Optimal. Leaf size=337 \[ \frac{2 a^{5/4} \sqrt{e} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (3 A b-a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 b^{7/4} \sqrt{a+b x^2}}-\frac{4 a^{5/4} \sqrt{e} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (3 A b-a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 b^{7/4} \sqrt{a+b x^2}}+\frac{4 a \sqrt{e x} \sqrt{a+b x^2} (3 A b-a B)}{15 b^{3/2} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{2 (e x)^{3/2} \sqrt{a+b x^2} (3 A b-a B)}{15 b e}+\frac{2 B (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 b e} \]
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Rubi [A] time = 0.634259, 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 \[ \frac{2 a^{5/4} \sqrt{e} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (3 A b-a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 b^{7/4} \sqrt{a+b x^2}}-\frac{4 a^{5/4} \sqrt{e} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (3 A b-a B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{15 b^{7/4} \sqrt{a+b x^2}}+\frac{4 a \sqrt{e x} \sqrt{a+b x^2} (3 A b-a B)}{15 b^{3/2} \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{2 (e x)^{3/2} \sqrt{a+b x^2} (3 A b-a B)}{15 b e}+\frac{2 B (e x)^{3/2} \left (a+b x^2\right )^{3/2}}{9 b e} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[e*x]*Sqrt[a + b*x^2]*(A + B*x^2),x]
[Out]
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Rubi in Sympy [A] time = 62.874, size = 308, normalized size = 0.91 \[ \frac{2 B \left (e x\right )^{\frac{3}{2}} \left (a + b x^{2}\right )^{\frac{3}{2}}}{9 b e} - \frac{4 a^{\frac{5}{4}} \sqrt{e} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (3 A b - B a\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{15 b^{\frac{7}{4}} \sqrt{a + b x^{2}}} + \frac{2 a^{\frac{5}{4}} \sqrt{e} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (3 A b - B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{15 b^{\frac{7}{4}} \sqrt{a + b x^{2}}} + \frac{4 a \sqrt{e x} \sqrt{a + b x^{2}} \left (3 A b - B a\right )}{15 b^{\frac{3}{2}} \left (\sqrt{a} + \sqrt{b} x\right )} + \frac{2 \left (e x\right )^{\frac{3}{2}} \sqrt{a + b x^{2}} \left (3 A b - B a\right )}{15 b e} \]
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)**(1/2),x)
[Out]
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Mathematica [C] time = 1.66605, size = 234, normalized size = 0.69 \[ \frac{2 e \left (b x^2 \left (a+b x^2\right ) \left (2 a B+9 A b+5 b B x^2\right )-\frac{6 a (a B-3 A b) \left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (a+b x^2\right )+\sqrt{a} \sqrt{b} x^{3/2} \sqrt{\frac{a}{b x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )-\sqrt{a} \sqrt{b} x^{3/2} \sqrt{\frac{a}{b x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}\right )}{45 b^2 \sqrt{e x} \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[e*x]*Sqrt[a + b*x^2]*(A + B*x^2),x]
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Maple [A] time = 0.047, size = 414, normalized size = 1.2 \[{\frac{2}{45\,{b}^{2}x}\sqrt{ex} \left ( 5\,B{x}^{6}{b}^{3}+18\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){a}^{2}b-9\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){a}^{2}b-6\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){a}^{3}+3\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){a}^{3}+9\,A{x}^{4}{b}^{3}+7\,B{x}^{4}a{b}^{2}+9\,A{x}^{2}a{b}^{2}+2\,B{x}^{2}{a}^{2}b \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)*(e*x)^(1/2)*(b*x^2+a)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (B x^{2} + A\right )} \sqrt{b x^{2} + a} \sqrt{e x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*sqrt(b*x^2 + a)*sqrt(e*x),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (B x^{2} + A\right )} \sqrt{b x^{2} + a} \sqrt{e x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*sqrt(b*x^2 + a)*sqrt(e*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 9.64481, size = 95, normalized size = 0.28 \[ \frac{A \sqrt{a} \left (e x\right )^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 e \Gamma \left (\frac{7}{4}\right )} + \frac{B \sqrt{a} \left (e x\right )^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 e^{3} \Gamma \left (\frac{11}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**2+A)*(e*x)**(1/2)*(b*x**2+a)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (B x^{2} + A\right )} \sqrt{b x^{2} + a} \sqrt{e x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*sqrt(b*x^2 + a)*sqrt(e*x),x, algorithm="giac")
[Out]