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

Optimal. Leaf size=346 \[ \frac{(11 b c-3 a d) (b c-a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} \sqrt [4]{c} d^{15/4}}-\frac{(11 b c-3 a d) (b c-a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} \sqrt [4]{c} d^{15/4}}-\frac{(11 b c-3 a d) (b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} \sqrt [4]{c} d^{15/4}}+\frac{(11 b c-3 a d) (b c-a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} \sqrt [4]{c} d^{15/4}}-\frac{x^{3/2} (11 b c-3 a d) (b c-a d)}{6 c d^3}+\frac{x^{7/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac{2 b^2 x^{7/2}}{7 d^2} \]

[Out]

-((11*b*c - 3*a*d)*(b*c - a*d)*x^(3/2))/(6*c*d^3) + (2*b^2*x^(7/2))/(7*d^2) + ((
b*c - a*d)^2*x^(7/2))/(2*c*d^2*(c + d*x^2)) - ((11*b*c - 3*a*d)*(b*c - a*d)*ArcT
an[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(1/4)*d^(15/4)) + ((11*b
*c - 3*a*d)*(b*c - a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2
]*c^(1/4)*d^(15/4)) + ((11*b*c - 3*a*d)*(b*c - a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4
)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(1/4)*d^(15/4)) - ((11*b*c - 3*a*d)
*(b*c - a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt
[2]*c^(1/4)*d^(15/4))

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Rubi [A]  time = 0.697139, antiderivative size = 346, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ \frac{(11 b c-3 a d) (b c-a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} \sqrt [4]{c} d^{15/4}}-\frac{(11 b c-3 a d) (b c-a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} \sqrt [4]{c} d^{15/4}}-\frac{(11 b c-3 a d) (b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} \sqrt [4]{c} d^{15/4}}+\frac{(11 b c-3 a d) (b c-a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} \sqrt [4]{c} d^{15/4}}-\frac{x^{3/2} (11 b c-3 a d) (b c-a d)}{6 c d^3}+\frac{x^{7/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac{2 b^2 x^{7/2}}{7 d^2} \]

Antiderivative was successfully verified.

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

[Out]

-((11*b*c - 3*a*d)*(b*c - a*d)*x^(3/2))/(6*c*d^3) + (2*b^2*x^(7/2))/(7*d^2) + ((
b*c - a*d)^2*x^(7/2))/(2*c*d^2*(c + d*x^2)) - ((11*b*c - 3*a*d)*(b*c - a*d)*ArcT
an[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*c^(1/4)*d^(15/4)) + ((11*b
*c - 3*a*d)*(b*c - a*d)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2
]*c^(1/4)*d^(15/4)) + ((11*b*c - 3*a*d)*(b*c - a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4
)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt[2]*c^(1/4)*d^(15/4)) - ((11*b*c - 3*a*d)
*(b*c - a*d)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(8*Sqrt
[2]*c^(1/4)*d^(15/4))

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Rubi in Sympy [A]  time = 111.288, size = 321, normalized size = 0.93 \[ \frac{2 b^{2} x^{\frac{7}{2}}}{7 d^{2}} + \frac{x^{\frac{7}{2}} \left (a d - b c\right )^{2}}{2 c d^{2} \left (c + d x^{2}\right )} - \frac{x^{\frac{3}{2}} \left (a d - b c\right ) \left (3 a d - 11 b c\right )}{6 c d^{3}} + \frac{\sqrt{2} \left (a d - b c\right ) \left (3 a d - 11 b c\right ) \log{\left (- \sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{16 \sqrt [4]{c} d^{\frac{15}{4}}} - \frac{\sqrt{2} \left (a d - b c\right ) \left (3 a d - 11 b c\right ) \log{\left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{16 \sqrt [4]{c} d^{\frac{15}{4}}} - \frac{\sqrt{2} \left (a d - b c\right ) \left (3 a d - 11 b c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{8 \sqrt [4]{c} d^{\frac{15}{4}}} + \frac{\sqrt{2} \left (a d - b c\right ) \left (3 a d - 11 b c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{8 \sqrt [4]{c} d^{\frac{15}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)*(b*x**2+a)**2/(d*x**2+c)**2,x)

[Out]

