Integrand size = 31, antiderivative size = 553 \[ \int \frac {1}{(d+e x) (f+g x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\frac {2 g^2 \sqrt {a+b x+c x^2}}{(e f-d g) \left (c f^2-b f g+a g^2\right ) \sqrt {f+g x}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} g \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{(e f-d g) \left (c f^2-b f g+a g^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} e \sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \operatorname {EllipticPi}\left (\frac {e \left (2 c f-b g+\sqrt {b^2-4 a c} g\right )}{2 c (e f-d g)},\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {f+g x}}{\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}\right ),\frac {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}}{b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}}\right )}{\sqrt {c} (e f-d g)^2 \sqrt {a+b x+c x^2}} \]
[Out]
Time = 0.97 (sec) , antiderivative size = 553, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {971, 758, 21, 732, 435, 948, 175, 552, 551} \[ \int \frac {1}{(d+e x) (f+g x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=-\frac {\sqrt {2} g \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{\sqrt {a+b x+c x^2} (e f-d g) \left (a g^2-b f g+c f^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {\sqrt {2} e \sqrt {2 c f-g \left (b-\sqrt {b^2-4 a c}\right )} \sqrt {1-\frac {2 c (f+g x)}{2 c f-g \left (b-\sqrt {b^2-4 a c}\right )}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticPi}\left (\frac {e \left (2 c f-b g+\sqrt {b^2-4 a c} g\right )}{2 c (e f-d g)},\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {f+g x}}{\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}\right ),\frac {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}}{b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}}\right )}{\sqrt {c} \sqrt {a+b x+c x^2} (e f-d g)^2}+\frac {2 g^2 \sqrt {a+b x+c x^2}}{\sqrt {f+g x} (e f-d g) \left (a g^2-b f g+c f^2\right )} \]
[In]
[Out]
Rule 21
Rule 175
Rule 435
Rule 551
Rule 552
Rule 732
Rule 758
Rule 948
Rule 971
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {g}{(e f-d g) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}+\frac {e}{(e f-d g) (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}\right ) \, dx \\ & = \frac {e \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{e f-d g}-\frac {g \int \frac {1}{(f+g x)^{3/2} \sqrt {a+b x+c x^2}} \, dx}{e f-d g} \\ & = \frac {2 g^2 \sqrt {a+b x+c x^2}}{(e f-d g) \left (c f^2-b f g+a g^2\right ) \sqrt {f+g x}}+\frac {(2 g) \int \frac {-\frac {c f}{2}-\frac {c g x}{2}}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{(e f-d g) \left (c f^2-b f g+a g^2\right )}+\frac {\left (e \sqrt {b-\sqrt {b^2-4 a c}+2 c x} \sqrt {b+\sqrt {b^2-4 a c}+2 c x}\right ) \int \frac {1}{\sqrt {b-\sqrt {b^2-4 a c}+2 c x} \sqrt {b+\sqrt {b^2-4 a c}+2 c x} (d+e x) \sqrt {f+g x}} \, dx}{(e f-d g) \sqrt {a+b x+c x^2}} \\ & = \frac {2 g^2 \sqrt {a+b x+c x^2}}{(e f-d g) \left (c f^2-b f g+a g^2\right ) \sqrt {f+g x}}-\frac {(c g) \int \frac {\sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx}{(e f-d g) \left (c f^2-b f g+a g^2\right )}-\frac {\left (2 e \sqrt {b-\sqrt {b^2-4 a c}+2 c x} \sqrt {b+\sqrt {b^2-4 a c}+2 c x}\right ) \text {Subst}\left (\int \frac {1}{\left (e f-d g-e x^2\right ) \sqrt {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}+\frac {2 c x^2}{g}} \sqrt {b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}+\frac {2 c x^2}{g}}} \, dx,x,\sqrt {f+g x}\right )}{(e f-d g) \sqrt {a+b x+c x^2}} \\ & = \frac {2 g^2 \sqrt {a+b x+c x^2}}{(e f-d g) \left (c f^2-b f g+a g^2\right ) \sqrt {f+g x}}-\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} g \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {b^2-4 a c} g x^2}{2 c f-b g-\sqrt {b^2-4 a c} g}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{(e f-d g) \left (c f^2-b f g+a g^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-b g-\sqrt {b^2-4 a c} g}} \sqrt {a+b x+c x^2}}-\frac {\left (2 e \sqrt {b+\sqrt {b^2-4 a c}+2 c x} \sqrt {1+\frac {2 c (f+g x)}{\left (b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}}\right ) \text {Subst}\left (\int \frac {1}{\left (e f-d g-e x^2\right ) \sqrt {b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}+\frac {2 c x^2}{g}} \sqrt {1+\frac {2 c x^2}{\left (b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}}} \, dx,x,\sqrt {f+g x}\right )}{(e f-d g) \sqrt {a+b x+c x^2}} \\ & = \frac {2 g^2 \sqrt {a+b x+c x^2}}{(e f-d g) \left (c f^2-b f g+a g^2\right ) \sqrt {f+g