3.725 \(\int \frac{1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx\)

Optimal. Leaf size=404 \[ -\frac{13923 \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{25/4} d^{3/2}}+\frac{13923 \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{25/4} d^{3/2}}+\frac{13923 \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{25/4} d^{3/2}}-\frac{13923 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} a^{25/4} d^{3/2}}-\frac{13923}{4096 a^6 d \sqrt{d x}}+\frac{13923}{20480 a^5 d \sqrt{d x} \left (a+b x^2\right )}+\frac{1547}{5120 a^4 d \sqrt{d x} \left (a+b x^2\right )^2}+\frac{119}{640 a^3 d \sqrt{d x} \left (a+b x^2\right )^3}+\frac{21}{160 a^2 d \sqrt{d x} \left (a+b x^2\right )^4}+\frac{1}{10 a d \sqrt{d x} \left (a+b x^2\right )^5} \]

[Out]

-13923/(4096*a^6*d*Sqrt[d*x]) + 1/(10*a*d*Sqrt[d*x]*(a + b*x^2)^5) + 21/(160*a^2
*d*Sqrt[d*x]*(a + b*x^2)^4) + 119/(640*a^3*d*Sqrt[d*x]*(a + b*x^2)^3) + 1547/(51
20*a^4*d*Sqrt[d*x]*(a + b*x^2)^2) + 13923/(20480*a^5*d*Sqrt[d*x]*(a + b*x^2)) +
(13923*b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*
Sqrt[2]*a^(25/4)*d^(3/2)) - (13923*b^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x]
)/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(25/4)*d^(3/2)) - (13923*b^(1/4)*Log[Sqrt[
a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqrt
[2]*a^(25/4)*d^(3/2)) + (13923*b^(1/4)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x +
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqrt[2]*a^(25/4)*d^(3/2))

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Rubi [A]  time = 1.08302, antiderivative size = 404, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ -\frac{13923 \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{25/4} d^{3/2}}+\frac{13923 \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{25/4} d^{3/2}}+\frac{13923 \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{25/4} d^{3/2}}-\frac{13923 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} a^{25/4} d^{3/2}}-\frac{13923}{4096 a^6 d \sqrt{d x}}+\frac{13923}{20480 a^5 d \sqrt{d x} \left (a+b x^2\right )}+\frac{1547}{5120 a^4 d \sqrt{d x} \left (a+b x^2\right )^2}+\frac{119}{640 a^3 d \sqrt{d x} \left (a+b x^2\right )^3}+\frac{21}{160 a^2 d \sqrt{d x} \left (a+b x^2\right )^4}+\frac{1}{10 a d \sqrt{d x} \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]  Int[1/((d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]

[Out]

-13923/(4096*a^6*d*Sqrt[d*x]) + 1/(10*a*d*Sqrt[d*x]*(a + b*x^2)^5) + 21/(160*a^2
*d*Sqrt[d*x]*(a + b*x^2)^4) + 119/(640*a^3*d*Sqrt[d*x]*(a + b*x^2)^3) + 1547/(51
20*a^4*d*Sqrt[d*x]*(a + b*x^2)^2) + 13923/(20480*a^5*d*Sqrt[d*x]*(a + b*x^2)) +
(13923*b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*
Sqrt[2]*a^(25/4)*d^(3/2)) - (13923*b^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x]
)/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(25/4)*d^(3/2)) - (13923*b^(1/4)*Log[Sqrt[
a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqrt
[2]*a^(25/4)*d^(3/2)) + (13923*b^(1/4)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x +
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqrt[2]*a^(25/4)*d^(3/2))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(d*x)**(3/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.41306, size = 313, normalized size = 0.77 \[ \frac{x \left (-\frac{16384 a^{17/4} b x^2}{\left (a+b x^2\right )^5}-\frac{37888 a^{13/4} b x^2}{\left (a+b x^2\right )^4}-\frac{68352 a^{9/4} b x^2}{\left (a+b x^2\right )^3}-\frac{117856 a^{5/4} b x^2}{\left (a+b x^2\right )^2}-\frac{229240 \sqrt [4]{a} b x^2}{a+b x^2}-69615 \sqrt{2} \sqrt [4]{b} \sqrt{x} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+69615 \sqrt{2} \sqrt [4]{b} \sqrt{x} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+139230 \sqrt{2} \sqrt [4]{b} \sqrt{x} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )-139230 \sqrt{2} \sqrt [4]{b} \sqrt{x} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )-327680 \sqrt [4]{a}\right )}{163840 a^{25/4} (d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]

[Out]

