Integrand size = 22, antiderivative size = 129 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 (d+e x) (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 e (2 c d-b e) \sqrt {a+b x+c x^2}}{c \left (b^2-4 a c\right )}+\frac {e^2 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2}} \]
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Time = 0.06 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {752, 654, 635, 212} \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {e^2 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2}}+\frac {2 e \sqrt {a+b x+c x^2} (2 c d-b e)}{c \left (b^2-4 a c\right )}-\frac {2 (d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \]
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Rule 212
Rule 635
Rule 654
Rule 752
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (d+e x) (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 {-e (b d-2 a e)-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) (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) \sqrt {a+b x+c x^2}}{c \left (b^2-4 a c\right )}+\frac {e^2 \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{c} \\ & = -\frac {2 (d+e x) (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) \sqrt {a+b x+c x^2}}{c \left (b^2-4 a c\right )}+\frac {\left (2 e^2\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 )}{c} \\ & = -\frac {2 (d+e x) (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) \sqrt {a+b x+c x^2}}{c \left (b^2-4 a c\right )}+\frac {e^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2}} \\ \end{align*}
Time = 0.67 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.91 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (a b e^2+2 c^2 d^2 x+b^2 e^2 x+b c d (d-2 e x)-2 a c e (2 d+e x)\right )}{c \left (-b^2+4 a c\right ) \sqrt {a+x (b+c x)}}+\frac {2 e^2 \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{c^{3/2}} \]
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Time = 0.38 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.60
method | result | size |
default | \(\frac {2 d^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+e^{2} \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )+2 d e \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )\) | \(207\) |
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Time = 0.39 (sec) , antiderivative size = 461, normalized size of antiderivative = 3.57 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\left [\frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} e^{2} x^{2} + {\left (b^{3} - 4 \, a b c\right )} e^{2} x + {\left (a b^{2} - 4 \, a^{2} c\right )} e^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (b c^{2} d^{2} - 4 \, a c^{2} d e + a b c e^{2} + {\left (2 \, c^{3} d^{2} - 2 \, b c^{2} d e + {\left (b^{2} c - 2 \, a c^{2}\right )} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{2 \, {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )}}, -\frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} e^{2} x^{2} + {\left (b^{3} - 4 \, a b c\right )} e^{2} x + {\left (a b^{2} - 4 \, a^{2} c\right )} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (b c^{2} d^{2} - 4 \, a c^{2} d e + a b c e^{2} + {\left (2 \, c^{3} d^{2} - 2 \, b c^{2} d e + {\left (b^{2} c - 2 \, a c^{2}\right )} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{a b^{2} c^{2} - 4 \, a^{2} c^{3} + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x}\right ] \]
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\[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{2}}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.29 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.02 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {e^{2} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{c^{\frac {3}{2}}} - \frac {2 \, {\left (\frac {{\left (2 \, c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2} - 2 \, a c e^{2}\right )} x}{b^{2} c - 4 \, a c^{2}} + \frac {b c d^{2} - 4 \, a c d e + a b e^{2}}{b^{2} c - 4 \, a c^{2}}\right )}}{\sqrt {c x^{2} + b x + a}} \]
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Time = 10.43 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.16 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {e^2\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )}{c^{3/2}}+\frac {d^2\,\left (\frac {b}{2}+c\,x\right )}{\left (a\,c-\frac {b^2}{4}\right )\,\sqrt {c\,x^2+b\,x+a}}-\frac {2\,d\,e\,\left (4\,a+2\,b\,x\right )}{\left (4\,a\,c-b^2\right )\,\sqrt {c\,x^2+b\,x+a}}+\frac {e^2\,\left (\frac {a\,b}{2}-x\,\left (a\,c-\frac {b^2}{2}\right )\right )}{c\,\left (a\,c-\frac {b^2}{4}\right )\,\sqrt {c\,x^2+b\,x+a}} \]
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