3.454 \(\int \frac{x^{3/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=386 \[ -\frac{(b c-13 a d) (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{3/4} b^{17/4}}+\frac{(b c-13 a d) (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{3/4} b^{17/4}}-\frac{(b c-13 a d) (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{3/4} b^{17/4}}+\frac{(b c-13 a d) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{3/4} b^{17/4}}+\frac{d \sqrt{x} \left (585 a^2 d^2-1098 a b c d+497 b^2 c^2\right )}{90 b^4}+\frac{d \sqrt{x} \left (c+d x^2\right ) (113 b c-117 a d)}{90 b^3}-\frac{\sqrt{x} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{13 d \sqrt{x} \left (c+d x^2\right )^2}{18 b^2} \]

[Out]

(d*(497*b^2*c^2 - 1098*a*b*c*d + 585*a^2*d^2)*Sqrt[x])/(90*b^4) + (d*(113*b*c -
117*a*d)*Sqrt[x]*(c + d*x^2))/(90*b^3) + (13*d*Sqrt[x]*(c + d*x^2)^2)/(18*b^2) -
(Sqrt[x]*(c + d*x^2)^3)/(2*b*(a + b*x^2)) - ((b*c - 13*a*d)*(b*c - a*d)^2*ArcTa
n[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(3/4)*b^(17/4)) + ((b*c -
13*a*d)*(b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]
*a^(3/4)*b^(17/4)) - ((b*c - 13*a*d)*(b*c - a*d)^2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)
*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(3/4)*b^(17/4)) + ((b*c - 13*a*d)*(b
*c - a*d)^2*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[
2]*a^(3/4)*b^(17/4))

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Rubi [A]  time = 1.07515, antiderivative size = 386, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ -\frac{(b c-13 a d) (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{3/4} b^{17/4}}+\frac{(b c-13 a d) (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{3/4} b^{17/4}}-\frac{(b c-13 a d) (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{3/4} b^{17/4}}+\frac{(b c-13 a d) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{3/4} b^{17/4}}+\frac{d \sqrt{x} \left (585 a^2 d^2-1098 a b c d+497 b^2 c^2\right )}{90 b^4}+\frac{d \sqrt{x} \left (c+d x^2\right ) (113 b c-117 a d)}{90 b^3}-\frac{\sqrt{x} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{13 d \sqrt{x} \left (c+d x^2\right )^2}{18 b^2} \]

Antiderivative was successfully verified.

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

[Out]

(d*(497*b^2*c^2 - 1098*a*b*c*d + 585*a^2*d^2)*Sqrt[x])/(90*b^4) + (d*(113*b*c -
117*a*d)*Sqrt[x]*(c + d*x^2))/(90*b^3) + (13*d*Sqrt[x]*(c + d*x^2)^2)/(18*b^2) -
(Sqrt[x]*(c + d*x^2)^3)/(2*b*(a + b*x^2)) - ((b*c - 13*a*d)*(b*c - a*d)^2*ArcTa
n[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(3/4)*b^(17/4)) + ((b*c -
13*a*d)*(b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]
*a^(3/4)*b^(17/4)) - ((b*c - 13*a*d)*(b*c - a*d)^2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)
*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(3/4)*b^(17/4)) + ((b*c - 13*a*d)*(b
*c - a*d)^2*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[
2]*a^(3/4)*b^(17/4))

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Rubi in Sympy [A]  time = 172.812, size = 372, normalized size = 0.96 \[ - \frac{\sqrt{x} \left (c + d x^{2}\right )^{3}}{2 b \left (a + b x^{2}\right )} + \frac{13 d \sqrt{x} \left (c + d x^{2}\right )^{2}}{18 b^{2}} - \frac{d \sqrt{x} \left (c \left (13 a d - 9 b c\right ) + d x^{2} \left (117 a d - 113 b c\right )\right )}{90 b^{3}} + \frac{d \sqrt{x} \left (585 a^{2} d^{2} - 1202 a b c d + 601 b^{2} c^{2}\right )}{90 b^{4}} + \frac{\sqrt{2} \left (a d - b c\right )^{2} \left (13 a d - b c\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{3}{4}} b^{\frac{17}{4}}} - \frac{\sqrt{2} \left (a d - b c\right )^{2} \left (13 a d - b c\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{3}{4}} b^{\frac{17}{4}}} + \frac{\sqrt{2} \left (a d - b c\right )^{2} \left (13 a d - b c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{3}{4}} b^{\frac{17}{4}}} - \frac{\sqrt{2} \left (a d - b c\right )^{2} \left (13 a d - b c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{3}{4}} b^{\frac{17}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3/2)*(d*x**2+c)**3/(b*x**2+a)**2,x)

