Integrand size = 21, antiderivative size = 216 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\frac {\left (8 c^2 d^2-10 b c d e+b^2 e^2-2 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{8 c e^3}+\frac {\left (b x+c x^2\right )^{3/2}}{3 e}-\frac {(2 c d-b e) \left (8 c^2 d^2-8 b c d e-b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{3/2} e^4}+\frac {d^{3/2} (c d-b e)^{3/2} \text {arctanh}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{e^4} \]
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Time = 0.16 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {748, 828, 857, 634, 212, 738} \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (2 c d-b e) \left (-b^2 e^2-8 b c d e+8 c^2 d^2\right )}{8 c^{3/2} e^4}+\frac {d^{3/2} (c d-b e)^{3/2} \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{e^4}+\frac {\sqrt {b x+c x^2} \left (b^2 e^2-2 c e x (2 c d-b e)-10 b c d e+8 c^2 d^2\right )}{8 c e^3}+\frac {\left (b x+c x^2\right )^{3/2}}{3 e} \]
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Rule 212
Rule 634
Rule 738
Rule 748
Rule 828
Rule 857
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b x+c x^2\right )^{3/2}}{3 e}-\frac {\int \frac {(b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{d+e x} \, dx}{2 e} \\ & = \frac {\left (8 c^2 d^2-10 b c d e+b^2 e^2-2 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{8 c e^3}+\frac {\left (b x+c x^2\right )^{3/2}}{3 e}+\frac {\int \frac {-\frac {1}{2} b d \left (8 c^2 d^2-10 b c d e+b^2 e^2\right )-\frac {1}{2} (2 c d-b e) \left (8 c^2 d^2-8 b c d e-b^2 e^2\right ) x}{(d+e x) \sqrt {b x+c x^2}} \, dx}{8 c e^3} \\ & = \frac {\left (8 c^2 d^2-10 b c d e+b^2 e^2-2 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{8 c e^3}+\frac {\left (b x+c x^2\right )^{3/2}}{3 e}+\frac {\left (d^2 (c d-b e)^2\right ) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{e^4}-\frac {\left ((2 c d-b e) \left (8 c^2 d^2-8 b c d e-b^2 e^2\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{16 c e^4} \\ & = \frac {\left (8 c^2 d^2-10 b c d e+b^2 e^2-2 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{8 c e^3}+\frac {\left (b x+c x^2\right )^{3/2}}{3 e}-\frac {\left (2 d^2 (c d-b e)^2\right ) \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac {-b d-(2 c d-b e) x}{\sqrt {b x+c x^2}}\right )}{e^4}-\frac {\left ((2 c d-b e) \left (8 c^2 d^2-8 b c d e-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 )}{8 c e^4} \\ & = \frac {\left (8 c^2 d^2-10 b c d e+b^2 e^2-2 c e (2 c d-b e) x\right ) \sqrt {b x+c x^2}}{8 c e^3}+\frac {\left (b x+c x^2\right )^{3/2}}{3 e}-\frac {(2 c d-b e) \left (8 c^2 d^2-8 b c d e-b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{3/2} e^4}+\frac {d^{3/2} (c d-b e)^{3/2} \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{e^4} \\ \end{align*}
Result contains complex when optimal does not.