x}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} g \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{(e f-d g) \left (c f^2-b f g+a g^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}-\frac {\left (2 e \sqrt {1+\frac {2 c (f+g x)}{\left (b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}} \sqrt {1+\frac {2 c (f+g x)}{\left (b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}}\right ) \text {Subst}\left (\int \frac {1}{\left (e f-d g-e x^2\right ) \sqrt {1+\frac {2 c x^2}{\left (b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}} \sqrt {1+\frac {2 c x^2}{\left (b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}\right ) g}}} \, dx,x,\sqrt {f+g x}\right )}{(e f-d g) \sqrt {a+b x+c x^2}} \\ & = \frac {2 g^2 \sqrt {a+b x+c x^2}}{(e f-d g) \left (c f^2-b f g+a g^2\right ) \sqrt {f+g x}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} g \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{(e f-d g) \left (c f^2-b f g+a g^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} e \sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}} \sqrt {1-\frac {2 c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \Pi \left (\frac {e \left (2 c f-b g+\sqrt {b^2-4 a c} g\right )}{2 c (e f-d g)};\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {f+g x}}{\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g}}\right )|\frac {b-\sqrt {b^2-4 a c}-\frac {2 c f}{g}}{b+\sqrt {b^2-4 a c}-\frac {2 c f}{g}}\right )}{\sqrt {c} (e f-d g)^2 \sqrt {a+b x+c x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 26.22 (sec) , antiderivative size = 950, normalized size of antiderivative = 1.72 \[ \int \frac {1}{(d+e x) (f+g x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\frac {2 \left (\frac {g^2 (a+x (b+c x))}{e f-d g}+\frac {(f+g x)^2 \left (c+\frac {c f^2}{(f+g x)^2}-\frac {b f g}{(f+g x)^2}+\frac {a g^2}{(f+g x)^2}-\frac {2 c f}{f+g x}+\frac {b g}{f+g x}-\frac {i \sqrt {1-\frac {2 \left (c f^2+g (-b f+a g)\right )}{\left (2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} \sqrt {2+\frac {4 \left (c f^2+g (-b f+a g)\right )}{\left (-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} \left ((e f-d g) \left (2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) \left (E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c f^2-b f g+a g^2}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}}}{\sqrt {f+g x}}\right )|-\frac {-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c f^2-b f g+a g^2}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}}}{\sqrt {f+g x}}\right ),-\frac {-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}}\right )\right )-2 e \left (c f^2+g (-b f+a g)\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c f^2-b f g+a g^2}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}}}{\sqrt {f+g x}}\right ),-\frac {-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}}\right )+2 e \left (c f^2+g (-b f+a g)\right ) \operatorname {EllipticPi}\left (\frac {(e f-d g) \left (2 c f-b g-\sqrt {\left (b^2-4 a c\right ) g^2}\right )}{2 e \left (c f^2+g (-b f+a g)\right )},i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c f^2-b f g+a g^2}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}}}{\sqrt {f+g x}}\right ),-\frac {-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}}\right )\right )}{4 (e f-d g) \sqrt {\frac {c f^2+g (-b f+a g)}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}} \sqrt {f+g x}}\right )}{-e f+d g}\right )}{\left (c f^2+g (-b f+a g)\right ) \sqrt {f+g x} \sqrt {a+x (b+c x)}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(1263\) vs. \(2(490)=980\).
Time = 3.23 (sec) , antiderivative size = 1264, normalized size of antiderivative = 2.29
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1264\) |
default | \(\text {Expression too large to display}\) | \(4757\) |
[In]
[Out]
Timed out. \[ \int \frac {1}{(d+e x) (f+g x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {1}{(d+e x) (f+g x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{\left (d + e x\right ) \left (f + g x\right )^{\frac {3}{2}} \sqrt {a + b x + c x^{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{(d+e x) (f+g x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )} {\left (g x + f\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{(d+e x) (f+g x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )} {\left (g x + f\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(d+e x) (f+g x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{{\left (f+g\,x\right )}^{3/2}\,\left (d+e\,x\right )\,\sqrt {c\,x^2+b\,x+a}} \,d x \]
[In]
[Out]