Integrand size = 18, antiderivative size = 142 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^5} \, dx=\frac {5 b^2 (A b-8 a B) \sqrt {a+b x}}{64 a x}+\frac {5 b (A b-8 a B) (a+b x)^{3/2}}{96 a x^2}+\frac {(A b-8 a B) (a+b x)^{5/2}}{24 a x^3}-\frac {A (a+b x)^{7/2}}{4 a x^4}+\frac {5 b^3 (A b-8 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{64 a^{3/2}} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 142, 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, 43, 65, 214} \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^5} \, dx=\frac {5 b^3 (A b-8 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{64 a^{3/2}}+\frac {5 b^2 \sqrt {a+b x} (A b-8 a B)}{64 a x}+\frac {(a+b x)^{5/2} (A b-8 a B)}{24 a x^3}+\frac {5 b (a+b x)^{3/2} (A b-8 a B)}{96 a x^2}-\frac {A (a+b x)^{7/2}}{4 a x^4} \]
[In]
[Out]
Rule 43
Rule 65
Rule 79
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {A (a+b x)^{7/2}}{4 a x^4}+\frac {\left (-\frac {A b}{2}+4 a B\right ) \int \frac {(a+b x)^{5/2}}{x^4} \, dx}{4 a} \\ & = \frac {(A b-8 a B) (a+b x)^{5/2}}{24 a x^3}-\frac {A (a+b x)^{7/2}}{4 a x^4}-\frac {(5 b (A b-8 a B)) \int \frac {(a+b x)^{3/2}}{x^3} \, dx}{48 a} \\ & = \frac {5 b (A b-8 a B) (a+b x)^{3/2}}{96 a x^2}+\frac {(A b-8 a B) (a+b x)^{5/2}}{24 a x^3}-\frac {A (a+b x)^{7/2}}{4 a x^4}-\frac {\left (5 b^2 (A b-8 a B)\right ) \int \frac {\sqrt {a+b x}}{x^2} \, dx}{64 a} \\ & = \frac {5 b^2 (A b-8 a B) \sqrt {a+b x}}{64 a x}+\frac {5 b (A b-8 a B) (a+b x)^{3/2}}{96 a x^2}+\frac {(A b-8 a B) (a+b x)^{5/2}}{24 a x^3}-\frac {A (a+b x)^{7/2}}{4 a x^4}-\frac {\left (5 b^3 (A b-8 a B)\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx}{128 a} \\ & = \frac {5 b^2 (A b-8 a B) \sqrt {a+b x}}{64 a x}+\frac {5 b (A b-8 a B) (a+b x)^{3/2}}{96 a x^2}+\frac {(A b-8 a B) (a+b x)^{5/2}}{24 a x^3}-\frac {A (a+b x)^{7/2}}{4 a x^4}-\frac {\left (5 b^2 (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} \\ & = \frac {5 b^2 (A b-8 a B) \sqrt {a+b x}}{64 a x}+\frac {5 b (A b-8 a B) (a+b x)^{3/2}}{96 a x^2}+\frac {(A b-8 a B) (a+b x)^{5/2}}{24 a x^3}-\frac {A (a+b x)^{7/2}}{4 a x^4}+\frac {5 b^3 (A b-8 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{64 a^{3/2}} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.78 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^5} \, dx=-\frac {\sqrt {a+b x} \left (15 A b^3 x^3+16 a^3 (3 A+4 B x)+8 a^2 b x (17 A+26 B x)+2 a b^2 x^2 (59 A+132 B x)\right )}{192 a x^4}+\frac {5 b^3 (A b-8 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{64 a^{3/2}} \]
[In]
[Out]
Time = 0.54 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.71
method | result | size |
pseudoelliptic | \(-\frac {17 \left (-\frac {15 b^{3} x^{4} \left (A b -8 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{136}+\sqrt {b x +a}\, \left (\frac {59 \left (\frac {132 B x}{59}+A \right ) x^{2} b^{2} a^{\frac {3}{2}}}{68}+b x \left (\frac {26 B x}{17}+A \right ) a^{\frac {5}{2}}+\frac {2 \left (4 B x +3 A \right ) a^{\frac {7}{2}}}{17}+\frac {15 A \sqrt {a}\, b^{3} x^{3}}{136}\right )\right )}{24 a^{\frac {3}{2}} x^{4}}\) | \(101\) |
risch | \(-\frac {\sqrt {b x +a}\, \left (15 A \,b^{3} x^{3}+264 B a \,b^{2} x^{3}+118 a A \,b^{2} x^{2}+208 B \,a^{2} b \,x^{2}+136 a^{2} A b x +64 a^{3} B x +48 a^{3} A \right )}{192 x^{4} a}+\frac {5 b^{3} \left (A b -8 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{64 a^{\frac {3}{2}}}\) | \(106\) |
derivativedivides | \(2 b^{3} \left (-\frac {\frac {\left (5 A b +88 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{128 a}+\left (-\frac {73 B a}{48}+\frac {73 A b}{384}\right ) \left (b x +a \right )^{\frac {5}{2}}-\frac {55 a \left (A b -8 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{384}+\left (-\frac {5}{16} a^{3} B +\frac {5}{128} a^{2} b A \right ) \sqrt {b x +a}}{b^{4} x^{4}}+\frac {5 \left (A b -8 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{128 a^{\frac {3}{2}}}\right )\) | \(119\) |
default | \(2 b^{3} \left (-\frac {\frac {\left (5 A b +88 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{128 a}+\left (-\frac {73 B a}{48}+\frac {73 A b}{384}\right ) \left (b x +a \right )^{\frac {5}{2}}-\frac {55 a \left (A b -8 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{384}+\left (-\frac {5}{16} a^{3} B +\frac {5}{128} a^{2} b A \right ) \sqrt {b x +a}}{b^{4} x^{4}}+\frac {5 \left (A b -8 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{128 a^{\frac {3}{2}}}\right )\) | \(119\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.83 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^5} \, dx=\left [-\frac {15 \, {\left (8 \, B a b^{3} - A b^{4}\right )} \sqrt {a} x^{4} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (48 \, A a^{4} + 3 \, {\left (88 \, B a^{2} b^{2} + 5 \, A a b^{3}\right )} x^{3} + 2 \, {\left (104 \, B a^{3} b + 59 \, A a^{2} b^{2}\right )} x^{2} + 8 \, {\left (8 \, B a^{4} + 17 \, A a^{3} b\right )} x\right )} \sqrt {b x + a}}{384 \, a^{2} x^{4}}, \frac {15 \, {\left (8 \, B a b^{3} - A b^{4}\right )} \sqrt {-a} x^{4} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left (48 \, A a^{4} + 3 \, {\left (88 \, B a^{2} b^{2} + 5 \, A a b^{3}\right )} x^{3} + 2 \, {\left (104 \, B a^{3} b + 59 \, A a^{2} b^{2}\right )} x^{2} + 8 \, {\left (8 \, B a^{4} + 17 \, A a^{3} b\right )} x\right )} \sqrt {b x + a}}{192 \, a^{2} x^{4}}\right ] \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (129) = 258\).
