Optimal. Leaf size=317 \[ -\frac{\sqrt [4]{c} \left (5 \sqrt{a} B-7 A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{11/4}}+\frac{\sqrt [4]{c} \left (5 \sqrt{a} B-7 A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{11/4}}+\frac{\sqrt [4]{c} \left (5 \sqrt{a} B+7 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{11/4}}-\frac{\sqrt [4]{c} \left (5 \sqrt{a} B+7 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{11/4}}-\frac{7 A}{6 a^2 x^{3/2}}-\frac{5 B}{2 a^2 \sqrt{x}}+\frac{A+B x}{2 a x^{3/2} \left (a+c x^2\right )} \]
[Out]
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Rubi [A] time = 0.717654, antiderivative size = 317, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45 \[ -\frac{\sqrt [4]{c} \left (5 \sqrt{a} B-7 A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{11/4}}+\frac{\sqrt [4]{c} \left (5 \sqrt{a} B-7 A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{11/4}}+\frac{\sqrt [4]{c} \left (5 \sqrt{a} B+7 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{11/4}}-\frac{\sqrt [4]{c} \left (5 \sqrt{a} B+7 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{11/4}}-\frac{7 A}{6 a^2 x^{3/2}}-\frac{5 B}{2 a^2 \sqrt{x}}+\frac{A+B x}{2 a x^{3/2} \left (a+c x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(x^(5/2)*(a + c*x^2)^2),x]
[Out]
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Rubi in Sympy [A] time = 130.275, size = 301, normalized size = 0.95 \[ - \frac{7 A}{6 a^{2} x^{\frac{3}{2}}} - \frac{5 B}{2 a^{2} \sqrt{x}} + \frac{A + B x}{2 a x^{\frac{3}{2}} \left (a + c x^{2}\right )} + \frac{\sqrt{2} \sqrt [4]{c} \left (7 A \sqrt{c} - 5 B \sqrt{a}\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{16 a^{\frac{11}{4}}} - \frac{\sqrt{2} \sqrt [4]{c} \left (7 A \sqrt{c} - 5 B \sqrt{a}\right ) \log{\left (\sqrt{2} \sqrt [4]{a} c^{\frac{3}{4}} \sqrt{x} + \sqrt{a} \sqrt{c} + c x \right )}}{16 a^{\frac{11}{4}}} + \frac{\sqrt{2} \sqrt [4]{c} \left (7 A \sqrt{c} + 5 B \sqrt{a}\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{11}{4}}} - \frac{\sqrt{2} \sqrt [4]{c} \left (7 A \sqrt{c} + 5 B \sqrt{a}\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{11}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/x**(5/2)/(c*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.787859, size = 305, normalized size = 0.96 \[ \frac{3 \sqrt{2} \sqrt [4]{c} \left (7 \sqrt [4]{a} A \sqrt{c}-5 a^{3/4} B\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )+3 \sqrt{2} \sqrt [4]{c} \left (5 a^{3/4} B-7 \sqrt [4]{a} A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )-\frac{24 a c \sqrt{x} (A+B x)}{a+c x^2}+6 \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \left (5 \sqrt{a} B+7 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )-6 \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \left (5 \sqrt{a} B+7 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )-\frac{32 a A}{x^{3/2}}-\frac{96 a B}{\sqrt{x}}}{48 a^3} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(x^(5/2)*(a + c*x^2)^2),x]
[Out]
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Maple [A] time = 0.026, size = 327, normalized size = 1. \[ -{\frac{2\,A}{3\,{a}^{2}}{x}^{-{\frac{3}{2}}}}-2\,{\frac{B}{{a}^{2}\sqrt{x}}}-{\frac{Bc}{2\,{a}^{2} \left ( c{x}^{2}+a \right ) }{x}^{{\frac{3}{2}}}}-{\frac{Ac}{2\,{a}^{2} \left ( c{x}^{2}+a \right ) }\sqrt{x}}-{\frac{7\,Ac\sqrt{2}}{16\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }-{\frac{7\,Ac\sqrt{2}}{8\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }-{\frac{7\,Ac\sqrt{2}}{8\,{a}^{3}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }-{\frac{5\,B\sqrt{2}}{16\,{a}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{5\,B\sqrt{2}}{8\,{a}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-{\frac{5\,B\sqrt{2}}{8\,{a}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/x^(5/2)/(c*x^2+a)^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 + A)/((c*x^2 + a)^2*x^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.333849, size = 1235, normalized size = 3.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + a)^2*x^(5/2)),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((B*x+A)/x**(5/2)/(c*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.284282, size = 393, normalized size = 1.24 \[ -\frac{B c x^{\frac{3}{2}} + A c \sqrt{x}}{2 \,{\left (c x^{2} + a\right )} a^{2}} - \frac{2 \,{\left (3 \, B x + A\right )}}{3 \, a^{2} x^{\frac{3}{2}}} - \frac{\sqrt{2}{\left (7 \, \left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + 5 \, \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{8 \, a^{3} c^{2}} - \frac{\sqrt{2}{\left (7 \, \left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + 5 \, \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{8 \, a^{3} c^{2}} - \frac{\sqrt{2}{\left (7 \, \left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - 5 \, \left (a c^{3}\right )^{\frac{3}{4}} B\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{16 \, a^{3} c^{2}} + \frac{\sqrt{2}{\left (7 \, \left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - 5 \, \left (a c^{3}\right )^{\frac{3}{4}} B\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{16 \, a^{3} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + a)^2*x^(5/2)),x, algorithm="giac")
[Out]