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

Optimal. Leaf size=387 \[ -\frac{4389 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{23/4} \sqrt [4]{b} \sqrt{d}}+\frac{4389 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{23/4} \sqrt [4]{b} \sqrt{d}}-\frac{4389 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{23/4} \sqrt [4]{b} \sqrt{d}}+\frac{4389 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} a^{23/4} \sqrt [4]{b} \sqrt{d}}+\frac{1463 \sqrt{d x}}{4096 a^5 d \left (a+b x^2\right )}+\frac{209 \sqrt{d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac{19 \sqrt{d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac{19 \sqrt{d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac{\sqrt{d x}}{10 a d \left (a+b x^2\right )^5} \]

[Out]

Sqrt[d*x]/(10*a*d*(a + b*x^2)^5) + (19*Sqrt[d*x])/(160*a^2*d*(a + b*x^2)^4) + (1
9*Sqrt[d*x])/(128*a^3*d*(a + b*x^2)^3) + (209*Sqrt[d*x])/(1024*a^4*d*(a + b*x^2)
^2) + (1463*Sqrt[d*x])/(4096*a^5*d*(a + b*x^2)) - (4389*ArcTan[1 - (Sqrt[2]*b^(1
/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(23/4)*b^(1/4)*Sqrt[d]) + (43
89*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(2
3/4)*b^(1/4)*Sqrt[d]) - (4389*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^(23/4)*b^(1/4)*Sqrt[d]) + (4389*Log
[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(1638
4*Sqrt[2]*a^(23/4)*b^(1/4)*Sqrt[d])

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Rubi [A]  time = 0.960465, antiderivative size = 387, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.321 \[ -\frac{4389 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{23/4} \sqrt [4]{b} \sqrt{d}}+\frac{4389 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{23/4} \sqrt [4]{b} \sqrt{d}}-\frac{4389 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{23/4} \sqrt [4]{b} \sqrt{d}}+\frac{4389 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} a^{23/4} \sqrt [4]{b} \sqrt{d}}+\frac{1463 \sqrt{d x}}{4096 a^5 d \left (a+b x^2\right )}+\frac{209 \sqrt{d x}}{1024 a^4 d \left (a+b x^2\right )^2}+\frac{19 \sqrt{d x}}{128 a^3 d \left (a+b x^2\right )^3}+\frac{19 \sqrt{d x}}{160 a^2 d \left (a+b x^2\right )^4}+\frac{\sqrt{d x}}{10 a d \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[d*x]/(10*a*d*(a + b*x^2)^5) + (19*Sqrt[d*x])/(160*a^2*d*(a + b*x^2)^4) + (1
9*Sqrt[d*x])/(128*a^3*d*(a + b*x^2)^3) + (209*Sqrt[d*x])/(1024*a^4*d*(a + b*x^2)
^2) + (1463*Sqrt[d*x])/(4096*a^5*d*(a + b*x^2)) - (4389*ArcTan[1 - (Sqrt[2]*b^(1
/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(23/4)*b^(1/4)*Sqrt[d]) + (43
89*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(2
3/4)*b^(1/4)*Sqrt[d]) - (4389*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^(23/4)*b^(1/4)*Sqrt[d]) + (4389*Log
[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(1638
4*Sqrt[2]*a^(23/4)*b^(1/4)*Sqrt[d])

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

[Out]

Timed out

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Mathematica [A]  time = 0.474855, size = 295, normalized size = 0.76 \[ \frac{\sqrt{x} \left (\frac{16384 a^{19/4} \sqrt{x}}{\left (a+b x^2\right )^5}+\frac{19456 a^{15/4} \sqrt{x}}{\left (a+b x^2\right )^4}+\frac{24320 a^{11/4} \sqrt{x}}{\left (a+b x^2\right )^3}+\frac{33440 a^{7/4} \sqrt{x}}{\left (a+b x^2\right )^2}+\frac{58520 a^{3/4} \sqrt{x}}{a+b x^2}-\frac{21945 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{b}}+\frac{21945 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{b}}-\frac{43890 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{b}}+\frac{43890 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{b}}\right )}{163840 a^{23/4} \sqrt{d x}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*((16384*a^(19/4)*Sqrt[x])/(a + b*x^2)^5 + (19456*a^(15/4)*Sqrt[x])/(a +
b*x^2)^4 + (24320*a^(11/4)*Sqrt[x])/(a + b*x^2)^3 + (33440*a^(7/4)*Sqrt[x])/(a
+ b*x^2)^2 + (58520*a^(3/4)*Sqrt[x])/(a + b*x^2) - (43890*Sqrt[2]*ArcTan[1 - (Sq
rt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(1/4) + (43890*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^
(1/4)*Sqrt[x])/a^(1/4)])/b^(1/4) - (21945*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*
b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(1/4) + (21945*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(
1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(1/4)))/(163840*a^(23/4)*Sqrt[d*x])

