3.782 \(\int \frac{1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx\)

Optimal. Leaf size=602 \[ \frac{17}{96 a^2 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^2}+\frac{1}{8 a d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^3}-\frac{3315 \sqrt [4]{b} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{21/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3315 \sqrt [4]{b} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{21/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3315 \sqrt [4]{b} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} a^{21/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3315 \sqrt [4]{b} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{2048 \sqrt{2} a^{21/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3315 \left (a+b x^2\right )}{1024 a^5 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{663}{1024 a^4 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{221}{768 a^3 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )} \]

[Out]

663/(1024*a^4*d*Sqrt[d*x]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 1/(8*a*d*Sqrt[d*x]*
(a + b*x^2)^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 17/(96*a^2*d*Sqrt[d*x]*(a + b*x
^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 221/(768*a^3*d*Sqrt[d*x]*(a + b*x^2)*Sq
rt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (3315*(a + b*x^2))/(1024*a^5*d*Sqrt[d*x]*Sqrt[a
^2 + 2*a*b*x^2 + b^2*x^4]) + (3315*b^(1/4)*(a + b*x^2)*ArcTan[1 - (Sqrt[2]*b^(1/
4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(21/4)*d^(3/2)*Sqrt[a^2 + 2*a*
b*x^2 + b^2*x^4]) - (3315*b^(1/4)*(a + b*x^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d
*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(21/4)*d^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b
^2*x^4]) - (3315*b^(1/4)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - S
qrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(4096*Sqrt[2]*a^(21/4)*d^(3/2)*Sqrt[a^2 + 2*a
*b*x^2 + b^2*x^4]) + (3315*b^(1/4)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqr
t[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(4096*Sqrt[2]*a^(21/4)*d^(3/2)*Sqrt
[a^2 + 2*a*b*x^2 + b^2*x^4])

_______________________________________________________________________________________

Rubi [A]  time = 1.10559, antiderivative size = 602, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{17}{96 a^2 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^2}+\frac{1}{8 a d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^3}-\frac{3315 \sqrt [4]{b} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{21/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3315 \sqrt [4]{b} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} a^{21/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3315 \sqrt [4]{b} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} a^{21/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3315 \sqrt [4]{b} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{2048 \sqrt{2} a^{21/4} d^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3315 \left (a+b x^2\right )}{1024 a^5 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{663}{1024 a^4 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{221}{768 a^3 d \sqrt{d x} \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]

[Out]

663/(1024*a^4*d*Sqrt[d*x]*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 1/(8*a*d*Sqrt[d*x]*
(a + b*x^2)^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 17/(96*a^2*d*Sqrt[d*x]*(a + b*x
^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 221/(768*a^3*d*Sqrt[d*x]*(a + b*x^2)*Sq
rt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (3315*(a + b*x^2))/(1024*a^5*d*Sqrt[d*x]*Sqrt[a
^2 + 2*a*b*x^2 + b^2*x^4]) + (3315*b^(1/4)*(a + b*x^2)*ArcTan[1 - (Sqrt[2]*b^(1/
4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(21/4)*d^(3/2)*Sqrt[a^2 + 2*a*
b*x^2 + b^2*x^4]) - (3315*b^(1/4)*(a + b*x^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d
*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*a^(21/4)*d^(3/2)*Sqrt[a^2 + 2*a*b*x^2 + b
^2*x^4]) - (3315*b^(1/4)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - S
qrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(4096*Sqrt[2]*a^(21/4)*d^(3/2)*Sqrt[a^2 + 2*a
*b*x^2 + b^2*x^4]) + (3315*b^(1/4)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqr
t[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(4096*Sqrt[2]*a^(21/4)*d^(3/2)*Sqrt
[a^2 + 2*a*b*x^2 + b^2*x^4])

_______________________________________________________________________________________

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)**(5/2),x)

[Out]

Exception raised: RecursionError

_______________________________________________________________________________________

Mathematica [A]  time = 0.32584, size = 347, normalized size = 0.58 \[ \frac{x \left (a+b x^2\right ) \left (-14496 a^{5/4} b x^2 \left (a+b x^2\right )^2-7424 a^{9/4} b x^2 \left (a+b x^2\right )-3072 a^{13/4} b x^2-49152 \sqrt [4]{a} \left (a+b x^2\right )^4-30408 \sqrt [4]{a} b x^2 \left (a+b x^2\right )^3-9945 \sqrt{2} \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )^4 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+9945 \sqrt{2} \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )^4 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+19890 \sqrt{2} \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )^4 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )-19890 \sqrt{2} \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )^4 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )\right )}{24576 a^{21/4} (d x)^{3/2} \left (\left (a+b x^2\right )^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]

[Out]

