Optimal. Leaf size=337 \[ \frac{3 \sqrt [4]{a} e^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 A b-7 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{10 b^{11/4} \sqrt{a+b x^2}}-\frac{3 \sqrt [4]{a} e^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 A b-7 a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 b^{11/4} \sqrt{a+b x^2}}+\frac{3 e^2 \sqrt{e x} \sqrt{a+b x^2} (5 A b-7 a B)}{5 b^{5/2} \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{e (e x)^{3/2} (5 A b-7 a B)}{5 b^2 \sqrt{a+b x^2}}+\frac{2 B (e x)^{7/2}}{5 b e \sqrt{a+b x^2}} \]
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Rubi [A] time = 0.645411, antiderivative size = 337, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{3 \sqrt [4]{a} e^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 A b-7 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{10 b^{11/4} \sqrt{a+b x^2}}-\frac{3 \sqrt [4]{a} e^{5/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 A b-7 a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 b^{11/4} \sqrt{a+b x^2}}+\frac{3 e^2 \sqrt{e x} \sqrt{a+b x^2} (5 A b-7 a B)}{5 b^{5/2} \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{e (e x)^{3/2} (5 A b-7 a B)}{5 b^2 \sqrt{a+b x^2}}+\frac{2 B (e x)^{7/2}}{5 b e \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^(5/2)*(A + B*x^2))/(a + b*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 65.6036, size = 316, normalized size = 0.94 \[ \frac{2 B \left (e x\right )^{\frac{7}{2}}}{5 b e \sqrt{a + b x^{2}}} - \frac{3 \sqrt [4]{a} e^{\frac{5}{2}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (5 A b - 7 B a\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{5 b^{\frac{11}{4}} \sqrt{a + b x^{2}}} + \frac{3 \sqrt [4]{a} e^{\frac{5}{2}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (5 A b - 7 B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{10 b^{\frac{11}{4}} \sqrt{a + b x^{2}}} - \frac{e \left (e x\right )^{\frac{3}{2}} \left (5 A b - 7 B a\right )}{5 b^{2} \sqrt{a + b x^{2}}} + \frac{3 e^{2} \sqrt{e x} \sqrt{a + b x^{2}} \left (5 A b - 7 B a\right )}{5 b^{\frac{5}{2}} \left (\sqrt{a} + \sqrt{b} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**(5/2)*(B*x**2+A)/(b*x**2+a)**(3/2),x)
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Mathematica [C] time = 1.44467, size = 229, normalized size = 0.68 \[ \frac{(e x)^{5/2} \left (b x^2 \left (7 a B-5 A b+2 b B x^2\right )+\frac{3 (5 A b-7 a B) \left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (a+b x^2\right )+\sqrt{a} \sqrt{b} x^{3/2} \sqrt{\frac{a}{b x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )-\sqrt{a} \sqrt{b} x^{3/2} \sqrt{\frac{a}{b x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}\right )}{5 b^3 x^3 \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[((e*x)^(5/2)*(A + B*x^2))/(a + b*x^2)^(3/2),x]
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Maple [A] time = 0.05, size = 391, normalized size = 1.2 \[{\frac{{e}^{2}}{10\,x{b}^{3}}\sqrt{ex} \left ( 30\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) ab-15\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) ab-42\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){a}^{2}+21\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){a}^{2}+4\,{b}^{2}B{x}^{4}-10\,A{x}^{2}{b}^{2}+14\,B{x}^{2}ab \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^(5/2)*(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. \[ \int \frac{{\left (B x^{2} + A\right )} \left (e x\right )^{\frac{5}{2}}}{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(e*x)^(5/2)/(b*x^2 + a)^(3/2),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B e^{2} x^{4} + A e^{2} x^{2}\right )} \sqrt{e x}}{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(e*x)^(5/2)/(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)**(5/2)*(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. \[ \int \frac{{\left (B x^{2} + A\right )} \left (e x\right )^{\frac{5}{2}}}{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(e*x)^(5/2)/(b*x^2 + a)^(3/2),x, algorithm="giac")
[Out]