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

Optimal. Leaf size=387 \[ \frac{663 \sqrt{d} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{21/4} b^{3/4}}-\frac{663 \sqrt{d} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{21/4} b^{3/4}}-\frac{663 \sqrt{d} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{21/4} b^{3/4}}+\frac{663 \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} a^{21/4} b^{3/4}}+\frac{663 (d x)^{3/2}}{4096 a^5 d \left (a+b x^2\right )}+\frac{663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac{221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac{17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac{(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5} \]

[Out]

(d*x)^(3/2)/(10*a*d*(a + b*x^2)^5) + (17*(d*x)^(3/2))/(160*a^2*d*(a + b*x^2)^4)
+ (221*(d*x)^(3/2))/(1920*a^3*d*(a + b*x^2)^3) + (663*(d*x)^(3/2))/(5120*a^4*d*(
a + b*x^2)^2) + (663*(d*x)^(3/2))/(4096*a^5*d*(a + b*x^2)) - (663*Sqrt[d]*ArcTan
[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(21/4)*b^(3
/4)) + (663*Sqrt[d]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(
8192*Sqrt[2]*a^(21/4)*b^(3/4)) + (663*Sqrt[d]*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^(21/4)*b^(3/4)) - (
663*Sqrt[d]*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sq
rt[d*x]])/(16384*Sqrt[2]*a^(21/4)*b^(3/4))

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Rubi [A]  time = 0.952312, 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 \[ \frac{663 \sqrt{d} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{21/4} b^{3/4}}-\frac{663 \sqrt{d} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{16384 \sqrt{2} a^{21/4} b^{3/4}}-\frac{663 \sqrt{d} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{8192 \sqrt{2} a^{21/4} b^{3/4}}+\frac{663 \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{8192 \sqrt{2} a^{21/4} b^{3/4}}+\frac{663 (d x)^{3/2}}{4096 a^5 d \left (a+b x^2\right )}+\frac{663 (d x)^{3/2}}{5120 a^4 d \left (a+b x^2\right )^2}+\frac{221 (d x)^{3/2}}{1920 a^3 d \left (a+b x^2\right )^3}+\frac{17 (d x)^{3/2}}{160 a^2 d \left (a+b x^2\right )^4}+\frac{(d x)^{3/2}}{10 a d \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

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

[Out]

(d*x)^(3/2)/(10*a*d*(a + b*x^2)^5) + (17*(d*x)^(3/2))/(160*a^2*d*(a + b*x^2)^4)
+ (221*(d*x)^(3/2))/(1920*a^3*d*(a + b*x^2)^3) + (663*(d*x)^(3/2))/(5120*a^4*d*(
a + b*x^2)^2) + (663*(d*x)^(3/2))/(4096*a^5*d*(a + b*x^2)) - (663*Sqrt[d]*ArcTan
[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(21/4)*b^(3
/4)) + (663*Sqrt[d]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(
8192*Sqrt[2]*a^(21/4)*b^(3/4)) + (663*Sqrt[d]*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^(21/4)*b^(3/4)) - (
663*Sqrt[d]*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sq
rt[d*x]])/(16384*Sqrt[2]*a^(21/4)*b^(3/4))

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

[Out]

Timed out

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Mathematica [A]  time = 0.38709, size = 295, normalized size = 0.76 \[ \frac{\sqrt{d x} \left (\frac{49152 a^{17/4} x^{3/2}}{\left (a+b x^2\right )^5}+\frac{52224 a^{13/4} x^{3/2}}{\left (a+b x^2\right )^4}+\frac{56576 a^{9/4} x^{3/2}}{\left (a+b x^2\right )^3}+\frac{63648 a^{5/4} x^{3/2}}{\left (a+b x^2\right )^2}+\frac{9945 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{b^{3/4}}-\frac{9945 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{b^{3/4}}-\frac{19890 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{b^{3/4}}+\frac{19890 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{b^{3/4}}+\frac{79560 \sqrt [4]{a} x^{3/2}}{a+b x^2}\right )}{491520 a^{21/4} \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d*x]*((49152*a^(17/4)*x^(3/2))/(a + b*x^2)^5 + (52224*a^(13/4)*x^(3/2))/(a
+ b*x^2)^4 + (56576*a^(9/4)*x^(3/2))/(a + b*x^2)^3 + (63648*a^(5/4)*x^(3/2))/(a
+ b*x^2)^2 + (79560*a^(1/4)*x^(3/2))/(a + b*x^2) - (19890*Sqrt[2]*ArcTan[1 - (S
qrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(3/4) + (19890*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b
^(1/4)*Sqrt[x])/a^(1/4)])/b^(3/4) + (9945*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*
b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(3/4) - (9945*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1
/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(3/4)))/(491520*a^(21/4)*Sqrt[x])

