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

Optimal. Leaf size=363 \[ -\frac{9 a^2 d^2-10 a b c d+5 b^2 c^2}{10 c^2 d \sqrt{x} \left (c+d x^2\right )}-\frac{2 a^2}{5 c x^{5/2} \left (c+d x^2\right )}+\frac{(b c-9 a d) (b c-a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{13/4} d^{3/4}}-\frac{(b c-9 a d) (b c-a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{13/4} d^{3/4}}-\frac{(b c-9 a d) (b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{13/4} d^{3/4}}+\frac{(b c-9 a d) (b c-a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{13/4} d^{3/4}}+\frac{(b c-9 a d) (b c-a d)}{2 c^3 d \sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.805869, antiderivative size = 363, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ -\frac{9 a^2 d^2-10 a b c d+5 b^2 c^2}{10 c^2 d \sqrt{x} \left (c+d x^2\right )}-\frac{2 a^2}{5 c x^{5/2} \left (c+d x^2\right )}+\frac{(b c-9 a d) (b c-a d) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{13/4} d^{3/4}}-\frac{(b c-9 a d) (b c-a d) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{8 \sqrt{2} c^{13/4} d^{3/4}}-\frac{(b c-9 a d) (b c-a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{4 \sqrt{2} c^{13/4} d^{3/4}}+\frac{(b c-9 a d) (b c-a d) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{4 \sqrt{2} c^{13/4} d^{3/4}}+\frac{(b c-9 a d) (b c-a d)}{2 c^3 d \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 108.071, size = 332, normalized size = 0.91 \[ - \frac{2 a^{2}}{5 c x^{\frac{5}{2}} \left (c + d x^{2}\right )} - \frac{a d \left (9 a d - 10 b c\right ) + 5 b^{2} c^{2}}{10 c^{2} d \sqrt{x} \left (c + d x^{2}\right )} + \frac{\left (a d - b c\right ) \left (9 a d - b c\right )}{2 c^{3} d \sqrt{x}} + \frac{\sqrt{2} \left (a d - b c\right ) \left (9 a d - b c\right ) \log{\left (- \sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{16 c^{\frac{13}{4}} d^{\frac{3}{4}}} - \frac{\sqrt{2} \left (a d - b c\right ) \left (9 a d - b c\right ) \log{\left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x} + \sqrt{c} + \sqrt{d} x \right )}}{16 c^{\frac{13}{4}} d^{\frac{3}{4}}} - \frac{\sqrt{2} \left (a d - b c\right ) \left (9 a d - b c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{8 c^{\frac{13}{4}} d^{\frac{3}{4}}} + \frac{\sqrt{2} \left (a d - b c\right ) \left (9 a d - b c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}}{8 c^{\frac{13}{4}} d^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.324893, size = 333, normalized size = 0.92 \[ \frac{\frac{5 \sqrt{2} \left (9 a^2 d^2-10 a b c d+b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{d^{3/4}}-\frac{5 \sqrt{2} \left (9 a^2 d^2-10 a b c d+b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{d^{3/4}}-\frac{10 \sqrt{2} \left (9 a^2 d^2-10 a b c d+b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{d^{3/4}}+\frac{10 \sqrt{2} \left (9 a^2 d^2-10 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{d^{3/4}}-\frac{32 a^2 c^{5/4}}{x^{5/2}}+\frac{40 \sqrt [4]{c} x^{3/2} (b c-a d)^2}{c+d x^2}+\frac{320 a \sqrt [4]{c} (a d-b c)}{\sqrt{x}}}{80 c^{13/4}} \]

Antiderivative was successfully verified.

