3.770 \(\int \frac{(d x)^{21/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx\)

Optimal. Leaf size=600 \[ -\frac{19 d^3 (d x)^{15/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{19/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{7315 d^9 (d x)^{3/2} \left (a+b x^2\right )}{3072 b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{1045 d^7 (d x)^{7/2}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{95 d^5 (d x)^{11/2}}{256 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 a^{3/4} d^{21/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} b^{23/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{7315 a^{3/4} d^{21/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} b^{23/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{7315 a^{3/4} d^{21/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} b^{23/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 a^{3/4} d^{21/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{2048 \sqrt{2} b^{23/4} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

(-1045*d^7*(d*x)^(7/2))/(1024*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(d*x)^(1
9/2))/(8*b*(a + b*x^2)^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (19*d^3*(d*x)^(15/2)
)/(96*b^2*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (95*d^5*(d*x)^(11/2))
/(256*b^3*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (7315*d^9*(d*x)^(3/2)*(
a + b*x^2))/(3072*b^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (7315*a^(3/4)*d^(21/2)*
(a + b*x^2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqr
t[2]*b^(23/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (7315*a^(3/4)*d^(21/2)*(a + b*x
^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*b^(
23/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (7315*a^(3/4)*d^(21/2)*(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^(23/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (7315*a^(3/4)*d^(21/2)*(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^(23/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

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Rubi [A]  time = 1.07388, 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 \[ -\frac{19 d^3 (d x)^{15/2}}{96 b^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{d (d x)^{19/2}}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{7315 d^9 (d x)^{3/2} \left (a+b x^2\right )}{3072 b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{1045 d^7 (d x)^{7/2}}{1024 b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{95 d^5 (d x)^{11/2}}{256 b^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 a^{3/4} d^{21/2} \left (a+b x^2\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} b^{23/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{7315 a^{3/4} d^{21/2} \left (a+b x^2\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{d x}+\sqrt{a} \sqrt{d}+\sqrt{b} \sqrt{d} x\right )}{4096 \sqrt{2} b^{23/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{7315 a^{3/4} d^{21/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}\right )}{2048 \sqrt{2} b^{23/4} \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{7315 a^{3/4} d^{21/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{d x}}{\sqrt [4]{a} \sqrt{d}}+1\right )}{2048 \sqrt{2} b^{23/4} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^(21/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(-1045*d^7*(d*x)^(7/2))/(1024*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(d*x)^(1
9/2))/(8*b*(a + b*x^2)^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (19*d^3*(d*x)^(15/2)
)/(96*b^2*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (95*d^5*(d*x)^(11/2))
/(256*b^3*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (7315*d^9*(d*x)^(3/2)*(
a + b*x^2))/(3072*b^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (7315*a^(3/4)*d^(21/2)*
(a + b*x^2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqr
t[2]*b^(23/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (7315*a^(3/4)*d^(21/2)*(a + b*x
^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(2048*Sqrt[2]*b^(
23/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (7315*a^(3/4)*d^(21/2)*(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^(23/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (7315*a^(3/4)*d^(21/2)*(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^(23/4)*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)**(21/2)/(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 = 0.326894, size = 350, normalized size = 0.58 \[ \frac{(d x)^{21/2} \left (a+b x^2\right ) \left (-21945 \sqrt{2} a^{3/4} \left (a+b x^2\right )^4 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+21945 \sqrt{2} a^{3/4} \left (a+b x^2\right )^4 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+43890 \sqrt{2} a^{3/4} \left (a+b x^2\right )^4 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )-43890 \sqrt{2} a^{3/4} \left (a+b x^2\right )^4 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )-3072 a^4 b^{3/4} x^{3/2}+17152 a^3 b^{3/4} x^{3/2} \left (a+b x^2\right )-42144 a^2 b^{3/4} x^{3/2} \left (a+b x^2\right )^2+70200 a b^{3/4} x^{3/2} \left (a+b x^2\right )^3+16384 b^{3/4} x^{3/2} \left (a+b x^2\right )^4\right )}{24576 b^{23/4} x^{21/2} \left (\left (a+b x^2\right )^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x)^(21/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