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Maple [A]  time = 0.032, size = 336, normalized size = 0.9 \[{\frac{7529\,{d}^{9}}{12288\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}a} \left ( dx \right ) ^{{\frac{3}{2}}}}+{\frac{527\,{d}^{7}b}{384\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{a}^{2}} \left ( dx \right ) ^{{\frac{7}{2}}}}+{\frac{9061\,{d}^{5}{b}^{2}}{6144\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{a}^{3}} \left ( dx \right ) ^{{\frac{11}{2}}}}+{\frac{1989\,{d}^{3}{b}^{3}}{2560\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{a}^{4}} \left ( dx \right ) ^{{\frac{15}{2}}}}+{\frac{663\,{b}^{4}d}{4096\, \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) ^{5}{a}^{5}} \left ( dx \right ) ^{{\frac{19}{2}}}}+{\frac{663\,d\sqrt{2}}{32768\,{a}^{5}b}\ln \left ({1 \left ( dx-\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) \left ( dx+\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{663\,d\sqrt{2}}{16384\,{a}^{5}b}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+{\frac{663\,d\sqrt{2}}{16384\,{a}^{5}b}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

7529/12288*d^9/(b*d^2*x^2+a*d^2)^5/a*(d*x)^(3/2)+527/384*d^7/(b*d^2*x^2+a*d^2)^5
/a^2*b*(d*x)^(7/2)+9061/6144*d^5/(b*d^2*x^2+a*d^2)^5/a^3*b^2*(d*x)^(11/2)+1989/2
560*d^3/(b*d^2*x^2+a*d^2)^5/a^4*b^3*(d*x)^(15/2)+663/4096*d/(b*d^2*x^2+a*d^2)^5/
a^5*b^4*(d*x)^(19/2)+663/32768*d/a^5/b/(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)))+663/16384*d/a^5/b/(a*d^2/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/
(a*d^2/b)^(1/4)*(d*x)^(1/2)+1)+663/16384*d/a^5/b/(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(sqrt(d*x)/(b^2*x^4 + 2*a*b*x^2 + a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.288543, size = 609, normalized size = 1.57 \[ \frac{39780 \,{\left (a^{5} b^{5} x^{10} + 5 \, a^{6} b^{4} x^{8} + 10 \, a^{7} b^{3} x^{6} + 10 \, a^{8} b^{2} x^{4} + 5 \, a^{9} b x^{2} + a^{10}\right )} \left (-\frac{d^{2}}{a^{21} b^{3}}\right )^{\frac{1}{4}} \arctan \left (\frac{291434247 \, a^{16} b^{2} \left (-\frac{d^{2}}{a^{21} b^{3}}\right )^{\frac{3}{4}}}{291434247 \, \sqrt{d x} d + \sqrt{-84933920324457009 \, a^{11} b d^{2} \sqrt{-\frac{d^{2}}{a^{21} b^{3}}} + 84933920324457009 \, d^{3} x}}\right ) + 9945 \,{\left (a^{5} b^{5} x^{10} + 5 \, a^{6} b^{4} x^{8} + 10 \, a^{7} b^{3} x^{6} + 10 \, a^{8} b^{2} x^{4} + 5 \, a^{9} b x^{2} + a^{10}\right )} \left (-\frac{d^{2}}{a^{21} b^{3}}\right )^{\frac{1}{4}} \log \left (291434247 \, a^{16} b^{2} \left (-\frac{d^{2}}{a^{21} b^{3}}\right )^{\frac{3}{4}} + 291434247 \, \sqrt{d x} d\right ) - 9945 \,{\left (a^{5} b^{5} x^{10} + 5 \, a^{6} b^{4} x^{8} + 10 \, a^{7} b^{3} x^{6} + 10 \, a^{8} b^{2} x^{4} + 5 \, a^{9} b x^{2} + a^{10}\right )} \left (-\frac{d^{2}}{a^{21} b^{3}}\right )^{\frac{1}{4}} \log \left (-291434247 \, a^{16} b^{2} \left (-\frac{d^{2}}{a^{21} b^{3}}\right )^{\frac{3}{4}} + 291434247 \, \sqrt{d x} d\right ) + 4 \,{\left (9945 \, b^{4} x^{9} + 47736 \, a b^{3} x^{7} + 90610 \, a^{2} b^{2} x^{5} + 84320 \, a^{3} b x^{3} + 37645 \, a^{4} x\right )} \sqrt{d x}}{245760 \,{\left (a^{5} b^{5} x^{10} + 5 \, a^{6} b^{4} x^{8} + 10 \, a^{7} b^{3} x^{6} + 10 \, a^{8} b^{2} x^{4} + 5 \, a^{9} b x^{2} + a^{10}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/245760*(39780*(a^5*b^5*x^10 + 5*a^6*b^4*x^8 + 10*a^7*b^3*x^6 + 10*a^8*b^2*x^4
+ 5*a^9*b*x^2 + a^10)*(-d^2/(a^21*b^3))^(1/4)*arctan(291434247*a^16*b^2*(-d^2/(a
^21*b^3))^(3/4)/(291434247*sqrt(d*x)*d + sqrt(-84933920324457009*a^11*b*d^2*sqrt
(-d^2/(a^21*b^3)) + 84933920324457009*d^3*x))) + 9945*(a^5*b^5*x^10 + 5*a^6*b^4*
x^8 + 10*a^7*b^3*x^6 + 10*a^8*b^2*x^4 + 5*a^9*b*x^2 + a^10)*(-d^2/(a^21*b^3))^(1
/4)*log(291434247*a^16*b^2*(-d^2/(a^21*b^3))^(3/4) + 291434247*sqrt(d*x)*d) - 99
45*(a^5*b^5*x^10 + 5*a^6*b^4*x^8 + 10*a^7*b^3*x^6 + 10*a^8*b^2*x^4 + 5*a^9*b*x^2
+ a^10)*(-d^2/(a^21*b^3))^(1/4)*log(-291434247*a^16*b^2*(-d^2/(a^21*b^3))^(3/4)
+ 291434247*sqrt(d*x)*d) + 4*(9945*b^4*x^9 + 47736*a*b^3*x^7 + 90610*a^2*b^2*x^
5 + 84320*a^3*b*x^3 + 37645*a^4*x)*sqrt(d*x))/(a^5*b^5*x^10 + 5*a^6*b^4*x^8 + 10
*a^7*b^3*x^6 + 10*a^8*b^2*x^4 + 5*a^9*b*x^2 + a^10)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.280483, size = 468, normalized size = 1.21 \[ \frac{663 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{16384 \, a^{6} b^{3} d} + \frac{663 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{16384 \, a^{6} b^{3} d} - \frac{663 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{32768 \, a^{6} b^{3} d} + \frac{663 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{32768 \, a^{6} b^{3} d} + \frac{9945 \, \sqrt{d x} b^{4} d^{10} x^{9} + 47736 \, \sqrt{d x} a b^{3} d^{10} x^{7} + 90610 \, \sqrt{d x} a^{2} b^{2} d^{10} x^{5} + 84320 \, \sqrt{d x} a^{3} b d^{10} x^{3} + 37645 \, \sqrt{d x} a^{4} d^{10} x}{61440 \,{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

663/16384*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^6*b^3*d) + 663/16384*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^6*b^3*d) - 663/32768*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^6*b^3*d) + 663/32768*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^6*b^3*d) + 1/61440
*(9945*sqrt(d*x)*b^4*d^10*x^9 + 47736*sqrt(d*x)*a*b^3*d^10*x^7 + 90610*sqrt(d*x)
*a^2*b^2*d^10*x^5 + 84320*sqrt(d*x)*a^3*b*d^10*x^3 + 37645*sqrt(d*x)*a^4*d^10*x)
/((b*d^2*x^2 + a*d^2)^5*a^5)