(x*(a + b*x^2)*(-3072*a^(13/4)*b*x^2 - 7424*a^(9/4)*b*x^2*(a + b*x^2) - 14496*a^
(5/4)*b*x^2*(a + b*x^2)^2 - 30408*a^(1/4)*b*x^2*(a + b*x^2)^3 - 49152*a^(1/4)*(a
+ b*x^2)^4 + 19890*Sqrt[2]*b^(1/4)*Sqrt[x]*(a + b*x^2)^4*ArcTan[1 - (Sqrt[2]*b^
(1/4)*Sqrt[x])/a^(1/4)] - 19890*Sqrt[2]*b^(1/4)*Sqrt[x]*(a + b*x^2)^4*ArcTan[1 +
(Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 9945*Sqrt[2]*b^(1/4)*Sqrt[x]*(a + b*x^2)^4
*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 9945*Sqrt[2]*b^(1/
4)*Sqrt[x]*(a + b*x^2)^4*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]
*x]))/(24576*a^(21/4)*(d*x)^(3/2)*((a + b*x^2)^2)^(5/2))

_______________________________________________________________________________________

Maple [B]  time = 0.036, size = 1076, normalized size = 1.8 \[ \text{result too large to display} \]

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)^(5/2),x)

[Out]

-1/24576/d*(9945*2^(1/2)*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)))*(d*x)^(1/2)*x^
8*b^4+19890*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4)
)*(d*x)^(1/2)*x^8*b^4-19890*2^(1/2)*arctan((-2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4)
)/(a*d^2/b)^(1/4))*(d*x)^(1/2)*x^8*b^4+79560*(a*d^2/b)^(1/4)*x^8*b^4+39780*2^(1/
2)*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)))*(d*x)^(1/2)*x^6*a*b^3+79560*2^(1/2)*
arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*(d*x)^(1/2)*x^6*a*
b^3-79560*2^(1/2)*arctan((-2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))
*(d*x)^(1/2)*x^6*a*b^3+302328*(a*d^2/b)^(1/4)*x^6*a*b^3+59670*2^(1/2)*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)))*(d*x)^(1/2)*x^4*a^2*b^2+119340*2^(1/2)*arctan((2^
(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*(d*x)^(1/2)*x^4*a^2*b^2-1193
40*2^(1/2)*arctan((-2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*(d*x)^
(1/2)*x^4*a^2*b^2+422552*(a*d^2/b)^(1/4)*x^4*a^2*b^2+39780*2^(1/2)*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)))*(d*x)^(1/2)*x^2*a^3*b+79560*2^(1/2)*arctan((2^(1/2)*
(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*(d*x)^(1/2)*x^2*a^3*b-79560*2^(1/2
)*arctan((-2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*(d*x)^(1/2)*x^2
*a^3*b+252008*(a*d^2/b)^(1/4)*x^2*a^3*b+9945*2^(1/2)*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)))*(d*x)^(1/2)*a^4+19890*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b
)^(1/4))/(a*d^2/b)^(1/4))*(d*x)^(1/2)*a^4-19890*2^(1/2)*arctan((-2^(1/2)*(d*x)^(
1/2)+(a*d^2/b)^(1/4))/(a*d^2/b)^(1/4))*(d*x)^(1/2)*a^4+49152*(a*d^2/b)^(1/4)*a^4
)*(b*x^2+a)/(a*d^2/b)^(1/4)/(d*x)^(1/2)/a^5/((b*x^2+a)^2)^(5/2)

_______________________________________________________________________________________

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)^(5/2)*(d*x)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.320028, size = 579, normalized size = 0.96 \[ -\frac{39780 \, b^{4} x^{8} + 151164 \, a b^{3} x^{6} + 211276 \, a^{2} b^{2} x^{4} + 126004 \, a^{3} b x^{2} + 24576 \, a^{4} + 39780 \,{\left (a^{5} b^{4} d x^{8} + 4 \, a^{6} b^{3} d x^{6} + 6 \, a^{7} b^{2} d x^{4} + 4 \, a^{8} b d x^{2} + a^{9} d\right )} \sqrt{d x} \left (-\frac{b}{a^{21} d^{6}}\right )^{\frac{1}{4}} \arctan \left (\frac{36429280875 \, a^{16} d^{5} \left (-\frac{b}{a^{21} d^{6}}\right )^{\frac{3}{4}}}{36429280875 \, \sqrt{d x} b + \sqrt{-1327092505069640765625 \, a^{11} b d^{4} \sqrt{-\frac{b}{a^{21} d^{6}}} + 1327092505069640765625 \, b^{2} d x}}\right ) + 9945 \,{\left (a^{5} b^{4} d x^{8} + 4 \, a^{6} b^{3} d x^{6} + 6 \, a^{7} b^{2} d x^{4} + 4 \, a^{8} b d x^{2} + a^{9} d\right )} \sqrt{d x} \left (-\frac{b}{a^{21} d^{6}}\right )^{\frac{1}{4}} \log \left (36429280875 \, a^{16} d^{5} \left (-\frac{b}{a^{21} d^{6}}\right )^{\frac{3}{4}} + 36429280875 \, \sqrt{d x} b\right ) - 9945 \,{\left (a^{5} b^{4} d x^{8} + 4 \, a^{6} b^{3} d x^{6} + 6 \, a^{7} b^{2} d x^{4} + 4 \, a^{8} b d x^{2} + a^{9} d\right )} \sqrt{d x} \left (-\frac{b}{a^{21} d^{6}}\right )^{\frac{1}{4}} \log \left (-36429280875 \, a^{16} d^{5} \left (-\frac{b}{a^{21} d^{6}}\right )^{\frac{3}{4}} + 36429280875 \, \sqrt{d x} b\right )}{12288 \,{\left (a^{5} b^{4} d x^{8} + 4 \, a^{6} b^{3} d x^{6} + 6 \, a^{7} b^{2} d x^{4} + 4 \, a^{8} b d x^{2} + a^{9} d\right )} \sqrt{d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)*(d*x)^(3/2)),x, algorithm="fricas")

