Integrand size = 13, antiderivative size = 95 \[ \int \frac {1}{x^{5/2} (a+b x)^3} \, dx=-\frac {35}{12 a^3 x^{3/2}}+\frac {35 b}{4 a^4 \sqrt {x}}+\frac {1}{2 a x^{3/2} (a+b x)^2}+\frac {7}{4 a^2 x^{3/2} (a+b x)}+\frac {35 b^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{9/2}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {44, 53, 65, 211} \[ \int \frac {1}{x^{5/2} (a+b x)^3} \, dx=\frac {35 b^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{9/2}}+\frac {35 b}{4 a^4 \sqrt {x}}-\frac {35}{12 a^3 x^{3/2}}+\frac {7}{4 a^2 x^{3/2} (a+b x)}+\frac {1}{2 a x^{3/2} (a+b x)^2} \]
[In]
[Out]
Rule 44
Rule 53
Rule 65
Rule 211
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2 a x^{3/2} (a+b x)^2}+\frac {7 \int \frac {1}{x^{5/2} (a+b x)^2} \, dx}{4 a} \\ & = \frac {1}{2 a x^{3/2} (a+b x)^2}+\frac {7}{4 a^2 x^{3/2} (a+b x)}+\frac {35 \int \frac {1}{x^{5/2} (a+b x)} \, dx}{8 a^2} \\ & = -\frac {35}{12 a^3 x^{3/2}}+\frac {1}{2 a x^{3/2} (a+b x)^2}+\frac {7}{4 a^2 x^{3/2} (a+b x)}-\frac {(35 b) \int \frac {1}{x^{3/2} (a+b x)} \, dx}{8 a^3} \\ & = -\frac {35}{12 a^3 x^{3/2}}+\frac {35 b}{4 a^4 \sqrt {x}}+\frac {1}{2 a x^{3/2} (a+b x)^2}+\frac {7}{4 a^2 x^{3/2} (a+b x)}+\frac {\left (35 b^2\right ) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{8 a^4} \\ & = -\frac {35}{12 a^3 x^{3/2}}+\frac {35 b}{4 a^4 \sqrt {x}}+\frac {1}{2 a x^{3/2} (a+b x)^2}+\frac {7}{4 a^2 x^{3/2} (a+b x)}+\frac {\left (35 b^2\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{4 a^4} \\ & = -\frac {35}{12 a^3 x^{3/2}}+\frac {35 b}{4 a^4 \sqrt {x}}+\frac {1}{2 a x^{3/2} (a+b x)^2}+\frac {7}{4 a^2 x^{3/2} (a+b x)}+\frac {35 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{9/2}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^{5/2} (a+b x)^3} \, dx=\frac {-8 a^3+56 a^2 b x+175 a b^2 x^2+105 b^3 x^3}{12 a^4 x^{3/2} (a+b x)^2}+\frac {35 b^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{9/2}} \]
[In]
[Out]
Time = 0.11 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.67
method | result | size |
risch | \(-\frac {2 \left (-9 b x +a \right )}{3 a^{4} x^{\frac {3}{2}}}+\frac {b^{2} \left (\frac {\frac {11 b \,x^{\frac {3}{2}}}{4}+\frac {13 a \sqrt {x}}{4}}{\left (b x +a \right )^{2}}+\frac {35 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}}\right )}{a^{4}}\) | \(64\) |
derivativedivides | \(\frac {2 b^{2} \left (\frac {\frac {11 b \,x^{\frac {3}{2}}}{8}+\frac {13 a \sqrt {x}}{8}}{\left (b x +a \right )^{2}}+\frac {35 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{4}}-\frac {2}{3 a^{3} x^{\frac {3}{2}}}+\frac {6 b}{a^{4} \sqrt {x}}\) | \(67\) |
default | \(\frac {2 b^{2} \left (\frac {\frac {11 b \,x^{\frac {3}{2}}}{8}+\frac {13 a \sqrt {x}}{8}}{\left (b x +a \right )^{2}}+\frac {35 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{4}}-\frac {2}{3 a^{3} x^{\frac {3}{2}}}+\frac {6 b}{a^{4} \sqrt {x}}\) | \(67\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 250, normalized size of antiderivative = 2.63 \[ \int \frac {1}{x^{5/2} (a+b x)^3} \, dx=\left [\frac {105 \, {\left (b^{3} x^{4} + 2 \, a b^{2} x^{3} + a^{2} b x^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x + 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) + 2 \, {\left (105 \, b^{3} x^{3} + 175 \, a b^{2} x^{2} + 56 \, a^{2} b x - 8 \, a^{3}\right )} \sqrt {x}}{24 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}, -\frac {105 \, {\left (b^{3} x^{4} + 2 \, a b^{2} x^{3} + a^{2} b x^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (105 \, b^{3} x^{3} + 175 \, a b^{2} x^{2} + 56 \, a^{2} b x - 8 \, a^{3}\right )} \sqrt {x}}{12 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}\right ] \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 869 vs. \(2 (88) = 176\).
Time = 65.48 (sec) , antiderivative size = 869, normalized size of antiderivative = 9.15 \[ \int \frac {1}{x^{5/2} (a+b x)^3} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {9}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{3 a^{3} x^{\frac {3}{2}}} & \text {for}\: b = 0 \\- \frac {2}{9 b^{3} x^{\frac {9}{2}}} & \text {for}\: a = 0 \\- \frac {16 a^{3} \sqrt {- \frac {a}{b}}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {105 a^{2} b x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} - \frac {105 a^{2} b x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {112 a^{2} b x \sqrt {- \frac {a}{b}}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {210 a b^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} - \frac {210 a b^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {350 a b^{2} x^{2} \sqrt {- \frac {a}{b}}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {105 b^{3} x^{\frac {7}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} - \frac {105 b^{3} x^{\frac {7}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {210 b^{3} x^{3} \sqrt {- \frac {a}{b}}}{24 a^{6} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 48 a^{5} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 24 a^{4} b^{2} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x^{5/2} (a+b x)^3} \, dx=\frac {105 \, b^{3} x^{3} + 175 \, a b^{2} x^{2} + 56 \, a^{2} b x - 8 \, a^{3}}{12 \, {\left (a^{4} b^{2} x^{\frac {7}{2}} + 2 \, a^{5} b x^{\frac {5}{2}} + a^{6} x^{\frac {3}{2}}\right )}} + \frac {35 \, b^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{4}} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^{5/2} (a+b x)^3} \, dx=\frac {35 \, b^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{4}} + \frac {2 \, {\left (9 \, b x - a\right )}}{3 \, a^{4} x^{\frac {3}{2}}} + \frac {11 \, b^{3} x^{\frac {3}{2}} + 13 \, a b^{2} \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} a^{4}} \]
[In]
[Out]
Time = 0.23 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^{5/2} (a+b x)^3} \, dx=\frac {\frac {175\,b^2\,x^2}{12\,a^3}-\frac {2}{3\,a}+\frac {35\,b^3\,x^3}{4\,a^4}+\frac {14\,b\,x}{3\,a^2}}{a^2\,x^{3/2}+b^2\,x^{7/2}+2\,a\,b\,x^{5/2}}+\frac {35\,b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{4\,a^{9/2}} \]
[In]
[Out]