Integrand size = 21, antiderivative size = 162 \[ \int (d+e x)^2 \sqrt {b x+c x^2} \, dx=\frac {\left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{64 c^3}+\frac {5 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{24 c^2}+\frac {e (d+e x) \left (b x+c x^2\right )^{3/2}}{4 c}-\frac {b^2 \left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{64 c^{7/2}} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {756, 654, 626, 634, 212} \[ \int (d+e x)^2 \sqrt {b x+c x^2} \, dx=-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right )}{64 c^{7/2}}+\frac {(b+2 c x) \sqrt {b x+c x^2} \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right )}{64 c^3}+\frac {5 e \left (b x+c x^2\right )^{3/2} (2 c d-b e)}{24 c^2}+\frac {e \left (b x+c x^2\right )^{3/2} (d+e x)}{4 c} \]
[In]
[Out]
Rule 212
Rule 626
Rule 634
Rule 654
Rule 756
Rubi steps \begin{align*} \text {integral}& = \frac {e (d+e x) \left (b x+c x^2\right )^{3/2}}{4 c}+\frac {\int \left (\frac {1}{2} d (8 c d-3 b e)+\frac {5}{2} e (2 c d-b e) x\right ) \sqrt {b x+c x^2} \, dx}{4 c} \\ & = \frac {5 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{24 c^2}+\frac {e (d+e x) \left (b x+c x^2\right )^{3/2}}{4 c}+\frac {\left (c d (8 c d-3 b e)-\frac {5}{2} b e (2 c d-b e)\right ) \int \sqrt {b x+c x^2} \, dx}{8 c^2} \\ & = \frac {\left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{64 c^3}+\frac {5 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{24 c^2}+\frac {e (d+e x) \left (b x+c x^2\right )^{3/2}}{4 c}-\frac {\left (b^2 \left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{128 c^3} \\ & = \frac {\left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{64 c^3}+\frac {5 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{24 c^2}+\frac {e (d+e x) \left (b x+c x^2\right )^{3/2}}{4 c}-\frac {\left (b^2 \left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{64 c^3} \\ & = \frac {\left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right ) (b+2 c x) \sqrt {b x+c x^2}}{64 c^3}+\frac {5 e (2 c d-b e) \left (b x+c x^2\right )^{3/2}}{24 c^2}+\frac {e (d+e x) \left (b x+c x^2\right )^{3/2}}{4 c}-\frac {b^2 \left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{64 c^{7/2}} \\ \end{align*}
Time = 1.11 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.07 \[ \int (d+e x)^2 \sqrt {b x+c x^2} \, dx=\frac {\sqrt {x (b+c x)} \left (\sqrt {c} \left (15 b^3 e^2-2 b^2 c e (24 d+5 e x)+8 b c^2 \left (6 d^2+4 d e x+e^2 x^2\right )+16 c^3 x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right )+\frac {6 b^2 \left (16 c^2 d^2-16 b c d e+5 b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}-\sqrt {b+c x}}\right )}{\sqrt {x} \sqrt {b+c x}}\right )}{192 c^{7/2}} \]
[In]
[Out]
Time = 2.05 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.81
method | result | size |
pseudoelliptic | \(-\frac {5 \left (b^{2} \left (b^{2} e^{2}-\frac {16}{5} b c d e +\frac {16}{5} c^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )-\sqrt {x \left (c x +b \right )}\, \left (\frac {16 b \left (\frac {1}{6} x^{2} e^{2}+\frac {2}{3} d e x +d^{2}\right ) c^{\frac {5}{2}}}{5}+\frac {32 \left (\frac {1}{2} x^{2} e^{2}+\frac {4}{3} d e x +d^{2}\right ) x \,c^{\frac {7}{2}}}{5}+e \left (\left (-\frac {2 e x}{3}-\frac {16 d}{5}\right ) c^{\frac {3}{2}}+\sqrt {c}\, b e \right ) b^{2}\right )\right )}{64 c^{\frac {7}{2}}}\) | \(132\) |
risch | \(\frac {\left (48 c^{3} e^{2} x^{3}+8 b \,c^{2} e^{2} x^{2}+128 c^{3} d e \,x^{2}-10 b^{2} c \,e^{2} x +32 b \,c^{2} d e x +96 c^{3} d^{2} x +15 e^{2} b^{3}-48 b^{2} d c e +48 b \,c^{2} d^{2}\right ) x \left (c x +b \right )}{192 c^{3} \sqrt {x \left (c x +b \right )}}-\frac {b^{2} \left (5 b^{2} e^{2}-16 b c d e +16 c^{2} d^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{128 c^{\frac {7}{2}}}\) | \(164\) |
