Integrand size = 18, antiderivative size = 146 \[ \int \frac {A+B x}{x^5 \sqrt {a+b x}} \, dx=-\frac {A \sqrt {a+b x}}{4 a x^4}+\frac {(7 A b-8 a B) \sqrt {a+b x}}{24 a^2 x^3}-\frac {5 b (7 A b-8 a B) \sqrt {a+b x}}{96 a^3 x^2}+\frac {5 b^2 (7 A b-8 a B) \sqrt {a+b x}}{64 a^4 x}-\frac {5 b^3 (7 A b-8 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{64 a^{9/2}} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {79, 44, 65, 214} \[ \int \frac {A+B x}{x^5 \sqrt {a+b x}} \, dx=-\frac {5 b^3 (7 A b-8 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{64 a^{9/2}}+\frac {5 b^2 \sqrt {a+b x} (7 A b-8 a B)}{64 a^4 x}-\frac {5 b \sqrt {a+b x} (7 A b-8 a B)}{96 a^3 x^2}+\frac {\sqrt {a+b x} (7 A b-8 a B)}{24 a^2 x^3}-\frac {A \sqrt {a+b x}}{4 a x^4} \]
[In]
[Out]
Rule 44
Rule 65
Rule 79
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {A \sqrt {a+b x}}{4 a x^4}+\frac {\left (-\frac {7 A b}{2}+4 a B\right ) \int \frac {1}{x^4 \sqrt {a+b x}} \, dx}{4 a} \\ & = -\frac {A \sqrt {a+b x}}{4 a x^4}+\frac {(7 A b-8 a B) \sqrt {a+b x}}{24 a^2 x^3}+\frac {(5 b (7 A b-8 a B)) \int \frac {1}{x^3 \sqrt {a+b x}} \, dx}{48 a^2} \\ & = -\frac {A \sqrt {a+b x}}{4 a x^4}+\frac {(7 A b-8 a B) \sqrt {a+b x}}{24 a^2 x^3}-\frac {5 b (7 A b-8 a B) \sqrt {a+b x}}{96 a^3 x^2}-\frac {\left (5 b^2 (7 A b-8 a B)\right ) \int \frac {1}{x^2 \sqrt {a+b x}} \, dx}{64 a^3} \\ & = -\frac {A \sqrt {a+b x}}{4 a x^4}+\frac {(7 A b-8 a B) \sqrt {a+b x}}{24 a^2 x^3}-\frac {5 b (7 A b-8 a B) \sqrt {a+b x}}{96 a^3 x^2}+\frac {5 b^2 (7 A b-8 a B) \sqrt {a+b x}}{64 a^4 x}+\frac {\left (5 b^3 (7 A b-8 a B)\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx}{128 a^4} \\ & = -\frac {A \sqrt {a+b x}}{4 a x^4}+\frac {(7 A b-8 a B) \sqrt {a+b x}}{24 a^2 x^3}-\frac {5 b (7 A b-8 a B) \sqrt {a+b x}}{96 a^3 x^2}+\frac {5 b^2 (7 A b-8 a B) \sqrt {a+b x}}{64 a^4 x}+\frac {\left (5 b^2 (7 A b-8 a B)\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{64 a^4} \\ & = -\frac {A \sqrt {a+b x}}{4 a x^4}+\frac {(7 A b-8 a B) \sqrt {a+b x}}{24 a^2 x^3}-\frac {5 b (7 A b-8 a B) \sqrt {a+b x}}{96 a^3 x^2}+\frac {5 b^2 (7 A b-8 a B) \sqrt {a+b x}}{64 a^4 x}-\frac {5 b^3 (7 A b-8 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{64 a^{9/2}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.77 \[ \int \frac {A+B x}{x^5 \sqrt {a+b x}} \, dx=\frac {\sqrt {a+b x} \left (105 A b^3 x^3-16 a^3 (3 A+4 B x)+8 a^2 b x (7 A+10 B x)-10 a b^2 x^2 (7 A+12 B x)\right )}{192 a^4 x^4}+\frac {5 b^3 (-7 A b+8 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{64 a^{9/2}} \]
[In]
[Out]
Time = 0.53 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.69
method | result | size |
pseudoelliptic | \(\frac {-\frac {35 x^{4} b^{3} \left (A b -\frac {8 B a}{7}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{64}+\frac {7 \sqrt {b x +a}\, \left (-\frac {5 x^{2} \left (\frac {12 B x}{7}+A \right ) b^{2} a^{\frac {3}{2}}}{4}+b x \left (\frac {10 B x}{7}+A \right ) a^{\frac {5}{2}}+\frac {2 \left (-4 B x -3 A \right ) a^{\frac {7}{2}}}{7}+\frac {15 A \sqrt {a}\, b^{3} x^{3}}{8}\right )}{24}}{a^{\frac {9}{2}} x^{4}}\) | \(101\) |
risch | \(-\frac {\sqrt {b x +a}\, \left (-105 A \,b^{3} x^{3}+120 B a \,b^{2} x^{3}+70 a A \,b^{2} x^{2}-80 B \,a^{2} b \,x^{2}-56 a^{2} A b x +64 a^{3} B x +48 a^{3} A \right )}{192 a^{4} x^{4}}-\frac {5 b^{3} \left (7 A b -8 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{64 a^{\frac {9}{2}}}\) | \(107\) |
derivativedivides | \(2 b^{3} \left (-\frac {-\frac {5 \left (7 A b -8 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{128 a^{4}}+\frac {55 \left (7 A b -8 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{384 a^{3}}-\frac {73 \left (7 A b -8 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{384 a^{2}}+\frac {\left (93 A b -88 B a \right ) \sqrt {b x +a}}{128 a}}{b^{4} x^{4}}-\frac {5 \left (7 A b -8 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{128 a^{\frac {9}{2}}}\right )\) | \(126\) |
default | \(2 b^{3} \left (-\frac {-\frac {5 \left (7 A b -8 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{128 a^{4}}+\frac {55 \left (7 A b -8 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{384 a^{3}}-\frac {73 \left (7 A b -8 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{384 a^{2}}+\frac {\left (93 A b -88 B a \right ) \sqrt {b x +a}}{128 a}}{b^{4} x^{4}}-\frac {5 \left (7 A b -8 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{128 a^{\frac {9}{2}}}\right )\) | \(126\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.77 \[ \int \frac {A+B x}{x^5 \sqrt {a+b x}} \, dx=\left [-\frac {15 \, {\left (8 \, B a b^{3} - 7 \, A b^{4}\right )} \sqrt {a} x^{4} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (48 \, A a^{4} + 15 \, {\left (8 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{3} - 10 \, {\left (8 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{2} + 8 \, {\left (8 \, B a^{4} - 7 \, A a^{3} b\right )} x\right )} \sqrt {b x + a}}{384 \, a^{5} x^{4}}, -\frac {15 \, {\left (8 \, B a b^{3} - 7 \, A b^{4}\right )} \sqrt {-a} x^{4} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (48 \, A a^{4} + 15 \, {\left (8 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{3} - 10 \, {\left (8 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{2} + 8 \, {\left (8 \, B a^{4} - 7 \, A a^{3} b\right )} x\right )} \sqrt {b x + a}}{192 \, a^{5} x^{4}}\right ] \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (141) = 282\).
