Optimal. Leaf size=255 \[ \frac{\sqrt [4]{b} (A b-a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4}}-\frac{\sqrt [4]{b} (A b-a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4}}-\frac{\sqrt [4]{b} (A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{9/4}}+\frac{\sqrt [4]{b} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{9/4}}+\frac{2 (A b-a B)}{a^2 \sqrt{x}}-\frac{2 A}{5 a x^{5/2}} \]
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Rubi [A] time = 0.444201, antiderivative size = 255, 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 \[ \frac{\sqrt [4]{b} (A b-a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4}}-\frac{\sqrt [4]{b} (A b-a B) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{9/4}}-\frac{\sqrt [4]{b} (A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{9/4}}+\frac{\sqrt [4]{b} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{9/4}}+\frac{2 (A b-a B)}{a^2 \sqrt{x}}-\frac{2 A}{5 a x^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/(x^(7/2)*(a + b*x^2)),x]
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Rubi in Sympy [A] time = 74.8395, size = 240, normalized size = 0.94 \[ - \frac{2 A}{5 a x^{\frac{5}{2}}} + \frac{2 \left (A b - B a\right )}{a^{2} \sqrt{x}} + \frac{\sqrt{2} \sqrt [4]{b} \left (A b - B a\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 a^{\frac{9}{4}}} - \frac{\sqrt{2} \sqrt [4]{b} \left (A b - B a\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 a^{\frac{9}{4}}} - \frac{\sqrt{2} \sqrt [4]{b} \left (A b - B a\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{9}{4}}} + \frac{\sqrt{2} \sqrt [4]{b} \left (A b - B a\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{9}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)/x**(7/2)/(b*x**2+a),x)
[Out]
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Mathematica [A] time = 0.391585, size = 243, normalized size = 0.95 \[ \frac{-\frac{8 a^{5/4} A}{x^{5/2}}-\frac{40 \sqrt [4]{a} (a B-A b)}{\sqrt{x}}+5 \sqrt{2} \sqrt [4]{b} (A b-a B) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+5 \sqrt{2} \sqrt [4]{b} (a B-A b) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-10 \sqrt{2} \sqrt [4]{b} (A b-a B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+10 \sqrt{2} \sqrt [4]{b} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{20 a^{9/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/(x^(7/2)*(a + b*x^2)),x]
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Maple [A] time = 0.017, size = 299, normalized size = 1.2 \[ -{\frac{2\,A}{5\,a}{x}^{-{\frac{5}{2}}}}+2\,{\frac{Ab}{\sqrt{x}{a}^{2}}}-2\,{\frac{B}{\sqrt{x}a}}+{\frac{\sqrt{2}Ab}{2\,{a}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{\sqrt{2}Ab}{2\,{a}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{\sqrt{2}Ab}{4\,{a}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{\sqrt{2}B}{2\,a}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{\sqrt{2}B}{2\,a}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{\sqrt{2}B}{4\,a}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)/x^(7/2)/(b*x^2+a),x)
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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)*x^(7/2)),x, algorithm="maxima")
