Optimal. Leaf size=397 \[ \frac{b^2 \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^2 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)}+\frac{b^2 \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^2 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)}+\frac{d^{3/4} \sqrt{1-\frac{d x^2}{c}} (2 b c-5 a d) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{3 a c^{7/4} e^{5/2} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt{c-d x^2} (2 b c-5 a d)}{3 a c^2 e (e x)^{3/2} (b c-a d)}-\frac{d}{c e (e x)^{3/2} \sqrt{c-d x^2} (b c-a d)} \]
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Rubi [A] time = 2.11053, antiderivative size = 397, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{b^2 \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^2 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)}+\frac{b^2 \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^2 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)}+\frac{d^{3/4} \sqrt{1-\frac{d x^2}{c}} (2 b c-5 a d) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{3 a c^{7/4} e^{5/2} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt{c-d x^2} (2 b c-5 a d)}{3 a c^2 e (e x)^{3/2} (b c-a d)}-\frac{d}{c e (e x)^{3/2} \sqrt{c-d x^2} (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[1/((e*x)^(5/2)*(a - b*x^2)*(c - d*x^2)^(3/2)),x]
[Out]
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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(1/(e*x)**(5/2)/(-b*x**2+a)/(-d*x**2+c)**(3/2),x)
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Mathematica [C] time = 1.60533, size = 413, normalized size = 1.04 \[ \frac{x \left (-\frac{25 c x^2 \left (-5 a^2 d^2+2 a b c d+6 b^2 c^2\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{\left (b x^2-a\right ) (b c-a d) \left (2 x^2 \left (2 b c F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}+\frac{9 b c d x^4 (2 b c-5 a d) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{\left (b x^2-a\right ) (b c-a d) \left (2 x^2 \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}+\frac{5 a d \left (5 d x^2-2 c\right )+10 b c \left (c-d x^2\right )}{a (a d-b c)}\right )}{15 c^2 (e x)^{5/2} \sqrt{c-d x^2}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/((e*x)^(5/2)*(a - b*x^2)*(c - d*x^2)^(3/2)),x]
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Maple [B] time = 0.046, size = 896, normalized size = 2.3 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x)^(5/2)/(-b*x^2+a)/(-d*x^2+c)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{1}{{\left (b x^{2} - a\right )}{\left (-d x^{2} + c\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((b*x^2 - a)*(-d*x^2 + c)^(3/2)*(e*x)^(5/2)),x, algorithm="maxima")
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((b*x^2 - a)*(-d*x^2 + c)^(3/2)*(e*x)^(5/2)),x, algorithm="fricas")
[Out]
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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(1/(e*x)**(5/2)/(-b*x**2+a)/(-d*x**2+c)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{1}{{\left (b x^{2} - a\right )}{\left (-d x^{2} + c\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((b*x^2 - a)*(-d*x^2 + c)^(3/2)*(e*x)^(5/2)),x, algorithm="giac")
[Out]