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

Optimal. Leaf size=460 \[ \frac{7 \sqrt{d x}}{16 a^2 d \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\sqrt{d x}}{4 a d \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{21 \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{21 \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{21 \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} a^{11/4} \sqrt [4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{21 \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} a^{11/4} \sqrt [4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

(7*Sqrt[d*x])/(16*a^2*d*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + Sqrt[d*x]/(4*a*d*(a +
b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (21*(a + b*x^2)*ArcTan[1 - (Sqrt[2]*b
^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*a^(11/4)*b^(1/4)*Sqrt[d]*Sqrt[
a^2 + 2*a*b*x^2 + b^2*x^4]) + (21*(a + b*x^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d
*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*a^(11/4)*b^(1/4)*Sqrt[d]*Sqrt[a^2 + 2*a*b*x
^2 + b^2*x^4]) - (21*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[
2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(64*Sqrt[2]*a^(11/4)*b^(1/4)*Sqrt[d]*Sqrt[a^2 + 2
*a*b*x^2 + b^2*x^4]) + (21*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x +
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(64*Sqrt[2]*a^(11/4)*b^(1/4)*Sqrt[d]*Sqrt[a
^2 + 2*a*b*x^2 + b^2*x^4])

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Rubi [A]  time = 0.767655, antiderivative size = 460, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{7 \sqrt{d x}}{16 a^2 d \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\sqrt{d x}}{4 a d \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{21 \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{21 \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{11/4} \sqrt [4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{21 \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} a^{11/4} \sqrt [4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{21 \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} a^{11/4} \sqrt [4]{b} \sqrt{d} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(7*Sqrt[d*x])/(16*a^2*d*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + Sqrt[d*x]/(4*a*d*(a +
b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (21*(a + b*x^2)*ArcTan[1 - (Sqrt[2]*b
^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*a^(11/4)*b^(1/4)*Sqrt[d]*Sqrt[
a^2 + 2*a*b*x^2 + b^2*x^4]) + (21*(a + b*x^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d
*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*a^(11/4)*b^(1/4)*Sqrt[d]*Sqrt[a^2 + 2*a*b*x
^2 + b^2*x^4]) - (21*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[
2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(64*Sqrt[2]*a^(11/4)*b^(1/4)*Sqrt[d]*Sqrt[a^2 + 2
*a*b*x^2 + b^2*x^4]) + (21*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x +
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(64*Sqrt[2]*a^(11/4)*b^(1/4)*Sqrt[d]*Sqrt[a
^2 + 2*a*b*x^2 + b^2*x^4])

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

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)**(3/2)/(d*x)**(1/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.38884, size = 272, normalized size = 0.59 \[ \frac{\sqrt{x} \left (a+b x^2\right ) \left (56 a^{3/4} \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )+32 a^{7/4} \sqrt [4]{b} \sqrt{x}-21 \sqrt{2} \left (a+b x^2\right )^2 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+21 \sqrt{2} \left (a+b x^2\right )^2 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-42 \sqrt{2} \left (a+b x^2\right )^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+42 \sqrt{2} \left (a+b x^2\right )^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )\right )}{128 a^{11/4} \sqrt [4]{b} \sqrt{d x} \left (\left (a+b x^2\right )^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x]*(a + b*x^2)*(32*a^(7/4)*b^(1/4)*Sqrt[x] + 56*a^(3/4)*b^(1/4)*Sqrt[x]*(a
+ b*x^2) - 42*Sqrt[2]*(a + b*x^2)^2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4
)] + 42*Sqrt[2]*(a + b*x^2)^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 21
*Sqrt[2]*(a + b*x^2)^2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x
] + 21*Sqrt[2]*(a + b*x^2)^2*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqr
t[b]*x]))/(128*a^(11/4)*b^(1/4)*Sqrt[d*x]*((a + b*x^2)^2)^(3/2))

