Integrand size = 18, antiderivative size = 115 \[ \int \frac {A+B x}{x^4 \sqrt {a+b x}} \, dx=-\frac {A \sqrt {a+b x}}{3 a x^3}+\frac {(5 A b-6 a B) \sqrt {a+b x}}{12 a^2 x^2}-\frac {b (5 A b-6 a B) \sqrt {a+b x}}{8 a^3 x}+\frac {b^2 (5 A b-6 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{7/2}} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {79, 44, 65, 214} \[ \int \frac {A+B x}{x^4 \sqrt {a+b x}} \, dx=\frac {b^2 (5 A b-6 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{7/2}}-\frac {b \sqrt {a+b x} (5 A b-6 a B)}{8 a^3 x}+\frac {\sqrt {a+b x} (5 A b-6 a B)}{12 a^2 x^2}-\frac {A \sqrt {a+b x}}{3 a x^3} \]
[In]
[Out]
Rule 44
Rule 65
Rule 79
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {A \sqrt {a+b x}}{3 a x^3}+\frac {\left (-\frac {5 A b}{2}+3 a B\right ) \int \frac {1}{x^3 \sqrt {a+b x}} \, dx}{3 a} \\ & = -\frac {A \sqrt {a+b x}}{3 a x^3}+\frac {(5 A b-6 a B) \sqrt {a+b x}}{12 a^2 x^2}+\frac {(b (5 A b-6 a B)) \int \frac {1}{x^2 \sqrt {a+b x}} \, dx}{8 a^2} \\ & = -\frac {A \sqrt {a+b x}}{3 a x^3}+\frac {(5 A b-6 a B) \sqrt {a+b x}}{12 a^2 x^2}-\frac {b (5 A b-6 a B) \sqrt {a+b x}}{8 a^3 x}-\frac {\left (b^2 (5 A b-6 a B)\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx}{16 a^3} \\ & = -\frac {A \sqrt {a+b x}}{3 a x^3}+\frac {(5 A b-6 a B) \sqrt {a+b x}}{12 a^2 x^2}-\frac {b (5 A b-6 a B) \sqrt {a+b x}}{8 a^3 x}-\frac {(b (5 A b-6 a B)) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{8 a^3} \\ & = -\frac {A \sqrt {a+b x}}{3 a x^3}+\frac {(5 A b-6 a B) \sqrt {a+b x}}{12 a^2 x^2}-\frac {b (5 A b-6 a B) \sqrt {a+b x}}{8 a^3 x}+\frac {b^2 (5 A b-6 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{7/2}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.81 \[ \int \frac {A+B x}{x^4 \sqrt {a+b x}} \, dx=\frac {\sqrt {a+b x} \left (-15 A b^2 x^2-4 a^2 (2 A+3 B x)+2 a b x (5 A+9 B x)\right )}{24 a^3 x^3}+\frac {b^2 (5 A b-6 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{7/2}} \]
[In]
[Out]
Time = 0.53 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.71
method | result | size |
pseudoelliptic | \(-\frac {-\frac {15 x^{3} b^{2} \left (A b -\frac {6 B a}{5}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8}+\left (-\frac {5 x b \left (\frac {9 B x}{5}+A \right ) a^{\frac {3}{2}}}{4}+\left (\frac {3 B x}{2}+A \right ) a^{\frac {5}{2}}+\frac {15 A \sqrt {a}\, b^{2} x^{2}}{8}\right ) \sqrt {b x +a}}{3 a^{\frac {7}{2}} x^{3}}\) | \(82\) |
risch | \(-\frac {\sqrt {b x +a}\, \left (15 A \,b^{2} x^{2}-18 B a b \,x^{2}-10 a A b x +12 a^{2} B x +8 a^{2} A \right )}{24 a^{3} x^{3}}+\frac {b^{2} \left (5 A b -6 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8 a^{\frac {7}{2}}}\) | \(83\) |
derivativedivides | \(2 b^{2} \left (-\frac {\frac {\left (5 A b -6 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{16 a^{3}}-\frac {\left (5 A b -6 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{6 a^{2}}+\frac {\left (11 A b -10 B a \right ) \sqrt {b x +a}}{16 a}}{b^{3} x^{3}}+\frac {\left (5 A b -6 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{16 a^{\frac {7}{2}}}\right )\) | \(105\) |
default | \(2 b^{2} \left (-\frac {\frac {\left (5 A b -6 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{16 a^{3}}-\frac {\left (5 A b -6 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{6 a^{2}}+\frac {\left (11 A b -10 B a \right ) \sqrt {b x +a}}{16 a}}{b^{3} x^{3}}+\frac {\left (5 A b -6 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{16 a^{\frac {7}{2}}}\right )\) | \(105\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.84 \[ \int \frac {A+B x}{x^4 \sqrt {a+b x}} \, dx=\left [-\frac {3 \, {\left (6 \, B a b^{2} - 5 \, A b^{3}\right )} \sqrt {a} x^{3} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (8 \, A a^{3} - 3 \, {\left (6 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{2} + 2 \, {\left (6 \, B a^{3} - 5 \, A a^{2} b\right )} x\right )} \sqrt {b x + a}}{48 \, a^{4} x^{3}}, \frac {3 \, {\left (6 \, B a b^{2} - 5 \, A b^{3}\right )} \sqrt {-a} x^{3} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left (8 \, A a^{3} - 3 \, {\left (6 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{2} + 2 \, {\left (6 \, B a^{3} - 5 \, A a^{2} b\right )} x\right )} \sqrt {b x + a}}{24 \, a^{4} x^{3}}\right ] \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (105) = 210\).