((d*x)^(21/2)*(a + b*x^2)*(-3072*a^4*b^(3/4)*x^(3/2) + 17152*a^3*b^(3/4)*x^(3/2)
*(a + b*x^2) - 42144*a^2*b^(3/4)*x^(3/2)*(a + b*x^2)^2 + 70200*a*b^(3/4)*x^(3/2)
*(a + b*x^2)^3 + 16384*b^(3/4)*x^(3/2)*(a + b*x^2)^4 + 43890*Sqrt[2]*a^(3/4)*(a
+ b*x^2)^4*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 43890*Sqrt[2]*a^(3/4)
*(a + b*x^2)^4*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 21945*Sqrt[2]*a^(
3/4)*(a + b*x^2)^4*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] +
21945*Sqrt[2]*a^(3/4)*(a + b*x^2)^4*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x
] + Sqrt[b]*x]))/(24576*b^(23/4)*x^(21/2)*((a + b*x^2)^2)^(5/2))

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Maple [B]  time = 0.035, size = 1166, normalized size = 1.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^(21/2)/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)

[Out]

1/24576*(16384*(a*d^2/b)^(1/4)*(d*x)^(3/2)*x^8*b^5*d^6-21945*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)))*x^8*a*b^4*d^8-43890*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*a*b^4*d^8+43890*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*a*b^4*d^8+70200*(a*d^2/b)
^(1/4)*(d*x)^(15/2)*a*b^4+65536*(a*d^2/b)^(1/4)*(d*x)^(3/2)*x^6*a*b^4*d^6-87780*
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)))*x^6*a^2*b^3*d^8-175560*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^2*b^3*d^8+17
5560*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^2*b^3*d^8+168456*(a*d^2/b)^(1/4)*(d*x)^(11/2)*a^2*b^3*d^2+98304*(a*d^2/b)^(1/4
)*(d*x)^(3/2)*x^4*a^2*b^3*d^6-131670*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)))*x^4*a^3*b^2*d^8-263340*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^3*b^2*d^8+263340*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^3*b^2*d^8+143464*(a*d^2/b)^(1/4)*(d*x)^(7
/2)*a^3*b^2*d^4+65536*(a*d^2/b)^(1/4)*(d*x)^(3/2)*x^2*a^3*b^2*d^6-87780*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)))*x^2*a^4*b*d^8-175560*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^4*b*d^8+175560*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^4*b*d^8+58
520*(a*d^2/b)^(1/4)*(d*x)^(3/2)*a^4*b*d^6-21945*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)))*a^5*d^8-43890*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a*d^2/b)^(1/
4))/(a*d^2/b)^(1/4))*a^5*d^8+43890*2^(1/2)*arctan((-2^(1/2)*(d*x)^(1/2)+(a*d^2/b
)^(1/4))/(a*d^2/b)^(1/4))*a^5*d^8)*d^3*(b*x^2+a)/(a*d^2/b)^(1/4)/b^6/((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)^(21/2)/(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 = 0.305634, size = 586, normalized size = 0.98 \[ -\frac{87780 \, \left (-\frac{a^{3} d^{42}}{b^{23}}\right )^{\frac{1}{4}}{\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )} \arctan \left (\frac{\left (-\frac{a^{3} d^{42}}{b^{23}}\right )^{\frac{3}{4}} b^{17}}{\sqrt{d x} a^{2} d^{31} + \sqrt{a^{4} d^{63} x - \sqrt{-\frac{a^{3} d^{42}}{b^{23}}} a^{3} b^{11} d^{42}}}\right ) + 21945 \, \left (-\frac{a^{3} d^{42}}{b^{23}}\right )^{\frac{1}{4}}{\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )} \log \left (391419980875 \, \sqrt{d x} a^{2} d^{31} + 391419980875 \, \left (-\frac{a^{3} d^{42}}{b^{23}}\right )^{\frac{3}{4}} b^{17}\right ) - 21945 \, \left (-\frac{a^{3} d^{42}}{b^{23}}\right )^{\frac{1}{4}}{\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )} \log \left (391419980875 \, \sqrt{d x} a^{2} d^{31} - 391419980875 \, \left (-\frac{a^{3} d^{42}}{b^{23}}\right )^{\frac{3}{4}} b^{17}\right ) - 4 \,{\left (2048 \, b^{4} d^{10} x^{9} + 16967 \, a b^{3} d^{10} x^{7} + 33345 \, a^{2} b^{2} d^{10} x^{5} + 26125 \, a^{3} b d^{10} x^{3} + 7315 \, a^{4} d^{10} x\right )} \sqrt{d x}}{12288 \,{\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(21/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2),x, algorithm="fricas")

