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

Optimal. Leaf size=300 \[ -\frac{5 \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} a^{9/4} d^{3/2}}+\frac{5 \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} a^{9/4} d^{3/2}}+\frac{5 \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} a^{9/4} d^{3/2}}-\frac{5 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{4 \sqrt{2} a^{9/4} d^{3/2}}-\frac{5}{2 a^2 d \sqrt{d x}}+\frac{1}{2 a d \sqrt{d x} \left (a+b x^2\right )} \]

[Out]

-5/(2*a^2*d*Sqrt[d*x]) + 1/(2*a*d*Sqrt[d*x]*(a + b*x^2)) + (5*b^(1/4)*ArcTan[1 -
(Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(4*Sqrt[2]*a^(9/4)*d^(3/2)) - (
5*b^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(4*Sqrt[2]*
a^(9/4)*d^(3/2)) - (5*b^(1/4)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*
a^(1/4)*b^(1/4)*Sqrt[d*x]])/(8*Sqrt[2]*a^(9/4)*d^(3/2)) + (5*b^(1/4)*Log[Sqrt[a]
*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(8*Sqrt[2]*a^
(9/4)*d^(3/2))

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Rubi [A]  time = 0.639628, antiderivative size = 300, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ -\frac{5 \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} a^{9/4} d^{3/2}}+\frac{5 \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{8 \sqrt{2} a^{9/4} d^{3/2}}+\frac{5 \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{4 \sqrt{2} a^{9/4} d^{3/2}}-\frac{5 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{4 \sqrt{2} a^{9/4} d^{3/2}}-\frac{5}{2 a^2 d \sqrt{d x}}+\frac{1}{2 a d \sqrt{d x} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-5/(2*a^2*d*Sqrt[d*x]) + 1/(2*a*d*Sqrt[d*x]*(a + b*x^2)) + (5*b^(1/4)*ArcTan[1 -
(Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(4*Sqrt[2]*a^(9/4)*d^(3/2)) - (
5*b^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(4*Sqrt[2]*
a^(9/4)*d^(3/2)) - (5*b^(1/4)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*
a^(1/4)*b^(1/4)*Sqrt[d*x]])/(8*Sqrt[2]*a^(9/4)*d^(3/2)) + (5*b^(1/4)*Log[Sqrt[a]
*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(8*Sqrt[2]*a^
(9/4)*d^(3/2))

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Rubi in Sympy [A]  time = 131.919, size = 279, normalized size = 0.93 \[ \frac{1}{2 a d \sqrt{d x} \left (a + b x^{2}\right )} - \frac{5}{2 a^{2} d \sqrt{d x}} - \frac{5 \sqrt{2} \sqrt [4]{b} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d} \sqrt{d x} + \sqrt{a} d + \sqrt{b} d x \right )}}{16 a^{\frac{9}{4}} d^{\frac{3}{2}}} + \frac{5 \sqrt{2} \sqrt [4]{b} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d} \sqrt{d x} + \sqrt{a} d + \sqrt{b} d x \right )}}{16 a^{\frac{9}{4}} d^{\frac{3}{2}}} + \frac{5 \sqrt{2} \sqrt [4]{b} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}} \right )}}{8 a^{\frac{9}{4}} d^{\frac{3}{2}}} - \frac{5 \sqrt{2} \sqrt [4]{b} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}} \right )}}{8 a^{\frac{9}{4}} d^{\frac{3}{2}}} \]

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),x)

[Out]

1/(2*a*d*sqrt(d*x)*(a + b*x**2)) - 5/(2*a**2*d*sqrt(d*x)) - 5*sqrt(2)*b**(1/4)*l
og(-sqrt(2)*a**(1/4)*b**(1/4)*sqrt(d)*sqrt(d*x) + sqrt(a)*d + sqrt(b)*d*x)/(16*a
**(9/4)*d**(3/2)) + 5*sqrt(2)*b**(1/4)*log(sqrt(2)*a**(1/4)*b**(1/4)*sqrt(d)*sqr
t(d*x) + sqrt(a)*d + sqrt(b)*d*x)/(16*a**(9/4)*d**(3/2)) + 5*sqrt(2)*b**(1/4)*at
an(1 - sqrt(2)*b**(1/4)*sqrt(d*x)/(a**(1/4)*sqrt(d)))/(8*a**(9/4)*d**(3/2)) - 5*
sqrt(2)*b**(1/4)*atan(1 + sqrt(2)*b**(1/4)*sqrt(d*x)/(a**(1/4)*sqrt(d)))/(8*a**(
9/4)*d**(3/2))

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Mathematica [A]  time = 0.297847, size = 233, normalized size = 0.78 \[ \frac{x \left (-\frac{8 \sqrt [4]{a} b x^2}{a+b x^2}-5 \sqrt{2} \sqrt [4]{b} \sqrt{x} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+5 \sqrt{2} \sqrt [4]{b} \sqrt{x} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+10 \sqrt{2} \sqrt [4]{b} \sqrt{x} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )-10 \sqrt{2} \sqrt [4]{b} \sqrt{x} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )-32 \sqrt [4]{a}\right )}{16 a^{9/4} (d x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(-32*a^(1/4) - (8*a^(1/4)*b*x^2)/(a + b*x^2) + 10*Sqrt[2]*b^(1/4)*Sqrt[x]*Arc
Tan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 10*Sqrt[2]*b^(1/4)*Sqrt[x]*ArcTan[1
+ (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 5*Sqrt[2]*b^(1/4)*Sqrt[x]*Log[Sqrt[a] -
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 5*Sqrt[2]*b^(1/4)*Sqrt[x]*Log[Sqr
t[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]))/(16*a^(9/4)*(d*x)^(3/2))

