Optimal. Leaf size=174 \[ \frac{e^{3/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (3 A b-5 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{6 \sqrt [4]{a} b^{9/4} \sqrt{a+b x^2}}-\frac{e \sqrt{e x} (3 A b-5 a B)}{3 b^2 \sqrt{a+b x^2}}+\frac{2 B (e x)^{5/2}}{3 b e \sqrt{a+b x^2}} \]
[Out]
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Rubi [A] time = 0.298376, antiderivative size = 174, 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 \[ \frac{e^{3/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (3 A b-5 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{6 \sqrt [4]{a} b^{9/4} \sqrt{a+b x^2}}-\frac{e \sqrt{e x} (3 A b-5 a B)}{3 b^2 \sqrt{a+b x^2}}+\frac{2 B (e x)^{5/2}}{3 b e \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^(3/2)*(A + B*x^2))/(a + b*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 29.3885, size = 160, normalized size = 0.92 \[ \frac{2 B \left (e x\right )^{\frac{5}{2}}}{3 b e \sqrt{a + b x^{2}}} - \frac{e \sqrt{e x} \left (3 A b - 5 B a\right )}{3 b^{2} \sqrt{a + b x^{2}}} + \frac{e^{\frac{3}{2}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (3 A b - 5 B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{6 \sqrt [4]{a} b^{\frac{9}{4}} \sqrt{a + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**(3/2)*(B*x**2+A)/(b*x**2+a)**(3/2),x)
[Out]
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Mathematica [C] time = 0.230373, size = 143, normalized size = 0.82 \[ \frac{e \sqrt{e x} \left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (5 a B-3 A b+2 b B x^2\right )+i \sqrt{x} \sqrt{\frac{a}{b x^2}+1} (3 A b-5 a B) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{3 b^2 \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[((e*x)^(3/2)*(A + B*x^2))/(a + b*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.027, size = 225, normalized size = 1.3 \[{\frac{e}{6\,x{b}^{3}}\sqrt{ex} \left ( 3\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}b-5\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}a+4\,{b}^{2}B{x}^{3}-6\,Ax{b}^{2}+10\,Bxab \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^(3/2)*(B*x^2+A)/(b*x^2+a)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} \left (e x\right )^{\frac{3}{2}}}{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(e*x)^(3/2)/(b*x^2 + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B e x^{3} + A e x\right )} \sqrt{e x}}{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(e*x)^(3/2)/(b*x^2 + a)^(3/2),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((e*x)**(3/2)*(B*x**2+A)/(b*x**2+a)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} \left (e x\right )^{\frac{3}{2}}}{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(e*x)^(3/2)/(b*x^2 + a)^(3/2),x, algorithm="giac")
[Out]