Integrand size = 33, antiderivative size = 181 \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\frac {2 b B d-5 A b e+3 a B e}{3 b (b d-a e)^2 (d+e x)^{3/2}}-\frac {A b-a B}{b (b d-a e) (a+b x) (d+e x)^{3/2}}+\frac {2 b B d-5 A b e+3 a B e}{(b d-a e)^3 \sqrt {d+e x}}-\frac {\sqrt {b} (2 b B d-5 A b e+3 a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{7/2}} \]
[Out]
Time = 0.11 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {27, 79, 53, 65, 214} \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=-\frac {\sqrt {b} (3 a B e-5 A b e+2 b B d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{7/2}}-\frac {A b-a B}{b (a+b x) (d+e x)^{3/2} (b d-a e)}+\frac {3 a B e-5 A b e+2 b B d}{\sqrt {d+e x} (b d-a e)^3}+\frac {3 a B e-5 A b e+2 b B d}{3 b (d+e x)^{3/2} (b d-a e)^2} \]
[In]
[Out]
Rule 27
Rule 53
Rule 65
Rule 79
Rule 214
Rubi steps \begin{align*} \text {integral}& = \int \frac {A+B x}{(a+b x)^2 (d+e x)^{5/2}} \, dx \\ & = -\frac {A b-a B}{b (b d-a e) (a+b x) (d+e x)^{3/2}}+\frac {(2 b B d-5 A b e+3 a B e) \int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx}{2 b (b d-a e)} \\ & = \frac {2 b B d-5 A b e+3 a B e}{3 b (b d-a e)^2 (d+e x)^{3/2}}-\frac {A b-a B}{b (b d-a e) (a+b x) (d+e x)^{3/2}}+\frac {(2 b B d-5 A b e+3 a B e) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{2 (b d-a e)^2} \\ & = \frac {2 b B d-5 A b e+3 a B e}{3 b (b d-a e)^2 (d+e x)^{3/2}}-\frac {A b-a B}{b (b d-a e) (a+b x) (d+e x)^{3/2}}+\frac {2 b B d-5 A b e+3 a B e}{(b d-a e)^3 \sqrt {d+e x}}+\frac {(b (2 b B d-5 A b e+3 a B e)) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{2 (b d-a e)^3} \\ & = \frac {2 b B d-5 A b e+3 a B e}{3 b (b d-a e)^2 (d+e x)^{3/2}}-\frac {A b-a B}{b (b d-a e) (a+b x) (d+e x)^{3/2}}+\frac {2 b B d-5 A b e+3 a B e}{(b d-a e)^3 \sqrt {d+e x}}+\frac {(b (2 b B d-5 A b e+3 a B e)) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{e (b d-a e)^3} \\ & = \frac {2 b B d-5 A b e+3 a B e}{3 b (b d-a e)^2 (d+e x)^{3/2}}-\frac {A b-a B}{b (b d-a e) (a+b x) (d+e x)^{3/2}}+\frac {2 b B d-5 A b e+3 a B e}{(b d-a e)^3 \sqrt {d+e x}}-\frac {\sqrt {b} (2 b B d-5 A b e+3 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{7/2}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.09 \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\frac {B \left (2 a^2 e (2 d+3 e x)+2 b^2 d x (4 d+3 e x)+a b \left (11 d^2+16 d e x+9 e^2 x^2\right )\right )-A \left (-2 a^2 e^2+2 a b e (7 d+5 e x)+b^2 \left (3 d^2+20 d e x+15 e^2 x^2\right )\right )}{3 (b d-a e)^3 (a+b x) (d+e x)^{3/2}}-\frac {\sqrt {b} (2 b B d-5 A b e+3 a B e) \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{7/2}} \]
[In]
[Out]
Time = 0.38 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(-\frac {2 \left (A e -B d \right )}{3 \left (a e -b d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (-2 A b e +B a e +B b d \right )}{\left (a e -b d \right )^{3} \sqrt {e x +d}}+\frac {2 b \left (\frac {\left (\frac {1}{2} A b e -\frac {1}{2} B a e \right ) \sqrt {e x +d}}{b \left (e x +d \right )+a e -b d}+\frac {\left (5 A b e -3 B a e -2 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{3}}\) | \(164\) |
