Optimal. Leaf size=420 \[ -\frac{d (e x)^{3/2}}{c e \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt [4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt [4]{c} \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt [4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt [4]{c} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt{b} \sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt{a} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt{b} \sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt{a} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 2.06945, antiderivative size = 420, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{d (e x)^{3/2}}{c e \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt [4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt [4]{c} \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt [4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt [4]{c} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt{b} \sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt{a} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt{b} \sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt{a} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[e*x]/((a - b*x^2)*(c - d*x^2)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**(1/2)/(-b*x**2+a)/(-d*x**2+c)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.798072, size = 356, normalized size = 0.85 \[ \frac{x \sqrt{e x} \left (\frac{33 a b d x^2 F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{\left (a-b x^2\right ) (a d-b c) \left (2 x^2 \left (2 b c F_1\left (\frac{11}{4};\frac{1}{2},2;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{11}{4};\frac{3}{2},1;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+11 a c F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}-\frac{49 a (a d+2 b c) F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{\left (a-b x^2\right ) (a d-b c) \left (2 x^2 \left (2 b c F_1\left (\frac{7}{4};\frac{1}{2},2;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{7}{4};\frac{3}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+7 a c F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}-\frac{21 d}{b c^2-a c d}\right )}{21 \sqrt{c-d x^2}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[Sqrt[e*x]/((a - b*x^2)*(c - d*x^2)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.038, size = 830, normalized size = 2. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^(1/2)/(-b*x^2+a)/(-d*x^2+c)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{\sqrt{e x}}{{\left (b x^{2} - a\right )}{\left (-d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(e*x)/((b*x^2 - a)*(-d*x^2 + c)^(3/2)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(e*x)/((b*x^2 - a)*(-d*x^2 + c)^(3/2)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)**(1/2)/(-b*x**2+a)/(-d*x**2+c)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{\sqrt{e x}}{{\left (b x^{2} - a\right )}{\left (-d x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(e*x)/((b*x^2 - a)*(-d*x^2 + c)^(3/2)),x, algorithm="giac")
[Out]