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

Optimal. Leaf size=335 \[ -\frac{77 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{15/4} \sqrt [4]{b} \sqrt{d}}+\frac{77 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{15/4} \sqrt [4]{b} \sqrt{d}}-\frac{77 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} a^{15/4} \sqrt [4]{b} \sqrt{d}}+\frac{77 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} a^{15/4} \sqrt [4]{b} \sqrt{d}}+\frac{77 \sqrt{d x}}{192 a^3 d \left (a+b x^2\right )}+\frac{11 \sqrt{d x}}{48 a^2 d \left (a+b x^2\right )^2}+\frac{\sqrt{d x}}{6 a d \left (a+b x^2\right )^3} \]

[Out]

Sqrt[d*x]/(6*a*d*(a + b*x^2)^3) + (11*Sqrt[d*x])/(48*a^2*d*(a + b*x^2)^2) + (77*
Sqrt[d*x])/(192*a^3*d*(a + b*x^2)) - (77*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/
(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*a^(15/4)*b^(1/4)*Sqrt[d]) + (77*ArcTan[1 + (Sqr
t[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*a^(15/4)*b^(1/4)*Sqrt[d
]) - (77*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[
d*x]])/(256*Sqrt[2]*a^(15/4)*b^(1/4)*Sqrt[d]) + (77*Log[Sqrt[a]*Sqrt[d] + Sqrt[b
]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(256*Sqrt[2]*a^(15/4)*b^(1/4)*
Sqrt[d])

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Rubi [A]  time = 0.741447, antiderivative size = 335, 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 \[ -\frac{77 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{15/4} \sqrt [4]{b} \sqrt{d}}+\frac{77 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{256 \sqrt{2} a^{15/4} \sqrt [4]{b} \sqrt{d}}-\frac{77 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{128 \sqrt{2} a^{15/4} \sqrt [4]{b} \sqrt{d}}+\frac{77 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{128 \sqrt{2} a^{15/4} \sqrt [4]{b} \sqrt{d}}+\frac{77 \sqrt{d x}}{192 a^3 d \left (a+b x^2\right )}+\frac{11 \sqrt{d x}}{48 a^2 d \left (a+b x^2\right )^2}+\frac{\sqrt{d x}}{6 a d \left (a+b x^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[d*x]/(6*a*d*(a + b*x^2)^3) + (11*Sqrt[d*x])/(48*a^2*d*(a + b*x^2)^2) + (77*
Sqrt[d*x])/(192*a^3*d*(a + b*x^2)) - (77*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/
(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*a^(15/4)*b^(1/4)*Sqrt[d]) + (77*ArcTan[1 + (Sqr
t[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*a^(15/4)*b^(1/4)*Sqrt[d
]) - (77*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[
d*x]])/(256*Sqrt[2]*a^(15/4)*b^(1/4)*Sqrt[d]) + (77*Log[Sqrt[a]*Sqrt[d] + Sqrt[b
]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(256*Sqrt[2]*a^(15/4)*b^(1/4)*
Sqrt[d])

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Rubi in Sympy [A]  time = 151.186, size = 309, normalized size = 0.92 \[ \frac{\sqrt{d x}}{6 a d \left (a + b x^{2}\right )^{3}} + \frac{11 \sqrt{d x}}{48 a^{2} d \left (a + b x^{2}\right )^{2}} + \frac{77 \sqrt{d x}}{192 a^{3} d \left (a + b x^{2}\right )} - \frac{77 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d} \sqrt{d x} + \sqrt{a} d + \sqrt{b} d x \right )}}{512 a^{\frac{15}{4}} \sqrt [4]{b} \sqrt{d}} + \frac{77 \sqrt{2} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d} \sqrt{d x} + \sqrt{a} d + \sqrt{b} d x \right )}}{512 a^{\frac{15}{4}} \sqrt [4]{b} \sqrt{d}} - \frac{77 \sqrt{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}} \right )}}{256 a^{\frac{15}{4}} \sqrt [4]{b} \sqrt{d}} + \frac{77 \sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}} \right )}}{256 a^{\frac{15}{4}} \sqrt [4]{b} \sqrt{d}} \]

