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

Optimal. Leaf size=506 \[ \frac{9}{16 a^2 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )}-\frac{45 \sqrt [4]{b} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{13/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt [4]{b} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{13/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt [4]{b} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} a^{13/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \sqrt [4]{b} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} a^{13/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \left (a+b x^2\right )}{16 a^3 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

9/(16*a^2*d*Sqrt[d*x]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 1/(4*a*d*Sqrt[d*x]*(a +
b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (45*(a + b*x^2))/(16*a^3*d*Sqrt[d*x]*
Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (45*b^(1/4)*(a + b*x^2)*ArcTan[1 - (Sqrt[2]*b
^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*a^(13/4)*d^(3/2)*Sqrt[a^2 + 2*
a*b*x^2 + b^2*x^4]) - (45*b^(1/4)*(a + b*x^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d
*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*a^(13/4)*d^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2
*x^4]) - (45*b^(1/4)*(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^(13/4)*d^(3/2)*Sqrt[a^2 + 2*a*b*x^2
+ b^2*x^4]) + (45*b^(1/4)*(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^(13/4)*d^(3/2)*Sqrt[a^2 + 2*a
*b*x^2 + b^2*x^4])

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Rubi [A]  time = 0.864768, antiderivative size = 506, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{9}{16 a^2 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )}-\frac{45 \sqrt [4]{b} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{13/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt [4]{b} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{64 \sqrt{2} a^{13/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{45 \sqrt [4]{b} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{32 \sqrt{2} a^{13/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \sqrt [4]{b} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{32 \sqrt{2} a^{13/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{45 \left (a+b x^2\right )}{16 a^3 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

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

[Out]

9/(16*a^2*d*Sqrt[d*x]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 1/(4*a*d*Sqrt[d*x]*(a +
b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (45*(a + b*x^2))/(16*a^3*d*Sqrt[d*x]*
Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (45*b^(1/4)*(a + b*x^2)*ArcTan[1 - (Sqrt[2]*b
^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*a^(13/4)*d^(3/2)*Sqrt[a^2 + 2*
a*b*x^2 + b^2*x^4]) - (45*b^(1/4)*(a + b*x^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d
*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*a^(13/4)*d^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2
*x^4]) - (45*b^(1/4)*(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^(13/4)*d^(3/2)*Sqrt[a^2 + 2*a*b*x^2
+ b^2*x^4]) + (45*b^(1/4)*(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^(13/4)*d^(3/2)*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/(d*x)**(3/2)/(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.385175, size = 307, normalized size = 0.61 \[ \frac{x \left (a+b x^2\right ) \left (-32 a^{5/4} b x^2-104 \sqrt [4]{a} b x^2 \left (a+b x^2\right )-256 \sqrt [4]{a} \left (a+b x^2\right )^2-45 \sqrt{2} \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )^2 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+45 \sqrt{2} \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )^2 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+90 \sqrt{2} \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )-90 \sqrt{2} \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )\right )}{128 a^{13/4} (d x)^{3/2} \left (\left (a+b x^2\right )^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(a + b*x^2)*(-32*a^(5/4)*b*x^2 - 104*a^(1/4)*b*x^2*(a + b*x^2) - 256*a^(1/4)*
(a + b*x^2)^2 + 90*Sqrt[2]*b^(1/4)*Sqrt[x]*(a + b*x^2)^2*ArcTan[1 - (Sqrt[2]*b^(
1/4)*Sqrt[x])/a^(1/4)] - 90*Sqrt[2]*b^(1/4)*Sqrt[x]*(a + b*x^2)^2*ArcTan[1 + (Sq
rt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 45*Sqrt[2]*b^(1/4)*Sqrt[x]*(a + b*x^2)^2*Log[S
qrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 45*Sqrt[2]*b^(1/4)*Sqrt[
x]*(a + b*x^2)^2*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]))/(1
28*a^(13/4)*(d*x)^(3/2)*((a + b*x^2)^2)^(3/2))

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Maple [A]  time = 0.03, size = 642, normalized size = 1.3 \[ \text{result too large to display} \]

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)^(3/2),x)

[Out]

