Optimal. Leaf size=333 \[ \frac{77 d^{13/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} \sqrt [4]{a} b^{15/4}}-\frac{77 d^{13/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} \sqrt [4]{a} b^{15/4}}-\frac{77 d^{13/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} \sqrt [4]{a} b^{15/4}}+\frac{77 d^{13/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} \sqrt [4]{a} b^{15/4}}-\frac{77 d^5 (d x)^{3/2}}{192 b^3 \left (a+b x^2\right )}-\frac{11 d^3 (d x)^{7/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac{d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3} \]
[Out]
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Rubi [A] time = 0.721298, antiderivative size = 333, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.321 \[ \frac{77 d^{13/2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} \sqrt [4]{a} b^{15/4}}-\frac{77 d^{13/2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} \sqrt [4]{a} b^{15/4}}-\frac{77 d^{13/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} \sqrt [4]{a} b^{15/4}}+\frac{77 d^{13/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} \sqrt [4]{a} b^{15/4}}-\frac{77 d^5 (d x)^{3/2}}{192 b^3 \left (a+b x^2\right )}-\frac{11 d^3 (d x)^{7/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac{d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
[In] Int[(d*x)^(13/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]
[Out]
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Rubi in Sympy [A] time = 147.988, size = 313, normalized size = 0.94 \[ - \frac{d \left (d x\right )^{\frac{11}{2}}}{6 b \left (a + b x^{2}\right )^{3}} - \frac{11 d^{3} \left (d x\right )^{\frac{7}{2}}}{48 b^{2} \left (a + b x^{2}\right )^{2}} - \frac{77 d^{5} \left (d x\right )^{\frac{3}{2}}}{192 b^{3} \left (a + b x^{2}\right )} + \frac{77 \sqrt{2} d^{\frac{13}{2}} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d} \sqrt{d x} + \sqrt{a} d + \sqrt{b} d x \right )}}{512 \sqrt [4]{a} b^{\frac{15}{4}}} - \frac{77 \sqrt{2} d^{\frac{13}{2}} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d} \sqrt{d x} + \sqrt{a} d + \sqrt{b} d x \right )}}{512 \sqrt [4]{a} b^{\frac{15}{4}}} - \frac{77 \sqrt{2} d^{\frac{13}{2}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}} \right )}}{256 \sqrt [4]{a} b^{\frac{15}{4}}} + \frac{77 \sqrt{2} d^{\frac{13}{2}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}} \right )}}{256 \sqrt [4]{a} b^{\frac{15}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x)**(13/2)/(b**2*x**4+2*a*b*x**2+a**2)**2,x)
[Out]
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Mathematica [A] time = 0.270199, size = 260, normalized size = 0.78 \[ \frac{d^6 \sqrt{d x} \left (-\frac{256 a^2 b^{3/4} x^{3/2}}{\left (a+b x^2\right )^3}+\frac{864 a b^{3/4} x^{3/2}}{\left (a+b x^2\right )^2}-\frac{1224 b^{3/4} x^{3/2}}{a+b x^2}+\frac{231 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{a}}-\frac{231 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{a}}-\frac{462 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a}}+\frac{462 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{a}}\right )}{1536 b^{15/4} \sqrt{x}} \]
Antiderivative was successfully verified.
[In] Integrate[(d*x)^(13/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]
[Out]
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Maple [A] time = 0.027, size = 271, normalized size = 0.8 \[ -{\frac{51\,{d}^{7}}{64\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}b} \left ( dx \right ) ^{{\frac{11}{2}}}}-{\frac{33\,{d}^{9}a}{32\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}{b}^{2}} \left ( dx \right ) ^{{\frac{7}{2}}}}-{\frac{77\,{d}^{11}{a}^{2}}{192\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}{b}^{3}} \left ( dx \right ) ^{{\frac{3}{2}}}}+{\frac{77\,{d}^{7}\sqrt{2}}{512\,{b}^{4}}\ln \left ({1 \left ( dx-\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) \left ( dx+\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{77\,{d}^{7}\sqrt{2}}{256\,{b}^{4}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{77\,{d}^{7}\sqrt{2}}{256\,{b}^{4}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x)^(13/2)/(b^2*x^4+2*a*b*x^2+a^2)^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)^(13/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.285922, size = 474, normalized size = 1.42 \[ \frac{924 \,{\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )} \left (-\frac{d^{26}}{a b^{15}}\right )^{\frac{1}{4}} \arctan \left (\frac{\left (-\frac{d^{26}}{a b^{15}}\right )^{\frac{3}{4}} a b^{11}}{\sqrt{d x} d^{19} + \sqrt{d^{39} x - \sqrt{-\frac{d^{26}}{a b^{15}}} a b^{7} d^{26}}}\right ) + 231 \,{\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )} \left (-\frac{d^{26}}{a b^{15}}\right )^{\frac{1}{4}} \log \left (456533 \, \sqrt{d x} d^{19} + 456533 \, \left (-\frac{d^{26}}{a b^{15}}\right )^{\frac{3}{4}} a b^{11}\right ) - 231 \,{\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )} \left (-\frac{d^{26}}{a b^{15}}\right )^{\frac{1}{4}} \log \left (456533 \, \sqrt{d x} d^{19} - 456533 \, \left (-\frac{d^{26}}{a b^{15}}\right )^{\frac{3}{4}} a b^{11}\right ) - 4 \,{\left (153 \, b^{2} d^{6} x^{5} + 198 \, a b d^{6} x^{3} + 77 \, a^{2} d^{6} x\right )} \sqrt{d x}}{768 \,{\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)^(13/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^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)**(13/2)/(b**2*x**4+2*a*b*x**2+a**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.278115, size = 408, normalized size = 1.23 \[ \frac{1}{1536} \, d^{5}{\left (\frac{462 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a b^{6}} + \frac{462 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a b^{6}} - \frac{231 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a b^{6}} + \frac{231 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a b^{6}} - \frac{8 \,{\left (153 \, \sqrt{d x} b^{2} d^{7} x^{5} + 198 \, \sqrt{d x} a b d^{7} x^{3} + 77 \, \sqrt{d x} a^{2} d^{7} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} b^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x)^(13/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^2,x, algorithm="giac")
[Out]