Optimal. Leaf size=176 \[ -\frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 A b-3 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{6 a^{9/4} \sqrt [4]{b} e^{5/2} \sqrt{a+b x^2}}-\frac{\sqrt{e x} (5 A b-3 a B)}{3 a^2 e^3 \sqrt{a+b x^2}}-\frac{2 A}{3 a e (e x)^{3/2} \sqrt{a+b x^2}} \]
[Out]
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Rubi [A] time = 0.303116, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{\left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 A b-3 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{6 a^{9/4} \sqrt [4]{b} e^{5/2} \sqrt{a+b x^2}}-\frac{\sqrt{e x} (5 A b-3 a B)}{3 a^2 e^3 \sqrt{a+b x^2}}-\frac{2 A}{3 a e (e x)^{3/2} \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/((e*x)^(5/2)*(a + b*x^2)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 30.44, size = 163, normalized size = 0.93 \[ - \frac{2 A}{3 a e \left (e x\right )^{\frac{3}{2}} \sqrt{a + b x^{2}}} - \frac{\sqrt{e x} \left (5 A b - 3 B a\right )}{3 a^{2} e^{3} \sqrt{a + b x^{2}}} - \frac{\sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (5 A b - 3 B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{6 a^{\frac{9}{4}} \sqrt [4]{b} e^{\frac{5}{2}} \sqrt{a + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)/(e*x)**(5/2)/(b*x**2+a)**(3/2),x)
[Out]
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Mathematica [C] time = 0.223779, size = 146, normalized size = 0.83 \[ \frac{x \left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (-2 a A+3 a B x^2-5 A b x^2\right )-i x^{5/2} \sqrt{\frac{a}{b x^2}+1} (5 A b-3 a B) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{3 a^2 \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} (e x)^{5/2} \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/((e*x)^(5/2)*(a + b*x^2)^(3/2)),x]
[Out]
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Maple [A] time = 0.032, size = 232, normalized size = 1.3 \[ -{\frac{1}{6\,bx{a}^{2}{e}^{2}} \left ( 5\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}xb-3\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}xa+10\,A{x}^{2}{b}^{2}-6\,B{x}^{2}ab+4\,abA \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)/(e*x)^(5/2)/(b*x^2+a)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{2} + A}{{\left (b x^{2} + a\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*(e*x)^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{B x^{2} + A}{{\left (b e^{2} x^{4} + a e^{2} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{e x}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)^(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((B*x**2+A)/(e*x)**(5/2)/(b*x**2+a)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{2} + A}{{\left (b x^{2} + a\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)^(3/2)*(e*x)^(5/2)),x, algorithm="giac")
[Out]