Integrand size = 24, antiderivative size = 122 \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{(d+e x)^{3/2}} \, dx=\frac {2 d (B d-A e) (c d-b e)}{e^4 \sqrt {d+e x}}+\frac {2 (B d (3 c d-2 b e)-A e (2 c d-b e)) \sqrt {d+e x}}{e^4}-\frac {2 (3 B c d-b B e-A c e) (d+e x)^{3/2}}{3 e^4}+\frac {2 B c (d+e x)^{5/2}}{5 e^4} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {785} \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{(d+e x)^{3/2}} \, dx=-\frac {2 (d+e x)^{3/2} (-A c e-b B e+3 B c d)}{3 e^4}+\frac {2 \sqrt {d+e x} (B d (3 c d-2 b e)-A e (2 c d-b e))}{e^4}+\frac {2 d (B d-A e) (c d-b e)}{e^4 \sqrt {d+e x}}+\frac {2 B c (d+e x)^{5/2}}{5 e^4} \]
[In]
[Out]
Rule 785
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {d (B d-A e) (c d-b e)}{e^3 (d+e x)^{3/2}}+\frac {B d (3 c d-2 b e)-A e (2 c d-b e)}{e^3 \sqrt {d+e x}}+\frac {(-3 B c d+b B e+A c e) \sqrt {d+e x}}{e^3}+\frac {B c (d+e x)^{3/2}}{e^3}\right ) \, dx \\ & = \frac {2 d (B d-A e) (c d-b e)}{e^4 \sqrt {d+e x}}+\frac {2 (B d (3 c d-2 b e)-A e (2 c d-b e)) \sqrt {d+e x}}{e^4}-\frac {2 (3 B c d-b B e-A c e) (d+e x)^{3/2}}{3 e^4}+\frac {2 B c (d+e x)^{5/2}}{5 e^4} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.90 \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{(d+e x)^{3/2}} \, dx=\frac {2 \left (5 A e \left (3 b e (2 d+e x)+c \left (-8 d^2-4 d e x+e^2 x^2\right )\right )+B \left (5 b e \left (-8 d^2-4 d e x+e^2 x^2\right )+3 c \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )\right )}{15 e^4 \sqrt {d+e x}} \]
[In]
[Out]
Time = 0.29 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.87
method | result | size |
pseudoelliptic | \(\frac {\frac {2 \left (3 \left (e^{3} x^{3}-2 d \,e^{2} x^{2}+8 d^{2} e x +16 d^{3}\right ) c -40 \left (-\frac {1}{8} e^{2} x^{2}+\frac {1}{2} d e x +d^{2}\right ) e b \right ) B}{15}+4 A \left (\frac {\left (\frac {1}{2} e^{2} x^{2}-2 d e x -4 d^{2}\right ) c}{3}+b e \left (\frac {e x}{2}+d \right )\right ) e}{\sqrt {e x +d}\, e^{4}}\) | \(106\) |
risch | \(\frac {2 \left (3 B c \,x^{2} e^{2}+5 A c x \,e^{2}+5 B b x \,e^{2}-9 B c d e x +15 A b \,e^{2}-25 A c d e -25 B b d e +33 B c \,d^{2}\right ) \sqrt {e x +d}}{15 e^{4}}+\frac {2 d \left (A b \,e^{2}-A c d e -B b d e +B c \,d^{2}\right )}{e^{4} \sqrt {e x +d}}\) | \(112\) |
gosper | \(\frac {\frac {2}{5} B c \,x^{3} e^{3}+\frac {2}{3} A c \,e^{3} x^{2}+\frac {2}{3} B \,x^{2} b \,e^{3}-\frac {4}{5} B \,x^{2} c d \,e^{2}+2 A b \,e^{3} x -\frac {8}{3} A c d \,e^{2} x -\frac {8}{3} B x b d \,e^{2}+\frac {16}{5} B c \,d^{2} e x +4 A b d \,e^{2}-\frac {16}{3} A c \,d^{2} e -\frac {16}{3} B b \,d^{2} e +\frac {32}{5} B c \,d^{3}}{\sqrt {e x +d}\, e^{4}}\) | \(121\) |
trager | \(\frac {\frac {2}{5} B c \,x^{3} e^{3}+\frac {2}{3} A c \,e^{3} x^{2}+\frac {2}{3} B \,x^{2} b \,e^{3}-\frac {4}{5} B \,x^{2} c d \,e^{2}+2 A b \,e^{3} x -\frac {8}{3} A c d \,e^{2} x -\frac {8}{3} B x b d \,e^{2}+\frac {16}{5} B c \,d^{2} e x +4 A b d \,e^{2}-\frac {16}{3} A c \,d^{2} e -\frac {16}{3} B b \,d^{2} e +\frac {32}{5} B c \,d^{3}}{\sqrt {e x +d}\, e^{4}}\) | \(121\) |
derivativedivides | \(\frac {\frac {2 B c \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 A c e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 B b e \left (e x +d \right )^{\frac {3}{2}}}{3}-2 B c d \left (e x +d \right )^{\frac {3}{2}}+2 A b \,e^{2} \sqrt {e x +d}-4 A c d e \sqrt {e x +d}-4 B b d e \sqrt {e x +d}+6 B c \,d^{2} \sqrt {e x +d}+\frac {2 d \left (A b \,e^{2}-A c d e -B b d e +B c \,d^{2}\right )}{\sqrt {e x +d}}}{e^{4}}\) | \(141\) |
