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

Optimal. Leaf size=288 \[ \frac{\sqrt [4]{c} (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} d^{13/4}}-\frac{\sqrt [4]{c} (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} d^{13/4}}+\frac{\sqrt [4]{c} (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt{2} d^{13/4}}-\frac{\sqrt [4]{c} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{\sqrt{2} d^{13/4}}+\frac{2 \sqrt{x} (b c-a d)^2}{d^3}-\frac{2 b x^{5/2} (b c-2 a d)}{5 d^2}+\frac{2 b^2 x^{9/2}}{9 d} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.506879, antiderivative size = 288, 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 \[ \frac{\sqrt [4]{c} (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} d^{13/4}}-\frac{\sqrt [4]{c} (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{2 \sqrt{2} d^{13/4}}+\frac{\sqrt [4]{c} (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{\sqrt{2} d^{13/4}}-\frac{\sqrt [4]{c} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{\sqrt{2} d^{13/4}}+\frac{2 \sqrt{x} (b c-a d)^2}{d^3}-\frac{2 b x^{5/2} (b c-2 a d)}{5 d^2}+\frac{2 b^2 x^{9/2}}{9 d} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 97.4904, size = 270, normalized size = 0.94 \[ \frac{2 b^{2} x^{\frac{9}{2}}}{9 d} + \frac{2 b x^{\frac{5}{2}} \left (2 a d - b c\right )}{5 d^{2}} + \frac{\sqrt{2} \sqrt [4]{c} \left (a d - b c\right )^{2} \log{\left (- \sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{4 d^{\frac{13}{4}}} - \frac{\sqrt{2} \sqrt [4]{c} \left (a d - b c\right )^{2} \log{\left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{4 d^{\frac{13}{4}}} + \frac{\sqrt{2} \sqrt [4]{c} \left (a d - b c\right )^{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{2 d^{\frac{13}{4}}} - \frac{\sqrt{2} \sqrt [4]{c} \left (a d - b c\right )^{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{2 d^{\frac{13}{4}}} + \frac{2 \sqrt{x} \left (a d - b c\right )^{2}}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.163061, size = 276, normalized size = 0.96 \[ \frac{-72 b d^{5/4} x^{5/2} (b c-2 a d)+360 \sqrt [4]{d} \sqrt{x} (b c-a d)^2+45 \sqrt{2} \sqrt [4]{c} (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )-45 \sqrt{2} \sqrt [4]{c} (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )+90 \sqrt{2} \sqrt [4]{c} (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )-90 \sqrt{2} \sqrt [4]{c} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )+40 b^2 d^{9/4} x^{9/2}}{180 d^{13/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.016, size = 495, normalized size = 1.7 \[{\frac{2\,{b}^{2}}{9\,d}{x}^{{\frac{9}{2}}}}+{\frac{4\,ab}{5\,d}{x}^{{\frac{5}{2}}}}-{\frac{2\,{b}^{2}c}{5\,{d}^{2}}{x}^{{\frac{5}{2}}}}+2\,{\frac{{a}^{2}\sqrt{x}}{d}}-4\,{\frac{abc\sqrt{x}}{{d}^{2}}}+2\,{\frac{{b}^{2}{c}^{2}\sqrt{x}}{{d}^{3}}}-{\frac{\sqrt{2}{a}^{2}}{2\,d}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }+{\frac{\sqrt{2}abc}{{d}^{2}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }-{\frac{\sqrt{2}{b}^{2}{c}^{2}}{2\,{d}^{3}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}-1 \right ) }-{\frac{\sqrt{2}{a}^{2}}{4\,d}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ( x-\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}abc}{2\,{d}^{2}}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ( x-\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }-{\frac{\sqrt{2}{b}^{2}{c}^{2}}{4\,{d}^{3}}\sqrt [4]{{\frac{c}{d}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) \left ( x-\sqrt [4]{{\frac{c}{d}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{c}{d}}} \right ) ^{-1}} \right ) }-{\frac{\sqrt{2}{a}^{2}}{2\,d}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }+{\frac{\sqrt{2}abc}{{d}^{2}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) }-{\frac{\sqrt{2}{b}^{2}{c}^{2}}{2\,{d}^{3}}\sqrt [4]{{\frac{c}{d}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{c}{d}}}}}}+1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.259936, size = 1401, normalized size = 4.86 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/90*(180*d^3*(-(b^8*c^9 - 8*a*b^7*c^8*d + 28*a^2*b^6*c^7*d^2 - 56*a^3*b^5*c^6*d