2*b**2*x**(7/2)/(7*d**2) + x**(7/2)*(a*d - b*c)**2/(2*c*d**2*(c + d*x**2)) - x**
(3/2)*(a*d - b*c)*(3*a*d - 11*b*c)/(6*c*d**3) + sqrt(2)*(a*d - b*c)*(3*a*d - 11*
b*c)*log(-sqrt(2)*c**(1/4)*d**(1/4)*sqrt(x) + sqrt(c) + sqrt(d)*x)/(16*c**(1/4)*
d**(15/4)) - sqrt(2)*(a*d - b*c)*(3*a*d - 11*b*c)*log(sqrt(2)*c**(1/4)*d**(1/4)*
sqrt(x) + sqrt(c) + sqrt(d)*x)/(16*c**(1/4)*d**(15/4)) - sqrt(2)*(a*d - b*c)*(3*
a*d - 11*b*c)*atan(1 - sqrt(2)*d**(1/4)*sqrt(x)/c**(1/4))/(8*c**(1/4)*d**(15/4))
+ sqrt(2)*(a*d - b*c)*(3*a*d - 11*b*c)*atan(1 + sqrt(2)*d**(1/4)*sqrt(x)/c**(1/
4))/(8*c**(1/4)*d**(15/4))

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Mathematica [A]  time = 0.311673, size = 337, normalized size = 0.97 \[ \frac{\frac{21 \sqrt{2} \left (3 a^2 d^2-14 a b c d+11 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{\sqrt [4]{c}}-\frac{21 \sqrt{2} \left (3 a^2 d^2-14 a b c d+11 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{\sqrt [4]{c}}-\frac{42 \sqrt{2} \left (3 a^2 d^2-14 a b c d+11 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt [4]{c}}+\frac{42 \sqrt{2} \left (3 a^2 d^2-14 a b c d+11 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{\sqrt [4]{c}}-448 b d^{3/4} x^{3/2} (b c-a d)-\frac{168 d^{3/4} x^{3/2} (b c-a d)^2}{c+d x^2}+96 b^2 d^{7/4} x^{7/2}}{336 d^{15/4}} \]

Antiderivative was successfully verified.

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

[Out]

(-448*b*d^(3/4)*(b*c - a*d)*x^(3/2) + 96*b^2*d^(7/4)*x^(7/2) - (168*d^(3/4)*(b*c
- a*d)^2*x^(3/2))/(c + d*x^2) - (42*Sqrt[2]*(11*b^2*c^2 - 14*a*b*c*d + 3*a^2*d^
2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(1/4) + (42*Sqrt[2]*(11*b^2*
c^2 - 14*a*b*c*d + 3*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(
1/4) + (21*Sqrt[2]*(11*b^2*c^2 - 14*a*b*c*d + 3*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c
^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(1/4) - (21*Sqrt[2]*(11*b^2*c^2 - 14*a*b*
c*d + 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(
1/4))/(336*d^(15/4))

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Maple [A]  time = 0.025, size = 523, normalized size = 1.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*b^2*x^(7/2)/d^2+4/3*b/d^2*x^(3/2)*a-4/3*b^2/d^3*x^(3/2)*c-1/2/d*x^(3/2)/(d*x
^2+c)*a^2+1/d^2*x^(3/2)/(d*x^2+c)*c*a*b-1/2/d^3*x^(3/2)/(d*x^2+c)*b^2*c^2-7/8/d^
3/(c/d)^(1/4)*2^(1/2)*c*a*b*ln((x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x+(c
/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))-7/4/d^3/(c/d)^(1/4)*2^(1/2)*c*a*b*arctan
(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)-7/4/d^3/(c/d)^(1/4)*2^(1/2)*c*a*b*arctan(2^(1/2)
/(c/d)^(1/4)*x^(1/2)-1)+11/16/d^4/(c/d)^(1/4)*2^(1/2)*b^2*c^2*ln((x-(c/d)^(1/4)*
x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))+11/8/d
^4/(c/d)^(1/4)*2^(1/2)*b^2*c^2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+11/8/d^4/(c
/d)^(1/4)*2^(1/2)*b^2*c^2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)+3/16/d^2/(c/d)^(
1/4)*2^(1/2)*a^2*ln((x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*x
^(1/2)*2^(1/2)+(c/d)^(1/2)))+3/8/d^2/(c/d)^(1/4)*2^(1/2)*a^2*arctan(2^(1/2)/(c/d
)^(1/4)*x^(1/2)+1)+3/8/d^2/(c/d)^(1/4)*2^(1/2)*a^2*arctan(2^(1/2)/(c/d)^(1/4)*x^
(1/2)-1)