-1/128/d*(45*ln(-((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(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)))*2^(1/2)*(d*x)^(1/2)*x^4*b^
2+90*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*2^(1/2)*(d*x)
^(1/2)*x^4*b^2-90*arctan((-2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))
*2^(1/2)*(d*x)^(1/2)*x^4*b^2+90*ln(-((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(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)))*2^(1/2)
*(d*x)^(1/2)*x^2*a*b+180*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^
(1/4))*2^(1/2)*(d*x)^(1/2)*x^2*a*b-180*arctan((-2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1
/4))/(a*d^2/b)^(1/4))*2^(1/2)*(d*x)^(1/2)*x^2*a*b+360*(a*d^2/b)^(1/4)*x^4*b^2+45
*ln(-((a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)-d*x-(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)))*2^(1/2)*(d*x)^(1/2)*a^2+90*arctan((2^(
1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*2^(1/2)*(d*x)^(1/2)*a^2-90*ar
ctan((-2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*2^(1/2)*(d*x)^(1/2)
*a^2+648*(a*d^2/b)^(1/4)*x^2*a*b+256*(a*d^2/b)^(1/4)*a^2)*(b*x^2+a)/(a*d^2/b)^(1
/4)/(d*x)^(1/2)/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)*(d*x)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.302468, size = 420, normalized size = 0.83 \[ -\frac{180 \, b^{2} x^{4} + 324 \, a b x^{2} + 180 \,{\left (a^{3} b^{2} d x^{4} + 2 \, a^{4} b d x^{2} + a^{5} d\right )} \sqrt{d x} \left (-\frac{b}{a^{13} d^{6}}\right )^{\frac{1}{4}} \arctan \left (\frac{91125 \, a^{10} d^{5} \left (-\frac{b}{a^{13} d^{6}}\right )^{\frac{3}{4}}}{91125 \, \sqrt{d x} b + \sqrt{-8303765625 \, a^{7} b d^{4} \sqrt{-\frac{b}{a^{13} d^{6}}} + 8303765625 \, b^{2} d x}}\right ) + 45 \,{\left (a^{3} b^{2} d x^{4} + 2 \, a^{4} b d x^{2} + a^{5} d\right )} \sqrt{d x} \left (-\frac{b}{a^{13} d^{6}}\right )^{\frac{1}{4}} \log \left (91125 \, a^{10} d^{5} \left (-\frac{b}{a^{13} d^{6}}\right )^{\frac{3}{4}} + 91125 \, \sqrt{d x} b\right ) - 45 \,{\left (a^{3} b^{2} d x^{4} + 2 \, a^{4} b d x^{2} + a^{5} d\right )} \sqrt{d x} \left (-\frac{b}{a^{13} d^{6}}\right )^{\frac{1}{4}} \log \left (-91125 \, a^{10} d^{5} \left (-\frac{b}{a^{13} d^{6}}\right )^{\frac{3}{4}} + 91125 \, \sqrt{d x} b\right ) + 128 \, a^{2}}{64 \,{\left (a^{3} b^{2} d x^{4} + 2 \, a^{4} b d x^{2} + a^{5} d\right )} \sqrt{d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/64*(180*b^2*x^4 + 324*a*b*x^2 + 180*(a^3*b^2*d*x^4 + 2*a^4*b*d*x^2 + a^5*d)*s
qrt(d*x)*(-b/(a^13*d^6))^(1/4)*arctan(91125*a^10*d^5*(-b/(a^13*d^6))^(3/4)/(9112
5*sqrt(d*x)*b + sqrt(-8303765625*a^7*b*d^4*sqrt(-b/(a^13*d^6)) + 8303765625*b^2*
d*x))) + 45*(a^3*b^2*d*x^4 + 2*a^4*b*d*x^2 + a^5*d)*sqrt(d*x)*(-b/(a^13*d^6))^(1
/4)*log(91125*a^10*d^5*(-b/(a^13*d^6))^(3/4) + 91125*sqrt(d*x)*b) - 45*(a^3*b^2*
d*x^4 + 2*a^4*b*d*x^2 + a^5*d)*sqrt(d*x)*(-b/(a^13*d^6))^(1/4)*log(-91125*a^10*d
^5*(-b/(a^13*d^6))^(3/4) + 91125*sqrt(d*x)*b) + 128*a^2)/((a^3*b^2*d*x^4 + 2*a^4
*b*d*x^2 + a^5*d)*sqrt(d*x))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.289165, size = 554, normalized size = 1.09 \[ -\frac{\frac{256}{\sqrt{d x} a^{3}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{8 \,{\left (13 \, \sqrt{d x} b^{2} d^{3} x^{3} + 17 \, \sqrt{d x} a b d^{3} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{2} a^{3}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{90 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a^{4} b^{2} d^{2}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{90 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a^{4} b^{2} d^{2}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{45 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a^{4} b^{2} d^{2}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{45 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a^{4} b^{2} d^{2}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )}}{128 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/128*(256/(sqrt(d*x)*a^3*sign(b*d^4*x^2 + a*d^4)) + 8*(13*sqrt(d*x)*b^2*d^3*x^
3 + 17*sqrt(d*x)*a*b*d^3*x)/((b*d^2*x^2 + a*d^2)^2*a^3*sign(b*d^4*x^2 + a*d^4))
+ 90*sqrt(2)*(a*b^3*d^2)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) + 2*s
qrt(d*x))/(a*d^2/b)^(1/4))/(a^4*b^2*d^2*sign(b*d^4*x^2 + a*d^4)) + 90*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^4*b^2*d^2*sign(b*d^4*x^2 + a*d^4)) - 45*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^4*b^2*d^2*sign(
b*d^4*x^2 + a*d^4)) + 45*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^4*b^2*d^2*sign(b*d^4*x^2 + a*d^4)))/d