[Out]

-sqrt(x)*(c + d*x**2)**3/(2*b*(a + b*x**2)) + 13*d*sqrt(x)*(c + d*x**2)**2/(18*b
**2) - d*sqrt(x)*(c*(13*a*d - 9*b*c) + d*x**2*(117*a*d - 113*b*c))/(90*b**3) + d
*sqrt(x)*(585*a**2*d**2 - 1202*a*b*c*d + 601*b**2*c**2)/(90*b**4) + sqrt(2)*(a*d
- b*c)**2*(13*a*d - b*c)*log(-sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqr
t(b)*x)/(16*a**(3/4)*b**(17/4)) - sqrt(2)*(a*d - b*c)**2*(13*a*d - b*c)*log(sqrt
(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(16*a**(3/4)*b**(17/4)) + s
qrt(2)*(a*d - b*c)**2*(13*a*d - b*c)*atan(1 - sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))
/(8*a**(3/4)*b**(17/4)) - sqrt(2)*(a*d - b*c)**2*(13*a*d - b*c)*atan(1 + sqrt(2)
*b**(1/4)*sqrt(x)/a**(1/4))/(8*a**(3/4)*b**(17/4))

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Mathematica [A]  time = 0.441102, size = 345, normalized size = 0.89 \[ \frac{\frac{45 \sqrt{2} (b c-a d)^2 (13 a d-b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{3/4}}+\frac{45 \sqrt{2} (b c-13 a d) (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{3/4}}+\frac{90 \sqrt{2} (b c-a d)^2 (13 a d-b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac{90 \sqrt{2} (b c-13 a d) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{3/4}}+288 b^{5/4} d^2 x^{5/2} (3 b c-2 a d)-\frac{360 \sqrt [4]{b} \sqrt{x} (b c-a d)^3}{a+b x^2}+4320 \sqrt [4]{b} d \sqrt{x} (b c-a d)^2+160 b^{9/4} d^3 x^{9/2}}{720 b^{17/4}} \]

Antiderivative was successfully verified.

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

[Out]

(4320*b^(1/4)*d*(b*c - a*d)^2*Sqrt[x] + 288*b^(5/4)*d^2*(3*b*c - 2*a*d)*x^(5/2)
+ 160*b^(9/4)*d^3*x^(9/2) - (360*b^(1/4)*(b*c - a*d)^3*Sqrt[x])/(a + b*x^2) + (9
0*Sqrt[2]*(b*c - a*d)^2*(-(b*c) + 13*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a
^(1/4)])/a^(3/4) + (90*Sqrt[2]*(b*c - 13*a*d)*(b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*
b^(1/4)*Sqrt[x])/a^(1/4)])/a^(3/4) + (45*Sqrt[2]*(b*c - a*d)^2*(-(b*c) + 13*a*d)
*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(3/4) + (45*Sqrt[
2]*(b*c - 13*a*d)*(b*c - a*d)^2*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] +
Sqrt[b]*x])/a^(3/4))/(720*b^(17/4))

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Maple [B]  time = 0.025, size = 748, normalized size = 1.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)*(d*x^2+c)^3/(b*x^2+a)^2,x)

[Out]

2/9*d^3/b^2*x^(9/2)-4/5*d^3/b^3*x^(5/2)*a+6/5*d^2/b^2*x^(5/2)*c+6*d^3/b^4*a^2*x^
(1/2)-12*d^2/b^3*a*c*x^(1/2)+6*d/b^2*c^2*x^(1/2)+1/2/b^4*x^(1/2)/(b*x^2+a)*a^3*d
^3-3/2/b^3*x^(1/2)/(b*x^2+a)*a^2*c*d^2+3/2/b^2*x^(1/2)/(b*x^2+a)*a*c^2*d-1/2/b*x
^(1/2)/(b*x^2+a)*c^3-13/8/b^4*(a/b)^(1/4)*a^2*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)
*x^(1/2)+1)*d^3+27/8/b^3*(a/b)^(1/4)*a*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2
)+1)*c*d^2-15/8/b^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*c^
2*d+1/8/b*(a/b)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*c^3-13/8/b
^4*(a/b)^(1/4)*a^2*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*d^3+27/8/b^3*(a
/b)^(1/4)*a*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*c*d^2-15/8/b^2*(a/b)^(
1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*c^2*d+1/8/b*(a/b)^(1/4)/a*2^(
1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*c^3-13/16/b^4*(a/b)^(1/4)*a^2*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+27/16/b^3*(a/b)^(1/4)*a*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^2-15/16/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^2*d+1/16/b*(a/b)^(1/4)/a*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