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Maple [A]  time = 0.032, size = 333, normalized size = 0.9 \[{\frac{3803\,{d}^{9}}{4096\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}a}\sqrt{dx}}+{\frac{6289\,{d}^{7}b}{2560\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{a}^{2}} \left ( dx \right ) ^{{\frac{5}{2}}}}+{\frac{5947\,{d}^{5}{b}^{2}}{2048\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{a}^{3}} \left ( dx \right ) ^{{\frac{9}{2}}}}+{\frac{209\,{d}^{3}{b}^{3}}{128\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{a}^{4}} \left ( dx \right ) ^{{\frac{13}{2}}}}+{\frac{1463\,{b}^{4}d}{4096\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{a}^{5}} \left ( dx \right ) ^{{\frac{17}{2}}}}+{\frac{4389\,\sqrt{2}}{32768\,d{a}^{6}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\ln \left ({1 \left ( dx+\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) \left ( dx-\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) ^{-1}} \right ) }+{\frac{4389\,\sqrt{2}}{16384\,d{a}^{6}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ) }+{\frac{4389\,\sqrt{2}}{16384\,d{a}^{6}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3803/4096*d^9/(b*d^2*x^2+a*d^2)^5/a*(d*x)^(1/2)+6289/2560*d^7/(b*d^2*x^2+a*d^2)^
5/a^2*b*(d*x)^(5/2)+5947/2048*d^5/(b*d^2*x^2+a*d^2)^5/a^3*b^2*(d*x)^(9/2)+209/12
8*d^3/(b*d^2*x^2+a*d^2)^5/a^4*b^3*(d*x)^(13/2)+1463/4096*d/(b*d^2*x^2+a*d^2)^5/a
^5*b^4*(d*x)^(17/2)+4389/32768/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)))+4389/16384/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)+4389/16384/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*sqrt(d*x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.289134, size = 616, normalized size = 1.59 \[ -\frac{87780 \,{\left (a^{5} b^{5} d x^{10} + 5 \, a^{6} b^{4} d x^{8} + 10 \, a^{7} b^{3} d x^{6} + 10 \, a^{8} b^{2} d x^{4} + 5 \, a^{9} b d x^{2} + a^{10} d\right )} \left (-\frac{1}{a^{23} b d^{2}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{6} d \left (-\frac{1}{a^{23} b d^{2}}\right )^{\frac{1}{4}}}{\sqrt{a^{12} d^{2} \sqrt{-\frac{1}{a^{23} b d^{2}}} + d x} + \sqrt{d x}}\right ) - 21945 \,{\left (a^{5} b^{5} d x^{10} + 5 \, a^{6} b^{4} d x^{8} + 10 \, a^{7} b^{3} d x^{6} + 10 \, a^{8} b^{2} d x^{4} + 5 \, a^{9} b d x^{2} + a^{10} d\right )} \left (-\frac{1}{a^{23} b d^{2}}\right )^{\frac{1}{4}} \log \left (a^{6} d \left (-\frac{1}{a^{23} b d^{2}}\right )^{\frac{1}{4}} + \sqrt{d x}\right ) + 21945 \,{\left (a^{5} b^{5} d x^{10} + 5 \, a^{6} b^{4} d x^{8} + 10 \, a^{7} b^{3} d x^{6} + 10 \, a^{8} b^{2} d x^{4} + 5 \, a^{9} b d x^{2} + a^{10} d\right )} \left (-\frac{1}{a^{23} b d^{2}}\right )^{\frac{1}{4}} \log \left (-a^{6} d \left (-\frac{1}{a^{23} b d^{2}}\right )^{\frac{1}{4}} + \sqrt{d x}\right ) - 4 \,{\left (7315 \, b^{4} x^{8} + 33440 \, a b^{3} x^{6} + 59470 \, a^{2} b^{2} x^{4} + 50312 \, a^{3} b x^{2} + 19015 \, a^{4}\right )} \sqrt{d x}}{81920 \,{\left (a^{5} b^{5} d x^{10} + 5 \, a^{6} b^{4} d x^{8} + 10 \, a^{7} b^{3} d x^{6} + 10 \, a^{8} b^{2} d x^{4} + 5 \, a^{9} b d x^{2} + a^{10} d\right )}} \]

