Optimal. Leaf size=458 \[ -\frac{5 d^3 \sqrt{d x}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{5/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 d^{7/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{3/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 d^{7/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{3/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 d^{7/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} a^{3/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 d^{7/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} a^{3/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
[Out]
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Rubi [A] time = 0.727506, antiderivative size = 458, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{5 d^3 \sqrt{d x}}{16 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{5/2}}{4 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 d^{7/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{3/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 d^{7/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{3/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{5 d^{7/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} a^{3/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 d^{7/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} a^{3/4} b^{9/4} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
[In] Int[(d*x)^(7/2)/(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)**(7/2)/(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.407838, size = 272, normalized size = 0.59 \[ \frac{(d x)^{7/2} \left (a+b x^2\right ) \left (-72 a^{3/4} \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )+32 a^{7/4} \sqrt [4]{b} \sqrt{x}-5 \sqrt{2} \left (a+b x^2\right )^2 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+5 \sqrt{2} \left (a+b x^2\right )^2 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-10 \sqrt{2} \left (a+b x^2\right )^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+10 \sqrt{2} \left (a+b x^2\right )^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )\right )}{128 a^{3/4} b^{9/4} x^{7/2} \left (\left (a+b x^2\right )^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d*x)^(7/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]
[Out]
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Maple [B] time = 0.024, size = 672, normalized size = 1.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x)^(7/2)/(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)^(7/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.300424, size = 392, normalized size = 0.86 \[ -\frac{20 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )} \left (-\frac{d^{14}}{a^{3} b^{9}}\right )^{\frac{1}{4}} \arctan \left (\frac{\left (-\frac{d^{14}}{a^{3} b^{9}}\right )^{\frac{1}{4}} a b^{2}}{\sqrt{d x} d^{3} + \sqrt{d^{7} x + \sqrt{-\frac{d^{14}}{a^{3} b^{9}}} a^{2} b^{4}}}\right ) - 5 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )} \left (-\frac{d^{14}}{a^{3} b^{9}}\right )^{\frac{1}{4}} \log \left (5 \, \sqrt{d x} d^{3} + 5 \, \left (-\frac{d^{14}}{a^{3} b^{9}}\right )^{\frac{1}{4}} a b^{2}\right ) + 5 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )} \left (-\frac{d^{14}}{a^{3} b^{9}}\right )^{\frac{1}{4}} \log \left (5 \, \sqrt{d x} d^{3} - 5 \, \left (-\frac{d^{14}}{a^{3} b^{9}}\right )^{\frac{1}{4}} a b^{2}\right ) + 4 \,{\left (9 \, b d^{3} x^{2} + 5 \, a d^{3}\right )} \sqrt{d x}}{64 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)^(7/2)/(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)**(7/2)/(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.288672, size = 501, normalized size = 1.09 \[ \frac{1}{128} \, d^{2}{\left (\frac{10 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a b^{3}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{10 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a b^{3}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{5 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a b^{3}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{5 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a b^{3}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{8 \,{\left (9 \, \sqrt{d x} b d^{5} x^{2} + 5 \, \sqrt{d x} a d^{5}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{2} b^{2}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)^(7/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2),x, algorithm="giac")
[Out]