[Out]

-1/12288*(39780*b^4*x^8 + 151164*a*b^3*x^6 + 211276*a^2*b^2*x^4 + 126004*a^3*b*x
^2 + 24576*a^4 + 39780*(a^5*b^4*d*x^8 + 4*a^6*b^3*d*x^6 + 6*a^7*b^2*d*x^4 + 4*a^
8*b*d*x^2 + a^9*d)*sqrt(d*x)*(-b/(a^21*d^6))^(1/4)*arctan(36429280875*a^16*d^5*(
-b/(a^21*d^6))^(3/4)/(36429280875*sqrt(d*x)*b + sqrt(-1327092505069640765625*a^1
1*b*d^4*sqrt(-b/(a^21*d^6)) + 1327092505069640765625*b^2*d*x))) + 9945*(a^5*b^4*
d*x^8 + 4*a^6*b^3*d*x^6 + 6*a^7*b^2*d*x^4 + 4*a^8*b*d*x^2 + a^9*d)*sqrt(d*x)*(-b
/(a^21*d^6))^(1/4)*log(36429280875*a^16*d^5*(-b/(a^21*d^6))^(3/4) + 36429280875*
sqrt(d*x)*b) - 9945*(a^5*b^4*d*x^8 + 4*a^6*b^3*d*x^6 + 6*a^7*b^2*d*x^4 + 4*a^8*b
*d*x^2 + a^9*d)*sqrt(d*x)*(-b/(a^21*d^6))^(1/4)*log(-36429280875*a^16*d^5*(-b/(a
^21*d^6))^(3/4) + 36429280875*sqrt(d*x)*b))/((a^5*b^4*d*x^8 + 4*a^6*b^3*d*x^6 +
6*a^7*b^2*d*x^4 + 4*a^8*b*d*x^2 + a^9*d)*sqrt(d*x))

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d x\right )^{\frac{3}{2}} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}}\, dx \]

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)**(5/2),x)

[Out]

Integral(1/((d*x)**(3/2)*((a + b*x**2)**2)**(5/2)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.29172, size = 605, normalized size = 1. \[ -\frac{\frac{49152}{\sqrt{d x} a^{5}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{19890 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a^{6} b^{2} d^{2}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{19890 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{a^{6} b^{2} d^{2}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{9945 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a^{6} b^{2} d^{2}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{9945 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{a^{6} b^{2} d^{2}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{8 \,{\left (3801 \, \sqrt{d x} b^{4} d^{7} x^{7} + 13215 \, \sqrt{d x} a b^{3} d^{7} x^{5} + 15955 \, \sqrt{d x} a^{2} b^{2} d^{7} x^{3} + 6925 \, \sqrt{d x} a^{3} b d^{7} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} a^{5}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )}}{24576 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)*(d*x)^(3/2)),x, algorithm="giac")

[Out]

-1/24576*(49152/(sqrt(d*x)*a^5*sign(b*d^4*x^2 + a*d^4)) + 19890*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^2*d^2*sign(b*d^4*x^2 + a*d^4)) + 19890*sqrt(2)*(a*b^3*d^2)^(3/4)*ar
ctan(-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^2*d^2*sign(b*d^4*x^2 + a*d^4)) - 9945*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^2*d^2*sign(b*d^4*x^2 + a*d^
4)) + 9945*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^2*d^2*sign(b*d^4*x^2 + a*d^4)) + 8*(3801*sqrt(d*x)*b^4*d
^7*x^7 + 13215*sqrt(d*x)*a*b^3*d^7*x^5 + 15955*sqrt(d*x)*a^2*b^2*d^7*x^3 + 6925*
sqrt(d*x)*a^3*b*d^7*x)/((b*d^2*x^2 + a*d^2)^4*a^5*sign(b*d^4*x^2 + a*d^4)))/d