(x*(-327680*a^(1/4) - (16384*a^(17/4)*b*x^2)/(a + b*x^2)^5 - (37888*a^(13/4)*b*x
^2)/(a + b*x^2)^4 - (68352*a^(9/4)*b*x^2)/(a + b*x^2)^3 - (117856*a^(5/4)*b*x^2)
/(a + b*x^2)^2 - (229240*a^(1/4)*b*x^2)/(a + b*x^2) + 139230*Sqrt[2]*b^(1/4)*Sqr
t[x]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 139230*Sqrt[2]*b^(1/4)*Sqrt
[x]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 69615*Sqrt[2]*b^(1/4)*Sqrt[x
]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 69615*Sqrt[2]*b^(
1/4)*Sqrt[x]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]))/(16384
0*a^(25/4)*(d*x)^(3/2))

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Maple [A]  time = 0.041, size = 349, normalized size = 0.9 \[ -2\,{\frac{1}{{a}^{6}d\sqrt{dx}}}-{\frac{11743\,{d}^{7}b}{4096\,{a}^{2} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}} \left ( dx \right ) ^{{\frac{3}{2}}}}-{\frac{1129\,{d}^{5}{b}^{2}}{128\,{a}^{3} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}} \left ( dx \right ) ^{{\frac{7}{2}}}}-{\frac{22467\,{d}^{3}{b}^{3}}{2048\,{a}^{4} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}} \left ( dx \right ) ^{{\frac{11}{2}}}}-{\frac{16169\,{b}^{4}d}{2560\,{a}^{5} \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}} \left ( dx \right ) ^{{\frac{15}{2}}}}-{\frac{5731\,{b}^{5}}{4096\,{a}^{6}d \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}} \left ( dx \right ) ^{{\frac{19}{2}}}}-{\frac{13923\,\sqrt{2}}{32768\,{a}^{6}d}\ln \left ({1 \left ( dx-\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) \left ( dx+\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-{\frac{13923\,\sqrt{2}}{16384\,{a}^{6}d}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-{\frac{13923\,\sqrt{2}}{16384\,{a}^{6}d}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

-2/a^6/d/(d*x)^(1/2)-11743/4096*d^7*b/a^2/(b*d^2*x^2+a*d^2)^5*(d*x)^(3/2)-1129/1
28*d^5*b^2/a^3/(b*d^2*x^2+a*d^2)^5*(d*x)^(7/2)-22467/2048*d^3*b^3/a^4/(b*d^2*x^2
+a*d^2)^5*(d*x)^(11/2)-16169/2560*d*b^4/a^5/(b*d^2*x^2+a*d^2)^5*(d*x)^(15/2)-573
1/4096/d*b^5/a^6/(b*d^2*x^2+a*d^2)^5*(d*x)^(19/2)-13923/32768/d/a^6/(a*d^2/b)^(1
/4)*2^(1/2)*ln((d*x-(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2))/(d*x+(a
*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2)))-13923/16384/d/a^6/(a*d^2/b)^
(1/4)*2^(1/2)*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)+1)-13923/16384/d/a^6/(a
*d^2/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)-1)

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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)^3*(d*x)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.326519, size = 659, normalized size = 1.63 \[ -\frac{278460 \, b^{5} x^{10} + 1336608 \, a b^{4} x^{8} + 2537080 \, a^{2} b^{3} x^{6} + 2360960 \, a^{3} b^{2} x^{4} + 1054060 \, a^{4} b x^{2} + 163840 \, a^{5} + 278460 \,{\left (a^{6} b^{5} d x^{10} + 5 \, a^{7} b^{4} d x^{8} + 10 \, a^{8} b^{3} d x^{6} + 10 \, a^{9} b^{2} d x^{4} + 5 \, a^{10} b d x^{2} + a^{11} d\right )} \sqrt{d x} \left (-\frac{b}{a^{25} d^{6}}\right )^{\frac{1}{4}} \arctan \left (\frac{2698972561467 \, a^{19} d^{5} \left (-\frac{b}{a^{25} d^{6}}\right )^{\frac{3}{4}}}{2698972561467 \, \sqrt{d x} b + \sqrt{-7284452887551739093192089 \, a^{13} b d^{4} \sqrt{-\frac{b}{a^{25} d^{6}}} + 7284452887551739093192089 \, b^{2} d x}}\right ) + 69615 \,{\left (a^{6} b^{5} d x^{10} + 5 \, a^{7} b^{4} d x^{8} + 10 \, a^{8} b^{3} d x^{6} + 10 \, a^{9} b^{2} d x^{4} + 5 \, a^{10} b d x^{2} + a^{11} d\right )} \sqrt{d x} \left (-\frac{b}{a^{25} d^{6}}\right )^{\frac{1}{4}} \log \left (2698972561467 \, a^{19} d^{5} \left (-\frac{b}{a^{25} d^{6}}\right )^{\frac{3}{4}} + 2698972561467 \, \sqrt{d x} b\right ) - 69615 \,{\left (a^{6} b^{5} d x^{10} + 5 \, a^{7} b^{4} d x^{8} + 10 \, a^{8} b^{3} d x^{6} + 10 \, a^{9} b^{2} d x^{4} + 5 \, a^{10} b d x^{2} + a^{11} d\right )} \sqrt{d x} \left (-\frac{b}{a^{25} d^{6}}\right )^{\frac{1}{4}} \log \left (-2698972561467 \, a^{19} d^{5} \left (-\frac{b}{a^{25} d^{6}}\right )^{\frac{3}{4}} + 2698972561467 \, \sqrt{d x} b\right )}{81920 \,{\left (a^{6} b^{5} d x^{10} + 5 \, a^{7} b^{4} d x^{8} + 10 \, a^{8} b^{3} d x^{6} + 10 \, a^{9} b^{2} d x^{4} + 5 \, a^{10} b d x^{2} + a^{11} d\right )} \sqrt{d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)^3*(d*x)^(3/2)),x, algorithm="fricas")