Time = 56.87 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.08 \[ \int \frac {A+B x}{x^5 \sqrt {a+b x}} \, dx=- \frac {A}{4 \sqrt {b} x^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {A \sqrt {b}}{24 a x^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {7 A b^{\frac {3}{2}}}{96 a^{2} x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {35 A b^{\frac {5}{2}}}{192 a^{3} x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {35 A b^{\frac {7}{2}}}{64 a^{4} \sqrt {x} \sqrt {\frac {a}{b x} + 1}} - \frac {35 A b^{4} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{64 a^{\frac {9}{2}}} - \frac {B}{3 \sqrt {b} x^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {B \sqrt {b}}{12 a x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {5 B b^{\frac {3}{2}}}{24 a^{2} x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {5 B b^{\frac {5}{2}}}{8 a^{3} \sqrt {x} \sqrt {\frac {a}{b x} + 1}} + \frac {5 B b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{8 a^{\frac {7}{2}}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.34 \[ \int \frac {A+B x}{x^5 \sqrt {a+b x}} \, dx=-\frac {1}{384} \, b^{4} {\left (\frac {2 \, {\left (15 \, {\left (8 \, B a - 7 \, A b\right )} {\left (b x + a\right )}^{\frac {7}{2}} - 55 \, {\left (8 \, B a^{2} - 7 \, A a b\right )} {\left (b x + a\right )}^{\frac {5}{2}} + 73 \, {\left (8 \, B a^{3} - 7 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {3}{2}} - 3 \, {\left (88 \, B a^{4} - 93 \, A a^{3} b\right )} \sqrt {b x + a}\right )}}{{\left (b x + a\right )}^{4} a^{4} b - 4 \, {\left (b x + a\right )}^{3} a^{5} b + 6 \, {\left (b x + a\right )}^{2} a^{6} b - 4 \, {\left (b x + a\right )} a^{7} b + a^{8} b} + \frac {15 \, {\left (8 \, B a - 7 \, A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {9}{2}} b}\right )} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.21 \[ \int \frac {A+B x}{x^5 \sqrt {a+b x}} \, dx=-\frac {\frac {15 \, {\left (8 \, B a b^{4} - 7 \, A b^{5}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{4}} + \frac {120 \, {\left (b x + a\right )}^{\frac {7}{2}} B a b^{4} - 440 \, {\left (b x + a\right )}^{\frac {5}{2}} B a^{2} b^{4} + 584 \, {\left (b x + a\right )}^{\frac {3}{2}} B a^{3} b^{4} - 264 \, \sqrt {b x + a} B a^{4} b^{4} - 105 \, {\left (b x + a\right )}^{\frac {7}{2}} A b^{5} + 385 \, {\left (b x + a\right )}^{\frac {5}{2}} A a b^{5} - 511 \, {\left (b x + a\right )}^{\frac {3}{2}} A a^{2} b^{5} + 279 \, \sqrt {b x + a} A a^{3} b^{5}}{a^{4} b^{4} x^{4}}}{192 \, b} \]
[In]
[Out]
Time = 0.48 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.24 \[ \int \frac {A+B x}{x^5 \sqrt {a+b x}} \, dx=\frac {\frac {73\,\left (7\,A\,b^4-8\,B\,a\,b^3\right )\,{\left (a+b\,x\right )}^{3/2}}{192\,a^2}-\frac {55\,\left (7\,A\,b^4-8\,B\,a\,b^3\right )\,{\left (a+b\,x\right )}^{5/2}}{192\,a^3}+\frac {5\,\left (7\,A\,b^4-8\,B\,a\,b^3\right )\,{\left (a+b\,x\right )}^{7/2}}{64\,a^4}-\frac {\left (93\,A\,b^4-88\,B\,a\,b^3\right )\,\sqrt {a+b\,x}}{64\,a}}{{\left (a+b\,x\right )}^4-4\,a^3\,\left (a+b\,x\right )-4\,a\,{\left (a+b\,x\right )}^3+6\,a^2\,{\left (a+b\,x\right )}^2+a^4}-\frac {5\,b^3\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (7\,A\,b-8\,B\,a\right )}{64\,a^{9/2}} \]
[In]
[Out]