Integrand size = 29, antiderivative size = 306 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{13}} \, dx=-\frac {a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{12 x^{12} (a+b x)}-\frac {a^4 (5 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac {a^3 b (2 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^{10} (a+b x)}-\frac {10 a^2 b^2 (A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac {5 a b^3 (A b+2 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)}-\frac {b^4 (A b+5 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)} \]
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Time = 0.08 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {784, 77} \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{13}} \, dx=-\frac {10 a^2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{9 x^9 (a+b x)}-\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{7 x^7 (a+b x)}-\frac {5 a b^3 \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{8 x^8 (a+b x)}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac {a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{12 x^{12} (a+b x)}-\frac {a^4 \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{11 x^{11} (a+b x)}-\frac {a^3 b \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{2 x^{10} (a+b x)} \]
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Rule 77
Rule 784
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5 (A+B x)}{x^{13}} \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a^5 A b^5}{x^{13}}+\frac {a^4 b^5 (5 A b+a B)}{x^{12}}+\frac {5 a^3 b^6 (2 A b+a B)}{x^{11}}+\frac {10 a^2 b^7 (A b+a B)}{x^{10}}+\frac {5 a b^8 (A b+2 a B)}{x^9}+\frac {b^9 (A b+5 a B)}{x^8}+\frac {b^{10} B}{x^7}\right ) \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = -\frac {a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{12 x^{12} (a+b x)}-\frac {a^4 (5 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac {a^3 b (2 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^{10} (a+b x)}-\frac {10 a^2 b^2 (A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac {5 a b^3 (A b+2 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)}-\frac {b^4 (A b+5 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)} \\ \end{align*}
Time = 0.94 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.40 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{13}} \, dx=\frac {\sqrt {(a+b x)^2} \left (\frac {(-6 A b+12 a B) \left (-\frac {a^5}{11 x^{11}}-\frac {a^4 b}{2 x^{10}}-\frac {10 a^3 b^2}{9 x^9}-\frac {5 a^2 b^3}{4 x^8}-\frac {5 a b^4}{7 x^7}-\frac {b^5}{6 x^6}\right )}{12 a}-\frac {A (a+b x)^6}{12 a x^{12}}\right )}{a+b x} \]
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Time = 2.17 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.44
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {B \,b^{5} x^{6}}{6}+\left (-\frac {1}{7} A \,b^{5}-\frac {5}{7} B a \,b^{4}\right ) x^{5}+\left (-\frac {5}{8} A a \,b^{4}-\frac {5}{4} B \,a^{2} b^{3}\right ) x^{4}+\left (-\frac {10}{9} A \,a^{2} b^{3}-\frac {10}{9} B \,a^{3} b^{2}\right ) x^{3}+\left (-A \,a^{3} b^{2}-\frac {1}{2} B \,a^{4} b \right ) x^{2}+\left (-\frac {5}{11} A \,a^{4} b -\frac {1}{11} a^{5} B \right ) x -\frac {A \,a^{5}}{12}\right )}{\left (b x +a \right ) x^{12}}\) | \(136\) |
gosper | \(-\frac {\left (924 B \,b^{5} x^{6}+792 A \,b^{5} x^{5}+3960 B a \,b^{4} x^{5}+3465 A a \,b^{4} x^{4}+6930 B \,a^{2} b^{3} x^{4}+6160 A \,a^{2} b^{3} x^{3}+6160 B \,a^{3} b^{2} x^{3}+5544 A \,a^{3} b^{2} x^{2}+2772 B \,a^{4} b \,x^{2}+2520 A \,a^{4} b x +504 a^{5} B x +462 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{5544 x^{12} \left (b x +a \right )^{5}}\) | \(140\) |
default | \(-\frac {\left (924 B \,b^{5} x^{6}+792 A \,b^{5} x^{5}+3960 B a \,b^{4} x^{5}+3465 A a \,b^{4} x^{4}+6930 B \,a^{2} b^{3} x^{4}+6160 A \,a^{2} b^{3} x^{3}+6160 B \,a^{3} b^{2} x^{3}+5544 A \,a^{3} b^{2} x^{2}+2772 B \,a^{4} b \,x^{2}+2520 A \,a^{4} b x +504 a^{5} B x +462 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{5544 x^{12} \left (b x +a \right )^{5}}\) | \(140\) |
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Time = 0.37 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.39 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{13}} \, dx=-\frac {924 \, B b^{5} x^{6} + 462 \, A a^{5} + 792 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 3465 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 6160 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 2772 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 504 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{5544 \, x^{12}} \]
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\[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{13}} \, dx=\int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{x^{13}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 735 vs. \(2 (215) = 430\).