Time = 20.20 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.13 \[ \int \frac {A+B x}{x^4 \sqrt {a+b x}} \, dx=- \frac {A}{3 \sqrt {b} x^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {A \sqrt {b}}{12 a x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {5 A b^{\frac {3}{2}}}{24 a^{2} x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {5 A b^{\frac {5}{2}}}{8 a^{3} \sqrt {x} \sqrt {\frac {a}{b x} + 1}} + \frac {5 A b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{8 a^{\frac {7}{2}}} - \frac {B}{2 \sqrt {b} x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {B \sqrt {b}}{4 a x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {3 B b^{\frac {3}{2}}}{4 a^{2} \sqrt {x} \sqrt {\frac {a}{b x} + 1}} - \frac {3 B b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{4 a^{\frac {5}{2}}} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.40 \[ \int \frac {A+B x}{x^4 \sqrt {a+b x}} \, dx=\frac {1}{48} \, b^{3} {\left (\frac {2 \, {\left (3 \, {\left (6 \, B a - 5 \, A b\right )} {\left (b x + a\right )}^{\frac {5}{2}} - 8 \, {\left (6 \, B a^{2} - 5 \, A a b\right )} {\left (b x + a\right )}^{\frac {3}{2}} + 3 \, {\left (10 \, B a^{3} - 11 \, A a^{2} b\right )} \sqrt {b x + a}\right )}}{{\left (b x + a\right )}^{3} a^{3} b - 3 \, {\left (b x + a\right )}^{2} a^{4} b + 3 \, {\left (b x + a\right )} a^{5} b - a^{6} b} + \frac {3 \, {\left (6 \, B a - 5 \, A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {7}{2}} b}\right )} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.25 \[ \int \frac {A+B x}{x^4 \sqrt {a+b x}} \, dx=\frac {\frac {3 \, {\left (6 \, B a b^{3} - 5 \, A b^{4}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{3}} + \frac {18 \, {\left (b x + a\right )}^{\frac {5}{2}} B a b^{3} - 48 \, {\left (b x + a\right )}^{\frac {3}{2}} B a^{2} b^{3} + 30 \, \sqrt {b x + a} B a^{3} b^{3} - 15 \, {\left (b x + a\right )}^{\frac {5}{2}} A b^{4} + 40 \, {\left (b x + a\right )}^{\frac {3}{2}} A a b^{4} - 33 \, \sqrt {b x + a} A a^{2} b^{4}}{a^{3} b^{3} x^{3}}}{24 \, b} \]
[In]
[Out]
Time = 0.17 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.26 \[ \int \frac {A+B x}{x^4 \sqrt {a+b x}} \, dx=\frac {\frac {\left (5\,A\,b^3-6\,B\,a\,b^2\right )\,{\left (a+b\,x\right )}^{5/2}}{8\,a^3}-\frac {\left (5\,A\,b^3-6\,B\,a\,b^2\right )\,{\left (a+b\,x\right )}^{3/2}}{3\,a^2}+\frac {\left (11\,A\,b^3-10\,B\,a\,b^2\right )\,\sqrt {a+b\,x}}{8\,a}}{3\,a\,{\left (a+b\,x\right )}^2-3\,a^2\,\left (a+b\,x\right )-{\left (a+b\,x\right )}^3+a^3}+\frac {b^2\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (5\,A\,b-6\,B\,a\right )}{8\,a^{7/2}} \]
[In]
[Out]