Optimal. Leaf size=392 \[ -\frac{\sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^{3/2} \sqrt{b} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^{3/2} \sqrt{b} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2}}+\frac{2 c^{3/4} \sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a e^{3/2} \sqrt{c-d x^2}}-\frac{2 c^{3/4} \sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a e^{3/2} \sqrt{c-d x^2}}-\frac{2 \sqrt{c-d x^2}}{a e \sqrt{e x}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 2.04357, antiderivative size = 392, 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 \[ -\frac{\sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^{3/2} \sqrt{b} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^{3/2} \sqrt{b} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2}}+\frac{2 c^{3/4} \sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a e^{3/2} \sqrt{c-d x^2}}-\frac{2 c^{3/4} \sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a e^{3/2} \sqrt{c-d x^2}}-\frac{2 \sqrt{c-d x^2}}{a e \sqrt{e x}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c - d*x^2]/((e*x)^(3/2)*(a - b*x^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((-d*x**2+c)**(1/2)/(e*x)**(3/2)/(-b*x**2+a),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.923681, size = 337, normalized size = 0.86 \[ \frac{2 x \left (\frac{49 c x^2 (b c-2 a d) F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{\left (a-b x^2\right ) \left (2 x^2 \left (2 b c F_1\left (\frac{7}{4};\frac{1}{2},2;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{7}{4};\frac{3}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+7 a c F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}-\frac{33 b c d x^4 F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{\left (b x^2-a\right ) \left (2 x^2 \left (2 b c F_1\left (\frac{11}{4};\frac{1}{2},2;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{11}{4};\frac{3}{2},1;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+11 a c F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}-\frac{21 \left (c-d x^2\right )}{a}\right )}{21 (e x)^{3/2} \sqrt{c-d x^2}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[Sqrt[c - d*x^2]/((e*x)^(3/2)*(a - b*x^2)),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.057, size = 1274, normalized size = 3.3 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-d*x^2+c)^(1/2)/(e*x)^(3/2)/(-b*x^2+a),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{\sqrt{-d x^{2} + c}}{{\left (b x^{2} - a\right )} \left (e x\right )^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(-d*x^2 + c)/((b*x^2 - a)*(e*x)^(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(-d*x^2 + c)/((b*x^2 - a)*(e*x)^(3/2)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{\sqrt{c - d x^{2}}}{- a \left (e x\right )^{\frac{3}{2}} + b x^{2} \left (e x\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-d*x**2+c)**(1/2)/(e*x)**(3/2)/(-b*x**2+a),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{\sqrt{-d x^{2} + c}}{{\left (b x^{2} - a\right )} \left (e x\right )^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(-d*x^2 + c)/((b*x^2 - a)*(e*x)^(3/2)),x, algorithm="giac")
[Out]