default | \(-\frac {2 \left (A e -B d \right )}{3 \left (a e -b d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (-2 A b e +B a e +B b d \right )}{\left (a e -b d \right )^{3} \sqrt {e x +d}}+\frac {2 b \left (\frac {\left (\frac {1}{2} A b e -\frac {1}{2} B a e \right ) \sqrt {e x +d}}{b \left (e x +d \right )+a e -b d}+\frac {\left (5 A b e -3 B a e -2 B b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{3}}\) | \(164\) |
pseudoelliptic | \(\frac {5 b \left (\left (A e -\frac {2 B d}{5}\right ) b -\frac {3 B a e}{5}\right ) \left (e x +d \right )^{\frac {3}{2}} \left (b x +a \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )-\frac {2 \sqrt {\left (a e -b d \right ) b}\, \left (\left (-\frac {15 A \,e^{2} x^{2}}{2}-10 x d \left (-\frac {3 B x}{10}+A \right ) e -\frac {3 d^{2} \left (-\frac {8 B x}{3}+A \right )}{2}\right ) b^{2}-7 \left (\frac {5 x \left (-\frac {9 B x}{10}+A \right ) e^{2}}{7}+d \left (-\frac {8 B x}{7}+A \right ) e -\frac {11 B \,d^{2}}{14}\right ) a b +\left (\left (3 B x +A \right ) e +2 B d \right ) e \,a^{2}\right )}{3}}{\left (a e -b d \right )^{3} \left (e x +d \right )^{\frac {3}{2}} \left (b x +a \right ) \sqrt {\left (a e -b d \right ) b}}\) | \(193\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 548 vs. \(2 (162) = 324\).
Time = 0.35 (sec) , antiderivative size = 1106, normalized size of antiderivative = 6.11 \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\text {Too large to display} \]
[In]
[Out]
\[ \int \frac {A+B x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\int \frac {A + B x}{\left (a + b x\right )^{2} \left (d + e x\right )^{\frac {5}{2}}}\, dx \]
[In]
[Out]
Timed out. \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.56 \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\frac {{\left (2 \, B b^{2} d + 3 \, B a b e - 5 \, A b^{2} e\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt {-b^{2} d + a b e}} + \frac {\sqrt {e x + d} B a b e - \sqrt {e x + d} A b^{2} e}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}} + \frac {2 \, {\left (3 \, {\left (e x + d\right )} B b d + B b d^{2} + 3 \, {\left (e x + d\right )} B a e - 6 \, {\left (e x + d\right )} A b e - B a d e - A b d e + A a e^{2}\right )}}{3 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {3}{2}}} \]
[In]
[Out]
Time = 10.86 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.16 \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=-\frac {\frac {2\,\left (A\,e-B\,d\right )}{3\,\left (a\,e-b\,d\right )}+\frac {2\,\left (d+e\,x\right )\,\left (3\,B\,a\,e-5\,A\,b\,e+2\,B\,b\,d\right )}{3\,{\left (a\,e-b\,d\right )}^2}+\frac {b\,{\left (d+e\,x\right )}^2\,\left (3\,B\,a\,e-5\,A\,b\,e+2\,B\,b\,d\right )}{{\left (a\,e-b\,d\right )}^3}}{b\,{\left (d+e\,x\right )}^{5/2}+\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{3/2}}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{{\left (a\,e-b\,d\right )}^{7/2}}\right )\,\left (3\,B\,a\,e-5\,A\,b\,e+2\,B\,b\,d\right )}{{\left (a\,e-b\,d\right )}^{7/2}} \]
[In]
[Out]