Optimal. Leaf size=344 \[ -\frac{\sqrt{e} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (a B+A b) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{4 a^{7/4} b^{7/4} \sqrt{a+b x^2}}+\frac{\sqrt{e} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (a B+A b) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{2 a^{7/4} b^{7/4} \sqrt{a+b x^2}}-\frac{\sqrt{e x} \sqrt{a+b x^2} (a B+A b)}{2 a^2 b^{3/2} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{(e x)^{3/2} (a B+A b)}{2 a^2 b e \sqrt{a+b x^2}}+\frac{(e x)^{3/2} (A b-a B)}{3 a b e \left (a+b x^2\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.641502, antiderivative size = 344, 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 \[ -\frac{\sqrt{e} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (a B+A b) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{4 a^{7/4} b^{7/4} \sqrt{a+b x^2}}+\frac{\sqrt{e} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (a B+A b) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{2 a^{7/4} b^{7/4} \sqrt{a+b x^2}}-\frac{\sqrt{e x} \sqrt{a+b x^2} (a B+A b)}{2 a^2 b^{3/2} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{(e x)^{3/2} (a B+A b)}{2 a^2 b e \sqrt{a+b x^2}}+\frac{(e x)^{3/2} (A b-a B)}{3 a b e \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[e*x]*(A + B*x^2))/(a + b*x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 67.1822, size = 304, normalized size = 0.88 \[ \frac{\left (e x\right )^{\frac{3}{2}} \left (A b - B a\right )}{3 a b e \left (a + b x^{2}\right )^{\frac{3}{2}}} + \frac{\left (e x\right )^{\frac{3}{2}} \left (A b + B a\right )}{2 a^{2} b e \sqrt{a + b x^{2}}} - \frac{\sqrt{e x} \sqrt{a + b x^{2}} \left (A b + B a\right )}{2 a^{2} b^{\frac{3}{2}} \left (\sqrt{a} + \sqrt{b} x\right )} + \frac{\sqrt{e} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (A b + B a\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{2 a^{\frac{7}{4}} b^{\frac{7}{4}} \sqrt{a + b x^{2}}} - \frac{\sqrt{e} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (A b + B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{4 a^{\frac{7}{4}} b^{\frac{7}{4}} \sqrt{a + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)*(e*x)**(1/2)/(b*x**2+a)**(5/2),x)
[Out]
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Mathematica [C] time = 0.940555, size = 247, normalized size = 0.72 \[ \frac{e \left (b x^2 \left (a^2 B+a b \left (5 A+3 B x^2\right )+3 A b^2 x^2\right )-\frac{3 \left (a+b x^2\right ) (a B+A b) \left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (a+b x^2\right )+\sqrt{a} \sqrt{b} x^{3/2} \sqrt{\frac{a}{b x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )-\sqrt{a} \sqrt{b} x^{3/2} \sqrt{\frac{a}{b x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}\right )}{6 a^2 b^2 \sqrt{e x} \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[e*x]*(A + B*x^2))/(a + b*x^2)^(5/2),x]
[Out]
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Maple [B] time = 0.027, size = 764, normalized size = 2.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)*(e*x)^(1/2)/(b*x^2+a)^(5/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} \sqrt{e x}}{{\left (b x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*sqrt(e*x)/(b*x^2 + a)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x^{2} + A\right )} \sqrt{e x}}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{b x^{2} + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*sqrt(e*x)/(b*x^2 + a)^(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)**(1/2)/(b*x**2+a)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} \sqrt{e x}}{{\left (b x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*sqrt(e*x)/(b*x^2 + a)^(5/2),x, algorithm="giac")
[Out]