^3 + 70*a^4*b^4*c^5*d^4 - 56*a^5*b^3*c^4*d^5 + 28*a^6*b^2*c^3*d^6 - 8*a^7*b*c^2*
d^7 + a^8*c*d^8)/d^13)^(1/4)*arctan(d^3*(-(b^8*c^9 - 8*a*b^7*c^8*d + 28*a^2*b^6*
c^7*d^2 - 56*a^3*b^5*c^6*d^3 + 70*a^4*b^4*c^5*d^4 - 56*a^5*b^3*c^4*d^5 + 28*a^6*
b^2*c^3*d^6 - 8*a^7*b*c^2*d^7 + a^8*c*d^8)/d^13)^(1/4)/((b^2*c^2 - 2*a*b*c*d + a
^2*d^2)*sqrt(x) + sqrt(d^6*sqrt(-(b^8*c^9 - 8*a*b^7*c^8*d + 28*a^2*b^6*c^7*d^2 -
56*a^3*b^5*c^6*d^3 + 70*a^4*b^4*c^5*d^4 - 56*a^5*b^3*c^4*d^5 + 28*a^6*b^2*c^3*d
^6 - 8*a^7*b*c^2*d^7 + a^8*c*d^8)/d^13) + (b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c
^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*x))) - 45*d^3*(-(b^8*c^9 - 8*a*b^7*c^8*d + 28*
a^2*b^6*c^7*d^2 - 56*a^3*b^5*c^6*d^3 + 70*a^4*b^4*c^5*d^4 - 56*a^5*b^3*c^4*d^5 +
28*a^6*b^2*c^3*d^6 - 8*a^7*b*c^2*d^7 + a^8*c*d^8)/d^13)^(1/4)*log(d^3*(-(b^8*c^
9 - 8*a*b^7*c^8*d + 28*a^2*b^6*c^7*d^2 - 56*a^3*b^5*c^6*d^3 + 70*a^4*b^4*c^5*d^4
- 56*a^5*b^3*c^4*d^5 + 28*a^6*b^2*c^3*d^6 - 8*a^7*b*c^2*d^7 + a^8*c*d^8)/d^13)^
(1/4) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(x)) + 45*d^3*(-(b^8*c^9 - 8*a*b^7*c
^8*d + 28*a^2*b^6*c^7*d^2 - 56*a^3*b^5*c^6*d^3 + 70*a^4*b^4*c^5*d^4 - 56*a^5*b^3
*c^4*d^5 + 28*a^6*b^2*c^3*d^6 - 8*a^7*b*c^2*d^7 + a^8*c*d^8)/d^13)^(1/4)*log(-d^
3*(-(b^8*c^9 - 8*a*b^7*c^8*d + 28*a^2*b^6*c^7*d^2 - 56*a^3*b^5*c^6*d^3 + 70*a^4*
b^4*c^5*d^4 - 56*a^5*b^3*c^4*d^5 + 28*a^6*b^2*c^3*d^6 - 8*a^7*b*c^2*d^7 + a^8*c*
d^8)/d^13)^(1/4) + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(x)) + 4*(5*b^2*d^2*x^4 +
45*b^2*c^2 - 90*a*b*c*d + 45*a^2*d^2 - 9*(b^2*c*d - 2*a*b*d^2)*x^2)*sqrt(x))/d^
3

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.251715, size = 520, normalized size = 1.81 \[ -\frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{2 \, d^{4}} - \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{2 \, d^{4}} - \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{4 \, d^{4}} + \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{1}{4}} b^{2} c^{2} - 2 \, \left (c d^{3}\right )^{\frac{1}{4}} a b c d + \left (c d^{3}\right )^{\frac{1}{4}} a^{2} d^{2}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{4 \, d^{4}} + \frac{2 \,{\left (5 \, b^{2} d^{8} x^{\frac{9}{2}} - 9 \, b^{2} c d^{7} x^{\frac{5}{2}} + 18 \, a b d^{8} x^{\frac{5}{2}} + 45 \, b^{2} c^{2} d^{6} \sqrt{x} - 90 \, a b c d^{7} \sqrt{x} + 45 \, a^{2} d^{8} \sqrt{x}\right )}}{45 \, d^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(2)*((c*d^3)^(1/4)*b^2*c^2 - 2*(c*d^3)^(1/4)*a*b*c*d + (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^4 - 1
/2*sqrt(2)*((c*d^3)^(1/4)*b^2*c^2 - 2*(c*d^3)^(1/4)*a*b*c*d + (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^4 - 1/
4*sqrt(2)*((c*d^3)^(1/4)*b^2*c^2 - 2*(c*d^3)^(1/4)*a*b*c*d + (c*d^3)^(1/4)*a^2*d
^2)*ln(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/d^4 + 1/4*sqrt(2)*((c*d^3)^(
1/4)*b^2*c^2 - 2*(c*d^3)^(1/4)*a*b*c*d + (c*d^3)^(1/4)*a^2*d^2)*ln(-sqrt(2)*sqrt
(x)*(c/d)^(1/4) + x + sqrt(c/d))/d^4 + 2/45*(5*b^2*d^8*x^(9/2) - 9*b^2*c*d^7*x^(
5/2) + 18*a*b*d^8*x^(5/2) + 45*b^2*c^2*d^6*sqrt(x) - 90*a*b*c*d^7*sqrt(x) + 45*a
^2*d^8*sqrt(x))/d^9