Optimal. Leaf size=284 \[ -\frac{(A b-5 a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{3/4} b^{9/4}}+\frac{(A b-5 a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{3/4} b^{9/4}}-\frac{(A b-5 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{3/4} b^{9/4}}+\frac{(A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{3/4} b^{9/4}}-\frac{\sqrt{x} (A b-5 a B)}{2 a b^2}+\frac{x^{5/2} (A b-a B)}{2 a b \left (a+b x^2\right )} \]
[Out]
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Rubi [A] time = 0.469691, antiderivative size = 284, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409 \[ -\frac{(A b-5 a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{3/4} b^{9/4}}+\frac{(A b-5 a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{3/4} b^{9/4}}-\frac{(A b-5 a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{3/4} b^{9/4}}+\frac{(A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{3/4} b^{9/4}}-\frac{\sqrt{x} (A b-5 a B)}{2 a b^2}+\frac{x^{5/2} (A b-a B)}{2 a b \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(x^(3/2)*(A + B*x^2))/(a + b*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 79.0688, size = 260, normalized size = 0.92 \[ \frac{x^{\frac{5}{2}} \left (A b - B a\right )}{2 a b \left (a + b x^{2}\right )} - \frac{\sqrt{x} \left (A b - 5 B a\right )}{2 a b^{2}} - \frac{\sqrt{2} \left (A b - 5 B a\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{3}{4}} b^{\frac{9}{4}}} + \frac{\sqrt{2} \left (A b - 5 B a\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{3}{4}} b^{\frac{9}{4}}} - \frac{\sqrt{2} \left (A b - 5 B a\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{3}{4}} b^{\frac{9}{4}}} + \frac{\sqrt{2} \left (A b - 5 B a\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{3}{4}} b^{\frac{9}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3/2)*(B*x**2+A)/(b*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.545892, size = 252, normalized size = 0.89 \[ \frac{\frac{\sqrt{2} (5 a B-A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{3/4}}+\frac{\sqrt{2} (A b-5 a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{3/4}}+\frac{2 \sqrt{2} (5 a B-A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac{2 \sqrt{2} (A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{3/4}}-\frac{8 \sqrt [4]{b} \sqrt{x} (A b-a B)}{a+b x^2}+32 \sqrt [4]{b} B \sqrt{x}}{16 b^{9/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^(3/2)*(A + B*x^2))/(a + b*x^2)^2,x]
[Out]
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Maple [A] time = 0.019, size = 323, normalized size = 1.1 \[ 2\,{\frac{B\sqrt{x}}{{b}^{2}}}-{\frac{A}{2\,b \left ( b{x}^{2}+a \right ) }\sqrt{x}}+{\frac{Ba}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }\sqrt{x}}+{\frac{\sqrt{2}A}{8\,ab}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{\sqrt{2}A}{8\,ab}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{\sqrt{2}A}{16\,ab}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }-{\frac{5\,\sqrt{2}B}{8\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }-{\frac{5\,\sqrt{2}B}{8\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }-{\frac{5\,\sqrt{2}B}{16\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(3/2)*(B*x^2+A)/(b*x^2+a)^2,x)
[Out]
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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)*x^(3/2)/(b*x^2 + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.255432, size = 817, normalized size = 2.88 \[ -\frac{4 \,{\left (b^{3} x^{2} + a b^{2}\right )} \left (-\frac{625 \, B^{4} a^{4} - 500 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 20 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{9}}\right )^{\frac{1}{4}} \arctan \left (-\frac{a b^{2} \left (-\frac{625 \, B^{4} a^{4} - 500 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 20 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{9}}\right )^{\frac{1}{4}}}{{\left (5 \, B a - A b\right )} \sqrt{x} - \sqrt{a^{2} b^{4} \sqrt{-\frac{625 \, B^{4} a^{4} - 500 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 20 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{9}}} +{\left (25 \, B^{2} a^{2} - 10 \, A B a b + A^{2} b^{2}\right )} x}}\right ) -{\left (b^{3} x^{2} + a b^{2}\right )} \left (-\frac{625 \, B^{4} a^{4} - 500 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 20 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{9}}\right )^{\frac{1}{4}} \log \left (a b^{2} \left (-\frac{625 \, B^{4} a^{4} - 500 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 20 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{9}}\right )^{\frac{1}{4}} -{\left (5 \, B a - A b\right )} \sqrt{x}\right ) +{\left (b^{3} x^{2} + a b^{2}\right )} \left (-\frac{625 \, B^{4} a^{4} - 500 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 20 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{9}}\right )^{\frac{1}{4}} \log \left (-a b^{2} \left (-\frac{625 \, B^{4} a^{4} - 500 \, A B^{3} a^{3} b + 150 \, A^{2} B^{2} a^{2} b^{2} - 20 \, A^{3} B a b^{3} + A^{4} b^{4}}{a^{3} b^{9}}\right )^{\frac{1}{4}} -{\left (5 \, B a - A b\right )} \sqrt{x}\right ) - 4 \,{\left (4 \, B b x^{2} + 5 \, B a - A b\right )} \sqrt{x}}{8 \,{\left (b^{3} x^{2} + a b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^(3/2)/(b*x^2 + a)^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(x**(3/2)*(B*x**2+A)/(b*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.254144, size = 382, normalized size = 1.35 \[ \frac{2 \, B \sqrt{x}}{b^{2}} - \frac{\sqrt{2}{\left (5 \, \left (a b^{3}\right )^{\frac{1}{4}} B a - \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a b^{3}} - \frac{\sqrt{2}{\left (5 \, \left (a b^{3}\right )^{\frac{1}{4}} B a - \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a b^{3}} - \frac{\sqrt{2}{\left (5 \, \left (a b^{3}\right )^{\frac{1}{4}} B a - \left (a b^{3}\right )^{\frac{1}{4}} A b\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a b^{3}} + \frac{\sqrt{2}{\left (5 \, \left (a b^{3}\right )^{\frac{1}{4}} B a - \left (a b^{3}\right )^{\frac{1}{4}} A b\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a b^{3}} + \frac{B a \sqrt{x} - A b \sqrt{x}}{2 \,{\left (b x^{2} + a\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^(3/2)/(b*x^2 + a)^2,x, algorithm="giac")
[Out]