Integrand size = 36, antiderivative size = 137 \[ \int \frac {c i+d i x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 \sqrt {-f g+e h} i \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\arcsin \left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {-f g+e h}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right )}{f \sqrt {h} \sqrt {-\frac {f (c+d x)}{d e-c f}} \sqrt {g+h x}} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {21, 115, 114} \[ \int \frac {c i+d i x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 i \sqrt {c+d x} \sqrt {e h-f g} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\arcsin \left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {e h-f g}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right )}{f \sqrt {h} \sqrt {g+h x} \sqrt {-\frac {f (c+d x)}{d e-c f}}} \]
[In]
[Out]
Rule 21
Rule 114
Rule 115
Rubi steps \begin{align*} \text {integral}& = i \int \frac {\sqrt {c+d x}}{\sqrt {e+f x} \sqrt {g+h x}} \, dx \\ & = \frac {\left (i \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}}\right ) \int \frac {\sqrt {\frac {c f}{-d e+c f}+\frac {d f x}{-d e+c f}}}{\sqrt {e+f x} \sqrt {\frac {f g}{f g-e h}+\frac {f h x}{f g-e h}}} \, dx}{\sqrt {\frac {f (c+d x)}{-d e+c f}} \sqrt {g+h x}} \\ & = \frac {2 \sqrt {-f g+e h} i \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {-f g+e h}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right )}{f \sqrt {h} \sqrt {-\frac {f (c+d x)}{d e-c f}} \sqrt {g+h x}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.74 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.31 \[ \int \frac {c i+d i x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 i i \sqrt {c+d x} \sqrt {g+h x} \left (E\left (i \text {arcsinh}\left (\sqrt {\frac {f (c+d x)}{d e-c f}}\right )|\frac {d e h-c f h}{d f g-c f h}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {f (c+d x)}{d e-c f}}\right ),\frac {d e h-c f h}{d f g-c f h}\right )\right )}{h \sqrt {\frac {f (c+d x)}{d (e+f x)}} \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}} \]
[In]
[Out]
Time = 1.63 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.53
method | result | size |
default | \(-\frac {2 i \left (c e \,h^{2}-c f g h -d e g h +d f \,g^{2}\right ) E\left (\sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}, \sqrt {\frac {\left (e h -f g \right ) d}{f \left (c h -d g \right )}}\right ) \sqrt {\frac {\left (f x +e \right ) h}{e h -f g}}\, \sqrt {\frac {\left (d x +c \right ) h}{c h -d g}}\, \sqrt {-\frac {\left (h x +g \right ) f}{e h -f g}}\, \sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}{h^{2} f \left (d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g \right )}\) | \(210\) |
elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (f x +e \right ) \left (h x +g \right )}\, \left (\frac {2 c i \left (\frac {g}{h}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, F\left (\sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{\sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}+\frac {2 d i \left (\frac {g}{h}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, \left (\left (-\frac {g}{h}+\frac {c}{d}\right ) E\left (\sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )-\frac {c F\left (\sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{d}\right )}{\sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}\right )}{\sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}\) | \(500\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 664, normalized size of antiderivative = 4.85 \[ \int \frac {c i+d i x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {d f h} d f h i {\rm weierstrassZeta}\left (\frac {4 \, {\left (d^{2} f^{2} g^{2} - {\left (d^{2} e f + c d f^{2}\right )} g h + {\left (d^{2} e^{2} - c d e f + c^{2} f^{2}\right )} h^{2}\right )}}{3 \, d^{2} f^{2} h^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} g^{3} - 3 \, {\left (d^{3} e f^{2} + c d^{2} f^{3}\right )} g^{2} h - 3 \, {\left (d^{3} e^{2} f - 4 \, c d^{2} e f^{2} + c^{2} d f^{3}\right )} g h^{2} + {\left (2 \, d^{3} e^{3} - 3 \, c d^{2} e^{2} f - 3 \, c^{2} d e f^{2} + 2 \, c^{3} f^{3}\right )} h^{3}\right )}}{27 \, d^{3} f^{3} h^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} f^{2} g^{2} - {\left (d^{2} e f + c d f^{2}\right )} g h + {\left (d^{2} e^{2} - c d e f + c^{2} f^{2}\right )} h^{2}\right )}}{3 \, d^{2} f^{2} h^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} g^{3} - 3 \, {\left (d^{3} e f^{2} + c d^{2} f^{3}\right )} g^{2} h - 3 \, {\left (d^{3} e^{2} f - 4 \, c d^{2} e f^{2} + c^{2} d f^{3}\right )} g h^{2} + {\left (2 \, d^{3} e^{3} - 3 \, c d^{2} e^{2} f - 3 \, c^{2} d e f^{2} + 2 \, c^{3} f^{3}\right )} h^{3}\right )}}{27 \, d^{3} f^{3} h^{3}}, \frac {3 \, d f h x + d f g + {\left (d e + c f\right )} h}{3 \, d f h}\right )\right ) + {\left (d f g + {\left (d e - 2 \, c f\right )} h\right )} \sqrt {d f h} i {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} f^{2} g^{2} - {\left (d^{2} e f + c d f^{2}\right )} g h + {\left (d^{2} e^{2} - c d e f + c^{2} f^{2}\right )} h^{2}\right )}}{3 \, d^{2} f^{2} h^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} g^{3} - 3 \, {\left (d^{3} e f^{2} + c d^{2} f^{3}\right )} g^{2} h - 3 \, {\left (d^{3} e^{2} f - 4 \, c d^{2} e f^{2} + c^{2} d f^{3}\right )} g h^{2} + {\left (2 \, d^{3} e^{3} - 3 \, c d^{2} e^{2} f - 3 \, c^{2} d e f^{2} + 2 \, c^{3} f^{3}\right )} h^{3}\right )}}{27 \, d^{3} f^{3} h^{3}}, \frac {3 \, d f h x + d f g + {\left (d e + c f\right )} h}{3 \, d f h}\right )\right )}}{3 \, d f^{2} h^{2}} \]
[In]
[Out]
\[ \int \frac {c i+d i x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=i \int \frac {\sqrt {c + d x}}{\sqrt {e + f x} \sqrt {g + h x}}\, dx \]
[In]
[Out]
\[ \int \frac {c i+d i x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {d i x + c i}{\sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
[In]
[Out]
\[ \int \frac {c i+d i x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {d i x + c i}{\sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {c i+d i x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {c\,i+d\,i\,x}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\sqrt {c+d\,x}} \,d x \]
[In]
[Out]