3.771 \(\int \frac{(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx\)

Optimal. Leaf size=600 \[ -\frac{17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{2048 \sqrt{2} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3315 d^9 \sqrt{d x} \left (a+b x^2\right )}{1024 b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

(-663*d^7*(d*x)^(5/2))/(1024*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(d*x)^(17
/2))/(8*b*(a + b*x^2)^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (17*d^3*(d*x)^(13/2))
/(96*b^2*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (221*d^5*(d*x)^(9/2))/
(768*b^3*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (3315*d^9*Sqrt[d*x]*(a +
b*x^2))/(1024*b^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (3315*a^(1/4)*d^(19/2)*(a
+ b*x^2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2
]*b^(21/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (3315*a^(1/4)*d^(19/2)*(a + b*x^2)
*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*b^(21/
4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (3315*a^(1/4)*d^(19/2)*(a + b*x^2)*Log[Sqr
t[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(4096*Sqr
t[2]*b^(21/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (3315*a^(1/4)*d^(19/2)*(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]]
)/(4096*Sqrt[2]*b^(21/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

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Rubi [A]  time = 1.04916, antiderivative size = 600, 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 \[ -\frac{17 d^3 (d x)^{13/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{17/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3315 \sqrt [4]{a} d^{19/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{2048 \sqrt{2} b^{21/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3315 d^9 \sqrt{d x} \left (a+b x^2\right )}{1024 b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{663 d^7 (d x)^{5/2}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{221 d^5 (d x)^{9/2}}{768 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^(19/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(-663*d^7*(d*x)^(5/2))/(1024*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(d*x)^(17
/2))/(8*b*(a + b*x^2)^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (17*d^3*(d*x)^(13/2))
/(96*b^2*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (221*d^5*(d*x)^(9/2))/
(768*b^3*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (3315*d^9*Sqrt[d*x]*(a +
b*x^2))/(1024*b^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (3315*a^(1/4)*d^(19/2)*(a
+ b*x^2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2
]*b^(21/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (3315*a^(1/4)*d^(19/2)*(a + b*x^2)
*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*b^(21/
4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (3315*a^(1/4)*d^(19/2)*(a + b*x^2)*Log[Sqr
t[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(4096*Sqr
t[2]*b^(21/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (3315*a^(1/4)*d^(19/2)*(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]]
)/(4096*Sqrt[2]*b^(21/4)*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((d*x)**(19/2)/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.303498, size = 350, normalized size = 0.58 \[ \frac{(d x)^{19/2} \left (a+b x^2\right ) \left (-3072 a^4 \sqrt [4]{b} \sqrt{x}+16640 a^3 \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )-38560 a^2 \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )^2+55400 a \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )^3+49152 \sqrt [4]{b} \sqrt{x} \left (a+b x^2\right )^4+9945 \sqrt{2} \sqrt [4]{a} \left (a+b x^2\right )^4 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-9945 \sqrt{2} \sqrt [4]{a} \left (a+b x^2\right )^4 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+19890 \sqrt{2} \sqrt [4]{a} \left (a+b x^2\right )^4 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )-19890 \sqrt{2} \sqrt [4]{a} \left (a+b x^2\right )^4 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )\right )}{24576 b^{21/4} x^{19/2} \left (\left (a+b x^2\right )^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^(19/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

((d*x)^(19/2)*(a + b*x^2)*(-3072*a^4*b^(1/4)*Sqrt[x] + 16640*a^3*b^(1/4)*Sqrt[x]
*(a + b*x^2) - 38560*a^2*b^(1/4)*Sqrt[x]*(a + b*x^2)^2 + 55400*a*b^(1/4)*Sqrt[x]
*(a + b*x^2)^3 + 49152*b^(1/4)*Sqrt[x]*(a + b*x^2)^4 + 19890*Sqrt[2]*a^(1/4)*(a
+ b*x^2)^4*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 19890*Sqrt[2]*a^(1/4)
*(a + b*x^2)^4*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 9945*Sqrt[2]*a^(1
/4)*(a + b*x^2)^4*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - 9
945*Sqrt[2]*a^(1/4)*(a + b*x^2)^4*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x]
+ Sqrt[b]*x]))/(24576*b^(21/4)*x^(19/2)*((a + b*x^2)^2)^(5/2))

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Maple [B]  time = 0.035, size = 1212, normalized size = 2. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^(19/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)

[Out]

