3.441 \(\int \frac{x^{5/2} \left (c+d x^2\right )^3}{a+b x^2} \, dx\)
Optimal. Leaf size=328 \[ -\frac{a^{3/4} (b c-a d)^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{19/4}}+\frac{a^{3/4} (b c-a d)^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{19/4}}+\frac{a^{3/4} (b c-a d)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{19/4}}-\frac{a^{3/4} (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} b^{19/4}}+\frac{2 d x^{7/2} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{7 b^3}+\frac{2 x^{3/2} (b c-a d)^3}{3 b^4}+\frac{2 d^2 x^{11/2} (3 b c-a d)}{11 b^2}+\frac{2 d^3 x^{15/2}}{15 b} \]
[Out]
(2*(b*c - a*d)^3*x^(3/2))/(3*b^4) + (2*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^(7/
2))/(7*b^3) + (2*d^2*(3*b*c - a*d)*x^(11/2))/(11*b^2) + (2*d^3*x^(15/2))/(15*b)
+ (a^(3/4)*(b*c - a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]
*b^(19/4)) - (a^(3/4)*(b*c - a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)
])/(Sqrt[2]*b^(19/4)) - (a^(3/4)*(b*c - a*d)^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(
1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^(19/4)) + (a^(3/4)*(b*c - a*d)^3*Log[Sqr
t[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^(19/4))
_______________________________________________________________________________________
Rubi [A] time = 0.638363, antiderivative size = 328, normalized size of antiderivative = 1.,
number of steps used = 13, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375
\[ -\frac{a^{3/4} (b c-a d)^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{19/4}}+\frac{a^{3/4} (b c-a d)^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} b^{19/4}}+\frac{a^{3/4} (b c-a d)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} b^{19/4}}-\frac{a^{3/4} (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} b^{19/4}}+\frac{2 d x^{7/2} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{7 b^3}+\frac{2 x^{3/2} (b c-a d)^3}{3 b^4}+\frac{2 d^2 x^{11/2} (3 b c-a d)}{11 b^2}+\frac{2 d^3 x^{15/2}}{15 b} \]
Antiderivative was successfully verified.
[In] Int[(x^(5/2)*(c + d*x^2)^3)/(a + b*x^2),x]
[Out]
(2*(b*c - a*d)^3*x^(3/2))/(3*b^4) + (2*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^(7/
2))/(7*b^3) + (2*d^2*(3*b*c - a*d)*x^(11/2))/(11*b^2) + (2*d^3*x^(15/2))/(15*b)
+ (a^(3/4)*(b*c - a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]
*b^(19/4)) - (a^(3/4)*(b*c - a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)
])/(Sqrt[2]*b^(19/4)) - (a^(3/4)*(b*c - a*d)^3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(
1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^(19/4)) + (a^(3/4)*(b*c - a*d)^3*Log[Sqr
t[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*b^(19/4))
_______________________________________________________________________________________
Rubi in Sympy [A] time = 106.645, size = 311, normalized size = 0.95 \[ \frac{\sqrt{2} a^{\frac{3}{4}} \left (a d - b c\right )^{3} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 b^{\frac{19}{4}}} - \frac{\sqrt{2} a^{\frac{3}{4}} \left (a d - b c\right )^{3} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 b^{\frac{19}{4}}} - \frac{\sqrt{2} a^{\frac{3}{4}} \left (a d - b c\right )^{3} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 b^{\frac{19}{4}}} + \frac{\sqrt{2} a^{\frac{3}{4}} \left (a d - b c\right )^{3} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 b^{\frac{19}{4}}} + \frac{2 d^{3} x^{\frac{15}{2}}}{15 b} - \frac{2 d^{2} x^{\frac{11}{2}} \left (a d - 3 b c\right )}{11 b^{2}} + \frac{2 d x^{\frac{7}{2}} \left (a^{2} d^{2} - 3 a b c d + 3 b^{2} c^{2}\right )}{7 b^{3}} - \frac{2 x^{\frac{3}{2}} \left (a d - b c\right )^{3}}{3 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(5/2)*(d*x**2+c)**3/(b*x**2+a),x)