default | \(\frac {\frac {2 B c \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 A c e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 B b e \left (e x +d \right )^{\frac {3}{2}}}{3}-2 B c d \left (e x +d \right )^{\frac {3}{2}}+2 A b \,e^{2} \sqrt {e x +d}-4 A c d e \sqrt {e x +d}-4 B b d e \sqrt {e x +d}+6 B c \,d^{2} \sqrt {e x +d}+\frac {2 d \left (A b \,e^{2}-A c d e -B b d e +B c \,d^{2}\right )}{\sqrt {e x +d}}}{e^{4}}\) | \(141\) |
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.97 \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (3 \, B c e^{3} x^{3} + 48 \, B c d^{3} + 30 \, A b d e^{2} - 40 \, {\left (B b + A c\right )} d^{2} e - {\left (6 \, B c d e^{2} - 5 \, {\left (B b + A c\right )} e^{3}\right )} x^{2} + {\left (24 \, B c d^{2} e + 15 \, A b e^{3} - 20 \, {\left (B b + A c\right )} d e^{2}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{5} x + d e^{4}\right )}} \]
[In]
[Out]
Time = 2.45 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.31 \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{(d+e x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {B c \left (d + e x\right )^{\frac {5}{2}}}{5 e^{3}} - \frac {d \left (- A e + B d\right ) \left (b e - c d\right )}{e^{3} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (A c e + B b e - 3 B c d\right )}{3 e^{3}} + \frac {\sqrt {d + e x} \left (A b e^{2} - 2 A c d e - 2 B b d e + 3 B c d^{2}\right )}{e^{3}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {\frac {A b x^{2}}{2} + \frac {B c x^{4}}{4} + \frac {x^{3} \left (A c + B b\right )}{3}}{d^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.98 \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {3 \, {\left (e x + d\right )}^{\frac {5}{2}} B c - 5 \, {\left (3 \, B c d - {\left (B b + A c\right )} e\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 15 \, {\left (3 \, B c d^{2} + A b e^{2} - 2 \, {\left (B b + A c\right )} d e\right )} \sqrt {e x + d}}{e^{3}} + \frac {15 \, {\left (B c d^{3} + A b d e^{2} - {\left (B b + A c\right )} d^{2} e\right )}}{\sqrt {e x + d} e^{3}}\right )}}{15 \, e} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.37 \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (B c d^{3} - B b d^{2} e - A c d^{2} e + A b d e^{2}\right )}}{\sqrt {e x + d} e^{4}} + \frac {2 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} B c e^{16} - 15 \, {\left (e x + d\right )}^{\frac {3}{2}} B c d e^{16} + 45 \, \sqrt {e x + d} B c d^{2} e^{16} + 5 \, {\left (e x + d\right )}^{\frac {3}{2}} B b e^{17} + 5 \, {\left (e x + d\right )}^{\frac {3}{2}} A c e^{17} - 30 \, \sqrt {e x + d} B b d e^{17} - 30 \, \sqrt {e x + d} A c d e^{17} + 15 \, \sqrt {e x + d} A b e^{18}\right )}}{15 \, e^{20}} \]
[In]
[Out]
Time = 10.40 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.02 \[ \int \frac {(A+B x) \left (b x+c x^2\right )}{(d+e x)^{3/2}} \, dx=\frac {\sqrt {d+e\,x}\,\left (2\,A\,b\,e^2+6\,B\,c\,d^2-4\,A\,c\,d\,e-4\,B\,b\,d\,e\right )}{e^4}+\frac {{\left (d+e\,x\right )}^{3/2}\,\left (2\,A\,c\,e+2\,B\,b\,e-6\,B\,c\,d\right )}{3\,e^4}+\frac {2\,B\,c\,d^3+2\,A\,b\,d\,e^2-2\,A\,c\,d^2\,e-2\,B\,b\,d^2\,e}{e^4\,\sqrt {d+e\,x}}+\frac {2\,B\,c\,{\left (d+e\,x\right )}^{5/2}}{5\,e^4} \]
[In]
[Out]