Optimal. Leaf size=185 \[ \frac{e^{3/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 a B+A b) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{12 a^{5/4} b^{9/4} \sqrt{a+b x^2}}-\frac{e \sqrt{e x} (5 a B+A b)}{6 a b^2 \sqrt{a+b x^2}}+\frac{(e x)^{5/2} (A b-a B)}{3 a b e \left (a+b x^2\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.306688, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{e^{3/2} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 a B+A b) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{12 a^{5/4} b^{9/4} \sqrt{a+b x^2}}-\frac{e \sqrt{e x} (5 a B+A b)}{6 a b^2 \sqrt{a+b x^2}}+\frac{(e x)^{5/2} (A b-a B)}{3 a b e \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^(3/2)*(A + B*x^2))/(a + b*x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 32.0885, size = 163, normalized size = 0.88 \[ \frac{\left (e x\right )^{\frac{5}{2}} \left (A b - B a\right )}{3 a b e \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{e \sqrt{e x} \left (A b + 5 B a\right )}{6 a b^{2} \sqrt{a + b x^{2}}} + \frac{e^{\frac{3}{2}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (A b + 5 B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{12 a^{\frac{5}{4}} b^{\frac{9}{4}} \sqrt{a + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**(3/2)*(B*x**2+A)/(b*x**2+a)**(5/2),x)
[Out]
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Mathematica [C] time = 0.319532, size = 163, normalized size = 0.88 \[ \frac{e \sqrt{e x} \left (\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (-5 a^2 B-a b \left (A+7 B x^2\right )+A b^2 x^2\right )+i \sqrt{x} \sqrt{\frac{a}{b x^2}+1} \left (a+b x^2\right ) (5 a B+A b) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{6 a b^2 \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((e*x)^(3/2)*(A + B*x^2))/(a + b*x^2)^(5/2),x]
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Maple [B] time = 0.028, size = 429, normalized size = 2.3 \[{\frac{e}{12\,ax{b}^{3}} \left ( A\sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{2}\sqrt{{1 \left ( -bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-ab}{x}^{2}{b}^{2}+5\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}{x}^{2}ab+A\sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{2}\sqrt{{1 \left ( -bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-ab}ab+5\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}{a}^{2}+2\,A{x}^{3}{b}^{3}-14\,B{x}^{3}a{b}^{2}-2\,Axa{b}^{2}-10\,Bx{a}^{2}b \right ) \sqrt{ex} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^(3/2)*(B*x^2+A)/(b*x^2+a)^(5/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} \left (e x\right )^{\frac{3}{2}}}{{\left (b x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(e*x)^(3/2)/(b*x^2 + a)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B e x^{3} + A e x\right )} \sqrt{e x}}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{b x^{2} + a}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(e*x)^(3/2)/(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((e*x)**(3/2)*(B*x**2+A)/(b*x**2+a)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} \left (e x\right )^{\frac{3}{2}}}{{\left (b x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*(e*x)^(3/2)/(b*x^2 + a)^(5/2),x, algorithm="giac")
[Out]