-1/24576*(9945*(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^8*b^4*d^6+19890*(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^8*b^4*d^6-19890*(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^8*b^4*d^6+39780*(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^6*a*b^3*d^6+79560*(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^
6*a*b^3*d^6-79560*(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^6*a*b^3*d^6-49152*(d*x)^(1/2)*x^8*b^4*d^6+59670*(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*a^2*b^2*d^6+1193
40*(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*a^2*b^2*d^6-119340*(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*a^2*b^2*d^6-55400*(d*x)^(13/2)*a*b^3-
196608*(d*x)^(1/2)*x^6*a*b^3*d^6+39780*(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^3*b*d^6+79560*(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^3*b*d^6-79560*(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^3*b*d^6-127640*(d*x)^(9/2)*a^2*b^2*d^2-294912*(d*x)^(1/2)*x^4*a^2*b^2*d^6+994
5*(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^4*d^6+19890
*(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^4*d^6-19890*(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^4*d^6-105720*(d*x)^(5/2)*a^3*b*d^4-196608*(d*x)^(1
/2)*x^2*a^3*b*d^6-79560*(d*x)^(1/2)*a^4*d^6)*d^3*(b*x^2+a)/b^5/((b*x^2+a)^2)^(5/
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((d*x)^(19/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.303305, size = 540, normalized size = 0.9 \[ \frac{39780 \, \left (-\frac{a d^{38}}{b^{21}}\right )^{\frac{1}{4}}{\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )} \arctan \left (\frac{\left (-\frac{a d^{38}}{b^{21}}\right )^{\frac{1}{4}} b^{5}}{\sqrt{d x} d^{9} + \sqrt{d^{19} x + \sqrt{-\frac{a d^{38}}{b^{21}}} b^{10}}}\right ) - 9945 \, \left (-\frac{a d^{38}}{b^{21}}\right )^{\frac{1}{4}}{\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )} \log \left (3315 \, \sqrt{d x} d^{9} + 3315 \, \left (-\frac{a d^{38}}{b^{21}}\right )^{\frac{1}{4}} b^{5}\right ) + 9945 \, \left (-\frac{a d^{38}}{b^{21}}\right )^{\frac{1}{4}}{\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )} \log \left (3315 \, \sqrt{d x} d^{9} - 3315 \, \left (-\frac{a d^{38}}{b^{21}}\right )^{\frac{1}{4}} b^{5}\right ) + 4 \,{\left (6144 \, b^{4} d^{9} x^{8} + 31501 \, a b^{3} d^{9} x^{6} + 52819 \, a^{2} b^{2} d^{9} x^{4} + 37791 \, a^{3} b d^{9} x^{2} + 9945 \, a^{4} d^{9}\right )} \sqrt{d x}}{12288 \,{\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(19/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2),x, algorithm="fricas")

[Out]

1/12288*(39780*(-a*d^38/b^21)^(1/4)*(b^9*x^8 + 4*a*b^8*x^6 + 6*a^2*b^7*x^4 + 4*a
^3*b^6*x^2 + a^4*b^5)*arctan((-a*d^38/b^21)^(1/4)*b^5/(sqrt(d*x)*d^9 + sqrt(d^19
*x + sqrt(-a*d^38/b^21)*b^10))) - 9945*(-a*d^38/b^21)^(1/4)*(b^9*x^8 + 4*a*b^8*x
^6 + 6*a^2*b^7*x^4 + 4*a^3*b^6*x^2 + a^4*b^5)*log(3315*sqrt(d*x)*d^9 + 3315*(-a*
d^38/b^21)^(1/4)*b^5) + 9945*(-a*d^38/b^21)^(1/4)*(b^9*x^8 + 4*a*b^8*x^6 + 6*a^2
*b^7*x^4 + 4*a^3*b^6*x^2 + a^4*b^5)*log(3315*sqrt(d*x)*d^9 - 3315*(-a*d^38/b^21)
^(1/4)*b^5) + 4*(6144*b^4*d^9*x^8 + 31501*a*b^3*d^9*x^6 + 52819*a^2*b^2*d^9*x^4
+ 37791*a^3*b*d^9*x^2 + 9945*a^4*d^9)*sqrt(d*x))/(b^9*x^8 + 4*a*b^8*x^6 + 6*a^2*
b^7*x^4 + 4*a^3*b^6*x^2 + a^4*b^5)

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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)**(19/2)/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.293601, size = 578, normalized size = 0.96 \[ -\frac{1}{24576} \, d^{8}{\left (\frac{19890 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{b^{6}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{19890 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{b^{6}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{9945 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{b^{6}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{9945 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{1}{4}} d{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{b^{6}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{49152 \, \sqrt{d x} d}{b^{5}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{8 \,{\left (6925 \, \sqrt{d x} a b^{3} d^{9} x^{6} + 15955 \, \sqrt{d x} a^{2} b^{2} d^{9} x^{4} + 13215 \, \sqrt{d x} a^{3} b d^{9} x^{2} + 3801 \, \sqrt{d x} a^{4} d^{9}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} b^{5}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(19/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2),x, algorithm="giac")

[Out]

-1/24576*d^8*(19890*sqrt(2)*(a*b^3*d^2)^(1/4)*d*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d
^2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(b^6*sign(b*d^4*x^2 + a*d^4)) + 1989
0*sqrt(2)*(a*b^3*d^2)^(1/4)*d*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*s
qrt(d*x))/(a*d^2/b)^(1/4))/(b^6*sign(b*d^4*x^2 + a*d^4)) + 9945*sqrt(2)*(a*b^3*d
^2)^(1/4)*d*ln(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(b^6*sig
n(b*d^4*x^2 + a*d^4)) - 9945*sqrt(2)*(a*b^3*d^2)^(1/4)*d*ln(d*x - sqrt(2)*(a*d^2
/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(b^6*sign(b*d^4*x^2 + a*d^4)) - 49152*sqrt(
d*x)*d/(b^5*sign(b*d^4*x^2 + a*d^4)) - 8*(6925*sqrt(d*x)*a*b^3*d^9*x^6 + 15955*s
qrt(d*x)*a^2*b^2*d^9*x^4 + 13215*sqrt(d*x)*a^3*b*d^9*x^2 + 3801*sqrt(d*x)*a^4*d^
9)/((b*d^2*x^2 + a*d^2)^4*b^5*sign(b*d^4*x^2 + a*d^4)))