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Fricas [A] time = 0.252987, size = 1031, normalized size = 4.04 \[ \frac{20 \, a^{2} x^{\frac{5}{2}} \left (-\frac{B^{4} a^{4} b - 4 \, A B^{3} a^{3} b^{2} + 6 \, A^{2} B^{2} a^{2} b^{3} - 4 \, A^{3} B a b^{4} + A^{4} b^{5}}{a^{9}}\right )^{\frac{1}{4}} \arctan \left (-\frac{a^{7} \left (-\frac{B^{4} a^{4} b - 4 \, A B^{3} a^{3} b^{2} + 6 \, A^{2} B^{2} a^{2} b^{3} - 4 \, A^{3} B a b^{4} + A^{4} b^{5}}{a^{9}}\right )^{\frac{3}{4}}}{{\left (B^{3} a^{3} b - 3 \, A B^{2} a^{2} b^{2} + 3 \, A^{2} B a b^{3} - A^{3} b^{4}\right )} \sqrt{x} - \sqrt{{\left (B^{6} a^{6} b^{2} - 6 \, A B^{5} a^{5} b^{3} + 15 \, A^{2} B^{4} a^{4} b^{4} - 20 \, A^{3} B^{3} a^{3} b^{5} + 15 \, A^{4} B^{2} a^{2} b^{6} - 6 \, A^{5} B a b^{7} + A^{6} b^{8}\right )} x -{\left (B^{4} a^{9} b - 4 \, A B^{3} a^{8} b^{2} + 6 \, A^{2} B^{2} a^{7} b^{3} - 4 \, A^{3} B a^{6} b^{4} + A^{4} a^{5} b^{5}\right )} \sqrt{-\frac{B^{4} a^{4} b - 4 \, A B^{3} a^{3} b^{2} + 6 \, A^{2} B^{2} a^{2} b^{3} - 4 \, A^{3} B a b^{4} + A^{4} b^{5}}{a^{9}}}}}\right ) + 5 \, a^{2} x^{\frac{5}{2}} \left (-\frac{B^{4} a^{4} b - 4 \, A B^{3} a^{3} b^{2} + 6 \, A^{2} B^{2} a^{2} b^{3} - 4 \, A^{3} B a b^{4} + A^{4} b^{5}}{a^{9}}\right )^{\frac{1}{4}} \log \left (a^{7} \left (-\frac{B^{4} a^{4} b - 4 \, A B^{3} a^{3} b^{2} + 6 \, A^{2} B^{2} a^{2} b^{3} - 4 \, A^{3} B a b^{4} + A^{4} b^{5}}{a^{9}}\right )^{\frac{3}{4}} -{\left (B^{3} a^{3} b - 3 \, A B^{2} a^{2} b^{2} + 3 \, A^{2} B a b^{3} - A^{3} b^{4}\right )} \sqrt{x}\right ) - 5 \, a^{2} x^{\frac{5}{2}} \left (-\frac{B^{4} a^{4} b - 4 \, A B^{3} a^{3} b^{2} + 6 \, A^{2} B^{2} a^{2} b^{3} - 4 \, A^{3} B a b^{4} + A^{4} b^{5}}{a^{9}}\right )^{\frac{1}{4}} \log \left (-a^{7} \left (-\frac{B^{4} a^{4} b - 4 \, A B^{3} a^{3} b^{2} + 6 \, A^{2} B^{2} a^{2} b^{3} - 4 \, A^{3} B a b^{4} + A^{4} b^{5}}{a^{9}}\right )^{\frac{3}{4}} -{\left (B^{3} a^{3} b - 3 \, A B^{2} a^{2} b^{2} + 3 \, A^{2} B a b^{3} - A^{3} b^{4}\right )} \sqrt{x}\right ) - 20 \,{\left (B a - A b\right )} x^{2} - 4 \, A a}{10 \, a^{2} x^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)*x^(7/2)),x, algorithm="fricas")
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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)/x**(7/2)/(b*x**2+a),x)
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GIAC/XCAS [A] time = 0.25155, size = 362, normalized size = 1.42 \[ -\frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} B a - \left (a b^{3}\right )^{\frac{3}{4}} A b\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, a^{3} b^{2}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} B a - \left (a b^{3}\right )^{\frac{3}{4}} A b\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, a^{3} b^{2}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} B a - \left (a b^{3}\right )^{\frac{3}{4}} A b\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, a^{3} b^{2}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} B a - \left (a b^{3}\right )^{\frac{3}{4}} A b\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, a^{3} b^{2}} - \frac{2 \,{\left (5 \, B a x^{2} - 5 \, A b x^{2} + A a\right )}}{5 \, a^{2} x^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)*x^(7/2)),x, algorithm="giac")
[Out]