default | \(d^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )+e^{2} \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{4 c}-\frac {5 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )}{8 c}\right )+2 d e \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )\) | \(249\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 337, normalized size of antiderivative = 2.08 \[ \int (d+e x)^2 \sqrt {b x+c x^2} \, dx=\left [\frac {3 \, {\left (16 \, b^{2} c^{2} d^{2} - 16 \, b^{3} c d e + 5 \, b^{4} e^{2}\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (48 \, c^{4} e^{2} x^{3} + 48 \, b c^{3} d^{2} - 48 \, b^{2} c^{2} d e + 15 \, b^{3} c e^{2} + 8 \, {\left (16 \, c^{4} d e + b c^{3} e^{2}\right )} x^{2} + 2 \, {\left (48 \, c^{4} d^{2} + 16 \, b c^{3} d e - 5 \, b^{2} c^{2} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{384 \, c^{4}}, \frac {3 \, {\left (16 \, b^{2} c^{2} d^{2} - 16 \, b^{3} c d e + 5 \, b^{4} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (48 \, c^{4} e^{2} x^{3} + 48 \, b c^{3} d^{2} - 48 \, b^{2} c^{2} d e + 15 \, b^{3} c e^{2} + 8 \, {\left (16 \, c^{4} d e + b c^{3} e^{2}\right )} x^{2} + 2 \, {\left (48 \, c^{4} d^{2} + 16 \, b c^{3} d e - 5 \, b^{2} c^{2} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{192 \, c^{4}}\right ] \]
[In]
[Out]
Time = 0.44 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.75 \[ \int (d+e x)^2 \sqrt {b x+c x^2} \, dx=\begin {cases} - \frac {b \left (b d^{2} - \frac {3 b \left (2 b d e - \frac {5 b \left (\frac {b e^{2}}{8} + 2 c d e\right )}{6 c} + c d^{2}\right )}{4 c}\right ) \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: \frac {b^{2}}{c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right )}{2 c} + \sqrt {b x + c x^{2}} \left (\frac {e^{2} x^{3}}{4} + \frac {x^{2} \left (\frac {b e^{2}}{8} + 2 c d e\right )}{3 c} + \frac {x \left (2 b d e - \frac {5 b \left (\frac {b e^{2}}{8} + 2 c d e\right )}{6 c} + c d^{2}\right )}{2 c} + \frac {b d^{2} - \frac {3 b \left (2 b d e - \frac {5 b \left (\frac {b e^{2}}{8} + 2 c d e\right )}{6 c} + c d^{2}\right )}{4 c}}{c}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (\frac {d^{2} \left (b x\right )^{\frac {3}{2}}}{3} + \frac {2 d e \left (b x\right )^{\frac {5}{2}}}{5 b} + \frac {e^{2} \left (b x\right )^{\frac {7}{2}}}{7 b^{2}}\right )}{b} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.75 \[ \int (d+e x)^2 \sqrt {b x+c x^2} \, dx=\frac {1}{2} \, \sqrt {c x^{2} + b x} d^{2} x - \frac {\sqrt {c x^{2} + b x} b d e x}{2 \, c} + \frac {5 \, \sqrt {c x^{2} + b x} b^{2} e^{2} x}{32 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} e^{2} x}{4 \, c} - \frac {b^{2} d^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{8 \, c^{\frac {3}{2}}} + \frac {b^{3} d e \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{8 \, c^{\frac {5}{2}}} - \frac {5 \, b^{4} e^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {7}{2}}} + \frac {\sqrt {c x^{2} + b x} b d^{2}}{4 \, c} - \frac {\sqrt {c x^{2} + b x} b^{2} d e}{4 \, c^{2}} + \frac {2 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} d e}{3 \, c} + \frac {5 \, \sqrt {c x^{2} + b x} b^{3} e^{2}}{64 \, c^{3}} - \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b e^{2}}{24 \, c^{2}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.06 \[ \int (d+e x)^2 \sqrt {b x+c x^2} \, dx=\frac {1}{192} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (6 \, e^{2} x + \frac {16 \, c^{3} d e + b c^{2} e^{2}}{c^{3}}\right )} x + \frac {48 \, c^{3} d^{2} + 16 \, b c^{2} d e - 5 \, b^{2} c e^{2}}{c^{3}}\right )} x + \frac {3 \, {\left (16 \, b c^{2} d^{2} - 16 \, b^{2} c d e + 5 \, b^{3} e^{2}\right )}}{c^{3}}\right )} + \frac {{\left (16 \, b^{2} c^{2} d^{2} - 16 \, b^{3} c d e + 5 \, b^{4} e^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{128 \, c^{\frac {7}{2}}} \]
[In]
[Out]
Time = 9.99 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.43 \[ \int (d+e x)^2 \sqrt {b x+c x^2} \, dx=d^2\,\sqrt {c\,x^2+b\,x}\,\left (\frac {x}{2}+\frac {b}{4\,c}\right )-\frac {5\,b\,e^2\,\left (\frac {b^3\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{16\,c^{5/2}}+\frac {\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{24\,c^2}\right )}{8\,c}-\frac {b^2\,d^2\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x}\right )}{8\,c^{3/2}}+\frac {e^2\,x\,{\left (c\,x^2+b\,x\right )}^{3/2}}{4\,c}+\frac {b^3\,d\,e\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{8\,c^{5/2}}+\frac {d\,e\,\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{12\,c^2} \]
[In]
[Out]