[Out]
sqrt(2)*a**(3/4)*(a*d - b*c)**3*log(-sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a)
+ sqrt(b)*x)/(4*b**(19/4)) - sqrt(2)*a**(3/4)*(a*d - b*c)**3*log(sqrt(2)*a**(1/
4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(4*b**(19/4)) - sqrt(2)*a**(3/4)*(a*d
- b*c)**3*atan(1 - sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(2*b**(19/4)) + sqrt(2)*a
**(3/4)*(a*d - b*c)**3*atan(1 + sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(2*b**(19/4))
+ 2*d**3*x**(15/2)/(15*b) - 2*d**2*x**(11/2)*(a*d - 3*b*c)/(11*b**2) + 2*d*x**(
7/2)*(a**2*d**2 - 3*a*b*c*d + 3*b**2*c**2)/(7*b**3) - 2*x**(3/2)*(a*d - b*c)**3/
(3*b**4)
_______________________________________________________________________________________
Mathematica [A] time = 0.259935, size = 314, normalized size = 0.96 \[ \frac{1155 \sqrt{2} a^{3/4} (a d-b c)^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-1155 \sqrt{2} a^{3/4} (a d-b c)^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-2310 \sqrt{2} a^{3/4} (a d-b c)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+2310 \sqrt{2} a^{3/4} (a d-b c)^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )+1320 b^{7/4} d x^{7/2} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )+840 b^{11/4} d^2 x^{11/2} (3 b c-a d)+3080 b^{3/4} x^{3/2} (b c-a d)^3+616 b^{15/4} d^3 x^{15/2}}{4620 b^{19/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^(5/2)*(c + d*x^2)^3)/(a + b*x^2),x]
[Out]
(3080*b^(3/4)*(b*c - a*d)^3*x^(3/2) + 1320*b^(7/4)*d*(3*b^2*c^2 - 3*a*b*c*d + a^
2*d^2)*x^(7/2) + 840*b^(11/4)*d^2*(3*b*c - a*d)*x^(11/2) + 616*b^(15/4)*d^3*x^(1
5/2) - 2310*Sqrt[2]*a^(3/4)*(-(b*c) + a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x]
)/a^(1/4)] + 2310*Sqrt[2]*a^(3/4)*(-(b*c) + a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*S
qrt[x])/a^(1/4)] + 1155*Sqrt[2]*a^(3/4)*(-(b*c) + a*d)^3*Log[Sqrt[a] - Sqrt[2]*a
^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - 1155*Sqrt[2]*a^(3/4)*(-(b*c) + a*d)^3*Log[
Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(4620*b^(19/4))
_______________________________________________________________________________________
Maple [B] time = 0.015, size = 721, normalized size = 2.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(5/2)*(d*x^2+c)^3/(b*x^2+a),x)
[Out]
2/15*d^3*x^(15/2)/b-2/11/b^2*x^(11/2)*a*d^3+6/11/b*x^(11/2)*c*d^2+2/7/b^3*x^(7/2
)*a^2*d^3-6/7/b^2*x^(7/2)*a*c*d^2+6/7/b*x^(7/2)*c^2*d-2/3/b^4*x^(3/2)*a^3*d^3+2/
b^3*x^(3/2)*a^2*c*d^2-2/b^2*x^(3/2)*a*c^2*d+2/3/b*x^(3/2)*c^3+1/2*a^4/b^5/(a/b)^
(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*d^3-3/2*a^3/b^4/(a/b)^(1/4)*
2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*c*d^2+3/2*a^2/b^3/(a/b)^(1/4)*2^(1
/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*c^2*d-1/2*a/b^2/(a/b)^(1/4)*2^(1/2)*ar
ctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*c^3+1/2*a^4/b^5/(a/b)^(1/4)*2^(1/2)*arctan(2
^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*d^3-3/2*a^3/b^4/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)
/(a/b)^(1/4)*x^(1/2)-1)*c*d^2+3/2*a^2/b^3/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/
b)^(1/4)*x^(1/2)-1)*c^2*d-1/2*a/b^2/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/
4)*x^(1/2)-1)*c^3+1/4*a^4/b^5/(a/b)^(1/4)*2^(1/2)*ln((x-(a/b)^(1/4)*x^(1/2)*2^(1
/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))*d^3-3/4*a^3/b^4/(a
/b)^(1/4)*2^(1/2)*ln((x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*
x^(1/2)*2^(1/2)+(a/b)^(1/2)))*c*d^2+3/4*a^2/b^3/(a/b)^(1/4)*2^(1/2)*ln((x-(a/b)^
(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))*
c^2*d-1/4*a/b^2/(a/b)^(1/4)*2^(1/2)*ln((x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2
))/(x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))*c^3
_______________________________________________________________________________________
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^2 + c)^3*x^(5/2)/(b*x^2 + a),x, algorithm="maxima")
[Out]
Exception raised: ValueError
_______________________________________________________________________________________
Fricas [A] time = 0.278634, size = 2951, normalized size = 9. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3*x^(5/2)/(b*x^2 + a),x, algorithm="fricas")
[Out]
1/2310*(4620*b^4*(-(a^3*b^12*c^12 - 12*a^4*b^11*c^11*d + 66*a^5*b^10*c^10*d^2 -
220*a^6*b^9*c^9*d^3 + 495*a^7*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^
6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11*b^4*c^4*d^8 - 220*a^12*b^3*c^3*d^9 + 66*
a^13*b^2*c^2*d^10 - 12*a^14*b*c*d^11 + a^15*d^12)/b^19)^(1/4)*arctan(-b^14*(-(a^
3*b^12*c^12 - 12*a^4*b^11*c^11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 +
495*a^7*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b^5*c
^5*d^7 + 495*a^11*b^4*c^4*d^8 - 220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12
*a^14*b*c*d^11 + a^15*d^12)/b^19)^(3/4)/((a^2*b^9*c^9 - 9*a^3*b^8*c^8*d + 36*a^4
*b^7*c^7*d^2 - 84*a^5*b^6*c^6*d^3 + 126*a^6*b^5*c^5*d^4 - 126*a^7*b^4*c^4*d^5 +
84*a^8*b^3*c^3*d^6 - 36*a^9*b^2*c^2*d^7 + 9*a^10*b*c*d^8 - a^11*d^9)*sqrt(x) - s
qrt((a^4*b^18*c^18 - 18*a^5*b^17*c^17*d + 153*a^6*b^16*c^16*d^2 - 816*a^7*b^15*c
^15*d^3 + 3060*a^8*b^14*c^14*d^4 - 8568*a^9*b^13*c^13*d^5 + 18564*a^10*b^12*c^12
*d^6 - 31824*a^11*b^11*c^11*d^7 + 43758*a^12*b^10*c^10*d^8 - 48620*a^13*b^9*c^9*
d^9 + 43758*a^14*b^8*c^8*d^10 - 31824*a^15*b^7*c^7*d^11 + 18564*a^16*b^6*c^6*d^1
2 - 8568*a^17*b^5*c^5*d^13 + 3060*a^18*b^4*c^4*d^14 - 816*a^19*b^3*c^3*d^15 + 15
3*a^20*b^2*c^2*d^16 - 18*a^21*b*c*d^17 + a^22*d^18)*x - (a^3*b^21*c^12 - 12*a^4*
b^20*c^11*d + 66*a^5*b^19*c^10*d^2 - 220*a^6*b^18*c^9*d^3 + 495*a^7*b^17*c^8*d^4
- 792*a^8*b^16*c^7*d^5 + 924*a^9*b^15*c^6*d^6 - 792*a^10*b^14*c^5*d^7 + 495*a^1
1*b^13*c^4*d^8 - 220*a^12*b^12*c^3*d^9 + 66*a^13*b^11*c^2*d^10 - 12*a^14*b^10*c*
d^11 + a^15*b^9*d^12)*sqrt(-(a^3*b^12*c^12 - 12*a^4*b^11*c^11*d + 66*a^5*b^10*c^
10*d^2 - 220*a^6*b^9*c^9*d^3 + 495*a^7*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a
^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11*b^4*c^4*d^8 - 220*a^12*b^3*c^3*
