Optimal. Leaf size=261 \[ \frac{(3 a B+A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} b^{7/4}}-\frac{(3 a B+A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} b^{7/4}}-\frac{(3 a B+A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{5/4} b^{7/4}}+\frac{(3 a B+A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{5/4} b^{7/4}}+\frac{x^{3/2} (A b-a B)}{2 a b \left (a+b x^2\right )} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.398931, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ \frac{(3 a B+A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} b^{7/4}}-\frac{(3 a B+A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} b^{7/4}}-\frac{(3 a B+A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{5/4} b^{7/4}}+\frac{(3 a B+A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{5/4} b^{7/4}}+\frac{x^{3/2} (A b-a B)}{2 a b \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[x]*(A + B*x^2))/(a + b*x^2)^2,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 71.6248, size = 240, normalized size = 0.92 \[ \frac{x^{\frac{3}{2}} \left (A b - B a\right )}{2 a b \left (a + b x^{2}\right )} + \frac{\sqrt{2} \left (A b + 3 B a\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{5}{4}} b^{\frac{7}{4}}} - \frac{\sqrt{2} \left (A b + 3 B a\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{5}{4}} b^{\frac{7}{4}}} - \frac{\sqrt{2} \left (A b + 3 B a\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{5}{4}} b^{\frac{7}{4}}} + \frac{\sqrt{2} \left (A b + 3 B a\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{5}{4}} b^{\frac{7}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)*x**(1/2)/(b*x**2+a)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.307772, size = 228, normalized size = 0.87 \[ \frac{-\frac{8 \sqrt [4]{a} b^{3/4} x^{3/2} (a B-A b)}{a+b x^2}+\sqrt{2} (3 a B+A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-\sqrt{2} (3 a B+A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-2 \sqrt{2} (3 a B+A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+2 \sqrt{2} (3 a B+A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{16 a^{5/4} b^{7/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[x]*(A + B*x^2))/(a + b*x^2)^2,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.019, size = 305, normalized size = 1.2 \[{\frac{Ab-Ba}{2\,ab \left ( b{x}^{2}+a \right ) }{x}^{{\frac{3}{2}}}}+{\frac{\sqrt{2}A}{8\,ab}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{\sqrt{2}A}{16\,ab}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{\sqrt{2}A}{8\,ab}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{3\,\sqrt{2}B}{8\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{3\,\sqrt{2}B}{16\,{b}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{3\,\sqrt{2}B}{8\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)*x^(1/2)/(b*x^2+a)^2,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*sqrt(x)/(b*x^2 + a)^2,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.244753, size = 1072, normalized size = 4.11 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*sqrt(x)/(b*x^2 + a)^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)*x**(1/2)/(b*x**2+a)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.244316, size = 369, normalized size = 1.41 \[ -\frac{B a x^{\frac{3}{2}} - A b x^{\frac{3}{2}}}{2 \,{\left (b x^{2} + a\right )} a b} + \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{3}{4}} B a + \left (a b^{3}\right )^{\frac{3}{4}} A b\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{2} b^{4}} + \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{3}{4}} B a + \left (a b^{3}\right )^{\frac{3}{4}} A b\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{2} b^{4}} - \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{3}{4}} B a + \left (a b^{3}\right )^{\frac{3}{4}} A b\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{2} b^{4}} + \frac{\sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{3}{4}} B a + \left (a b^{3}\right )^{\frac{3}{4}} A b\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{2} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*sqrt(x)/(b*x^2 + a)^2,x, algorithm="giac")
[Out]