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

[Out]

((-32*a^2*c^(5/4))/x^(5/2) + (320*a*c^(1/4)*(-(b*c) + a*d))/Sqrt[x] + (40*c^(1/4
)*(b*c - a*d)^2*x^(3/2))/(c + d*x^2) - (10*Sqrt[2]*(b^2*c^2 - 10*a*b*c*d + 9*a^2
*d^2)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(3/4) + (10*Sqrt[2]*(b^2*
c^2 - 10*a*b*c*d + 9*a^2*d^2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/d^(
3/4) + (5*Sqrt[2]*(b^2*c^2 - 10*a*b*c*d + 9*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/
4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/d^(3/4) - (5*Sqrt[2]*(b^2*c^2 - 10*a*b*c*d + 9*
a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/d^(3/4))/(8
0*c^(13/4))

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Maple [A]  time = 0.028, size = 524, normalized size = 1.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.276757, size = 2043, normalized size = 5.63 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/40*(20*(b^2*c^2 - 10*a*b*c*d + 9*a^2*d^2)*x^4 - 16*a^2*c^2 - 16*(10*a*b*c^2 -
9*a^2*c*d)*x^2 + 20*(c^3*d*x^4 + c^4*x^2)*sqrt(x)*(-(b^8*c^8 - 40*a*b^7*c^7*d +
636*a^2*b^6*c^6*d^2 - 5080*a^3*b^5*c^5*d^3 + 21286*a^4*b^4*c^4*d^4 - 45720*a^5*b
^3*c^3*d^5 + 51516*a^6*b^2*c^2*d^6 - 29160*a^7*b*c*d^7 + 6561*a^8*d^8)/(c^13*d^3
))^(1/4)*arctan(c^10*d^2*(-(b^8*c^8 - 40*a*b^7*c^7*d + 636*a^2*b^6*c^6*d^2 - 508
0*a^3*b^5*c^5*d^3 + 21286*a^4*b^4*c^4*d^4 - 45720*a^5*b^3*c^3*d^5 + 51516*a^6*b^
2*c^2*d^6 - 29160*a^7*b*c*d^7 + 6561*a^8*d^8)/(c^13*d^3))^(3/4)/((b^6*c^6 - 30*a
*b^5*c^5*d + 327*a^2*b^4*c^4*d^2 - 1540*a^3*b^3*c^3*d^3 + 2943*a^4*b^2*c^2*d^4 -
2430*a^5*b*c*d^5 + 729*a^6*d^6)*sqrt(x) + sqrt((b^12*c^12 - 60*a*b^11*c^11*d +
1554*a^2*b^10*c^10*d^2 - 22700*a^3*b^9*c^9*d^3 + 205215*a^4*b^8*c^8*d^4 - 118860
0*a^5*b^7*c^7*d^5 + 4443580*a^6*b^6*c^6*d^6 - 10697400*a^7*b^5*c^5*d^7 + 1662241
5*a^8*b^4*c^4*d^8 - 16548300*a^9*b^3*c^3*d^9 + 10195794*a^10*b^2*c^2*d^10 - 3542
940*a^11*b*c*d^11 + 531441*a^12*d^12)*x - (b^8*c^15*d - 40*a*b^7*c^14*d^2 + 636*
a^2*b^6*c^13*d^3 - 5080*a^3*b^5*c^12*d^4 + 21286*a^4*b^4*c^11*d^5 - 45720*a^5*b^
3*c^10*d^6 + 51516*a^6*b^2*c^9*d^7 - 29160*a^7*b*c^8*d^8 + 6561*a^8*c^7*d^9)*sqr
t(-(b^8*c^8 - 40*a*b^7*c^7*d + 636*a^2*b^6*c^6*d^2 - 5080*a^3*b^5*c^5*d^3 + 2128
6*a^4*b^4*c^4*d^4 - 45720*a^5*b^3*c^3*d^5 + 51516*a^6*b^2*c^2*d^6 - 29160*a^7*b*
c*d^7 + 6561*a^8*d^8)/(c^13*d^3))))) + 5*(c^3*d*x^4 + c^4*x^2)*sqrt(x)*(-(b^8*c^
8 - 40*a*b^7*c^7*d + 636*a^2*b^6*c^6*d^2 - 5080*a^3*b^5*c^5*d^3 + 21286*a^4*b^4*
c^4*d^4 - 45720*a^5*b^3*c^3*d^5 + 51516*a^6*b^2*c^2*d^6 - 29160*a^7*b*c*d^7 + 65
61*a^8*d^8)/(c^13*d^3))^(1/4)*log(c^10*d^2*(-(b^8*c^8 - 40*a*b^7*c^7*d + 636*a^2
*b^6*c^6*d^2 - 5080*a^3*b^5*c^5*d^3 + 21286*a^4*b^4*c^4*d^4 - 45720*a^5*b^3*c^3*
d^5 + 51516*a^6*b^2*c^2*d^6 - 29160*a^7*b*c*d^7 + 6561*a^8*d^8)/(c^13*d^3))^(3/4
) + (b^6*c^6 - 30*a*b^5*c^5*d + 327*a^2*b^4*c^4*d^2 - 1540*a^3*b^3*c^3*d^3 + 294
3*a^4*b^2*c^2*d^4 - 2430*a^5*b*c*d^5 + 729*a^6*d^6)*sqrt(x)) - 5*(c^3*d*x^4 + c^
4*x^2)*sqrt(x)*(-(b^8*c^8 - 40*a*b^7*c^7*d + 636*a^2*b^6*c^6*d^2 - 5080*a^3*b^5*
c^5*d^3 + 21286*a^4*b^4*c^4*d^4 - 45720*a^5*b^3*c^3*d^5 + 51516*a^6*b^2*c^2*d^6
- 29160*a^7*b*c*d^7 + 6561*a^8*d^8)/(c^13*d^3))^(1/4)*log(-c^10*d^2*(-(b^8*c^8 -
40*a*b^7*c^7*d + 636*a^2*b^6*c^6*d^2 - 5080*a^3*b^5*c^5*d^3 + 21286*a^4*b^4*c^4
*d^4 - 45720*a^5*b^3*c^3*d^5 + 51516*a^6*b^2*c^2*d^6 - 29160*a^7*b*c*d^7 + 6561*
a^8*d^8)/(c^13*d^3))^(3/4) + (b^6*c^6 - 30*a*b^5*c^5*d + 327*a^2*b^4*c^4*d^2 - 1
540*a^3*b^3*c^3*d^3 + 2943*a^4*b^2*c^2*d^4 - 2430*a^5*b*c*d^5 + 729*a^6*d^6)*sqr
t(x)))/((c^3*d*x^4 + c^4*x^2)*sqrt(x))