[Out]

-1/81920*(278460*b^5*x^10 + 1336608*a*b^4*x^8 + 2537080*a^2*b^3*x^6 + 2360960*a^
3*b^2*x^4 + 1054060*a^4*b*x^2 + 163840*a^5 + 278460*(a^6*b^5*d*x^10 + 5*a^7*b^4*
d*x^8 + 10*a^8*b^3*d*x^6 + 10*a^9*b^2*d*x^4 + 5*a^10*b*d*x^2 + a^11*d)*sqrt(d*x)
*(-b/(a^25*d^6))^(1/4)*arctan(2698972561467*a^19*d^5*(-b/(a^25*d^6))^(3/4)/(2698
972561467*sqrt(d*x)*b + sqrt(-7284452887551739093192089*a^13*b*d^4*sqrt(-b/(a^25
*d^6)) + 7284452887551739093192089*b^2*d*x))) + 69615*(a^6*b^5*d*x^10 + 5*a^7*b^
4*d*x^8 + 10*a^8*b^3*d*x^6 + 10*a^9*b^2*d*x^4 + 5*a^10*b*d*x^2 + a^11*d)*sqrt(d*
x)*(-b/(a^25*d^6))^(1/4)*log(2698972561467*a^19*d^5*(-b/(a^25*d^6))^(3/4) + 2698
972561467*sqrt(d*x)*b) - 69615*(a^6*b^5*d*x^10 + 5*a^7*b^4*d*x^8 + 10*a^8*b^3*d*
x^6 + 10*a^9*b^2*d*x^4 + 5*a^10*b*d*x^2 + a^11*d)*sqrt(d*x)*(-b/(a^25*d^6))^(1/4
)*log(-2698972561467*a^19*d^5*(-b/(a^25*d^6))^(3/4) + 2698972561467*sqrt(d*x)*b)
)/((a^6*b^5*d*x^10 + 5*a^7*b^4*d*x^8 + 10*a^8*b^3*d*x^6 + 10*a^9*b^2*d*x^4 + 5*a
^10*b*d*x^2 + a^11*d)*sqrt(d*x))

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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(1/(d*x)**(3/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.278729, size = 493, normalized size = 1.22 \[ -\frac{\frac{327680}{\sqrt{d x} a^{6}} + \frac{139230 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a^{7} b^{2} d^{2}} + \frac{139230 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a^{7} b^{2} d^{2}} - \frac{69615 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a^{7} b^{2} d^{2}} + \frac{69615 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a^{7} b^{2} d^{2}} + \frac{8 \,{\left (28655 \, \sqrt{d x} b^{5} d^{9} x^{9} + 129352 \, \sqrt{d x} a b^{4} d^{9} x^{7} + 224670 \, \sqrt{d x} a^{2} b^{3} d^{9} x^{5} + 180640 \, \sqrt{d x} a^{3} b^{2} d^{9} x^{3} + 58715 \, \sqrt{d x} a^{4} b d^{9} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{6}}}{163840 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)^3*(d*x)^(3/2)),x, algorithm="giac")

[Out]

-1/163840*(327680/(sqrt(d*x)*a^6) + 139230*sqrt(2)*(a*b^3*d^2)^(3/4)*arctan(1/2*
sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a^7*b^2*d^2) +
139230*sqrt(2)*(a*b^3*d^2)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) -
2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a^7*b^2*d^2) - 69615*sqrt(2)*(a*b^3*d^2)^(3/4)*l
n(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^7*b^2*d^2) + 69615
*sqrt(2)*(a*b^3*d^2)^(3/4)*ln(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d
^2/b))/(a^7*b^2*d^2) + 8*(28655*sqrt(d*x)*b^5*d^9*x^9 + 129352*sqrt(d*x)*a*b^4*d
^9*x^7 + 224670*sqrt(d*x)*a^2*b^3*d^9*x^5 + 180640*sqrt(d*x)*a^3*b^2*d^9*x^3 + 5
8715*sqrt(d*x)*a^4*b*d^9*x)/((b*d^2*x^2 + a*d^2)^5*a^6))/d