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)**2/(d*x)**(1/2),x)

[Out]

sqrt(d*x)/(6*a*d*(a + b*x**2)**3) + 11*sqrt(d*x)/(48*a**2*d*(a + b*x**2)**2) + 7
7*sqrt(d*x)/(192*a**3*d*(a + b*x**2)) - 77*sqrt(2)*log(-sqrt(2)*a**(1/4)*b**(1/4
)*sqrt(d)*sqrt(d*x) + sqrt(a)*d + sqrt(b)*d*x)/(512*a**(15/4)*b**(1/4)*sqrt(d))
+ 77*sqrt(2)*log(sqrt(2)*a**(1/4)*b**(1/4)*sqrt(d)*sqrt(d*x) + sqrt(a)*d + sqrt(
b)*d*x)/(512*a**(15/4)*b**(1/4)*sqrt(d)) - 77*sqrt(2)*atan(1 - sqrt(2)*b**(1/4)*
sqrt(d*x)/(a**(1/4)*sqrt(d)))/(256*a**(15/4)*b**(1/4)*sqrt(d)) + 77*sqrt(2)*atan
(1 + sqrt(2)*b**(1/4)*sqrt(d*x)/(a**(1/4)*sqrt(d)))/(256*a**(15/4)*b**(1/4)*sqrt
(d))

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Mathematica [A]  time = 0.270083, size = 253, normalized size = 0.76 \[ \frac{\sqrt{x} \left (\frac{256 a^{11/4} \sqrt{x}}{\left (a+b x^2\right )^3}+\frac{352 a^{7/4} \sqrt{x}}{\left (a+b x^2\right )^2}+\frac{616 a^{3/4} \sqrt{x}}{a+b x^2}-\frac{231 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{b}}+\frac{231 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{b}}-\frac{462 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{b}}+\frac{462 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{b}}\right )}{1536 a^{15/4} \sqrt{d x}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*((256*a^(11/4)*Sqrt[x])/(a + b*x^2)^3 + (352*a^(7/4)*Sqrt[x])/(a + b*x^
2)^2 + (616*a^(3/4)*Sqrt[x])/(a + b*x^2) - (462*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1
/4)*Sqrt[x])/a^(1/4)])/b^(1/4) + (462*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x
])/a^(1/4)])/b^(1/4) - (231*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x
] + Sqrt[b]*x])/b^(1/4) + (231*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqr
t[x] + Sqrt[b]*x])/b^(1/4)))/(1536*a^(15/4)*Sqrt[d*x])

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Maple [A]  time = 0.026, size = 269, normalized size = 0.8 \[{\frac{77\,{b}^{2}d}{192\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}{a}^{3}} \left ( dx \right ) ^{{\frac{9}{2}}}}+{\frac{33\,{d}^{3}b}{32\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}{a}^{2}} \left ( dx \right ) ^{{\frac{5}{2}}}}+{\frac{51\,{d}^{5}}{64\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{3}a}\sqrt{dx}}+{\frac{77\,\sqrt{2}}{512\,d{a}^{4}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\ln \left ({1 \left ( dx+\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) \left ( dx-\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) ^{-1}} \right ) }+{\frac{77\,\sqrt{2}}{256\,d{a}^{4}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ) }+{\frac{77\,\sqrt{2}}{256\,d{a}^{4}}\sqrt [4]{{\frac{a{d}^{2}}{b}}}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