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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^2 + c)^3*x^(3/2)/(b*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.271988, size = 2183, normalized size = 5.66 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/360*(180*(b^5*x^2 + a*b^4)*(-(b^12*c^12 - 60*a*b^11*c^11*d + 1458*a^2*b^10*c^1
0*d^2 - 18412*a^3*b^9*c^9*d^3 + 130239*a^4*b^8*c^8*d^4 - 535032*a^5*b^7*c^7*d^5
+ 1365756*a^6*b^6*c^6*d^6 - 2272824*a^7*b^5*c^5*d^7 + 2520207*a^8*b^4*c^4*d^8 -
1853644*a^9*b^3*c^3*d^9 + 871026*a^10*b^2*c^2*d^10 - 237276*a^11*b*c*d^11 + 2856
1*a^12*d^12)/(a^3*b^17))^(1/4)*arctan(-a*b^4*(-(b^12*c^12 - 60*a*b^11*c^11*d + 1
458*a^2*b^10*c^10*d^2 - 18412*a^3*b^9*c^9*d^3 + 130239*a^4*b^8*c^8*d^4 - 535032*
a^5*b^7*c^7*d^5 + 1365756*a^6*b^6*c^6*d^6 - 2272824*a^7*b^5*c^5*d^7 + 2520207*a^
8*b^4*c^4*d^8 - 1853644*a^9*b^3*c^3*d^9 + 871026*a^10*b^2*c^2*d^10 - 237276*a^11
*b*c*d^11 + 28561*a^12*d^12)/(a^3*b^17))^(1/4)/((b^3*c^3 - 15*a*b^2*c^2*d + 27*a
^2*b*c*d^2 - 13*a^3*d^3)*sqrt(x) - sqrt(a^2*b^8*sqrt(-(b^12*c^12 - 60*a*b^11*c^1
1*d + 1458*a^2*b^10*c^10*d^2 - 18412*a^3*b^9*c^9*d^3 + 130239*a^4*b^8*c^8*d^4 -
535032*a^5*b^7*c^7*d^5 + 1365756*a^6*b^6*c^6*d^6 - 2272824*a^7*b^5*c^5*d^7 + 252
0207*a^8*b^4*c^4*d^8 - 1853644*a^9*b^3*c^3*d^9 + 871026*a^10*b^2*c^2*d^10 - 2372
76*a^11*b*c*d^11 + 28561*a^12*d^12)/(a^3*b^17)) + (b^6*c^6 - 30*a*b^5*c^5*d + 27
9*a^2*b^4*c^4*d^2 - 836*a^3*b^3*c^3*d^3 + 1119*a^4*b^2*c^2*d^4 - 702*a^5*b*c*d^5
+ 169*a^6*d^6)*x))) - 45*(b^5*x^2 + a*b^4)*(-(b^12*c^12 - 60*a*b^11*c^11*d + 14
58*a^2*b^10*c^10*d^2 - 18412*a^3*b^9*c^9*d^3 + 130239*a^4*b^8*c^8*d^4 - 535032*a
^5*b^7*c^7*d^5 + 1365756*a^6*b^6*c^6*d^6 - 2272824*a^7*b^5*c^5*d^7 + 2520207*a^8
*b^4*c^4*d^8 - 1853644*a^9*b^3*c^3*d^9 + 871026*a^10*b^2*c^2*d^10 - 237276*a^11*
b*c*d^11 + 28561*a^12*d^12)/(a^3*b^17))^(1/4)*log(a*b^4*(-(b^12*c^12 - 60*a*b^11
*c^11*d + 1458*a^2*b^10*c^10*d^2 - 18412*a^3*b^9*c^9*d^3 + 130239*a^4*b^8*c^8*d^
4 - 535032*a^5*b^7*c^7*d^5 + 1365756*a^6*b^6*c^6*d^6 - 2272824*a^7*b^5*c^5*d^7 +
2520207*a^8*b^4*c^4*d^8 - 1853644*a^9*b^3*c^3*d^9 + 871026*a^10*b^2*c^2*d^10 -
237276*a^11*b*c*d^11 + 28561*a^12*d^12)/(a^3*b^17))^(1/4) - (b^3*c^3 - 15*a*b^2*
c^2*d + 27*a^2*b*c*d^2 - 13*a^3*d^3)*sqrt(x)) + 45*(b^5*x^2 + a*b^4)*(-(b^12*c^1
2 - 60*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 18412*a^3*b^9*c^9*d^3 + 130239*a
^4*b^8*c^8*d^4 - 535032*a^5*b^7*c^7*d^5 + 1365756*a^6*b^6*c^6*d^6 - 2272824*a^7*
b^5*c^5*d^7 + 2520207*a^8*b^4*c^4*d^8 - 1853644*a^9*b^3*c^3*d^9 + 871026*a^10*b^
2*c^2*d^10 - 237276*a^11*b*c*d^11 + 28561*a^12*d^12)/(a^3*b^17))^(1/4)*log(-a*b^
4*(-(b^12*c^12 - 60*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 18412*a^3*b^9*c^9*d
^3 + 130239*a^4*b^8*c^8*d^4 - 535032*a^5*b^7*c^7*d^5 + 1365756*a^6*b^6*c^6*d^6 -
2272824*a^7*b^5*c^5*d^7 + 2520207*a^8*b^4*c^4*d^8 - 1853644*a^9*b^3*c^3*d^9 + 8
71026*a^10*b^2*c^2*d^10 - 237276*a^11*b*c*d^11 + 28561*a^12*d^12)/(a^3*b^17))^(1
/4) - (b^3*c^3 - 15*a*b^2*c^2*d + 27*a^2*b*c*d^2 - 13*a^3*d^3)*sqrt(x)) + 4*(20*
b^3*d^3*x^6 - 45*b^3*c^3 + 675*a*b^2*c^2*d - 1215*a^2*b*c*d^2 + 585*a^3*d^3 + 4*
(27*b^3*c*d^2 - 13*a*b^2*d^3)*x^4 + 36*(15*b^3*c^2*d - 27*a*b^2*c*d^2 + 13*a^2*b
*d^3)*x^2)*sqrt(x))/(b^5*x^2 + a*b^4)

