Integrand size = 22, antiderivative size = 459 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{d+e x} \, dx=\frac {\left (128 c^4 d^4-3 b^4 e^4-2 b^2 c e^3 (5 b d-14 a e)-32 c^3 d^2 e (9 b d-8 a e)+8 c^2 e^2 \left (22 b^2 d^2-39 a b d e+16 a^2 e^2\right )-2 c e (2 c d-b e) \left (16 c^2 d^2-3 b^2 e^2-4 c e (4 b d-7 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{128 c^2 e^5}+\frac {\left (16 c^2 d^2+3 b^2 e^2-2 c e (11 b d-8 a e)-6 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c e^3}+\frac {\left (a+b x+c x^2\right )^{5/2}}{5 e}-\frac {(2 c d-b e) \left (128 c^4 d^4+3 b^4 e^4+8 b^2 c e^3 (2 b d-5 a e)-64 c^3 d^2 e (4 b d-5 a e)+16 c^2 e^2 \left (7 b^2 d^2-20 a b d e+15 a^2 e^2\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{5/2} e^6}+\frac {\left (c d^2-b d e+a e^2\right )^{5/2} \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e^6} \]
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Time = 0.48 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {748, 828, 857, 635, 212, 738} \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{d+e x} \, dx=-\frac {(2 c d-b e) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (16 c^2 e^2 \left (15 a^2 e^2-20 a b d e+7 b^2 d^2\right )+8 b^2 c e^3 (2 b d-5 a e)-64 c^3 d^2 e (4 b d-5 a e)+3 b^4 e^4+128 c^4 d^4\right )}{256 c^{5/2} e^6}+\frac {\sqrt {a+b x+c x^2} \left (8 c^2 e^2 \left (16 a^2 e^2-39 a b d e+22 b^2 d^2\right )-2 c e x (2 c d-b e) \left (-4 c e (4 b d-7 a e)-3 b^2 e^2+16 c^2 d^2\right )-2 b^2 c e^3 (5 b d-14 a e)-32 c^3 d^2 e (9 b d-8 a e)-3 b^4 e^4+128 c^4 d^4\right )}{128 c^2 e^5}+\frac {\left (a e^2-b d e+c d^2\right )^{5/2} \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e^6}+\frac {\left (a+b x+c x^2\right )^{3/2} \left (-2 c e (11 b d-8 a e)+3 b^2 e^2-6 c e x (2 c d-b e)+16 c^2 d^2\right )}{48 c e^3}+\frac {\left (a+b x+c x^2\right )^{5/2}}{5 e} \]
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Rule 212
Rule 635
Rule 738
Rule 748
Rule 828
Rule 857
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a+b x+c x^2\right )^{5/2}}{5 e}-\frac {\int \frac {(b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx}{2 e} \\ & = \frac {\left (16 c^2 d^2+3 b^2 e^2-2 c e (11 b d-8 a e)-6 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c e^3}+\frac {\left (a+b x+c x^2\right )^{5/2}}{5 e}+\frac {\int \frac {\left (\frac {1}{2} \left (8 c e (b d-2 a e)^2+2 (2 c d-b e) \left (2 a c d e-b d \left (4 c d-\frac {3 b e}{2}\right )\right )\right )-\frac {1}{2} (2 c d-b e) \left (16 c^2 d^2-3 b^2 e^2-4 c e (4 b d-7 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{d+e x} \, dx}{16 c e^3} \\ & = \frac {\left (128 c^4 d^4-3 b^4 e^4-2 b^2 c e^3 (5 b d-14 a e)-32 c^3 d^2 e (9 b d-8 a e)+8 c^2 e^2 \left (22 b^2 d^2-39 a b d e+16 a^2 e^2\right )-2 c e (2 c d-b e) \left (16 c^2 d^2-3 b^2 e^2-4 c e (4 b d-7 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{128 c^2 e^5}+\frac {\left (16 c^2 d^2+3 b^2 e^2-2 c e (11 b d-8 a e)-6 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c e^3}+\frac {\left (a+b x+c x^2\right )^{5/2}}{5 e}-\frac {\int \frac {\frac {1}{4} \left (d (2 c d-b e) \left (4 b c d-b^2 e-4 a c e\right ) \left (16 c^2 d^2-3 b^2 e^2-4 c e (4 b d-7 a e)\right )+4 c e (b d-2 a e) \left (8 c e (b d-2 a e)^2-d (2 c d-b e) \left (8 b c d-3 b^2 e-4 a c e\right )\right )\right )+\frac {1}{4} (2 c d-b e) \left (128 c^4 d^4+3 b^4 e^4+8 b^2 c e^3 (2 b d-5 a e)-64 c^3 d^2 e (4 b d-5 a e)+16 c^2 e^2 \left (7 b^2 d^2-20 a b d e+15 a^2 e^2\right )\right ) x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{64 c^2 e^5} \\ & = \frac {\left (128 c^4 d^4-3 b^4 e^4-2 b^2 c e^3 (5 b d-14 a e)-32 c^3 d^2 e (9 b d-8 a e)+8 c^2 e^2 \left (22 b^2 d^2-39 a b d e+16 a^2 e^2\right )-2 c e (2 c d-b e) \left (16 c^2 d^2-3 b^2 e^2-4 c e (4 b d-7 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{128 c^2 e^5}+\frac {\left (16 c^2 d^2+3 b^2 e^2-2 c e (11 b d-8 a e)-6 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c e^3}+\frac {\left (a+b x+c x^2\right )^{5/2}}{5 e}+\frac {\left (c d^2-b d e+a e^2\right )^3 \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{e^6}-\frac {\left ((2 c d-b e) \left (128 c^4 d^4+3 b^4 e^4+8 b^2 c e^3 (2 b d-5 a e)-64 c^3 d^2 e (4 b d-5 a e)+16 c^2 e^2 \left (7 b^2 d^2-20 a b d e+15 a^2 e^2\right )\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{256 c^2 e^6} \\ & = \frac {\left (128 c^4 d^4-3 b^4 e^4-2 b^2 c e^3 (5 b d-14 a e)-32 c^3 d^2 e (9 b d-8 a e)+8 c^2 e^2 \left (22 b^2 d^2-39 a b d e+16 a^2 e^2\right )-2 c e (2 c d-b e) \left (16 c^2 d^2-3 b^2 e^2-4 c e (4 b d-7 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{128 c^2 e^5}+\frac {\left (16 c^2 d^2+3 b^2 e^2-2 c e (11 b d-8 a e)-6 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c e^3}+\frac {\left (a+b x+c x^2\right )^{5/2}}{5 e}-\frac {\left (2 \left (c d^2-b d e+a e^2\right )^3\right ) \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac {-b d+2 a e-(2 c d-b e) x}{\sqrt {a+b x+c x^2}}\right )}{e^6}-\frac {\left ((2 c d-b e) \left (128 c^4 d^4+3 b^4 e^4+8 b^2 c e^3 (2 b d-5 a e)-64 c^3 d^2 e (4 b d-5 a e)+16 c^2 e^2 \left (7 b^2 d^2-20 a b d e+15 a^2 e^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{128 c^2 e^6} \\ & = \frac {\left (128 c^4 d^4-3 b^4 e^4-2 b^2 c e^3 (5 b d-14 a e)-32 c^3 d^2 e (9 b d-8 a e)+8 c^2 e^2 \left (22 b^2 d^2-39 a b d e+16 a^2 e^2\right )-2 c e (2 c d-b e) \left (16 c^2 d^2-3 b^2 e^2-4 c e (4 b d-7 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{128 c^2 e^5}+\frac {\left (16 c^2 d^2+3 b^2 e^2-2 c e (11 b d-8 a e)-6 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{48 c e^3}+\frac {\left (a+b x+c x^2\right )^{5/2}}{5 e}-\frac {(2 c d-b e) \left (128 c^4 d^4+3 b^4 e^4+8 b^2 c e^3 (2 b d-5 a e)-64 c^3 d^2 e (4 b d-5 a e)+16 c^2 e^2 \left (7 b^2 d^2-20 a b d e+15 a^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{5/2} e^6}+\frac {\left (c d^2-b d e+a e^2\right )^{5/2} \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e^6} \\ \end{align*}