Time = 3.74 (sec) , antiderivative size = 500, normalized size of antiderivative = 2.31 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\frac {(x (b+c x))^{3/2} \left (\sqrt {c} e \sqrt {x} \sqrt {b+c x} \left (3 b^2 e^2+2 b c e (-15 d+7 e x)+4 c^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+48 \sqrt {c} \sqrt {d} (c d-b e) \left (c d-b e-i \sqrt {b} \sqrt {e} \sqrt {c d-b e}\right ) \sqrt {-c d+2 b e-2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \arctan \left (\frac {\sqrt {-c d+2 b e-2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \sqrt {x}}{\sqrt {d} \left (\sqrt {b}-\sqrt {b+c x}\right )}\right )+48 \sqrt {c} \sqrt {d} (c d-b e) \left (c d-b e+i \sqrt {b} \sqrt {e} \sqrt {c d-b e}\right ) \sqrt {-c d+2 b e+2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \arctan \left (\frac {\sqrt {-c d+2 b e+2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \sqrt {x}}{\sqrt {d} \left (\sqrt {b}-\sqrt {b+c x}\right )}\right )+6 \left (16 c^3 d^3-24 b c^2 d^2 e+6 b^2 c d e^2+b^3 e^3\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}-\sqrt {b+c x}}\right )\right )}{24 c^{3/2} e^4 x^{3/2} (b+c x)^{3/2}} \]
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Time = 2.19 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.85
method | result | size |
pseudoelliptic | \(-\frac {\frac {2 \left (b e -c d \right )^{2} d^{2} \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )}{\sqrt {d \left (b e -c d \right )}}-\frac {e \sqrt {x \left (c x +b \right )}\, \left (8 c^{2} e^{2} x^{2}+14 b c \,e^{2} x -12 c^{2} d e x +3 b^{2} e^{2}-30 b c d e +24 c^{2} d^{2}\right )}{24 c}+\frac {\left (b^{3} e^{3}+6 b^{2} d \,e^{2} c -24 b \,c^{2} d^{2} e +16 c^{3} d^{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )}{8 c^{\frac {3}{2}}}}{e^{4}}\) | \(184\) |
risch | \(\frac {\left (8 c^{2} e^{2} x^{2}+14 b c \,e^{2} x -12 c^{2} d e x +3 b^{2} e^{2}-30 b c d e +24 c^{2} d^{2}\right ) x \left (c x +b \right )}{24 c \,e^{3} \sqrt {x \left (c x +b \right )}}-\frac {\frac {\left (b^{3} e^{3}+6 b^{2} d \,e^{2} c -24 b \,c^{2} d^{2} e +16 c^{3} d^{3}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{e \sqrt {c}}+\frac {16 d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) c \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}}{16 e^{3} c}\) | \(307\) |
default | \(\frac {\frac {\left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}\right )^{\frac {3}{2}}}{3}+\frac {\left (b e -2 c d \right ) \left (\frac {\left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{4 c}+\frac {\left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}\right )}{8 c^{\frac {3}{2}}}\right )}{2 e}-\frac {d \left (b e -c d \right ) \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}\right )}{2 e \sqrt {c}}+\frac {d \left (b e -c d \right ) \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{e^{2}}}{e}\) | \(549\) |
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Time = 0.51 (sec) , antiderivative size = 901, normalized size of antiderivative = 4.17 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\left [\frac {3 \, {\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} + b^{3} e^{3}\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 48 \, {\left (c^{3} d^{2} - b c^{2} d e\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {b d + {\left (2 \, c d - b e\right )} x - 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{e x + d}\right ) + 2 \, {\left (8 \, c^{3} e^{3} x^{2} + 24 \, c^{3} d^{2} e - 30 \, b c^{2} d e^{2} + 3 \, b^{2} c e^{3} - 2 \, {\left (6 \, c^{3} d e^{2} - 7 \, b c^{2} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{48 \, c^{2} e^{4}}, \frac {96 \, {\left (c^{3} d^{2} - b c^{2} d e\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) + 3 \, {\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} + b^{3} e^{3}\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (8 \, c^{3} e^{3} x^{2} + 24 \, c^{3} d^{2} e - 30 \, b c^{2} d e^{2} + 3 \, b^{2} c e^{3} - 2 \, {\left (6 \, c^{3} d e^{2} - 7 \, b c^{2} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{48 \, c^{2} e^{4}}, \frac {3 \, {\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} + b^{3} e^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - 24 \, {\left (c^{3} d^{2} - b c^{2} d e\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {b d + {\left (2 \, c d - b e\right )} x - 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{e x + d}\right ) + {\left (8 \, c^{3} e^{3} x^{2} + 24 \, c^{3} d^{2} e - 30 \, b c^{2} d e^{2} + 3 \, b^{2} c e^{3} - 2 \, {\left (6 \, c^{3} d e^{2} - 7 \, b c^{2} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{24 \, c^{2} e^{4}}, \frac {48 \, {\left (c^{3} d^{2} - b c^{2} d e\right )} \sqrt {-c d^{2} + b d e} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) + 3 \, {\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} + b^{3} e^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (8 \, c^{3} e^{3} x^{2} + 24 \, c^{3} d^{2} e - 30 \, b c^{2} d e^{2} + 3 \, b^{2} c e^{3} - 2 \, {\left (6 \, c^{3} d e^{2} - 7 \, b c^{2} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{24 \, c^{2} e^{4}}\right ] \]
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\[ \int \frac {\left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}{d + e x}\, dx \]
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Exception generated. \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}}{d+e\,x} \,d x \]
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