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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((b*x^2 + a)^2*x^(5/2)/(d*x^2 + c)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.278114, size = 2029, normalized size = 5.86 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/168*(84*(d^4*x^2 + c*d^3)*(-(14641*b^8*c^8 - 74536*a*b^7*c^7*d + 158268*a^2*b^
6*c^6*d^2 - 181720*a^3*b^5*c^5*d^3 + 122566*a^4*b^4*c^4*d^4 - 49560*a^5*b^3*c^3*
d^5 + 11772*a^6*b^2*c^2*d^6 - 1512*a^7*b*c*d^7 + 81*a^8*d^8)/(c*d^15))^(1/4)*arc
tan(c*d^11*(-(14641*b^8*c^8 - 74536*a*b^7*c^7*d + 158268*a^2*b^6*c^6*d^2 - 18172
0*a^3*b^5*c^5*d^3 + 122566*a^4*b^4*c^4*d^4 - 49560*a^5*b^3*c^3*d^5 + 11772*a^6*b
^2*c^2*d^6 - 1512*a^7*b*c*d^7 + 81*a^8*d^8)/(c*d^15))^(3/4)/((1331*b^6*c^6 - 508
2*a*b^5*c^5*d + 7557*a^2*b^4*c^4*d^2 - 5516*a^3*b^3*c^3*d^3 + 2061*a^4*b^2*c^2*d
^4 - 378*a^5*b*c*d^5 + 27*a^6*d^6)*sqrt(x) + sqrt((1771561*b^12*c^12 - 13528284*
a*b^11*c^11*d + 45943458*a^2*b^10*c^10*d^2 - 91492940*a^3*b^9*c^9*d^3 + 11865925
5*a^4*b^8*c^8*d^4 - 105323064*a^5*b^7*c^7*d^5 + 65490076*a^6*b^6*c^6*d^6 - 28724
472*a^7*b^5*c^5*d^7 + 8825895*a^8*b^4*c^4*d^8 - 1855980*a^9*b^3*c^3*d^9 + 254178
*a^10*b^2*c^2*d^10 - 20412*a^11*b*c*d^11 + 729*a^12*d^12)*x - (14641*b^8*c^9*d^7
- 74536*a*b^7*c^8*d^8 + 158268*a^2*b^6*c^7*d^9 - 181720*a^3*b^5*c^6*d^10 + 1225
66*a^4*b^4*c^5*d^11 - 49560*a^5*b^3*c^4*d^12 + 11772*a^6*b^2*c^3*d^13 - 1512*a^7
*b*c^2*d^14 + 81*a^8*c*d^15)*sqrt(-(14641*b^8*c^8 - 74536*a*b^7*c^7*d + 158268*a
^2*b^6*c^6*d^2 - 181720*a^3*b^5*c^5*d^3 + 122566*a^4*b^4*c^4*d^4 - 49560*a^5*b^3
*c^3*d^5 + 11772*a^6*b^2*c^2*d^6 - 1512*a^7*b*c*d^7 + 81*a^8*d^8)/(c*d^15))))) +
21*(d^4*x^2 + c*d^3)*(-(14641*b^8*c^8 - 74536*a*b^7*c^7*d + 158268*a^2*b^6*c^6*
d^2 - 181720*a^3*b^5*c^5*d^3 + 122566*a^4*b^4*c^4*d^4 - 49560*a^5*b^3*c^3*d^5 +
11772*a^6*b^2*c^2*d^6 - 1512*a^7*b*c*d^7 + 81*a^8*d^8)/(c*d^15))^(1/4)*log(c*d^1
1*(-(14641*b^8*c^8 - 74536*a*b^7*c^7*d + 158268*a^2*b^6*c^6*d^2 - 181720*a^3*b^5
*c^5*d^3 + 122566*a^4*b^4*c^4*d^4 - 49560*a^5*b^3*c^3*d^5 + 11772*a^6*b^2*c^2*d^
6 - 1512*a^7*b*c*d^7 + 81*a^8*d^8)/(c*d^15))^(3/4) + (1331*b^6*c^6 - 5082*a*b^5*
c^5*d + 7557*a^2*b^4*c^4*d^2 - 5516*a^3*b^3*c^3*d^3 + 2061*a^4*b^2*c^2*d^4 - 378
*a^5*b*c*d^5 + 27*a^6*d^6)*sqrt(x)) - 21*(d^4*x^2 + c*d^3)*(-(14641*b^8*c^8 - 74
536*a*b^7*c^7*d + 158268*a^2*b^6*c^6*d^2 - 181720*a^3*b^5*c^5*d^3 + 122566*a^4*b
^4*c^4*d^4 - 49560*a^5*b^3*c^3*d^5 + 11772*a^6*b^2*c^2*d^6 - 1512*a^7*b*c*d^7 +
81*a^8*d^8)/(c*d^15))^(1/4)*log(-c*d^11*(-(14641*b^8*c^8 - 74536*a*b^7*c^7*d + 1
58268*a^2*b^6*c^6*d^2 - 181720*a^3*b^5*c^5*d^3 + 122566*a^4*b^4*c^4*d^4 - 49560*
a^5*b^3*c^3*d^5 + 11772*a^6*b^2*c^2*d^6 - 1512*a^7*b*c*d^7 + 81*a^8*d^8)/(c*d^15
))^(3/4) + (1331*b^6*c^6 - 5082*a*b^5*c^5*d + 7557*a^2*b^4*c^4*d^2 - 5516*a^3*b^
3*c^3*d^3 + 2061*a^4*b^2*c^2*d^4 - 378*a^5*b*c*d^5 + 27*a^6*d^6)*sqrt(x)) + 4*(1
2*b^2*d^2*x^5 - 4*(11*b^2*c*d - 14*a*b*d^2)*x^3 - 7*(11*b^2*c^2 - 14*a*b*c*d + 3
*a^2*d^2)*x)*sqrt(x))/(d^4*x^2 + c*d^3)