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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**(3/2)*(d*x**2+c)**3/(b*x**2+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.292189, size = 745, normalized size = 1.93 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 15*(a*b^3)^(1/4)*a*b^2*c^2*d + 27*(a*b^3)^(
1/4)*a^2*b*c*d^2 - 13*(a*b^3)^(1/4)*a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(
1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a*b^5) + 1/8*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 15
*(a*b^3)^(1/4)*a*b^2*c^2*d + 27*(a*b^3)^(1/4)*a^2*b*c*d^2 - 13*(a*b^3)^(1/4)*a^3
*d^3)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a*b^5)
+ 1/16*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 15*(a*b^3)^(1/4)*a*b^2*c^2*d + 27*(a*b^
3)^(1/4)*a^2*b*c*d^2 - 13*(a*b^3)^(1/4)*a^3*d^3)*ln(sqrt(2)*sqrt(x)*(a/b)^(1/4)
+ x + sqrt(a/b))/(a*b^5) - 1/16*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 15*(a*b^3)^(1/4
)*a*b^2*c^2*d + 27*(a*b^3)^(1/4)*a^2*b*c*d^2 - 13*(a*b^3)^(1/4)*a^3*d^3)*ln(-sqr
t(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a*b^5) - 1/2*(b^3*c^3*sqrt(x) - 3*a*b
^2*c^2*d*sqrt(x) + 3*a^2*b*c*d^2*sqrt(x) - a^3*d^3*sqrt(x))/((b*x^2 + a)*b^4) +
2/45*(5*b^16*d^3*x^(9/2) + 27*b^16*c*d^2*x^(5/2) - 18*a*b^15*d^3*x^(5/2) + 135*b
^16*c^2*d*sqrt(x) - 270*a*b^15*c*d^2*sqrt(x) + 135*a^2*b^14*d^3*sqrt(x))/b^18