Optimal. Leaf size=298 \[ -\frac{(5 a B+3 A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{7/4} b^{9/4}}+\frac{(5 a B+3 A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{7/4} b^{9/4}}-\frac{(5 a B+3 A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{7/4} b^{9/4}}+\frac{(5 a B+3 A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{7/4} b^{9/4}}-\frac{\sqrt{x} (5 a B+3 A b)}{16 a b^2 \left (a+b x^2\right )}+\frac{x^{5/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2} \]
[Out]
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Rubi [A] time = 0.45225, antiderivative size = 298, 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 \[ -\frac{(5 a B+3 A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{7/4} b^{9/4}}+\frac{(5 a B+3 A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{7/4} b^{9/4}}-\frac{(5 a B+3 A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{7/4} b^{9/4}}+\frac{(5 a B+3 A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{7/4} b^{9/4}}-\frac{\sqrt{x} (5 a B+3 A b)}{16 a b^2 \left (a+b x^2\right )}+\frac{x^{5/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(x^(3/2)*(A + B*x^2))/(a + b*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 78.9186, size = 277, normalized size = 0.93 \[ \frac{x^{\frac{5}{2}} \left (A b - B a\right )}{4 a b \left (a + b x^{2}\right )^{2}} - \frac{\sqrt{x} \left (3 A b + 5 B a\right )}{16 a b^{2} \left (a + b x^{2}\right )} - \frac{\sqrt{2} \left (3 A b + 5 B a\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{128 a^{\frac{7}{4}} b^{\frac{9}{4}}} + \frac{\sqrt{2} \left (3 A b + 5 B a\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{128 a^{\frac{7}{4}} b^{\frac{9}{4}}} - \frac{\sqrt{2} \left (3 A b + 5 B a\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{7}{4}} b^{\frac{9}{4}}} + \frac{\sqrt{2} \left (3 A b + 5 B a\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{7}{4}} b^{\frac{9}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3/2)*(B*x**2+A)/(b*x**2+a)**3,x)
[Out]
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Mathematica [A] time = 0.498726, size = 274, normalized size = 0.92 \[ \frac{-\frac{\sqrt{2} (5 a B+3 A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{7/4}}+\frac{\sqrt{2} (5 a B+3 A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{7/4}}-\frac{2 \sqrt{2} (5 a B+3 A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac{2 \sqrt{2} (5 a B+3 A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{7/4}}+\frac{8 \sqrt [4]{b} \sqrt{x} (A b-9 a B)}{a \left (a+b x^2\right )}-\frac{32 \sqrt [4]{b} \sqrt{x} (A b-a B)}{\left (a+b x^2\right )^2}}{128 b^{9/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^(3/2)*(A + B*x^2))/(a + b*x^2)^3,x]
[Out]
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Maple [A] time = 0.023, size = 334, normalized size = 1.1 \[ 2\,{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{2}} \left ( 1/32\,{\frac{ \left ( Ab-9\,Ba \right ){x}^{5/2}}{ab}}-1/32\,{\frac{ \left ( 3\,Ab+5\,Ba \right ) \sqrt{x}}{{b}^{2}}} \right ) }+{\frac{3\,\sqrt{2}A}{64\,b{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{3\,\sqrt{2}A}{64\,b{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{3\,\sqrt{2}A}{128\,b{a}^{2}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{5\,\sqrt{2}B}{64\,{b}^{2}a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{5\,\sqrt{2}B}{64\,{b}^{2}a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{5\,\sqrt{2}B}{128\,{b}^{2}a}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(3/2)*(B*x^2+A)/(b*x^2+a)^3,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)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.257932, size = 921, normalized size = 3.09 \[ -\frac{4 \,{\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \left (-\frac{625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{2} b^{2} \left (-\frac{625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}\right )^{\frac{1}{4}}}{{\left (5 \, B a + 3 \, A b\right )} \sqrt{x} + \sqrt{a^{4} b^{4} \sqrt{-\frac{625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}} +{\left (25 \, B^{2} a^{2} + 30 \, A B a b + 9 \, A^{2} b^{2}\right )} x}}\right ) -{\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \left (-\frac{625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}\right )^{\frac{1}{4}} \log \left (a^{2} b^{2} \left (-\frac{625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}\right )^{\frac{1}{4}} +{\left (5 \, B a + 3 \, A b\right )} \sqrt{x}\right ) +{\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \left (-\frac{625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}\right )^{\frac{1}{4}} \log \left (-a^{2} b^{2} \left (-\frac{625 \, B^{4} a^{4} + 1500 \, A B^{3} a^{3} b + 1350 \, A^{2} B^{2} a^{2} b^{2} + 540 \, A^{3} B a b^{3} + 81 \, A^{4} b^{4}}{a^{7} b^{9}}\right )^{\frac{1}{4}} +{\left (5 \, B a + 3 \, A b\right )} \sqrt{x}\right ) + 4 \,{\left (5 \, B a^{2} + 3 \, A a b +{\left (9 \, B a b - A b^{2}\right )} x^{2}\right )} \sqrt{x}}{64 \,{\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^(3/2)/(b*x^2 + a)^3,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)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.25181, size = 402, normalized size = 1.35 \[ \frac{\sqrt{2}{\left (5 \, \left (a b^{3}\right )^{\frac{1}{4}} B a + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{2} b^{3}} + \frac{\sqrt{2}{\left (5 \, \left (a b^{3}\right )^{\frac{1}{4}} B a + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{2} b^{3}} + \frac{\sqrt{2}{\left (5 \, \left (a b^{3}\right )^{\frac{1}{4}} B a + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a^{2} b^{3}} - \frac{\sqrt{2}{\left (5 \, \left (a b^{3}\right )^{\frac{1}{4}} B a + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a^{2} b^{3}} - \frac{9 \, B a b x^{\frac{5}{2}} - A b^{2} x^{\frac{5}{2}} + 5 \, B a^{2} \sqrt{x} + 3 \, A a b \sqrt{x}}{16 \,{\left (b x^{2} + a\right )}^{2} a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^(3/2)/(b*x^2 + a)^3,x, algorithm="giac")
[Out]