Optimal. Leaf size=504 \[ -\frac{9 d^3 (d x)^{5/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{9/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 d^5 \sqrt{d x} \left (a+b x^2\right )}{16 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
[Out]
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Rubi [A] time = 0.819311, antiderivative size = 504, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{9 d^3 (d x)^{5/2}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{9/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \sqrt [4]{a} d^{11/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} b^{13/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 d^5 \sqrt{d x} \left (a+b x^2\right )}{16 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
[In] Int[(d*x)^(11/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]
[Out]
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Rubi in Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x)**(11/2)/(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.350338, size = 474, normalized size = 0.94 \[ -\frac{a^2 (d x)^{11/2} \left (a+b x^2\right )}{4 b^3 x^5 \left (\left (a+b x^2\right )^2\right )^{3/2}}+\frac{45 \sqrt [4]{a} (d x)^{11/2} \left (a+b x^2\right )^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} b^{13/4} x^{11/2} \left (\left (a+b x^2\right )^2\right )^{3/2}}-\frac{45 \sqrt [4]{a} (d x)^{11/2} \left (a+b x^2\right )^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{64 \sqrt{2} b^{13/4} x^{11/2} \left (\left (a+b x^2\right )^2\right )^{3/2}}-\frac{45 \sqrt [4]{a} (d x)^{11/2} \left (a+b x^2\right )^3 \tan ^{-1}\left (\frac{2 \sqrt [4]{b} \sqrt{x}-\sqrt{2} \sqrt [4]{a}}{\sqrt{2} \sqrt [4]{a}}\right )}{32 \sqrt{2} b^{13/4} x^{11/2} \left (\left (a+b x^2\right )^2\right )^{3/2}}-\frac{45 \sqrt [4]{a} (d x)^{11/2} \left (a+b x^2\right )^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a}+2 \sqrt [4]{b} \sqrt{x}}{\sqrt{2} \sqrt [4]{a}}\right )}{32 \sqrt{2} b^{13/4} x^{11/2} \left (\left (a+b x^2\right )^2\right )^{3/2}}+\frac{2 (d x)^{11/2} \left (a+b x^2\right )^3}{b^3 x^5 \left (\left (a+b x^2\right )^2\right )^{3/2}}+\frac{17 a (d x)^{11/2} \left (a+b x^2\right )^2}{16 b^3 x^5 \left (\left (a+b x^2\right )^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d*x)^(11/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]
[Out]
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Maple [B] time = 0.027, size = 702, normalized size = 1.4 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x)^(11/2)/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),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((d*x)^(11/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.295295, size = 383, normalized size = 0.76 \[ \frac{180 \, \left (-\frac{a d^{22}}{b^{13}}\right )^{\frac{1}{4}}{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \arctan \left (\frac{\left (-\frac{a d^{22}}{b^{13}}\right )^{\frac{1}{4}} b^{3}}{\sqrt{d x} d^{5} + \sqrt{d^{11} x + \sqrt{-\frac{a d^{22}}{b^{13}}} b^{6}}}\right ) - 45 \, \left (-\frac{a d^{22}}{b^{13}}\right )^{\frac{1}{4}}{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \log \left (45 \, \sqrt{d x} d^{5} + 45 \, \left (-\frac{a d^{22}}{b^{13}}\right )^{\frac{1}{4}} b^{3}\right ) + 45 \, \left (-\frac{a d^{22}}{b^{13}}\right )^{\frac{1}{4}}{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \log \left (45 \, \sqrt{d x} d^{5} - 45 \, \left (-\frac{a d^{22}}{b^{13}}\right )^{\frac{1}{4}} b^{3}\right ) + 4 \,{\left (32 \, b^{2} d^{5} x^{4} + 81 \, a b d^{5} x^{2} + 45 \, a^{2} d^{5}\right )} \sqrt{d x}}{64 \,{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)^(11/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2),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((d*x)**(11/2)/(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.287295, size = 527, normalized size = 1.05 \[ -\frac{1}{128} \, d^{4}{\left (\frac{90 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{b^{4}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{90 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{b^{4}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{45 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{b^{4}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{45 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{b^{4}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{256 \, \sqrt{d x} d}{b^{3}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{8 \,{\left (17 \, \sqrt{d x} a b d^{5} x^{2} + 13 \, \sqrt{d x} a^{2} d^{5}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{2} b^{3}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)^(11/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2),x, algorithm="giac")
[Out]