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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(x**(5/2)*(b*x**2+a)**2/(d*x**2+c)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.246771, size = 558, normalized size = 1.61 \[ -\frac{b^{2} c^{2} x^{\frac{3}{2}} - 2 \, a b c d x^{\frac{3}{2}} + a^{2} d^{2} x^{\frac{3}{2}}}{2 \,{\left (d x^{2} + c\right )} d^{3}} + \frac{\sqrt{2}{\left (11 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 14 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + 3 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{8 \, c d^{6}} + \frac{\sqrt{2}{\left (11 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 14 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + 3 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{8 \, c d^{6}} - \frac{\sqrt{2}{\left (11 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 14 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + 3 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{16 \, c d^{6}} + \frac{\sqrt{2}{\left (11 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 14 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + 3 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{16 \, c d^{6}} + \frac{2 \,{\left (3 \, b^{2} d^{12} x^{\frac{7}{2}} - 14 \, b^{2} c d^{11} x^{\frac{3}{2}} + 14 \, a b d^{12} x^{\frac{3}{2}}\right )}}{21 \, d^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(b^2*c^2*x^(3/2) - 2*a*b*c*d*x^(3/2) + a^2*d^2*x^(3/2))/((d*x^2 + c)*d^3) +
1/8*sqrt(2)*(11*(c*d^3)^(3/4)*b^2*c^2 - 14*(c*d^3)^(3/4)*a*b*c*d + 3*(c*d^3)^(3
/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(
c*d^6) + 1/8*sqrt(2)*(11*(c*d^3)^(3/4)*b^2*c^2 - 14*(c*d^3)^(3/4)*a*b*c*d + 3*(c
*d^3)^(3/4)*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)
^(1/4))/(c*d^6) - 1/16*sqrt(2)*(11*(c*d^3)^(3/4)*b^2*c^2 - 14*(c*d^3)^(3/4)*a*b*
c*d + 3*(c*d^3)^(3/4)*a^2*d^2)*ln(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(
c*d^6) + 1/16*sqrt(2)*(11*(c*d^3)^(3/4)*b^2*c^2 - 14*(c*d^3)^(3/4)*a*b*c*d + 3*(
c*d^3)^(3/4)*a^2*d^2)*ln(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c*d^6) +
2/21*(3*b^2*d^12*x^(7/2) - 14*b^2*c*d^11*x^(3/2) + 14*a*b*d^12*x^(3/2))/d^14