Integrand size = 25, antiderivative size = 126 \[ \int \frac {(A+B x) (d+e x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {B e \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2}} \]
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Time = 0.04 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {791, 635, 212} \[ \int \frac {(A+B x) (d+e x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {B e \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2}} \]
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Rule 212
Rule 635
Rule 791
Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {(B e) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{c} \\ & = \frac {2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {(2 B e) \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 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {B e \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 = 0.58 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.02 \[ \int \frac {(A+B x) (d+e x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {\frac {2 \sqrt {c} (A c (-2 a e+2 c d x+b (d-e x))+B (a b e+b (-c d+b e) x-2 a c (d+e x)))}{\sqrt {a+x (b+c x)}}-B \left (b^2-4 a c\right ) e \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{c^{3/2} \left (-b^2+4 a c\right )} \]
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Time = 0.51 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.66
method | result | size |
default | \(\frac {2 d A \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+B e \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )+\left (A e +B d \right ) \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )\) | \(209\) |
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Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (116) = 232\).
Time = 1.00 (sec) , antiderivative size = 489, normalized size of antiderivative = 3.88 \[ \int \frac {(A+B x) (d+e x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\left [\frac {{\left ({\left (B b^{2} c - 4 \, B a c^{2}\right )} e x^{2} + {\left (B b^{3} - 4 \, B a b c\right )} e x + {\left (B a b^{2} - 4 \, B a^{2} c\right )} e\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left ({\left (2 \, B a - A b\right )} c^{2} d - {\left (B a b c - 2 \, A a c^{2}\right )} e + {\left ({\left (B b c^{2} - 2 \, A c^{3}\right )} d - {\left (B b^{2} c - {\left (2 \, B a + A b\right )} c^{2}\right )} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{2 \, {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )}}, -\frac {{\left ({\left (B b^{2} c - 4 \, B a c^{2}\right )} e x^{2} + {\left (B b^{3} - 4 \, B a b c\right )} e x + {\left (B a b^{2} - 4 \, B a^{2} c\right )} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left ({\left (2 \, B a - A b\right )} c^{2} d - {\left (B a b c - 2 \, A a c^{2}\right )} e + {\left ({\left (B b c^{2} - 2 \, A c^{3}\right )} d - {\left (B b^{2} c - {\left (2 \, B a + A b\right )} c^{2}\right )} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{a b^{2} c^{2} - 4 \, a^{2} c^{3} + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x}\right ] \]
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\[ \int \frac {(A+B x) (d+e x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (A + B x\right ) \left (d + e x\right )}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {(A+B x) (d+e x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.30 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.10 \[ \int \frac {(A+B x) (d+e x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {B e \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{c^{\frac {3}{2}}} + \frac {2 \, {\left (\frac {{\left (B b c d - 2 \, A c^{2} d - B b^{2} e + 2 \, B a c e + A b c e\right )} x}{b^{2} c - 4 \, a c^{2}} + \frac {2 \, B a c d - A b c d - B a b e + 2 \, A a c e}{b^{2} c - 4 \, a c^{2}}\right )}}{\sqrt {c x^{2} + b x + a}} \]
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Time = 12.23 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.29 \[ \int \frac {(A+B x) (d+e x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {B\,e\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )}{c^{3/2}}-\frac {4\,A\,a\,e-2\,A\,b\,d+2\,A\,b\,e\,x-4\,A\,c\,d\,x}{\left (4\,a\,c-b^2\right )\,\sqrt {c\,x^2+b\,x+a}}-\frac {B\,d\,\left (4\,a+2\,b\,x\right )}{\left (4\,a\,c-b^2\right )\,\sqrt {c\,x^2+b\,x+a}}+\frac {B\,e\,\left (\frac {a\,b}{2}-x\,\left (a\,c-\frac {b^2}{2}\right )\right )}{c\,\left (a\,c-\frac {b^2}{4}\right )\,\sqrt {c\,x^2+b\,x+a}} \]
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