Optimal. Leaf size=152 \[ -\frac{2 b^{3/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (A b-7 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{21 a^{5/4} \sqrt{a+b x^2}}+\frac{2 \sqrt{a+b x^2} (A b-7 a B)}{21 a x^{3/2}}-\frac{2 A \left (a+b x^2\right )^{3/2}}{7 a x^{7/2}} \]
[Out]
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Rubi [A] time = 0.229706, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{2 b^{3/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (A b-7 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{21 a^{5/4} \sqrt{a+b x^2}}+\frac{2 \sqrt{a+b x^2} (A b-7 a B)}{21 a x^{3/2}}-\frac{2 A \left (a+b x^2\right )^{3/2}}{7 a x^{7/2}} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[a + b*x^2]*(A + B*x^2))/x^(9/2),x]
[Out]
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Rubi in Sympy [A] time = 20.3476, size = 139, normalized size = 0.91 \[ - \frac{2 A \left (a + b x^{2}\right )^{\frac{3}{2}}}{7 a x^{\frac{7}{2}}} + \frac{2 \sqrt{a + b x^{2}} \left (A b - 7 B a\right )}{21 a x^{\frac{3}{2}}} - \frac{2 b^{\frac{3}{4}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (A b - 7 B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{21 a^{\frac{5}{4}} \sqrt{a + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)*(b*x**2+a)**(1/2)/x**(9/2),x)
[Out]
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Mathematica [C] time = 0.247867, size = 139, normalized size = 0.91 \[ \sqrt{a+b x^2} \left (-\frac{2 (7 a B+2 A b)}{21 a x^{3/2}}-\frac{2 A}{7 x^{7/2}}\right )+\frac{4 i b x \sqrt{\frac{a}{b x^2}+1} (7 a B-A b) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )}{21 a \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/x^(9/2),x]
[Out]
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Maple [A] time = 0.058, size = 242, normalized size = 1.6 \[ -{\frac{2}{21\,a} \left ( A\sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{2}\sqrt{{1 \left ( -bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-ab}{x}^{3}b-7\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}{x}^{3}a+2\,A{b}^{2}{x}^{4}+7\,B{x}^{4}ab+5\,aAb{x}^{2}+7\,B{x}^{2}{a}^{2}+3\,A{a}^{2} \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{x}^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)*(b*x^2+a)^(1/2)/x^(9/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} \sqrt{b x^{2} + a}}{x^{\frac{9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^(9/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x^{2} + A\right )} \sqrt{b x^{2} + a}}{x^{\frac{9}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^(9/2),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)*(b*x**2+a)**(1/2)/x**(9/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} \sqrt{b x^{2} + a}}{x^{\frac{9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^(9/2),x, algorithm="giac")
[Out]