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*sqrt(d*x)),x, algorithm="fricas")

[Out]

-1/81920*(87780*(a^5*b^5*d*x^10 + 5*a^6*b^4*d*x^8 + 10*a^7*b^3*d*x^6 + 10*a^8*b^
2*d*x^4 + 5*a^9*b*d*x^2 + a^10*d)*(-1/(a^23*b*d^2))^(1/4)*arctan(a^6*d*(-1/(a^23
*b*d^2))^(1/4)/(sqrt(a^12*d^2*sqrt(-1/(a^23*b*d^2)) + d*x) + sqrt(d*x))) - 21945
*(a^5*b^5*d*x^10 + 5*a^6*b^4*d*x^8 + 10*a^7*b^3*d*x^6 + 10*a^8*b^2*d*x^4 + 5*a^9
*b*d*x^2 + a^10*d)*(-1/(a^23*b*d^2))^(1/4)*log(a^6*d*(-1/(a^23*b*d^2))^(1/4) + s
qrt(d*x)) + 21945*(a^5*b^5*d*x^10 + 5*a^6*b^4*d*x^8 + 10*a^7*b^3*d*x^6 + 10*a^8*
b^2*d*x^4 + 5*a^9*b*d*x^2 + a^10*d)*(-1/(a^23*b*d^2))^(1/4)*log(-a^6*d*(-1/(a^23
*b*d^2))^(1/4) + sqrt(d*x)) - 4*(7315*b^4*x^8 + 33440*a*b^3*x^6 + 59470*a^2*b^2*
x^4 + 50312*a^3*b*x^2 + 19015*a^4)*sqrt(d*x))/(a^5*b^5*d*x^10 + 5*a^6*b^4*d*x^8
+ 10*a^7*b^3*d*x^6 + 10*a^8*b^2*d*x^4 + 5*a^9*b*d*x^2 + a^10*d)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.274431, size = 467, normalized size = 1.21 \[ \frac{4389 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{16384 \, a^{6} b d} + \frac{4389 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{16384 \, a^{6} b d} + \frac{4389 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}}{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{32768 \, a^{6} b d} - \frac{4389 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}}{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{32768 \, a^{6} b d} + \frac{7315 \, \sqrt{d x} b^{4} d^{9} x^{8} + 33440 \, \sqrt{d x} a b^{3} d^{9} x^{6} + 59470 \, \sqrt{d x} a^{2} b^{2} d^{9} x^{4} + 50312 \, \sqrt{d x} a^{3} b d^{9} x^{2} + 19015 \, \sqrt{d x} a^{4} d^{9}}{20480 \,{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{5}} \]

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*sqrt(d*x)),x, algorithm="giac")

[Out]

4389/16384*sqrt(2)*(a*b^3*d^2)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4)
+ 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a^6*b*d) + 4389/16384*sqrt(2)*(a*b^3*d^2)^(1/4
)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(
a^6*b*d) + 4389/32768*sqrt(2)*(a*b^3*d^2)^(1/4)*ln(d*x + sqrt(2)*(a*d^2/b)^(1/4)
*sqrt(d*x) + sqrt(a*d^2/b))/(a^6*b*d) - 4389/32768*sqrt(2)*(a*b^3*d^2)^(1/4)*ln(
d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^6*b*d) + 1/20480*(73
15*sqrt(d*x)*b^4*d^9*x^8 + 33440*sqrt(d*x)*a*b^3*d^9*x^6 + 59470*sqrt(d*x)*a^2*b
^2*d^9*x^4 + 50312*sqrt(d*x)*a^3*b*d^9*x^2 + 19015*sqrt(d*x)*a^4*d^9)/((b*d^2*x^
2 + a*d^2)^5*a^5)