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Maple [B]  time = 0.014, size = 644, normalized size = 1.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/128*(21*(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))/((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(a*d^2/b)^(1/2)))*x^4*b
^2+42*(a*d^2/b)^(1/4)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^
2/b)^(1/4))*x^4*b^2-42*(a*d^2/b)^(1/4)*2^(1/2)*arctan((-2^(1/2)*(d*x)^(1/2)+(a*d
^2/b)^(1/4))/(a*d^2/b)^(1/4))*x^4*b^2+42*(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))/((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1
/2)-d*x-(a*d^2/b)^(1/2)))*x^2*a*b+84*(a*d^2/b)^(1/4)*2^(1/2)*arctan((2^(1/2)*(d*
x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*x^2*a*b-84*(a*d^2/b)^(1/4)*2^(1/2)*ar
ctan((-2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*x^2*a*b+21*(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))/((
a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(a*d^2/b)^(1/2)))*a^2+42*(a*d^2/b)^(1/4)*
2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*a^2-42*(a*
d^2/b)^(1/4)*2^(1/2)*arctan((-2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/
4))*a^2+56*(d*x)^(1/2)*x^2*a*b+88*(d*x)^(1/2)*a^2)/d*(b*x^2+a)/a^3/((b*x^2+a)^2)
^(3/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.295579, size = 377, normalized size = 0.82 \[ -\frac{84 \,{\left (a^{2} b^{2} d x^{4} + 2 \, a^{3} b d x^{2} + a^{4} d\right )} \left (-\frac{1}{a^{11} b d^{2}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{3} d \left (-\frac{1}{a^{11} b d^{2}}\right )^{\frac{1}{4}}}{\sqrt{a^{6} d^{2} \sqrt{-\frac{1}{a^{11} b d^{2}}} + d x} + \sqrt{d x}}\right ) - 21 \,{\left (a^{2} b^{2} d x^{4} + 2 \, a^{3} b d x^{2} + a^{4} d\right )} \left (-\frac{1}{a^{11} b d^{2}}\right )^{\frac{1}{4}} \log \left (a^{3} d \left (-\frac{1}{a^{11} b d^{2}}\right )^{\frac{1}{4}} + \sqrt{d x}\right ) + 21 \,{\left (a^{2} b^{2} d x^{4} + 2 \, a^{3} b d x^{2} + a^{4} d\right )} \left (-\frac{1}{a^{11} b d^{2}}\right )^{\frac{1}{4}} \log \left (-a^{3} d \left (-\frac{1}{a^{11} b d^{2}}\right )^{\frac{1}{4}} + \sqrt{d x}\right ) - 4 \,{\left (7 \, b x^{2} + 11 \, a\right )} \sqrt{d x}}{64 \,{\left (a^{2} b^{2} d x^{4} + 2 \, a^{3} b d x^{2} + a^{4} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/64*(84*(a^2*b^2*d*x^4 + 2*a^3*b*d*x^2 + a^4*d)*(-1/(a^11*b*d^2))^(1/4)*arctan
(a^3*d*(-1/(a^11*b*d^2))^(1/4)/(sqrt(a^6*d^2*sqrt(-1/(a^11*b*d^2)) + d*x) + sqrt
(d*x))) - 21*(a^2*b^2*d*x^4 + 2*a^3*b*d*x^2 + a^4*d)*(-1/(a^11*b*d^2))^(1/4)*log
(a^3*d*(-1/(a^11*b*d^2))^(1/4) + sqrt(d*x)) + 21*(a^2*b^2*d*x^4 + 2*a^3*b*d*x^2
+ a^4*d)*(-1/(a^11*b*d^2))^(1/4)*log(-a^3*d*(-1/(a^11*b*d^2))^(1/4) + sqrt(d*x))
- 4*(7*b*x^2 + 11*a)*sqrt(d*x))/(a^2*b^2*d*x^4 + 2*a^3*b*d*x^2 + a^4*d)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.282072, size = 505, normalized size = 1.1 \[ \frac{7 \, \sqrt{d x} b d^{3} x^{2} + 11 \, \sqrt{d x} a d^{3}}{16 \,{\left (b d^{2} x^{2} + a d^{2}\right )}^{2} a^{2}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{21 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{3} b d{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{21 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{64 \, a^{3} b d{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{21 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}}{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{128 \, a^{3} b d{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{21 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}}{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{128 \, a^{3} b d{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*(7*sqrt(d*x)*b*d^3*x^2 + 11*sqrt(d*x)*a*d^3)/((b*d^2*x^2 + a*d^2)^2*a^2*sig
n(b*d^4*x^2 + a*d^4)) + 21/64*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^3*b*d*sign(b*d^4*x^2 + a*
d^4)) + 21/64*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^3*b*d*sign(b*d^4*x^2 + a*d^4)) + 21/128*
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^3*b*d*sign(b*d^4*x^2 + a*d^4)) - 21/128*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^3*b*d*sign(b*d^4*x^2 +
a*d^4))