Time = 124.43 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.30 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^5} \, dx=- \frac {A a^{3}}{4 \sqrt {b} x^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {23 A a^{2} \sqrt {b}}{24 x^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {127 A a b^{\frac {3}{2}}}{96 x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {133 A b^{\frac {5}{2}}}{192 x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {5 A b^{\frac {7}{2}}}{64 a \sqrt {x} \sqrt {\frac {a}{b x} + 1}} + \frac {5 A b^{4} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{64 a^{\frac {3}{2}}} - \frac {B a^{3}}{3 \sqrt {b} x^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {17 B a^{2} \sqrt {b}}{12 x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {35 B a b^{\frac {3}{2}}}{24 x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {B b^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}}{\sqrt {x}} - \frac {3 B b^{\frac {5}{2}}}{8 \sqrt {x} \sqrt {\frac {a}{b x} + 1}} - \frac {5 B b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{8 \sqrt {a}} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.37 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^5} \, dx=-\frac {1}{384} \, b^{4} {\left (\frac {2 \, {\left (3 \, {\left (88 \, B a + 5 \, A b\right )} {\left (b x + a\right )}^{\frac {7}{2}} - 73 \, {\left (8 \, B a^{2} - A a b\right )} {\left (b x + a\right )}^{\frac {5}{2}} + 55 \, {\left (8 \, B a^{3} - A a^{2} b\right )} {\left (b x + a\right )}^{\frac {3}{2}} - 15 \, {\left (8 \, B a^{4} - A a^{3} b\right )} \sqrt {b x + a}\right )}}{{\left (b x + a\right )}^{4} a b - 4 \, {\left (b x + a\right )}^{3} a^{2} b + 6 \, {\left (b x + a\right )}^{2} a^{3} b - 4 \, {\left (b x + a\right )} a^{4} b + a^{5} b} - \frac {15 \, {\left (8 \, B a - A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}} b}\right )} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.25 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^5} \, dx=\frac {\frac {15 \, {\left (8 \, B a b^{4} - A b^{5}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} - \frac {264 \, {\left (b x + a\right )}^{\frac {7}{2}} B a b^{4} - 584 \, {\left (b x + a\right )}^{\frac {5}{2}} B a^{2} b^{4} + 440 \, {\left (b x + a\right )}^{\frac {3}{2}} B a^{3} b^{4} - 120 \, \sqrt {b x + a} B a^{4} b^{4} + 15 \, {\left (b x + a\right )}^{\frac {7}{2}} A b^{5} + 73 \, {\left (b x + a\right )}^{\frac {5}{2}} A a b^{5} - 55 \, {\left (b x + a\right )}^{\frac {3}{2}} A a^{2} b^{5} + 15 \, \sqrt {b x + a} A a^{3} b^{5}}{a b^{4} x^{4}}}{192 \, b} \]
[In]
[Out]
Time = 0.55 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.25 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^5} \, dx=\frac {5\,b^3\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (A\,b-8\,B\,a\right )}{64\,a^{3/2}}-\frac {\left (\frac {73\,A\,b^4}{192}-\frac {73\,B\,a\,b^3}{24}\right )\,{\left (a+b\,x\right )}^{5/2}+\left (\frac {5\,A\,a^2\,b^4}{64}-\frac {5\,B\,a^3\,b^3}{8}\right )\,\sqrt {a+b\,x}+\left (\frac {55\,B\,a^2\,b^3}{24}-\frac {55\,A\,a\,b^4}{192}\right )\,{\left (a+b\,x\right )}^{3/2}+\frac {\left (5\,A\,b^4+88\,B\,a\,b^3\right )\,{\left (a+b\,x\right )}^{7/2}}{64\,a}}{{\left (a+b\,x\right )}^4-4\,a^3\,\left (a+b\,x\right )-4\,a\,{\left (a+b\,x\right )}^3+6\,a^2\,{\left (a+b\,x\right )}^2+a^4} \]
[In]
[Out]