Optimal. Leaf size=293 \[ -\frac{3 (a B+7 A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{11/4} b^{5/4}}+\frac{3 (a B+7 A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{11/4} b^{5/4}}-\frac{3 (a B+7 A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{11/4} b^{5/4}}+\frac{3 (a B+7 A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{11/4} b^{5/4}}+\frac{\sqrt{x} (a B+7 A b)}{16 a^2 b \left (a+b x^2\right )}+\frac{\sqrt{x} (A b-a B)}{4 a b \left (a+b x^2\right )^2} \]
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Rubi [A] time = 0.454032, antiderivative size = 293, 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 \[ -\frac{3 (a B+7 A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{11/4} b^{5/4}}+\frac{3 (a B+7 A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} a^{11/4} b^{5/4}}-\frac{3 (a B+7 A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{11/4} b^{5/4}}+\frac{3 (a B+7 A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{11/4} b^{5/4}}+\frac{\sqrt{x} (a B+7 A b)}{16 a^2 b \left (a+b x^2\right )}+\frac{\sqrt{x} (A b-a B)}{4 a b \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/(Sqrt[x]*(a + b*x^2)^3),x]
[Out]
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Rubi in Sympy [A] time = 78.2893, size = 275, normalized size = 0.94 \[ \frac{\sqrt{x} \left (A b - B a\right )}{4 a b \left (a + b x^{2}\right )^{2}} + \frac{\sqrt{x} \left (7 A b + B a\right )}{16 a^{2} b \left (a + b x^{2}\right )} - \frac{3 \sqrt{2} \left (7 A b + B a\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{128 a^{\frac{11}{4}} b^{\frac{5}{4}}} + \frac{3 \sqrt{2} \left (7 A b + B a\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{128 a^{\frac{11}{4}} b^{\frac{5}{4}}} - \frac{3 \sqrt{2} \left (7 A b + B a\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{11}{4}} b^{\frac{5}{4}}} + \frac{3 \sqrt{2} \left (7 A b + B a\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{11}{4}} b^{\frac{5}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)/(b*x**2+a)**3/x**(1/2),x)
[Out]
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Mathematica [A] time = 0.36774, size = 263, normalized size = 0.9 \[ \frac{-\frac{32 a^{7/4} \sqrt [4]{b} \sqrt{x} (a B-A b)}{\left (a+b x^2\right )^2}+\frac{8 a^{3/4} \sqrt [4]{b} \sqrt{x} (a B+7 A b)}{a+b x^2}-3 \sqrt{2} (a B+7 A b) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+3 \sqrt{2} (a B+7 A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-6 \sqrt{2} (a B+7 A b) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+6 \sqrt{2} (a B+7 A b) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{128 a^{11/4} b^{5/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/(Sqrt[x]*(a + b*x^2)^3),x]
[Out]
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Maple [A] time = 0.022, size = 325, normalized size = 1.1 \[ 2\,{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{2}} \left ( 1/32\,{\frac{ \left ( 7\,Ab+Ba \right ){x}^{5/2}}{{a}^{2}}}+1/32\,{\frac{ \left ( 11\,Ab-3\,Ba \right ) \sqrt{x}}{ab}} \right ) }+{\frac{21\,\sqrt{2}A}{64\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{21\,\sqrt{2}A}{64\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{21\,\sqrt{2}A}{128\,{a}^{3}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{3\,\sqrt{2}B}{64\,{a}^{2}b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }+{\frac{3\,\sqrt{2}B}{64\,{a}^{2}b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) }+{\frac{3\,\sqrt{2}B}{128\,{a}^{2}b}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)/(b*x^2+a)^3/x^(1/2),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)/((b*x^2 + a)^3*sqrt(x)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.25412, size = 903, normalized size = 3.08 \[ -\frac{12 \,{\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )} \left (-\frac{B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{3} b \left (-\frac{B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}\right )^{\frac{1}{4}}}{{\left (B a + 7 \, A b\right )} \sqrt{x} + \sqrt{a^{6} b^{2} \sqrt{-\frac{B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}} +{\left (B^{2} a^{2} + 14 \, A B a b + 49 \, A^{2} b^{2}\right )} x}}\right ) - 3 \,{\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )} \left (-\frac{B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}\right )^{\frac{1}{4}} \log \left (3 \, a^{3} b \left (-\frac{B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}\right )^{\frac{1}{4}} + 3 \,{\left (B a + 7 \, A b\right )} \sqrt{x}\right ) + 3 \,{\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )} \left (-\frac{B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}\right )^{\frac{1}{4}} \log \left (-3 \, a^{3} b \left (-\frac{B^{4} a^{4} + 28 \, A B^{3} a^{3} b + 294 \, A^{2} B^{2} a^{2} b^{2} + 1372 \, A^{3} B a b^{3} + 2401 \, A^{4} b^{4}}{a^{11} b^{5}}\right )^{\frac{1}{4}} + 3 \,{\left (B a + 7 \, A b\right )} \sqrt{x}\right ) + 4 \,{\left (3 \, B a^{2} - 11 \, A a b -{\left (B a b + 7 \, A b^{2}\right )} x^{2}\right )} \sqrt{x}}{64 \,{\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)^3*sqrt(x)),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)**3/x**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.244603, size = 396, normalized size = 1.35 \[ \frac{3 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{3} b^{2}} + \frac{3 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{3} b^{2}} + \frac{3 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a^{3} b^{2}} - \frac{3 \, \sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} B a + 7 \, \left (a b^{3}\right )^{\frac{1}{4}} A b\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{128 \, a^{3} b^{2}} + \frac{B a b x^{\frac{5}{2}} + 7 \, A b^{2} x^{\frac{5}{2}} - 3 \, B a^{2} \sqrt{x} + 11 \, A a b \sqrt{x}}{16 \,{\left (b x^{2} + a\right )}^{2} a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)^3*sqrt(x)),x, algorithm="giac")
[Out]