Optimal. Leaf size=187 \[ \frac{2 b^{7/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 A b-11 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{231 a^{9/4} \sqrt{a+b x^2}}+\frac{4 b \sqrt{a+b x^2} (5 A b-11 a B)}{231 a^2 x^{3/2}}+\frac{2 \sqrt{a+b x^2} (5 A b-11 a B)}{77 a x^{7/2}}-\frac{2 A \left (a+b x^2\right )^{3/2}}{11 a x^{11/2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.278419, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{2 b^{7/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 A b-11 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{231 a^{9/4} \sqrt{a+b x^2}}+\frac{4 b \sqrt{a+b x^2} (5 A b-11 a B)}{231 a^2 x^{3/2}}+\frac{2 \sqrt{a+b x^2} (5 A b-11 a B)}{77 a x^{7/2}}-\frac{2 A \left (a+b x^2\right )^{3/2}}{11 a x^{11/2}} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[a + b*x^2]*(A + B*x^2))/x^(13/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 25.4306, size = 177, normalized size = 0.95 \[ - \frac{2 A \left (a + b x^{2}\right )^{\frac{3}{2}}}{11 a x^{\frac{11}{2}}} + \frac{2 \sqrt{a + b x^{2}} \left (5 A b - 11 B a\right )}{77 a x^{\frac{7}{2}}} + \frac{4 b \sqrt{a + b x^{2}} \left (5 A b - 11 B a\right )}{231 a^{2} x^{\frac{3}{2}}} + \frac{2 b^{\frac{7}{4}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (5 A b - 11 B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{231 a^{\frac{9}{4}} \sqrt{a + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)*(b*x**2+a)**(1/2)/x**(13/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.319543, size = 163, normalized size = 0.87 \[ \sqrt{a+b x^2} \left (-\frac{4 b (11 a B-5 A b)}{231 a^2 x^{3/2}}-\frac{2 (11 a B+2 A b)}{77 a x^{7/2}}-\frac{2 A}{11 x^{11/2}}\right )-\frac{4 i b^2 x \sqrt{\frac{a}{b x^2}+1} (11 a B-5 A b) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )}{231 a^2 \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/x^(13/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.046, size = 270, normalized size = 1.4 \[{\frac{2}{231\,{a}^{2}} \left ( 5\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}{x}^{5}{b}^{2}-11\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}{x}^{5}ab+10\,A{x}^{6}{b}^{3}-22\,B{x}^{6}a{b}^{2}+4\,A{x}^{4}a{b}^{2}-55\,B{x}^{4}{a}^{2}b-27\,A{x}^{2}{a}^{2}b-33\,B{x}^{2}{a}^{3}-21\,A{a}^{3} \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{x}^{-{\frac{11}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)*(b*x^2+a)^(1/2)/x^(13/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} \sqrt{b x^{2} + a}}{x^{\frac{13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^(13/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x^{2} + A\right )} \sqrt{b x^{2} + a}}{x^{\frac{13}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^(13/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
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**(13/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} \sqrt{b x^{2} + a}}{x^{\frac{13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^(13/2),x, algorithm="giac")
[Out]