3.425 \(\int \frac{x^{7/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^2} \, dx\)

Optimal. Leaf size=375 \[ \frac{\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} d^{17/4}}-\frac{\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} d^{17/4}}+\frac{\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} d^{17/4}}-\frac{\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} d^{17/4}}+\frac{\sqrt{x} (13 b c-5 a d) (b c-a d)}{2 d^4}-\frac{x^{5/2} (13 b c-5 a d) (b c-a d)}{10 c d^3}+\frac{x^{9/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac{2 b^2 x^{9/2}}{9 d^2} \]

[Out]

((13*b*c - 5*a*d)*(b*c - a*d)*Sqrt[x])/(2*d^4) - ((13*b*c - 5*a*d)*(b*c - a*d)*x
^(5/2))/(10*c*d^3) + (2*b^2*x^(9/2))/(9*d^2) + ((b*c - a*d)^2*x^(9/2))/(2*c*d^2*
(c + d*x^2)) + (c^(1/4)*(13*b*c - 5*a*d)*(b*c - a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)
*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*d^(17/4)) - (c^(1/4)*(13*b*c - 5*a*d)*(b*c - a*d)
*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*d^(17/4)) + (c^(1/4)*
(13*b*c - 5*a*d)*(b*c - a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqr
t[d]*x])/(8*Sqrt[2]*d^(17/4)) - (c^(1/4)*(13*b*c - 5*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]*d^(17/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.949134, antiderivative size = 375, 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 \[ \frac{\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} d^{17/4}}-\frac{\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} d^{17/4}}+\frac{\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} d^{17/4}}-\frac{\sqrt [4]{c} (13 b c-5 a d) (b c-a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} d^{17/4}}+\frac{\sqrt{x} (13 b c-5 a d) (b c-a d)}{2 d^4}-\frac{x^{5/2} (13 b c-5 a d) (b c-a d)}{10 c d^3}+\frac{x^{9/2} (b c-a d)^2}{2 c d^2 \left (c+d x^2\right )}+\frac{2 b^2 x^{9/2}}{9 d^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^2,x]

[Out]

((13*b*c - 5*a*d)*(b*c - a*d)*Sqrt[x])/(2*d^4) - ((13*b*c - 5*a*d)*(b*c - a*d)*x
^(5/2))/(10*c*d^3) + (2*b^2*x^(9/2))/(9*d^2) + ((b*c - a*d)^2*x^(9/2))/(2*c*d^2*
(c + d*x^2)) + (c^(1/4)*(13*b*c - 5*a*d)*(b*c - a*d)*ArcTan[1 - (Sqrt[2]*d^(1/4)
*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*d^(17/4)) - (c^(1/4)*(13*b*c - 5*a*d)*(b*c - a*d)
*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(4*Sqrt[2]*d^(17/4)) + (c^(1/4)*
(13*b*c - 5*a*d)*(b*c - a*d)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqr
t[d]*x])/(8*Sqrt[2]*d^(17/4)) - (c^(1/4)*(13*b*c - 5*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]*d^(17/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 114.981, size = 348, normalized size = 0.93 \[ \frac{2 b^{2} x^{\frac{9}{2}}}{9 d^{2}} + \frac{\sqrt{2} \sqrt [4]{c} \left (a d - b c\right ) \left (5 a d - 13 b c\right ) \log{\left (- \sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{16 d^{\frac{17}{4}}} - \frac{\sqrt{2} \sqrt [4]{c} \left (a d - b c\right ) \left (5 a d - 13 b c\right ) \log{\left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{16 d^{\frac{17}{4}}} + \frac{\sqrt{2} \sqrt [4]{c} \left (a d - b c\right ) \left (5 a d - 13 b c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{8 d^{\frac{17}{4}}} - \frac{\sqrt{2} \sqrt [4]{c} \left (a d - b c\right ) \left (5 a d - 13 b c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{8 d^{\frac{17}{4}}} + \frac{\sqrt{x} \left (a d - b c\right ) \left (5 a d - 13 b c\right )}{2 d^{4}} + \frac{x^{\frac{9}{2}} \left (a d - b c\right )^{2}}{2 c d^{2} \left (c + d x^{2}\right )} - \frac{x^{\frac{5}{2}} \left (a d - b c\right ) \left (5 a d - 13 b c\right )}{10 c d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)*(b*x**2+a)**2/(d*x**2+c)**2,x)

[Out]

