Optimal. Leaf size=252 \[ -\frac{4 a^{11/4} e^{3/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (3 A b-a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{231 b^{9/4} \sqrt{a+b x^2}}+\frac{8 a^2 e \sqrt{e x} \sqrt{a+b x^2} (3 A b-a B)}{231 b^2}+\frac{4 a (e x)^{5/2} \sqrt{a+b x^2} (3 A b-a B)}{77 b e}+\frac{2 (e x)^{5/2} \left (a+b x^2\right )^{3/2} (3 A b-a B)}{33 b e}+\frac{2 B (e x)^{5/2} \left (a+b x^2\right )^{5/2}}{15 b e} \]
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Rubi [A] time = 0.431151, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{4 a^{11/4} e^{3/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (3 A b-a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{231 b^{9/4} \sqrt{a+b x^2}}+\frac{8 a^2 e \sqrt{e x} \sqrt{a+b x^2} (3 A b-a B)}{231 b^2}+\frac{4 a (e x)^{5/2} \sqrt{a+b x^2} (3 A b-a B)}{77 b e}+\frac{2 (e x)^{5/2} \left (a+b x^2\right )^{3/2} (3 A b-a B)}{33 b e}+\frac{2 B (e x)^{5/2} \left (a+b x^2\right )^{5/2}}{15 b e} \]
Antiderivative was successfully verified.
[In] Int[(e*x)^(3/2)*(a + b*x^2)^(3/2)*(A + B*x^2),x]
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Rubi in Sympy [A] time = 41.8079, size = 230, normalized size = 0.91 \[ \frac{2 B \left (e x\right )^{\frac{5}{2}} \left (a + b x^{2}\right )^{\frac{5}{2}}}{15 b e} - \frac{4 a^{\frac{11}{4}} e^{\frac{3}{2}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (3 A b - B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{231 b^{\frac{9}{4}} \sqrt{a + b x^{2}}} + \frac{8 a^{2} e \sqrt{e x} \sqrt{a + b x^{2}} \left (3 A b - B a\right )}{231 b^{2}} + \frac{4 a \left (e x\right )^{\frac{5}{2}} \sqrt{a + b x^{2}} \left (3 A b - B a\right )}{77 b e} + \frac{2 \left (e x\right )^{\frac{5}{2}} \left (a + b x^{2}\right )^{\frac{3}{2}} \left (3 A b - B a\right )}{33 b e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**(3/2)*(b*x**2+a)**(3/2)*(B*x**2+A),x)
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Mathematica [C] time = 0.585068, size = 178, normalized size = 0.71 \[ \frac{2 e \sqrt{e x} \left (-\left (a+b x^2\right ) \left (20 a^3 B-12 a^2 b \left (5 A+B x^2\right )-a b^2 x^2 \left (195 A+119 B x^2\right )-7 b^3 x^4 \left (15 A+11 B x^2\right )\right )+\frac{20 i a^3 \sqrt{x} \sqrt{\frac{a}{b x^2}+1} (a B-3 A b) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}\right )}{1155 b^2 \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(e*x)^(3/2)*(a + b*x^2)^(3/2)*(A + B*x^2),x]
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Maple [A] time = 0.037, size = 300, normalized size = 1.2 \[ -{\frac{2\,e}{1155\,x{b}^{3}}\sqrt{ex} \left ( -77\,{b}^{5}B{x}^{9}-105\,A{x}^{7}{b}^{5}-196\,B{x}^{7}a{b}^{4}+30\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}{a}^{3}b-300\,A{x}^{5}a{b}^{4}-10\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}{a}^{4}-131\,B{x}^{5}{a}^{2}{b}^{3}-255\,A{x}^{3}{a}^{2}{b}^{3}+8\,B{x}^{3}{a}^{3}{b}^{2}-60\,Ax{a}^{3}{b}^{2}+20\,Bx{a}^{4}b \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^(3/2)*(b*x^2+a)^(3/2)*(B*x^2+A),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (B x^{2} + A\right )}{\left (b x^{2} + a\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)*(e*x)^(3/2),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (B b e x^{5} +{\left (B a + A b\right )} e x^{3} + A a e x\right )} \sqrt{b x^{2} + a} \sqrt{e x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)*(e*x)^(3/2),x, algorithm="fricas")
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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)**(3/2)*(B*x**2+A),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (B x^{2} + A\right )}{\left (b x^{2} + a\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)*(e*x)^(3/2),x, algorithm="giac")
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