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

Optimal. Leaf size=304 \[ -\frac{(b c-a d)^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{3/4} b^{13/4}}+\frac{(b c-a d)^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{3/4} b^{13/4}}-\frac{(b c-a d)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{3/4} b^{13/4}}+\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{3/4} b^{13/4}}+\frac{2 d \sqrt{x} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3}+\frac{2 d^2 x^{5/2} (3 b c-a d)}{5 b^2}+\frac{2 d^3 x^{9/2}}{9 b} \]

[Out]

(2*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*Sqrt[x])/b^3 + (2*d^2*(3*b*c - a*d)*x^(5/
2))/(5*b^2) + (2*d^3*x^(9/2))/(9*b) - ((b*c - a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)
*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(3/4)*b^(13/4)) + ((b*c - a*d)^3*ArcTan[1 + (Sqrt
[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(3/4)*b^(13/4)) - ((b*c - a*d)^3*Log[S
qrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(3/4)*b^(13/
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]*a^(3/4)*b^(13/4))

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Rubi [A]  time = 0.506925, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{(b c-a d)^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{3/4} b^{13/4}}+\frac{(b c-a d)^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} a^{3/4} b^{13/4}}-\frac{(b c-a d)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} a^{3/4} b^{13/4}}+\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} a^{3/4} b^{13/4}}+\frac{2 d \sqrt{x} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3}+\frac{2 d^2 x^{5/2} (3 b c-a d)}{5 b^2}+\frac{2 d^3 x^{9/2}}{9 b} \]

Antiderivative was successfully verified.

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

[Out]

(2*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*Sqrt[x])/b^3 + (2*d^2*(3*b*c - a*d)*x^(5/
2))/(5*b^2) + (2*d^3*x^(9/2))/(9*b) - ((b*c - a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)
*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(3/4)*b^(13/4)) + ((b*c - a*d)^3*ArcTan[1 + (Sqrt
[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(3/4)*b^(13/4)) - ((b*c - a*d)^3*Log[S
qrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(3/4)*b^(13/
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]*a^(3/4)*b^(13/4))

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Rubi in Sympy [A]  time = 96.7473, size = 289, normalized size = 0.95 \[ \frac{2 d^{3} x^{\frac{9}{2}}}{9 b} - \frac{2 d^{2} x^{\frac{5}{2}} \left (a d - 3 b c\right )}{5 b^{2}} + \frac{2 d \sqrt{x} \left (a^{2} d^{2} - 3 a b c d + 3 b^{2} c^{2}\right )}{b^{3}} + \frac{\sqrt{2} \left (a d - b c\right )^{3} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 a^{\frac{3}{4}} b^{\frac{13}{4}}} - \frac{\sqrt{2} \left (a d - b c\right )^{3} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{4 a^{\frac{3}{4}} b^{\frac{13}{4}}} + \frac{\sqrt{2} \left (a d - b c\right )^{3} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{3}{4}} b^{\frac{13}{4}}} - \frac{\sqrt{2} \left (a d - b c\right )^{3} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{3}{4}} b^{\frac{13}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)**3/(b*x**2+a)/x**(1/2),x)

[Out]

2*d**3*x**(9/2)/(9*b) - 2*d**2*x**(5/2)*(a*d - 3*b*c)/(5*b**2) + 2*d*sqrt(x)*(a*
*2*d**2 - 3*a*b*c*d + 3*b**2*c**2)/b**3 + sqrt(2)*(a*d - b*c)**3*log(-sqrt(2)*a*
*(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(4*a**(3/4)*b**(13/4)) - sqrt(2)*
(a*d - b*c)**3*log(sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(4*a
**(3/4)*b**(13/4)) + sqrt(2)*(a*d - b*c)**3*atan(1 - sqrt(2)*b**(1/4)*sqrt(x)/a*
*(1/4))/(2*a**(3/4)*b**(13/4)) - sqrt(2)*(a*d - b*c)**3*atan(1 + sqrt(2)*b**(1/4
)*sqrt(x)/a**(1/4))/(2*a**(3/4)*b**(13/4))

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Mathematica [A]  time = 0.260934, size = 291, normalized size = 0.96 \[ \frac{-72 a^{3/4} b^{5/4} d^2 x^{5/2} (a d-3 b c)+40 a^{3/4} b^{9/4} d^3 x^{9/2}+360 a^{3/4} \sqrt [4]{b} d \sqrt{x} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )-45 \sqrt{2} (b c-a d)^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+45 \sqrt{2} (b c-a d)^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-90 \sqrt{2} (b c-a d)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+90 \sqrt{2} (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{180 a^{3/4} b^{13/4}} \]

Antiderivative was successfully verified.