2*b**2*x**(9/2)/(9*d**2) + sqrt(2)*c**(1/4)*(a*d - b*c)*(5*a*d - 13*b*c)*log(-sq
rt(2)*c**(1/4)*d**(1/4)*sqrt(x) + sqrt(c) + sqrt(d)*x)/(16*d**(17/4)) - sqrt(2)*
c**(1/4)*(a*d - b*c)*(5*a*d - 13*b*c)*log(sqrt(2)*c**(1/4)*d**(1/4)*sqrt(x) + sq
rt(c) + sqrt(d)*x)/(16*d**(17/4)) + sqrt(2)*c**(1/4)*(a*d - b*c)*(5*a*d - 13*b*c
)*atan(1 - sqrt(2)*d**(1/4)*sqrt(x)/c**(1/4))/(8*d**(17/4)) - sqrt(2)*c**(1/4)*(
a*d - b*c)*(5*a*d - 13*b*c)*atan(1 + sqrt(2)*d**(1/4)*sqrt(x)/c**(1/4))/(8*d**(1
7/4)) + sqrt(x)*(a*d - b*c)*(5*a*d - 13*b*c)/(2*d**4) + x**(9/2)*(a*d - b*c)**2/
(2*c*d**2*(c + d*x**2)) - x**(5/2)*(a*d - b*c)*(5*a*d - 13*b*c)/(10*c*d**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.698379, size = 372, normalized size = 0.99 \[ \frac{1440 \sqrt [4]{d} \sqrt{x} \left (a^2 d^2-4 a b c d+3 b^2 c^2\right )+45 \sqrt{2} \sqrt [4]{c} \left (5 a^2 d^2-18 a b c d+13 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )-45 \sqrt{2} \sqrt [4]{c} \left (5 a^2 d^2-18 a b c d+13 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )+90 \sqrt{2} \sqrt [4]{c} \left (5 a^2 d^2-18 a b c d+13 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )-90 \sqrt{2} \sqrt [4]{c} \left (5 a^2 d^2-18 a b c d+13 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )-576 b d^{5/4} x^{5/2} (b c-a d)+\frac{360 c \sqrt [4]{d} \sqrt{x} (b c-a d)^2}{c+d x^2}+160 b^2 d^{9/4} x^{9/2}}{720 d^{17/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(7/2)*(a + b*x^2)^2)/(c + d*x^2)^2,x]

[Out]

(1440*d^(1/4)*(3*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*Sqrt[x] - 576*b*d^(5/4)*(b*c - a
*d)*x^(5/2) + 160*b^2*d^(9/4)*x^(9/2) + (360*c*d^(1/4)*(b*c - a*d)^2*Sqrt[x])/(c
+ d*x^2) + 90*Sqrt[2]*c^(1/4)*(13*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*ArcTan[1 -
(Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] - 90*Sqrt[2]*c^(1/4)*(13*b^2*c^2 - 18*a*b*c*d
+ 5*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)] + 45*Sqrt[2]*c^(1/4)
*(13*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqr
t[x] + Sqrt[d]*x] - 45*Sqrt[2]*c^(1/4)*(13*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*Log
[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(720*d^(17/4))

_______________________________________________________________________________________

Maple [A]  time = 0.024, size = 563, normalized size = 1.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)*(b*x^2+a)^2/(d*x^2+c)^2,x)

[Out]

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

_______________________________________________________________________________________

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^(7/2)/(d*x^2 + c)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.264339, size = 1538, normalized size = 4.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*x^(7/2)/(d*x^2 + c)^2,x, algorithm="fricas")

[Out]