[Out]

-1/12288*(87780*(-a^3*d^42/b^23)^(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^3*d^42/b^23)^(3/4)*b^17/(sqrt(d*x)*a^2*d^31
+ sqrt(a^4*d^63*x - sqrt(-a^3*d^42/b^23)*a^3*b^11*d^42))) + 21945*(-a^3*d^42/b^2
3)^(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(3
91419980875*sqrt(d*x)*a^2*d^31 + 391419980875*(-a^3*d^42/b^23)^(3/4)*b^17) - 219
45*(-a^3*d^42/b^23)^(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(391419980875*sqrt(d*x)*a^2*d^31 - 391419980875*(-a^3*d^42/b^23)^
(3/4)*b^17) - 4*(2048*b^4*d^10*x^9 + 16967*a*b^3*d^10*x^7 + 33345*a^2*b^2*d^10*x
^5 + 26125*a^3*b*d^10*x^3 + 7315*a^4*d^10*x)*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)**(21/2)/(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 = 0.296215, size = 575, normalized size = 0.96 \[ \frac{1}{24576} \, d^{9}{\left (\frac{16384 \, \sqrt{d x} d x}{b^{5}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{43890 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{b^{8}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{43890 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{d x}\right )}}{2 \, \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}}}\right )}{b^{8}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{21945 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x + \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{b^{8}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac{21945 \, \sqrt{2} \left (a b^{3} d^{2}\right )^{\frac{3}{4}}{\rm ln}\left (d x - \sqrt{2} \left (\frac{a d^{2}}{b}\right )^{\frac{1}{4}} \sqrt{d x} + \sqrt{\frac{a d^{2}}{b}}\right )}{b^{8}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac{8 \,{\left (8775 \, \sqrt{d x} a b^{3} d^{9} x^{7} + 21057 \, \sqrt{d x} a^{2} b^{2} d^{9} x^{5} + 17933 \, \sqrt{d x} a^{3} b d^{9} x^{3} + 5267 \, \sqrt{d x} a^{4} d^{9} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{4} b^{5}{\rm sign}\left (b d^{4} x^{2} + a d^{4}\right )}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)^(21/2)/(b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2),x, algorithm="giac")

[Out]

1/24576*d^9*(16384*sqrt(d*x)*d*x/(b^5*sign(b*d^4*x^2 + a*d^4)) - 43890*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))/(b^8*sign(b*d^4*x^2 + a*d^4)) - 43890*sqrt(2)*(a*b^3*d^2)^(3/4)*arc
tan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(b^8*s
ign(b*d^4*x^2 + a*d^4)) + 21945*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))/(b^8*sign(b*d^4*x^2 + a*d^4)) - 21945*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)
)/(b^8*sign(b*d^4*x^2 + a*d^4)) + 8*(8775*sqrt(d*x)*a*b^3*d^9*x^7 + 21057*sqrt(d
*x)*a^2*b^2*d^9*x^5 + 17933*sqrt(d*x)*a^3*b*d^9*x^3 + 5267*sqrt(d*x)*a^4*d^9*x)/
((b*d^2*x^2 + a*d^2)^4*b^5*sign(b*d^4*x^2 + a*d^4)))