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

[Out]

(360*a^(3/4)*b^(1/4)*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*Sqrt[x] - 72*a^(3/4)*b^
(5/4)*d^2*(-3*b*c + a*d)*x^(5/2) + 40*a^(3/4)*b^(9/4)*d^3*x^(9/2) - 90*Sqrt[2]*(
b*c - a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 90*Sqrt[2]*(b*c - a
*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 45*Sqrt[2]*(b*c - a*d)^3*L
og[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 45*Sqrt[2]*(b*c - a*
d)^3*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(180*a^(3/4)*b^
(13/4))

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Maple [B]  time = 0.014, size = 650, normalized size = 2.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)^3/(b*x^2+a)/x^(1/2),x)

[Out]

2/9*d^3*x^(9/2)/b-2/5*d^3/b^2*x^(5/2)*a+6/5*d^2/b*x^(5/2)*c+2*d^3/b^3*a^2*x^(1/2
)-6*d^2/b^2*a*c*x^(1/2)+6*d/b*c^2*x^(1/2)-1/2/b^3*(a/b)^(1/4)*a^2*2^(1/2)*arctan
(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*d^3+3/2/b^2*(a/b)^(1/4)*a*2^(1/2)*arctan(2^(1/2)
/(a/b)^(1/4)*x^(1/2)-1)*c*d^2-3/2/b*(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)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)
-1)*c^3-1/4/b^3*(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+3/4/b^2*(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^2-3/4/b*(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)^(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^3-1/2/b^3*(a/b)^(1/4)*a^2*2^(1/2)*arctan(2^(1/2)/(a/
b)^(1/4)*x^(1/2)+1)*d^3+3/2/b^2*(a/b)^(1/4)*a*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)
*x^(1/2)+1)*c*d^2-3/2/b*(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)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*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/((b*x^2 + a)*sqrt(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.254648, size = 2052, normalized size = 6.75 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/90*(180*b^3*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b
^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 7
92*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2
*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^3*b^13))^(1/4)*arctan(-a*b^3*(-(b^12*c^
12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8
*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495
*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11
+ a^12*d^12)/(a^3*b^13))^(1/4)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*
d^3)*sqrt(x) - sqrt(a^2*b^6*sqrt(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^
10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a
^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9
+ 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^3*b^13)) + (b^6*c^6 -
6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 -
6*a^5*b*c*d^5 + a^6*d^6)*x))) - 45*b^3*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*
b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5
+ 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*
c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^3*b^13))^(1/4)
*log(a*b^3*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*
c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*
a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^
10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^3*b^13))^(1/4) - (b^3*c^3 - 3*a*b^2*c^2*d
+ 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(x)) + 45*b^3*(-(b^12*c^12 - 12*a*b^11*c^11*d + 6
6*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^
7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^
9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^3*b^13))
^(1/4)*log(-a*b^3*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a
^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6
- 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2
*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a^3*b^13))^(1/4) - (b^3*c^3 - 3*a*b^2
*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(x)) + 4*(5*b^2*d^3*x^4 + 135*b^2*c^2*d -
135*a*b*c*d^2 + 45*a^2*d^3 + 9*(3*b^2*c*d^2 - a*b*d^3)*x^2)*sqrt(x))/b^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((d*x**2+c)**3/(b*x**2+a)/x**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.284875, size = 662, normalized size = 2.18 \[ \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, a b^{4}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, a b^{4}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, a b^{4}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 3 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} - \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{4 \, a b^{4}} + \frac{2 \,{\left (5 \, b^{8} d^{3} x^{\frac{9}{2}} + 27 \, b^{8} c d^{2} x^{\frac{5}{2}} - 9 \, a b^{7} d^{3} x^{\frac{5}{2}} + 135 \, b^{8} c^{2} d \sqrt{x} - 135 \, a b^{7} c d^{2} \sqrt{x} + 45 \, a^{2} b^{6} d^{3} \sqrt{x}\right )}}{45 \, b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 3*(a*b^3)^(1/4)*a*b^2*c^2*d + 3*(a*b^3)^(1/
4)*a^2*b*c*d^2 - (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^4) + 1/2*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 3*(a*b^
3)^(1/4)*a*b^2*c^2*d + 3*(a*b^3)^(1/4)*a^2*b*c*d^2 - (a*b^3)^(1/4)*a^3*d^3)*arct
an(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a*b^4) + 1/4*sqr
t(2)*((a*b^3)^(1/4)*b^3*c^3 - 3*(a*b^3)^(1/4)*a*b^2*c^2*d + 3*(a*b^3)^(1/4)*a^2*
b*c*d^2 - (a*b^3)^(1/4)*a^3*d^3)*ln(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))
/(a*b^4) - 1/4*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 3*(a*b^3)^(1/4)*a*b^2*c^2*d + 3*
(a*b^3)^(1/4)*a^2*b*c*d^2 - (a*b^3)^(1/4)*a^3*d^3)*ln(-sqrt(2)*sqrt(x)*(a/b)^(1/
4) + x + sqrt(a/b))/(a*b^4) + 2/45*(5*b^8*d^3*x^(9/2) + 27*b^8*c*d^2*x^(5/2) - 9
*a*b^7*d^3*x^(5/2) + 135*b^8*c^2*d*sqrt(x) - 135*a*b^7*c*d^2*sqrt(x) + 45*a^2*b^
6*d^3*sqrt(x))/b^9