Optimal. Leaf size=94 \[ \frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} \sqrt {b x+c x^2} \sqrt {d+e x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {715, 117, 116} \[ \frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} \sqrt {b x+c x^2} \sqrt {d+e x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 116
Rule 117
Rule 715
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx &=\frac {\left (\sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{\sqrt {b x+c x^2}}\\ &=\frac {\left (\sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}} \, dx}{\sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=\frac {2 \sqrt {-b} \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} \sqrt {d+e x} \sqrt {b x+c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 94, normalized size = 1.00 \[ -\frac {2 x^{3/2} \sqrt {\frac {\frac {b}{x}+c}{c}} \sqrt {\frac {\frac {d}{x}+e}{e}} F\left (\sin ^{-1}\left (\frac {\sqrt {-\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )}{\sqrt {-\frac {b}{c}} \sqrt {x (b+c x)} \sqrt {d+e x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.22, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x} \sqrt {e x + d}}{c e x^{3} + b d x + {\left (c d + b e\right )} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c x^{2} + b x} \sqrt {e x + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 113, normalized size = 1.20 \[ \frac {2 \sqrt {-\frac {c x}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {\frac {c x +b}{b}}\, \sqrt {e x +d}\, \sqrt {\left (c x +b \right ) x}\, b \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )}{\left (c e \,x^{2}+b e x +c d x +b d \right ) c x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {c x^{2} + b x} \sqrt {e x + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {c\,x^2+b\,x}\,\sqrt {d+e\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x \left (b + c x\right )} \sqrt {d + e x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________