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Maple [A]  time = 0.027, size = 223, normalized size = 0.7 \[ -2\,{\frac{1}{{a}^{2}d\sqrt{dx}}}-{\frac{b}{2\,{a}^{2}d \left ( b{d}^{2}{x}^{2}+a{d}^{2} \right ) } \left ( dx \right ) ^{{\frac{3}{2}}}}-{\frac{5\,\sqrt{2}}{16\,{a}^{2}d}\ln \left ({1 \left ( dx-\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) \left ( dx+\sqrt [4]{{\frac{a{d}^{2}}{b}}}\sqrt{dx}\sqrt{2}+\sqrt{{\frac{a{d}^{2}}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-{\frac{5\,\sqrt{2}}{8\,{a}^{2}d}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-{\frac{5\,\sqrt{2}}{8\,{a}^{2}d}\arctan \left ({\sqrt{2}\sqrt{dx}{\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a{d}^{2}}{b}}}}}} \]

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),x)

[Out]

-2/a^2/d/(d*x)^(1/2)-1/2/d*b/a^2*(d*x)^(3/2)/(b*d^2*x^2+a*d^2)-5/16/d/a^2/(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)))-5/8/d/a^2/(a*d^2/b)^(1
/4)*2^(1/2)*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)+1)-5/8/d/a^2/(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)*(d*x)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.289882, size = 340, normalized size = 1.13 \[ -\frac{20 \, b x^{2} + 20 \,{\left (a^{2} b d x^{2} + a^{3} d\right )} \sqrt{d x} \left (-\frac{b}{a^{9} d^{6}}\right )^{\frac{1}{4}} \arctan \left (\frac{125 \, a^{7} d^{5} \left (-\frac{b}{a^{9} d^{6}}\right )^{\frac{3}{4}}}{125 \, \sqrt{d x} b + \sqrt{-15625 \, a^{5} b d^{4} \sqrt{-\frac{b}{a^{9} d^{6}}} + 15625 \, b^{2} d x}}\right ) + 5 \,{\left (a^{2} b d x^{2} + a^{3} d\right )} \sqrt{d x} \left (-\frac{b}{a^{9} d^{6}}\right )^{\frac{1}{4}} \log \left (125 \, a^{7} d^{5} \left (-\frac{b}{a^{9} d^{6}}\right )^{\frac{3}{4}} + 125 \, \sqrt{d x} b\right ) - 5 \,{\left (a^{2} b d x^{2} + a^{3} d\right )} \sqrt{d x} \left (-\frac{b}{a^{9} d^{6}}\right )^{\frac{1}{4}} \log \left (-125 \, a^{7} d^{5} \left (-\frac{b}{a^{9} d^{6}}\right )^{\frac{3}{4}} + 125 \, \sqrt{d x} b\right ) + 16 \, a}{8 \,{\left (a^{2} b d x^{2} + a^{3} d\right )} \sqrt{d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(20*b*x^2 + 20*(a^2*b*d*x^2 + a^3*d)*sqrt(d*x)*(-b/(a^9*d^6))^(1/4)*arctan(
125*a^7*d^5*(-b/(a^9*d^6))^(3/4)/(125*sqrt(d*x)*b + sqrt(-15625*a^5*b*d^4*sqrt(-
b/(a^9*d^6)) + 15625*b^2*d*x))) + 5*(a^2*b*d*x^2 + a^3*d)*sqrt(d*x)*(-b/(a^9*d^6
))^(1/4)*log(125*a^7*d^5*(-b/(a^9*d^6))^(3/4) + 125*sqrt(d*x)*b) - 5*(a^2*b*d*x^
2 + a^3*d)*sqrt(d*x)*(-b/(a^9*d^6))^(1/4)*log(-125*a^7*d^5*(-b/(a^9*d^6))^(3/4)
+ 125*sqrt(d*x)*b) + 16*a)/((a^2*b*d*x^2 + a^3*d)*sqrt(d*x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d x\right )^{\frac{3}{2}} \left (a + b x^{2}\right )^{2}}\, dx \]

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),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.273224, size = 397, normalized size = 1.32 \[ -\frac{\frac{8 \,{\left (5 \, b d^{2} x^{2} + 4 \, a d^{2}\right )}}{{\left (\sqrt{d x} b d^{2} x^{2} + \sqrt{d x} a d^{2}\right )} a^{2}} + \frac{10 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a^{3} b^{2} d^{2}} + \frac{10 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a^{3} b^{2} d^{2}} - \frac{5 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a^{3} b^{2} d^{2}} + \frac{5 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a^{3} b^{2} d^{2}}}{16 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/16*(8*(5*b*d^2*x^2 + 4*a*d^2)/((sqrt(d*x)*b*d^2*x^2 + sqrt(d*x)*a*d^2)*a^2) +
10*sqrt(2)*(a*b^3*d^2)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) + 2*sq
rt(d*x))/(a*d^2/b)^(1/4))/(a^3*b^2*d^2) + 10*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^3*b^2*d^2
) - 5*sqrt(2)*(a*b^3*d^2)^(3/4)*ln(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqr
t(a*d^2/b))/(a^3*b^2*d^2) + 5*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^3*b^2*d^2))/d