Optimal. Leaf size=137 \[ \frac {2 i \sqrt {c+d x} \sqrt {e h-f g} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {e h-f g}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right )}{f \sqrt {h} \sqrt {g+h x} \sqrt {-\frac {f (c+d x)}{d e-c f}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {21, 114, 113} \[ \frac {2 i \sqrt {c+d x} \sqrt {e h-f g} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {e h-f g}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right )}{f \sqrt {h} \sqrt {g+h x} \sqrt {-\frac {f (c+d x)}{d e-c f}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 21
Rule 113
Rule 114
Rubi steps
\begin {align*} \int \frac {68 c+68 d x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx &=68 \int \frac {\sqrt {c+d x}}{\sqrt {e+f x} \sqrt {g+h x}} \, dx\\ &=\frac {\left (68 \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 {136 \sqrt {-f g+e h} \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*}
________________________________________________________________________________________
Mathematica [C] time = 0.60, size = 180, normalized size = 1.31 \[ -\frac {2 i i \sqrt {c+d x} \sqrt {g+h x} \left (E\left (i \sinh ^{-1}\left (\sqrt {\frac {f (c+d x)}{d e-c f}}\right )|\frac {d e h-c f h}{d f g-c f h}\right )-\operatorname {EllipticF}\left (i \sinh ^{-1}\left (\sqrt {\frac {f (c+d x)}{d e-c f}}\right ),\frac {d e h-c f h}{d f g-c f h}\right )\right )}{h \sqrt {e+f x} \sqrt {\frac {f (c+d x)}{d (e+f x)}} \sqrt {\frac {d (g+h x)}{d g-c h}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g} i}{f h x^{2} + e g + {\left (f g + e h\right )} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {d i x + c i}{\sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.03, size = 552, normalized size = 4.03 \[ \frac {2 \left (-c^{2} f h \EllipticE \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )+c^{2} f h \EllipticF \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )+c d e h \EllipticE \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )-c d e h \EllipticF \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )+c d f g \EllipticE \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )-c d f g \EllipticF \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )-d^{2} e g \EllipticE \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )+d^{2} e g \EllipticF \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )\right ) \sqrt {-\frac {\left (f x +e \right ) d}{c f -d e}}\, \sqrt {-\frac {\left (h x +g \right ) d}{c h -d g}}\, \sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}\, \sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}\, i}{\left (d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g \right ) d f h} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {d i x + c i}{\sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {c\,i+d\,i\,x}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\sqrt {c+d\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ i \int \frac {\sqrt {c + d x}}{\sqrt {e + f x} \sqrt {g + h x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________