Optimal. Leaf size=278 \[ -\frac{\sqrt [4]{c} \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{7/4}}+\frac{\sqrt [4]{c} \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{7/4}}+\frac{\sqrt [4]{c} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{7/4}}-\frac{\sqrt [4]{c} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{7/4}}-\frac{2 A}{3 a x^{3/2}}-\frac{2 B}{a \sqrt{x}} \]
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Rubi [A] time = 0.56495, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{\sqrt [4]{c} \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{7/4}}+\frac{\sqrt [4]{c} \left (\sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{2 \sqrt{2} a^{7/4}}+\frac{\sqrt [4]{c} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{7/4}}-\frac{\sqrt [4]{c} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{7/4}}-\frac{2 A}{3 a x^{3/2}}-\frac{2 B}{a \sqrt{x}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(x^(5/2)*(a + c*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 103.115, size = 262, normalized size = 0.94 \[ - \frac{2 A}{3 a x^{\frac{3}{2}}} - \frac{2 B}{a \sqrt{x}} + \frac{\sqrt{2} \sqrt [4]{c} \left (A \sqrt{c} - B \sqrt{a}\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{4 a^{\frac{7}{4}}} - \frac{\sqrt{2} \sqrt [4]{c} \left (A \sqrt{c} - B \sqrt{a}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{4 a^{\frac{7}{4}}} + \frac{\sqrt{2} \sqrt [4]{c} \left (A \sqrt{c} + B \sqrt{a}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{7}{4}}} - \frac{\sqrt{2} \sqrt [4]{c} \left (A \sqrt{c} + B \sqrt{a}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{7}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/x**(5/2)/(c*x**2+a),x)
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Mathematica [A] time = 0.776322, size = 276, normalized size = 0.99 \[ \frac{3 \sqrt{2} \sqrt [4]{c} \left (\sqrt [4]{a} A \sqrt{c}-a^{3/4} B\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )+3 \sqrt{2} \sqrt [4]{c} \left (a^{3/4} B-\sqrt [4]{a} A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )+6 \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )-6 \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \left (\sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )-\frac{8 a A}{x^{3/2}}-\frac{24 a B}{\sqrt{x}}}{12 a^2} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(x^(5/2)*(a + c*x^2)),x]
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Maple [A] time = 0.017, size = 289, normalized size = 1. \[ -{\frac{2\,A}{3\,a}{x}^{-{\frac{3}{2}}}}-2\,{\frac{B}{\sqrt{x}a}}-{\frac{Ac\sqrt{2}}{4\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }-{\frac{Ac\sqrt{2}}{2\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }-{\frac{Ac\sqrt{2}}{2\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }-{\frac{B\sqrt{2}}{4\,a}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{B\sqrt{2}}{2\,a}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{B\sqrt{2}}{2\,a}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/x^(5/2)/(c*x^2+a),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 + A)/((c*x^2 + a)*x^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.299191, size = 1077, normalized size = 3.87 \[ -\frac{3 \, a x^{\frac{3}{2}} \sqrt{-\frac{a^{3} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} + 2 \, A B c}{a^{3}}} \log \left (-{\left (B^{4} a^{2} c - A^{4} c^{3}\right )} \sqrt{x} +{\left (B a^{6} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} - A B^{2} a^{3} c + A^{3} a^{2} c^{2}\right )} \sqrt{-\frac{a^{3} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} + 2 \, A B c}{a^{3}}}\right ) - 3 \, a x^{\frac{3}{2}} \sqrt{-\frac{a^{3} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} + 2 \, A B c}{a^{3}}} \log \left (-{\left (B^{4} a^{2} c - A^{4} c^{3}\right )} \sqrt{x} -{\left (B a^{6} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} - A B^{2} a^{3} c + A^{3} a^{2} c^{2}\right )} \sqrt{-\frac{a^{3} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} + 2 \, A B c}{a^{3}}}\right ) - 3 \, a x^{\frac{3}{2}} \sqrt{\frac{a^{3} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} - 2 \, A B c}{a^{3}}} \log \left (-{\left (B^{4} a^{2} c - A^{4} c^{3}\right )} \sqrt{x} +{\left (B a^{6} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} + A B^{2} a^{3} c - A^{3} a^{2} c^{2}\right )} \sqrt{\frac{a^{3} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} - 2 \, A B c}{a^{3}}}\right ) + 3 \, a x^{\frac{3}{2}} \sqrt{\frac{a^{3} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} - 2 \, A B c}{a^{3}}} \log \left (-{\left (B^{4} a^{2} c - A^{4} c^{3}\right )} \sqrt{x} -{\left (B a^{6} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} + A B^{2} a^{3} c - A^{3} a^{2} c^{2}\right )} \sqrt{\frac{a^{3} \sqrt{-\frac{B^{4} a^{2} c - 2 \, A^{2} B^{2} a c^{2} + A^{4} c^{3}}{a^{7}}} - 2 \, A B c}{a^{3}}}\right ) + 12 \, B x + 4 \, A}{6 \, a x^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + a)*x^(5/2)),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+A)/x**(5/2)/(c*x**2+a),x)
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GIAC/XCAS [A] time = 0.283472, size = 348, normalized size = 1.25 \[ -\frac{2 \,{\left (3 \, B x + A\right )}}{3 \, a x^{\frac{3}{2}}} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{2 \, a^{2} c^{2}} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{2 \, a^{2} c^{2}} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - \left (a c^{3}\right )^{\frac{3}{4}} B\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{4 \, a^{2} c^{2}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - \left (a c^{3}\right )^{\frac{3}{4}} B\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{4 \, a^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + a)*x^(5/2)),x, algorithm="giac")
[Out]