Optimal. Leaf size=300 \[ \frac{7 b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} a^{11/4} d^{5/2}}-\frac{7 b^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} a^{11/4} d^{5/2}}+\frac{7 b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} a^{11/4} d^{5/2}}-\frac{7 b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{4 \sqrt{2} a^{11/4} d^{5/2}}-\frac{7}{6 a^2 d (d x)^{3/2}}+\frac{1}{2 a d (d x)^{3/2} \left (a+b x^2\right )} \]
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Rubi [A] time = 0.644911, antiderivative size = 300, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ \frac{7 b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} a^{11/4} d^{5/2}}-\frac{7 b^{3/4} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} a^{11/4} d^{5/2}}+\frac{7 b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} a^{11/4} d^{5/2}}-\frac{7 b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{4 \sqrt{2} a^{11/4} d^{5/2}}-\frac{7}{6 a^2 d (d x)^{3/2}}+\frac{1}{2 a d (d x)^{3/2} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[1/((d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)),x]
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Rubi in Sympy [A] time = 135.371, size = 279, normalized size = 0.93 \[ \frac{1}{2 a d \left (d x\right )^{\frac{3}{2}} \left (a + b x^{2}\right )} - \frac{7}{6 a^{2} d \left (d x\right )^{\frac{3}{2}}} + \frac{7 \sqrt{2} b^{\frac{3}{4}} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d} \sqrt{d x} + \sqrt{a} d + \sqrt{b} d x \right )}}{16 a^{\frac{11}{4}} d^{\frac{5}{2}}} - \frac{7 \sqrt{2} b^{\frac{3}{4}} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d} \sqrt{d x} + \sqrt{a} d + \sqrt{b} d x \right )}}{16 a^{\frac{11}{4}} d^{\frac{5}{2}}} + \frac{7 \sqrt{2} b^{\frac{3}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}} \right )}}{8 a^{\frac{11}{4}} d^{\frac{5}{2}}} - \frac{7 \sqrt{2} b^{\frac{3}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}} \right )}}{8 a^{\frac{11}{4}} d^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(d*x)**(5/2)/(b**2*x**4+2*a*b*x**2+a**2),x)
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Mathematica [A] time = 0.312241, size = 233, normalized size = 0.78 \[ \frac{x \left (-\frac{24 a^{3/4} b x^2}{a+b x^2}-32 a^{3/4}+21 \sqrt{2} b^{3/4} x^{3/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-21 \sqrt{2} b^{3/4} x^{3/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+42 \sqrt{2} b^{3/4} x^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )-42 \sqrt{2} b^{3/4} x^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )\right )}{48 a^{11/4} (d x)^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)),x]
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Maple [A] time = 0.023, size = 226, normalized size = 0.8 \[ -{\frac{2}{3\,{a}^{2}d} \left ( dx \right ) ^{-{\frac{3}{2}}}}-{\frac{b}{2\,{a}^{2}d \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) }\sqrt{dx}}-{\frac{7\,b\sqrt{2}}{16\,{d}^{3}{a}^{3}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\ln \left ({1 \left ( dx+\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) \left ( dx-\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) ^{-1}} \right ) }-{\frac{7\,b\sqrt{2}}{8\,{d}^{3}{a}^{3}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ) }-{\frac{7\,b\sqrt{2}}{8\,{d}^{3}{a}^{3}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2),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(1/((b^2*x^4 + 2*a*b*x^2 + a^2)*(d*x)^(5/2)),x, algorithm="maxima")
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Fricas [A] time = 0.289106, size = 379, normalized size = 1.26 \[ -\frac{28 \, b x^{2} - 84 \,{\left (a^{2} b d^{2} x^{3} + a^{3} d^{2} x\right )} \sqrt{d x} \left (-\frac{b^{3}}{a^{11} d^{10}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{3} d^{3} \left (-\frac{b^{3}}{a^{11} d^{10}}\right )^{\frac{1}{4}}}{\sqrt{d x} b + \sqrt{a^{6} d^{6} \sqrt{-\frac{b^{3}}{a^{11} d^{10}}} + b^{2} d x}}\right ) + 21 \,{\left (a^{2} b d^{2} x^{3} + a^{3} d^{2} x\right )} \sqrt{d x} \left (-\frac{b^{3}}{a^{11} d^{10}}\right )^{\frac{1}{4}} \log \left (7 \, a^{3} d^{3} \left (-\frac{b^{3}}{a^{11} d^{10}}\right )^{\frac{1}{4}} + 7 \, \sqrt{d x} b\right ) - 21 \,{\left (a^{2} b d^{2} x^{3} + a^{3} d^{2} x\right )} \sqrt{d x} \left (-\frac{b^{3}}{a^{11} d^{10}}\right )^{\frac{1}{4}} \log \left (-7 \, a^{3} d^{3} \left (-\frac{b^{3}}{a^{11} d^{10}}\right )^{\frac{1}{4}} + 7 \, \sqrt{d x} b\right ) + 16 \, a}{24 \,{\left (a^{2} b d^{2} x^{3} + a^{3} d^{2} x\right )} \sqrt{d x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)*(d*x)^(5/2)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d x\right )^{\frac{5}{2}} \left (a + b x^{2}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(d*x)**(5/2)/(b**2*x**4+2*a*b*x**2+a**2),x)
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GIAC/XCAS [A] time = 0.271856, size = 373, normalized size = 1.24 \[ -\frac{\sqrt{d x} b}{2 \,{\left (b d^{2} x^{2} + a d^{2}\right )} a^{2} d} - \frac{7 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{3} d^{3}} - \frac{7 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{3} d^{3}} - \frac{7 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}}{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{16 \, a^{3} d^{3}} + \frac{7 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}}{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{16 \, a^{3} d^{3}} - \frac{2}{3 \, \sqrt{d x} a^{2} d^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)*(d*x)^(5/2)),x, algorithm="giac")
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