77/192*d/(b*d^2*x^2+a*d^2)^3/a^3*b^2*(d*x)^(9/2)+33/32*d^3/(b*d^2*x^2+a*d^2)^3/a
^2*b*(d*x)^(5/2)+51/64*d^5/(b*d^2*x^2+a*d^2)^3/a*(d*x)^(1/2)+77/512/d/a^4*(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)))+77/256/d/a^4*(a*d^2/b)
^(1/4)*2^(1/2)*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)+1)+77/256/d/a^4*(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)^2*sqrt(d*x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.286165, size = 456, normalized size = 1.36 \[ -\frac{924 \,{\left (a^{3} b^{3} d x^{6} + 3 \, a^{4} b^{2} d x^{4} + 3 \, a^{5} b d x^{2} + a^{6} d\right )} \left (-\frac{1}{a^{15} b d^{2}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{4} d \left (-\frac{1}{a^{15} b d^{2}}\right )^{\frac{1}{4}}}{\sqrt{a^{8} d^{2} \sqrt{-\frac{1}{a^{15} b d^{2}}} + d x} + \sqrt{d x}}\right ) - 231 \,{\left (a^{3} b^{3} d x^{6} + 3 \, a^{4} b^{2} d x^{4} + 3 \, a^{5} b d x^{2} + a^{6} d\right )} \left (-\frac{1}{a^{15} b d^{2}}\right )^{\frac{1}{4}} \log \left (a^{4} d \left (-\frac{1}{a^{15} b d^{2}}\right )^{\frac{1}{4}} + \sqrt{d x}\right ) + 231 \,{\left (a^{3} b^{3} d x^{6} + 3 \, a^{4} b^{2} d x^{4} + 3 \, a^{5} b d x^{2} + a^{6} d\right )} \left (-\frac{1}{a^{15} b d^{2}}\right )^{\frac{1}{4}} \log \left (-a^{4} d \left (-\frac{1}{a^{15} b d^{2}}\right )^{\frac{1}{4}} + \sqrt{d x}\right ) - 4 \,{\left (77 \, b^{2} x^{4} + 198 \, a b x^{2} + 153 \, a^{2}\right )} \sqrt{d x}}{768 \,{\left (a^{3} b^{3} d x^{6} + 3 \, a^{4} b^{2} d x^{4} + 3 \, a^{5} b d x^{2} + a^{6} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/768*(924*(a^3*b^3*d*x^6 + 3*a^4*b^2*d*x^4 + 3*a^5*b*d*x^2 + a^6*d)*(-1/(a^15*
b*d^2))^(1/4)*arctan(a^4*d*(-1/(a^15*b*d^2))^(1/4)/(sqrt(a^8*d^2*sqrt(-1/(a^15*b
*d^2)) + d*x) + sqrt(d*x))) - 231*(a^3*b^3*d*x^6 + 3*a^4*b^2*d*x^4 + 3*a^5*b*d*x
^2 + a^6*d)*(-1/(a^15*b*d^2))^(1/4)*log(a^4*d*(-1/(a^15*b*d^2))^(1/4) + sqrt(d*x
)) + 231*(a^3*b^3*d*x^6 + 3*a^4*b^2*d*x^4 + 3*a^5*b*d*x^2 + a^6*d)*(-1/(a^15*b*d
^2))^(1/4)*log(-a^4*d*(-1/(a^15*b*d^2))^(1/4) + sqrt(d*x)) - 4*(77*b^2*x^4 + 198
*a*b*x^2 + 153*a^2)*sqrt(d*x))/(a^3*b^3*d*x^6 + 3*a^4*b^2*d*x^4 + 3*a^5*b*d*x^2
+ a^6*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{d x} \left (a + b x^{2}\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.274027, size = 416, normalized size = 1.24 \[ \frac{77 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{256 \, a^{4} b d} + \frac{77 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{256 \, a^{4} b d} + \frac{77 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}}{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{512 \, a^{4} b d} - \frac{77 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}}{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{512 \, a^{4} b d} + \frac{77 \, \sqrt{d x} b^{2} d^{5} x^{4} + 198 \, \sqrt{d x} a b d^{5} x^{2} + 153 \, \sqrt{d x} a^{2} d^{5}}{192 \,{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

77/256*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^4*b*d) + 77/256*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^4*b*d)
+ 77/512*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^4*b*d) - 77/512*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^4*b*d) + 1/192*(77*sqrt(d*x)*b^2*d^5
*x^4 + 198*sqrt(d*x)*a*b*d^5*x^2 + 153*sqrt(d*x)*a^2*d^5)/((b*d^2*x^2 + a*d^2)^3
*a^3)