Time = 3.06 (sec) , antiderivative size = 443, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{d+e x} \, dx=\frac {\frac {e \sqrt {a+x (b+c x)} \left (-45 b^4 e^4+30 b^2 c e^3 (-5 b d+18 a e+b e x)+32 c^4 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )+4 c^2 e^2 \left (736 a^2 e^2+2 a b e (-695 d+311 e x)+b^2 \left (660 d^2-295 d e x+186 e^2 x^2\right )\right )-16 c^3 e \left (a e \left (-280 d^2+135 d e x-88 e^2 x^2\right )+b \left (270 d^3-130 d^2 e x+85 d e^2 x^2-63 e^3 x^3\right )\right )\right )}{c^2}+3840 \sqrt {-c d^2+e (b d-a e)} \left (c d^2+e (-b d+a e)\right )^2 \arctan \left (\frac {\sqrt {-c d^2+e (b d-a e)} x}{\sqrt {a} (d+e x)-d \sqrt {a+x (b+c x)}}\right )-\frac {15 (2 c d-b e) \left (128 c^4 d^4+3 b^4 e^4+8 b^2 c e^3 (2 b d-5 a e)-64 c^3 d^2 e (4 b d-5 a e)+16 c^2 e^2 \left (7 b^2 d^2-20 a b d e+15 a^2 e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{c^{5/2}}}{1920 e^6} \]
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Time = 0.36 (sec) , antiderivative size = 715, normalized size of antiderivative = 1.56
method | result | size |
risch | \(\frac {\left (384 c^{4} e^{4} x^{4}+1008 b \,c^{3} e^{4} x^{3}-480 c^{4} d \,e^{3} x^{3}+1408 a \,c^{3} e^{4} x^{2}+744 b^{2} c^{2} e^{4} x^{2}-1360 b \,c^{3} d \,e^{3} x^{2}+640 c^{4} d^{2} e^{2} x^{2}+2488 a b \,c^{2} e^{4} x -2160 a \,c^{3} d \,e^{3} x +30 b^{3} c \,e^{4} x -1180 b^{2} c^{2} d \,e^{3} x +2080 b \,c^{3} d^{2} e^{2} x -960 c^{4} d^{3} e x +2944 a^{2} c^{2} e^{4}+540 a \,b^{2} e^{4} c -5560 a b \,c^{2} d \,e^{3}+4480 a \,c^{3} d^{2} e^{2}-45 b^{4} e^{4}-150 b^{3} c d \,e^{3}+2640 b^{2} c^{2} d^{2} e^{2}-4320 b \,c^{3} d^{3} e +1920 c^{4} d^{4}\right ) \sqrt {c \,x^{2}+b x +a}}{1920 c^{2} e^{5}}+\frac {-\frac {256 \left (a^{3} e^{6}-3 d \,a^{2} b \,e^{5}+3 a^{2} c \,d^{2} e^{4}+3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 a \,c^{2} d^{4} e^{2}-b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 d^{5} b \,c^{2} e +c^{3} d^{6}\right ) c^{2} \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}+\frac {\left (240 a^{2} b \,c^{2} e^{5}-480 a^{2} c^{3} d \,e^{4}-40 a \,b^{3} c \,e^{5}-240 a \,b^{2} c^{2} d \,e^{4}+960 a b \,c^{3} d^{2} e^{3}-640 a \,c^{4} d^{3} e^{2}+3 b^{5} e^{5}+10 b^{4} c d \,e^{4}+80 b^{3} c^{2} d^{2} e^{3}-480 b^{2} c^{3} d^{3} e^{2}+640 b \,c^{4} d^{4} e -256 c^{5} d^{5}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{e \sqrt {c}}}{256 e^{5} c^{2}}\) | \(715\) |
default | \(\text {Expression too large to display}\) | \(1046\) |
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Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{d+e x} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{d+e x} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{d + e x}\, dx \]
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Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{d+e x} \, dx=\text {Exception raised: ValueError} \]
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Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{d+e x} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{d+e x} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{d+e\,x} \,d x \]
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