2 c x^n\right )}{\left (b+c x^n\right )^8} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{b+2 c x}{x^8 (b+c x)^8} \, dx,x,x^n\right )}{n}\\ &=-\frac{x^{-7 n}}{7 n \left (b+c x^n\right )^7}\\ \end{align*}
Mathematica [A] time = 0.181327, size = 21, normalized size = 1. \[ -\frac{x^{-7 n}}{7 n \left (b+c x^n\right )^7} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.05, size = 203, normalized size = 9.7 \begin{align*} -132\,{\frac{{c}^{6}}{{b}^{13}n{x}^{n}}}+66\,{\frac{{c}^{5}}{{b}^{12}n \left ({x}^{n} \right ) ^{2}}}-30\,{\frac{{c}^{4}}{{b}^{11}n \left ({x}^{n} \right ) ^{3}}}+12\,{\frac{{c}^{3}}{{b}^{10}n \left ({x}^{n} \right ) ^{4}}}-4\,{\frac{{c}^{2}}{{b}^{9}n \left ({x}^{n} \right ) ^{5}}}+{\frac{c}{{b}^{8}n \left ({x}^{n} \right ) ^{6}}}-{\frac{1}{7\,{b}^{7}n \left ({x}^{n} \right ) ^{7}}}+{\frac{{c}^{7} \left ( 924\, \left ({x}^{n} \right ) ^{6}{c}^{6}+6006\,b{c}^{5} \left ({x}^{n} \right ) ^{5}+16380\,{b}^{2}{c}^{4} \left ({x}^{n} \right ) ^{4}+24024\,{b}^{3}{c}^{3} \left ({x}^{n} \right ) ^{3}+20020\,{b}^{4}{c}^{2} \left ({x}^{n} \right ) ^{2}+9009\,{b}^{5}c{x}^{n}+1716\,{b}^{6} \right ) }{7\,{b}^{13}n \left ( b+c{x}^{n} \right ) ^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.21517, size = 826, normalized size = 39.33 \begin{align*} -\frac{1}{105} \, b{\left (\frac{360360 \, c^{13} x^{13 \, n} + 2342340 \, b c^{12} x^{12 \, n} + 6426420 \, b^{2} c^{11} x^{11 \, n} + 9579570 \, b^{3} c^{10} x^{10 \, n} + 8270262 \, b^{4} c^{9} x^{9 \, n} + 4018014 \, b^{5} c^{8} x^{8 \, n} + 934362 \, b^{6} c^{7} x^{7 \, n} + 45045 \, b^{7} c^{6} x^{6 \, n} - 5005 \, b^{8} c^{5} x^{5 \, n} + 1001 \, b^{9} c^{4} x^{4 \, n} - 273 \, b^{10} c^{3} x^{3 \, n} + 91 \, b^{11} c^{2} x^{2 \, n} - 35 \, b^{12} c x^{n} + 15 \, b^{13}}{b^{14} c^{7} n x^{14 \, n} + 7 \, b^{15} c^{6} n x^{13 \, n} + 21 \, b^{16} c^{5} n x^{12 \, n} + 35 \, b^{17} c^{4} n x^{11 \, n} + 35 \, b^{18} c^{3} n x^{10 \, n} + 21 \, b^{19} c^{2} n x^{9 \, n} + 7 \, b^{20} c n x^{8 \, n} + b^{21} n x^{7 \, n}} + \frac{360360 \, c^{7} \log \left (x\right )}{b^{15}} - \frac{360360 \, c^{7} \log \left (\frac{c x^{n} + b}{c}\right )}{b^{15} n}\right )} + \frac{1}{105} \, c{\left (\frac{360360 \, c^{12} x^{12 \, n} + 2342340 \, b c^{11} x^{11 \, n} + 6426420 \, b^{2} c^{10} x^{10 \, n} + 9579570 \, b^{3} c^{9} x^{9 \, n} + 8270262 \, b^{4} c^{8} x^{8 \, n} + 4018014 \, b^{5} c^{7} x^{7 \, n} + 934362 \, b^{6} c^{6} x^{6 \, n} + 45045 \, b^{7} c^{5} x^{5 \, n} - 5005 \, b^{8} c^{4} x^{4 \, n} + 1001 \, b^{9} c^{3} x^{3 \, n} - 273 \, b^{10} c^{2} x^{2 \, n} + 91 \, b^{11} c x^{n} - 35 \, b^{12}}{b^{13} c^{7} n x^{13 \, n} + 7 \, b^{14} c^{6} n x^{12 \, n} + 21 \, b^{15} c^{5} n x^{11 \, n} + 35 \, b^{16} c^{4} n x^{10 \, n} + 35 \, b^{17} c^{3} n x^{9 \, n} + 21 \, b^{18} c^{2} n x^{8 \, n} + 7 \, b^{19} c n x^{7 \, n} + b^{20} n x^{6 \, n}} + \frac{360360 \, c^{6} \log \left (x\right )}{b^{14}} - \frac{360360 \, c^{6} \log \left (\frac{c x^{n} + b}{c}\right )}{b^{14} n}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.31501, size = 236, normalized size = 11.24 \begin{align*} -\frac{1}{7 \,{\left (c^{7} n x^{14 \, n} + 7 \, b c^{6} n x^{13 \, n} + 21 \, b^{2} c^{5} n x^{12 \, n} + 35 \, b^{3} c^{4} n x^{11 \, n} + 35 \, b^{4} c^{3} n x^{10 \, n} + 21 \, b^{5} c^{2} n x^{9 \, n} + 7 \, b^{6} c n x^{8 \, n} + b^{7} n x^{7 \, n}\right )}} \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 [A] time = 1.13892, size = 27, normalized size = 1.29 \begin{align*} -\frac{1}{7 \,{\left (c x^{2 \, n} + b x^{n}\right )}^{7} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]