d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b*c*d^11 + a^15*d^12)/b^19)))) + 1155*b^4*(
-(a^3*b^12*c^12 - 12*a^4*b^11*c^11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^
3 + 495*a^7*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b
^5*c^5*d^7 + 495*a^11*b^4*c^4*d^8 - 220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10
- 12*a^14*b*c*d^11 + a^15*d^12)/b^19)^(1/4)*log(b^14*(-(a^3*b^12*c^12 - 12*a^4*b
^11*c^11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 495*a^7*b^8*c^8*d^4 -
792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11*b^4*
c^4*d^8 - 220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b*c*d^11 + a^15*
d^12)/b^19)^(3/4) - (a^2*b^9*c^9 - 9*a^3*b^8*c^8*d + 36*a^4*b^7*c^7*d^2 - 84*a^5
*b^6*c^6*d^3 + 126*a^6*b^5*c^5*d^4 - 126*a^7*b^4*c^4*d^5 + 84*a^8*b^3*c^3*d^6 -
36*a^9*b^2*c^2*d^7 + 9*a^10*b*c*d^8 - a^11*d^9)*sqrt(x)) - 1155*b^4*(-(a^3*b^12*
c^12 - 12*a^4*b^11*c^11*d + 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 495*a^7
*b^8*c^8*d^4 - 792*a^8*b^7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7
+ 495*a^11*b^4*c^4*d^8 - 220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b
*c*d^11 + a^15*d^12)/b^19)^(1/4)*log(-b^14*(-(a^3*b^12*c^12 - 12*a^4*b^11*c^11*d
+ 66*a^5*b^10*c^10*d^2 - 220*a^6*b^9*c^9*d^3 + 495*a^7*b^8*c^8*d^4 - 792*a^8*b^
7*c^7*d^5 + 924*a^9*b^6*c^6*d^6 - 792*a^10*b^5*c^5*d^7 + 495*a^11*b^4*c^4*d^8 -
220*a^12*b^3*c^3*d^9 + 66*a^13*b^2*c^2*d^10 - 12*a^14*b*c*d^11 + a^15*d^12)/b^19
)^(3/4) - (a^2*b^9*c^9 - 9*a^3*b^8*c^8*d + 36*a^4*b^7*c^7*d^2 - 84*a^5*b^6*c^6*d
^3 + 126*a^6*b^5*c^5*d^4 - 126*a^7*b^4*c^4*d^5 + 84*a^8*b^3*c^3*d^6 - 36*a^9*b^2
*c^2*d^7 + 9*a^10*b*c*d^8 - a^11*d^9)*sqrt(x)) + 4*(77*b^3*d^3*x^7 + 105*(3*b^3*
c*d^2 - a*b^2*d^3)*x^5 + 165*(3*b^3*c^2*d - 3*a*b^2*c*d^2 + a^2*b*d^3)*x^3 + 385
*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x)*sqrt(x))/b^4
_______________________________________________________________________________________
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)*(d*x**2+c)**3/(b*x**2+a),x)
[Out]
Timed out
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.293454, size = 717, normalized size = 2.19 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3*x^(5/2)/(b*x^2 + a),x, algorithm="giac")
[Out]
-1/2*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d + 3*(a*b^3)^(3
/4)*a^2*b*c*d^2 - (a*b^3)^(3/4)*a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4)
+ 2*sqrt(x))/(a/b)^(1/4))/b^7 - 1/2*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^
(3/4)*a*b^2*c^2*d + 3*(a*b^3)^(3/4)*a^2*b*c*d^2 - (a*b^3)^(3/4)*a^3*d^3)*arctan(
-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/b^7 + 1/4*sqrt(2)*((
a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d + 3*(a*b^3)^(3/4)*a^2*b*c*d^2
- (a*b^3)^(3/4)*a^3*d^3)*ln(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/b^7 -
1/4*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d + 3*(a*b^3)^(3/
4)*a^2*b*c*d^2 - (a*b^3)^(3/4)*a^3*d^3)*ln(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sq
rt(a/b))/b^7 + 2/1155*(77*b^14*d^3*x^(15/2) + 315*b^14*c*d^2*x^(11/2) - 105*a*b^
13*d^3*x^(11/2) + 495*b^14*c^2*d*x^(7/2) - 495*a*b^13*c*d^2*x^(7/2) + 165*a^2*b^
12*d^3*x^(7/2) + 385*b^14*c^3*x^(3/2) - 1155*a*b^13*c^2*d*x^(3/2) + 1155*a^2*b^1
2*c*d^2*x^(3/2) - 385*a^3*b^11*d^3*x^(3/2))/b^15