Integrand size = 29, antiderivative size = 210 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^{12}} \, dx=-\frac {a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac {a^2 (3 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{10 x^{10} (a+b x)}-\frac {a b (A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^9 (a+b x)}-\frac {b^2 (A b+3 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)}-\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)} \]
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Time = 0.05 (sec) , antiderivative size = 210, 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} \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^{12}} \, dx=-\frac {a^2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+3 A b)}{10 x^{10} (a+b x)}-\frac {a b \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{3 x^9 (a+b x)}-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (3 a B+A b)}{8 x^8 (a+b x)}-\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac {a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)} \]
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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 )^3 (A+B x)}{x^{12}} \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a^3 A b^3}{x^{12}}+\frac {a^2 b^3 (3 A b+a B)}{x^{11}}+\frac {3 a b^4 (A b+a B)}{x^{10}}+\frac {b^5 (A b+3 a B)}{x^9}+\frac {b^6 B}{x^8}\right ) \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = -\frac {a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac {a^2 (3 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{10 x^{10} (a+b x)}-\frac {a b (A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^9 (a+b x)}-\frac {b^2 (A b+3 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)}-\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)} \\ \end{align*}
Time = 0.84 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.41 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^{12}} \, dx=-\frac {\sqrt {(a+b x)^2} \left (165 b^3 x^3 (7 A+8 B x)+385 a b^2 x^2 (8 A+9 B x)+308 a^2 b x (9 A+10 B x)+84 a^3 (10 A+11 B x)\right )}{9240 x^{11} (a+b x)} \]
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Time = 1.85 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.43
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {x^{4} B \,b^{3}}{7}+\left (-\frac {1}{8} A \,b^{3}-\frac {3}{8} B a \,b^{2}\right ) x^{3}+\left (-\frac {1}{3} A a \,b^{2}-\frac {1}{3} B b \,a^{2}\right ) x^{2}+\left (-\frac {3}{10} A \,a^{2} b -\frac {1}{10} B \,a^{3}\right ) x -\frac {A \,a^{3}}{11}\right )}{\left (b x +a \right ) x^{11}}\) | \(90\) |
gosper | \(-\frac {\left (1320 x^{4} B \,b^{3}+1155 A \,b^{3} x^{3}+3465 B a \,b^{2} x^{3}+3080 A a \,b^{2} x^{2}+3080 B \,a^{2} b \,x^{2}+2772 A \,a^{2} b x +924 a^{3} B x +840 A \,a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{9240 x^{11} \left (b x +a \right )^{3}}\) | \(92\) |
default | \(-\frac {\left (1320 x^{4} B \,b^{3}+1155 A \,b^{3} x^{3}+3465 B a \,b^{2} x^{3}+3080 A a \,b^{2} x^{2}+3080 B \,a^{2} b \,x^{2}+2772 A \,a^{2} b x +924 a^{3} B x +840 A \,a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{9240 x^{11} \left (b x +a \right )^{3}}\) | \(92\) |
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Time = 0.26 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.35 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^{12}} \, dx=-\frac {1320 \, B b^{3} x^{4} + 840 \, A a^{3} + 1155 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 3080 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} + 924 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{9240 \, x^{11}} \]
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\[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^{12}} \, dx=\int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{x^{12}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 675 vs. \(2 (145) = 290\).
Time = 0.23 (sec) , antiderivative size = 675, normalized size of antiderivative = 3.21 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^{12}} \, dx=\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b^{10}}{4 \, a^{10}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{11}}{4 \, a^{11}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b^{9}}{4 \, a^{9} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{10}}{4 \, a^{10} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{8}}{4 \, a^{10} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{9}}{4 \, a^{11} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{7}}{4 \, a^{9} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{8}}{4 \, a^{10} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{6}}{4 \, a^{8} x^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{7}}{4 \, a^{9} x^{4}} + \frac {209 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{5}}{840 \, a^{7} x^{5}} - \frac {329 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{6}}{1320 \, a^{8} x^{5}} - \frac {41 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{4}}{168 \, a^{6} x^{6}} + \frac {65 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{5}}{264 \, a^{7} x^{6}} + \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{3}}{56 \, a^{5} x^{7}} - \frac {21 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{4}}{88 \, a^{6} x^{7}} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{2}}{24 \, a^{4} x^{8}} + \frac {59 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{3}}{264 \, a^{5} x^{8}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b}{6 \, a^{3} x^{9}} - \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{2}}{66 \, a^{4} x^{9}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B}{10 \, a^{2} x^{10}} + \frac {17 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b}{110 \, a^{3} x^{10}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A}{11 \, a^{2} x^{11}} \]
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Time = 0.26 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.71 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^{12}} \, dx=-\frac {{\left (11 \, B a b^{10} - 7 \, A b^{11}\right )} \mathrm {sgn}\left (b x + a\right )}{9240 \, a^{8}} - \frac {1320 \, B b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 3465 \, B a b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 1155 \, A b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 3080 \, B a^{2} b x^{2} \mathrm {sgn}\left (b x + a\right ) + 3080 \, A a b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 924 \, B a^{3} x \mathrm {sgn}\left (b x + a\right ) + 2772 \, A a^{2} b x \mathrm {sgn}\left (b x + a\right ) + 840 \, A a^{3} \mathrm {sgn}\left (b x + a\right )}{9240 \, x^{11}} \]
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Time = 10.02 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.93 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^{12}} \, dx=-\frac {\left (\frac {B\,a^3}{10}+\frac {3\,A\,b\,a^2}{10}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^{10}\,\left (a+b\,x\right )}-\frac {\left (\frac {A\,b^3}{8}+\frac {3\,B\,a\,b^2}{8}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^8\,\left (a+b\,x\right )}-\frac {A\,a^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{11\,x^{11}\,\left (a+b\,x\right )}-\frac {B\,b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{7\,x^7\,\left (a+b\,x\right )}-\frac {a\,b\,\left (A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{3\,x^9\,\left (a+b\,x\right )} \]
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