Integrand size = 22, antiderivative size = 286 \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 (d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 e (2 c d-b e) (d+e x)^2 \sqrt {a+b x+c x^2}}{c \left (b^2-4 a c\right )}+\frac {e \left (32 c^3 d^3-15 b^3 e^3+4 b c e^2 (12 b d+13 a e)-8 c^2 d e (5 b d+16 a e)+2 c e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{4 c^3 \left (b^2-4 a c\right )}+\frac {3 e^2 \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{7/2}} \]
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Time = 0.21 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {752, 846, 793, 635, 212} \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {3 e^2 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right )}{8 c^{7/2}}-\frac {2 (d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 e (d+e x)^2 \sqrt {a+b x+c x^2} (2 c d-b e)}{c \left (b^2-4 a c\right )}+\frac {e \sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )-8 c^2 d e (16 a e+5 b d)+4 b c e^2 (13 a e+12 b d)-15 b^3 e^3+32 c^3 d^3\right )}{4 c^3 \left (b^2-4 a c\right )} \]
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Rule 212
Rule 635
Rule 752
Rule 793
Rule 846
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 \int \frac {(d+e x)^2 (-3 e (b d-2 a e)-3 e (2 c d-b e) x)}{\sqrt {a+b x+c x^2}} \, dx}{b^2-4 a c} \\ & = -\frac {2 (d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 e (2 c d-b e) (d+e x)^2 \sqrt {a+b x+c x^2}}{c \left (b^2-4 a c\right )}-\frac {2 \int \frac {(d+e x) \left (-\frac {3}{2} e \left (b^2 d e-20 a c d e+4 b \left (c d^2+a e^2\right )\right )-\frac {3}{2} e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{3 c \left (b^2-4 a c\right )} \\ & = -\frac {2 (d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 e (2 c d-b e) (d+e x)^2 \sqrt {a+b x+c x^2}}{c \left (b^2-4 a c\right )}+\frac {e \left (32 c^3 d^3-15 b^3 e^3+4 b c e^2 (12 b d+13 a e)-8 c^2 d e (5 b d+16 a e)+2 c e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{4 c^3 \left (b^2-4 a c\right )}+\frac {\left (3 e^2 \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{8 c^3} \\ & = -\frac {2 (d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 e (2 c d-b e) (d+e x)^2 \sqrt {a+b x+c x^2}}{c \left (b^2-4 a c\right )}+\frac {e \left (32 c^3 d^3-15 b^3 e^3+4 b c e^2 (12 b d+13 a e)-8 c^2 d e (5 b d+16 a e)+2 c e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{4 c^3 \left (b^2-4 a c\right )}+\frac {\left (3 e^2 \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\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 )}{4 c^3} \\ & = -\frac {2 (d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 e (2 c d-b e) (d+e x)^2 \sqrt {a+b x+c x^2}}{c \left (b^2-4 a c\right )}+\frac {e \left (32 c^3 d^3-15 b^3 e^3+4 b c e^2 (12 b d+13 a e)-8 c^2 d e (5 b d+16 a e)+2 c e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{4 c^3 \left (b^2-4 a c\right )}+\frac {3 e^2 \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{7/2}} \\ \end{align*}
Time = 1.51 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.11 \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {\sqrt {c} \left (15 b^4 e^4 x+b^3 e^3 (15 a e+c x (-48 d+5 e x))+4 b c \left (-13 a^2 e^4+2 c^2 d^3 (d-4 e x)+a c e^2 \left (12 d^2+40 d e x-5 e^2 x^2\right )\right )-2 b^2 c e^2 \left (a e (24 d+31 e x)+c x \left (-24 d^2+8 d e x+e^2 x^2\right )\right )+8 c^2 \left (2 c^2 d^4 x+a^2 e^3 (16 d+3 e x)+a c e \left (-8 d^3-12 d^2 e x+8 d e^2 x^2+e^3 x^3\right )\right )\right )-3 \left (b^2-4 a c\right ) e^2 \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) \sqrt {a+x (b+c x)} \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{4 c^{7/2} \left (-b^2+4 a c\right ) \sqrt {a+x (b+c x)}} \]
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Time = 0.50 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.56
method | result | size |
risch | \(-\frac {e^{3} \left (-2 c e x +7 b e -16 c d \right ) \sqrt {c \,x^{2}+b x +a}}{4 c^{3}}-\frac {-\frac {16 c^{3} d^{4} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {8 a^{2} c \,e^{4} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {14 a \,b^{2} e^{4} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {32 a b c d \,e^{3} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\left (12 a \,c^{2} e^{4}-15 b^{2} c \,e^{4}+48 b \,c^{2} d \,e^{3}-48 c^{3} d^{2} e^{2}\right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )+\left (-4 a b c \,e^{4}+32 a \,c^{2} d \,e^{3}-7 b^{3} e^{4}+16 b^{2} c d \,e^{3}-32 c^{3} d^{3} e \right ) \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{8 c^{3}}\) | \(447\) |
default | \(\frac {2 d^{4} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+e^{4} \left (\frac {x^{3}}{2 c \sqrt {c \,x^{2}+b x +a}}-\frac {5 b \left (\frac {x^{2}}{c \sqrt {c \,x^{2}+b x +a}}-\frac {3 b \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}-\frac {2 a \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{c}\right )}{4 c}-\frac {3 a \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}\right )+4 d \,e^{3} \left (\frac {x^{2}}{c \sqrt {c \,x^{2}+b x +a}}-\frac {3 b \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}-\frac {2 a \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{c}\right )+6 d^{2} e^{2} \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )+4 d^{3} e \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )\) | \(752\) |
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Leaf count of result is larger than twice the leaf count of optimal. 585 vs. \(2 (268) = 536\).
Time = 0.48 (sec) , antiderivative size = 1173, normalized size of antiderivative = 4.10 \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{4}}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.30 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.38 \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (\frac {2 \, {\left (b^{2} c^{2} e^{4} - 4 \, a c^{3} e^{4}\right )} x}{b^{2} c^{3} - 4 \, a c^{4}} + \frac {16 \, b^{2} c^{2} d e^{3} - 64 \, a c^{3} d e^{3} - 5 \, b^{3} c e^{4} + 20 \, a b c^{2} e^{4}}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x - \frac {16 \, c^{4} d^{4} - 32 \, b c^{3} d^{3} e + 48 \, b^{2} c^{2} d^{2} e^{2} - 96 \, a c^{3} d^{2} e^{2} - 48 \, b^{3} c d e^{3} + 160 \, a b c^{2} d e^{3} + 15 \, b^{4} e^{4} - 62 \, a b^{2} c e^{4} + 24 \, a^{2} c^{2} e^{4}}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x - \frac {8 \, b c^{3} d^{4} - 64 \, a c^{3} d^{3} e + 48 \, a b c^{2} d^{2} e^{2} - 48 \, a b^{2} c d e^{3} + 128 \, a^{2} c^{2} d e^{3} + 15 \, a b^{3} e^{4} - 52 \, a^{2} b c e^{4}}{b^{2} c^{3} - 4 \, a c^{4}}}{4 \, \sqrt {c x^{2} + b x + a}} - \frac {3 \, {\left (16 \, c^{2} d^{2} e^{2} - 16 \, b c d e^{3} + 5 \, b^{2} e^{4} - 4 \, a c e^{4}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{8 \, c^{\frac {7}{2}}} \]
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Timed out. \[ \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^4}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]
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