Time = 0.21 (sec) , antiderivative size = 735, normalized size of antiderivative = 2.40 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{13}} \, dx=-\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{11}}{6 \, a^{11}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{12}}{6 \, a^{12}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{10}}{6 \, a^{10} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{11}}{6 \, a^{11} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{9}}{6 \, a^{11} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{10}}{6 \, a^{12} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{8}}{6 \, a^{10} x^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{9}}{6 \, a^{11} x^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{7}}{6 \, a^{9} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{8}}{6 \, a^{10} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{6}}{6 \, a^{8} x^{5}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{7}}{6 \, a^{9} x^{5}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{5}}{6 \, a^{7} x^{6}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{6}}{6 \, a^{8} x^{6}} - \frac {461 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{4}}{2772 \, a^{6} x^{7}} + \frac {923 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{5}}{5544 \, a^{7} x^{7}} + \frac {65 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{3}}{396 \, a^{5} x^{8}} - \frac {131 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{4}}{792 \, a^{6} x^{8}} - \frac {31 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{2}}{198 \, a^{4} x^{9}} + \frac {16 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{3}}{99 \, a^{5} x^{9}} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b}{22 \, a^{3} x^{10}} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{2}}{33 \, a^{4} x^{10}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B}{11 \, a^{2} x^{11}} + \frac {17 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b}{132 \, a^{3} x^{11}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A}{12 \, a^{2} x^{12}} \]
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Time = 0.27 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.72 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{13}} \, dx=\frac {{\left (2 \, B a b^{11} - A b^{12}\right )} \mathrm {sgn}\left (b x + a\right )}{5544 \, a^{7}} - \frac {924 \, B b^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) + 3960 \, B a b^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + 792 \, A b^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 6930 \, B a^{2} b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 3465 \, A a b^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 6160 \, B a^{3} b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 6160 \, A a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 2772 \, B a^{4} b x^{2} \mathrm {sgn}\left (b x + a\right ) + 5544 \, A a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 504 \, B a^{5} x \mathrm {sgn}\left (b x + a\right ) + 2520 \, A a^{4} b x \mathrm {sgn}\left (b x + a\right ) + 462 \, A a^{5} \mathrm {sgn}\left (b x + a\right )}{5544 \, x^{12}} \]
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Time = 10.05 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.93 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{13}} \, dx=-\frac {\left (\frac {B\,a^5}{11}+\frac {5\,A\,b\,a^4}{11}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^{11}\,\left (a+b\,x\right )}-\frac {\left (\frac {A\,b^5}{7}+\frac {5\,B\,a\,b^4}{7}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^7\,\left (a+b\,x\right )}-\frac {A\,a^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{12\,x^{12}\,\left (a+b\,x\right )}-\frac {B\,b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{6\,x^6\,\left (a+b\,x\right )}-\frac {5\,a\,b^3\,\left (A\,b+2\,B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{8\,x^8\,\left (a+b\,x\right )}-\frac {a^3\,b\,\left (2\,A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,x^{10}\,\left (a+b\,x\right )}-\frac {10\,a^2\,b^2\,\left (A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{9\,x^9\,\left (a+b\,x\right )} \]
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