Optimal. Leaf size=342 \[ -\frac{\sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (3 A b-5 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 a^{7/4} e^{7/2} \sqrt{a+b x^2}}+\frac{2 \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (3 A b-5 a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 a^{7/4} e^{7/2} \sqrt{a+b x^2}}-\frac{2 \sqrt{b} \sqrt{e x} \sqrt{a+b x^2} (3 A b-5 a B)}{5 a^2 e^4 \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{2 \sqrt{a+b x^2} (3 A b-5 a B)}{5 a^2 e^3 \sqrt{e x}}-\frac{2 A \sqrt{a+b x^2}}{5 a e (e x)^{5/2}} \]
[Out]
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Rubi [A] time = 0.638127, antiderivative size = 342, 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 \[ -\frac{\sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (3 A b-5 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 a^{7/4} e^{7/2} \sqrt{a+b x^2}}+\frac{2 \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (3 A b-5 a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 a^{7/4} e^{7/2} \sqrt{a+b x^2}}-\frac{2 \sqrt{b} \sqrt{e x} \sqrt{a+b x^2} (3 A b-5 a B)}{5 a^2 e^4 \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{2 \sqrt{a+b x^2} (3 A b-5 a B)}{5 a^2 e^3 \sqrt{e x}}-\frac{2 A \sqrt{a+b x^2}}{5 a e (e x)^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/((e*x)^(7/2)*Sqrt[a + b*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 68.6799, size = 321, normalized size = 0.94 \[ - \frac{2 A \sqrt{a + b x^{2}}}{5 a e \left (e x\right )^{\frac{5}{2}}} - \frac{2 \sqrt{b} \sqrt{e x} \sqrt{a + b x^{2}} \left (3 A b - 5 B a\right )}{5 a^{2} e^{4} \left (\sqrt{a} + \sqrt{b} x\right )} + \frac{2 \sqrt{a + b x^{2}} \left (3 A b - 5 B a\right )}{5 a^{2} e^{3} \sqrt{e x}} + \frac{2 \sqrt [4]{b} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (3 A b - 5 B a\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{5 a^{\frac{7}{4}} e^{\frac{7}{2}} \sqrt{a + b x^{2}}} - \frac{\sqrt [4]{b} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (3 A b - 5 B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{5 a^{\frac{7}{4}} e^{\frac{7}{2}} \sqrt{a + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)/(e*x)**(7/2)/(b*x**2+a)**(1/2),x)
[Out]
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Mathematica [C] time = 0.486475, size = 221, normalized size = 0.65 \[ \frac{x \left (2 \sqrt{a} \sqrt{b} x^3 \sqrt{\frac{b x^2}{a}+1} (5 a B-3 A b) E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )-2 \left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \left (a+b x^2\right ) \left (a \left (A+5 B x^2\right )-3 A b x^2\right )+\sqrt{a} \sqrt{b} x^3 \sqrt{\frac{b x^2}{a}+1} (5 a B-3 A b) F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )\right )\right )}{5 a^2 (e x)^{7/2} \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/((e*x)^(7/2)*Sqrt[a + b*x^2]),x]
[Out]
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Maple [A] time = 0.027, size = 417, normalized size = 1.2 \[ -{\frac{1}{5\,{x}^{2}{e}^{3}{a}^{2}} \left ( 6\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){x}^{2}ab-3\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){x}^{2}ab-10\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}+5\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}-6\,A{b}^{2}{x}^{4}+10\,B{x}^{4}ab-4\,aAb{x}^{2}+10\,B{x}^{2}{a}^{2}+2\,A{a}^{2} \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)/(e*x)^(7/2)/(b*x^2+a)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{2} + A}{\sqrt{b x^{2} + a} \left (e x\right )^{\frac{7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/(sqrt(b*x^2 + a)*(e*x)^(7/2)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{B x^{2} + A}{\sqrt{b x^{2} + a} \sqrt{e x} e^{3} x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/(sqrt(b*x^2 + a)*(e*x)^(7/2)),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)**(7/2)/(b*x**2+a)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{2} + A}{\sqrt{b x^{2} + a} \left (e x\right )^{\frac{7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/(sqrt(b*x^2 + a)*(e*x)^(7/2)),x, algorithm="giac")
[Out]