Integrand size = 31, antiderivative size = 429 \[ \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \left (a+b x+c x^2\right )} \, dx=\frac {4 c e \sqrt {f+g x}}{\sqrt {b^2-4 a c} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) (e f-d g) \sqrt {d+e x}}-\frac {4 c e \sqrt {f+g x}}{\sqrt {b^2-4 a c} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (e f-d g) \sqrt {d+e x}}-\frac {8 c^2 \text {arctanh}\left (\frac {\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \sqrt {f+g x}}\right )}{\sqrt {b^2-4 a c} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right )^{3/2} \sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}+\frac {8 c^2 \text {arctanh}\left (\frac {\sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \sqrt {f+g x}}\right )}{\sqrt {b^2-4 a c} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right )^{3/2} \sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \]
[Out]
Time = 0.68 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {925, 98, 95, 214} \[ \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \left (a+b x+c x^2\right )} \, dx=-\frac {8 c^2 \text {arctanh}\left (\frac {\sqrt {d+e x} \sqrt {2 c f-g \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {f+g x} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {b^2-4 a c} \left (2 c d-e \left (b-\sqrt {b^2-4 a c}\right )\right )^{3/2} \sqrt {2 c f-g \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {8 c^2 \text {arctanh}\left (\frac {\sqrt {d+e x} \sqrt {2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {f+g x} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {b^2-4 a c} \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )^{3/2} \sqrt {2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {4 c e \sqrt {f+g x}}{\sqrt {b^2-4 a c} \sqrt {d+e x} (e f-d g) \left (2 c d-e \left (b-\sqrt {b^2-4 a c}\right )\right )}-\frac {4 c e \sqrt {f+g x}}{\sqrt {b^2-4 a c} \sqrt {d+e x} (e f-d g) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )} \]
[In]
[Out]
Rule 95
Rule 98
Rule 214
Rule 925
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2 c}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}+2 c x\right ) (d+e x)^{3/2} \sqrt {f+g x}}-\frac {2 c}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}+2 c x\right ) (d+e x)^{3/2} \sqrt {f+g x}}\right ) \, dx \\ & = \frac {(2 c) \int \frac {1}{\left (b-\sqrt {b^2-4 a c}+2 c x\right ) (d+e x)^{3/2} \sqrt {f+g x}} \, dx}{\sqrt {b^2-4 a c}}-\frac {(2 c) \int \frac {1}{\left (b+\sqrt {b^2-4 a c}+2 c x\right ) (d+e x)^{3/2} \sqrt {f+g x}} \, dx}{\sqrt {b^2-4 a c}} \\ & = \frac {4 c e \sqrt {f+g x}}{\sqrt {b^2-4 a c} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) (e f-d g) \sqrt {d+e x}}-\frac {4 c e \sqrt {f+g x}}{\sqrt {b^2-4 a c} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (e f-d g) \sqrt {d+e x}}+\frac {\left (4 c^2\right ) \int \frac {1}{\left (b-\sqrt {b^2-4 a c}+2 c x\right ) \sqrt {d+e x} \sqrt {f+g x}} \, dx}{\sqrt {b^2-4 a c} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right )}-\frac {\left (4 c^2\right ) \int \frac {1}{\left (b+\sqrt {b^2-4 a c}+2 c x\right ) \sqrt {d+e x} \sqrt {f+g x}} \, dx}{\sqrt {b^2-4 a c} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right )} \\ & = \frac {4 c e \sqrt {f+g x}}{\sqrt {b^2-4 a c} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) (e f-d g) \sqrt {d+e x}}-\frac {4 c e \sqrt {f+g x}}{\sqrt {b^2-4 a c} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (e f-d g) \sqrt {d+e x}}+\frac {\left (8 c^2\right ) \text {Subst}\left (\int \frac {1}{-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e-\left (-2 c f+\left (b-\sqrt {b^2-4 a c}\right ) g\right ) x^2} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{\sqrt {b^2-4 a c} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right )}-\frac {\left (8 c^2\right ) \text {Subst}\left (\int \frac {1}{-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e-\left (-2 c f+\left (b+\sqrt {b^2-4 a c}\right ) g\right ) x^2} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{\sqrt {b^2-4 a c} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right )} \\ & = \frac {4 c e \sqrt {f+g x}}{\sqrt {b^2-4 a c} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) (e f-d g) \sqrt {d+e x}}-\frac {4 c e \sqrt {f+g x}}{\sqrt {b^2-4 a c} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) (e f-d g) \sqrt {d+e x}}-\frac {8 c^2 \tanh ^{-1}\left (\frac {\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \sqrt {f+g x}}\right )}{\sqrt {b^2-4 a c} \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right )^{3/2} \sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}+\frac {8 c^2 \tanh ^{-1}\left (\frac {\sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \sqrt {f+g x}}\right )}{\sqrt {b^2-4 a c} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right )^{3/2} \sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \\ \end{align*}
Time = 3.47 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.27 \[ \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \left (a+b x+c x^2\right )} \, dx=\frac {2 e^2 \sqrt {f+g x}}{\left (c d^2+e (-b d+a e)\right ) (-e f+d g) \sqrt {d+e x}}+\frac {\sqrt {2} \left (2 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (b d+\sqrt {b^2-4 a c} d+a e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c d^2-b d e+a e^2} \sqrt {f+g x}}{\sqrt {-2 c d f+b e f+\sqrt {b^2-4 a c} e f+b d g-\sqrt {b^2-4 a c} d g-2 a e g} \sqrt {d+e x}}\right )}{\sqrt {b^2-4 a c} \left (c d^2+e (-b d+a e)\right )^{3/2} \sqrt {-2 c d f+b e f+\sqrt {b^2-4 a c} e f+b d g-\sqrt {b^2-4 a c} d g-2 a e g}}+\frac {\sqrt {2} \left (-2 c^2 d^2+b \left (-b+\sqrt {b^2-4 a c}\right ) e^2+2 c e \left (b d-\sqrt {b^2-4 a c} d+a e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c d^2-b d e+a e^2} \sqrt {f+g x}}{\sqrt {-2 c d f+b e f-\sqrt {b^2-4 a c} e f+b d g+\sqrt {b^2-4 a c} d g-2 a e g} \sqrt {d+e x}}\right )}{\sqrt {b^2-4 a c} \left (c d^2+e (-b d+a e)\right )^{3/2} \sqrt {-2 c d f+b e f-\sqrt {b^2-4 a c} e f+b d g+\sqrt {b^2-4 a c} d g-2 a e g}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(47350\) vs. \(2(369)=738\).
Time = 0.68 (sec) , antiderivative size = 47351, normalized size of antiderivative = 110.38
[In]
[Out]
Timed out. \[ \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \left (a+b x+c x^2\right )} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \left (a+b x+c x^2\right )} \, dx=\int \frac {1}{\left (d + e x\right )^{\frac {3}{2}} \sqrt {f + g x} \left (a + b x + c x^{2}\right )}\, dx \]
[In]
[Out]
\[ \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \left (a+b x+c x^2\right )} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )} {\left (e x + d\right )}^{\frac {3}{2}} \sqrt {g x + f}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \left (a+b x+c x^2\right )} \, dx=\text {Timed out} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \left (a+b x+c x^2\right )} \, dx=\int \frac {1}{\sqrt {f+g\,x}\,{\left (d+e\,x\right )}^{3/2}\,\left (c\,x^2+b\,x+a\right )} \,d x \]
[In]
[Out]