1/360*(180*(d^5*x^2 + c*d^4)*(-(28561*b^8*c^9 - 158184*a*b^7*c^8*d + 372476*a^2*
b^6*c^7*d^2 - 485784*a^3*b^5*c^6*d^3 + 383046*a^4*b^4*c^5*d^4 - 186840*a^5*b^3*c
^4*d^5 + 55100*a^6*b^2*c^3*d^6 - 9000*a^7*b*c^2*d^7 + 625*a^8*c*d^8)/d^17)^(1/4)
*arctan(d^4*(-(28561*b^8*c^9 - 158184*a*b^7*c^8*d + 372476*a^2*b^6*c^7*d^2 - 485
784*a^3*b^5*c^6*d^3 + 383046*a^4*b^4*c^5*d^4 - 186840*a^5*b^3*c^4*d^5 + 55100*a^
6*b^2*c^3*d^6 - 9000*a^7*b*c^2*d^7 + 625*a^8*c*d^8)/d^17)^(1/4)/((13*b^2*c^2 - 1
8*a*b*c*d + 5*a^2*d^2)*sqrt(x) + sqrt(d^8*sqrt(-(28561*b^8*c^9 - 158184*a*b^7*c^
8*d + 372476*a^2*b^6*c^7*d^2 - 485784*a^3*b^5*c^6*d^3 + 383046*a^4*b^4*c^5*d^4 -
186840*a^5*b^3*c^4*d^5 + 55100*a^6*b^2*c^3*d^6 - 9000*a^7*b*c^2*d^7 + 625*a^8*c
*d^8)/d^17) + (169*b^4*c^4 - 468*a*b^3*c^3*d + 454*a^2*b^2*c^2*d^2 - 180*a^3*b*c
*d^3 + 25*a^4*d^4)*x))) - 45*(d^5*x^2 + c*d^4)*(-(28561*b^8*c^9 - 158184*a*b^7*c
^8*d + 372476*a^2*b^6*c^7*d^2 - 485784*a^3*b^5*c^6*d^3 + 383046*a^4*b^4*c^5*d^4
- 186840*a^5*b^3*c^4*d^5 + 55100*a^6*b^2*c^3*d^6 - 9000*a^7*b*c^2*d^7 + 625*a^8*
c*d^8)/d^17)^(1/4)*log(d^4*(-(28561*b^8*c^9 - 158184*a*b^7*c^8*d + 372476*a^2*b^
6*c^7*d^2 - 485784*a^3*b^5*c^6*d^3 + 383046*a^4*b^4*c^5*d^4 - 186840*a^5*b^3*c^4
*d^5 + 55100*a^6*b^2*c^3*d^6 - 9000*a^7*b*c^2*d^7 + 625*a^8*c*d^8)/d^17)^(1/4) +
(13*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*sqrt(x)) + 45*(d^5*x^2 + c*d^4)*(-(28561*
b^8*c^9 - 158184*a*b^7*c^8*d + 372476*a^2*b^6*c^7*d^2 - 485784*a^3*b^5*c^6*d^3 +
383046*a^4*b^4*c^5*d^4 - 186840*a^5*b^3*c^4*d^5 + 55100*a^6*b^2*c^3*d^6 - 9000*
a^7*b*c^2*d^7 + 625*a^8*c*d^8)/d^17)^(1/4)*log(-d^4*(-(28561*b^8*c^9 - 158184*a*
b^7*c^8*d + 372476*a^2*b^6*c^7*d^2 - 485784*a^3*b^5*c^6*d^3 + 383046*a^4*b^4*c^5
*d^4 - 186840*a^5*b^3*c^4*d^5 + 55100*a^6*b^2*c^3*d^6 - 9000*a^7*b*c^2*d^7 + 625
*a^8*c*d^8)/d^17)^(1/4) + (13*b^2*c^2 - 18*a*b*c*d + 5*a^2*d^2)*sqrt(x)) + 4*(20
*b^2*d^3*x^6 + 585*b^2*c^3 - 810*a*b*c^2*d + 225*a^2*c*d^2 - 4*(13*b^2*c*d^2 - 1
8*a*b*d^3)*x^4 + 36*(13*b^2*c^2*d - 18*a*b*c*d^2 + 5*a^2*d^3)*x^2)*sqrt(x))/(d^5
*x^2 + c*d^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**(7/2)*(b*x**2+a)**2/(d*x**2+c)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.25398, size = 594, normalized size = 1.58 \[ -\frac{\sqrt{2}{\left (13 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 18 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{8 \, d^{5}} - \frac{\sqrt{2}{\left (13 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 18 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{8 \, d^{5}} - \frac{\sqrt{2}{\left (13 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 18 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{16 \, d^{5}} + \frac{\sqrt{2}{\left (13 \, \left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 18 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + 5 \, \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{16 \, d^{5}} + \frac{b^{2} c^{3} \sqrt{x} - 2 \, a b c^{2} d \sqrt{x} + a^{2} c d^{2} \sqrt{x}}{2 \,{\left (d x^{2} + c\right )} d^{4}} + \frac{2 \,{\left (5 \, b^{2} d^{16} x^{\frac{9}{2}} - 18 \, b^{2} c d^{15} x^{\frac{5}{2}} + 18 \, a b d^{16} x^{\frac{5}{2}} + 135 \, b^{2} c^{2} d^{14} \sqrt{x} - 180 \, a b c d^{15} \sqrt{x} + 45 \, a^{2} d^{16} \sqrt{x}\right )}}{45 \, d^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*x^(7/2)/(d*x^2 + c)^2,x, algorithm="giac")

[Out]

-1/8*sqrt(2)*(13*(c*d^3)^(1/4)*b^2*c^2 - 18*(c*d^3)^(1/4)*a*b*c*d + 5*(c*d^3)^(1
/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/d
^5 - 1/8*sqrt(2)*(13*(c*d^3)^(1/4)*b^2*c^2 - 18*(c*d^3)^(1/4)*a*b*c*d + 5*(c*d^3
)^(1/4)*a^2*d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/
4))/d^5 - 1/16*sqrt(2)*(13*(c*d^3)^(1/4)*b^2*c^2 - 18*(c*d^3)^(1/4)*a*b*c*d + 5*
(c*d^3)^(1/4)*a^2*d^2)*ln(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/d^5 + 1/1
6*sqrt(2)*(13*(c*d^3)^(1/4)*b^2*c^2 - 18*(c*d^3)^(1/4)*a*b*c*d + 5*(c*d^3)^(1/4)
*a^2*d^2)*ln(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/d^5 + 1/2*(b^2*c^3*sq
rt(x) - 2*a*b*c^2*d*sqrt(x) + a^2*c*d^2*sqrt(x))/((d*x^2 + c)*d^4) + 2/45*(5*b^2
*d^16*x^(9/2) - 18*b^2*c*d^15*x^(5/2) + 18*a*b*d^16*x^(5/2) + 135*b^2*c^2*d^14*s
qrt(x) - 180*a*b*c*d^15*sqrt(x) + 45*a^2*d^16*sqrt(x))/d^18