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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((b*x**2+a)**2/x**(7/2)/(d*x**2+c)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.258771, size = 541, normalized size = 1.49 \[ \frac{b^{2} c^{2} x^{\frac{3}{2}} - 2 \, a b c d x^{\frac{3}{2}} + a^{2} d^{2} x^{\frac{3}{2}}}{2 \,{\left (d x^{2} + c\right )} c^{3}} - \frac{2 \,{\left (10 \, a b c x^{2} - 10 \, a^{2} d x^{2} + a^{2} c\right )}}{5 \, c^{3} x^{\frac{5}{2}}} + \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 10 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + 9 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{8 \, c^{4} d^{3}} + \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 10 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + 9 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{8 \, c^{4} d^{3}} - \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 10 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + 9 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{16 \, c^{4} d^{3}} + \frac{\sqrt{2}{\left (\left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 10 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d + 9 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{16 \, c^{4} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b^2*c^2*x^(3/2) - 2*a*b*c*d*x^(3/2) + a^2*d^2*x^(3/2))/((d*x^2 + c)*c^3) -
2/5*(10*a*b*c*x^2 - 10*a^2*d*x^2 + a^2*c)/(c^3*x^(5/2)) + 1/8*sqrt(2)*((c*d^3)^(
3/4)*b^2*c^2 - 10*(c*d^3)^(3/4)*a*b*c*d + 9*(c*d^3)^(3/4)*a^2*d^2)*arctan(1/2*sq
rt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(c^4*d^3) + 1/8*sqrt(2)*((c
*d^3)^(3/4)*b^2*c^2 - 10*(c*d^3)^(3/4)*a*b*c*d + 9*(c*d^3)^(3/4)*a^2*d^2)*arctan
(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(c^4*d^3) - 1/16*sq
rt(2)*((c*d^3)^(3/4)*b^2*c^2 - 10*(c*d^3)^(3/4)*a*b*c*d + 9*(c*d^3)^(3/4)*a^2*d^
2)*ln(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c^4*d^3) + 1/16*sqrt(2)*((c*
d^3)^(3/4)*b^2*c^2 - 10*(c*d^3)^(3/4)*a*b*c*d + 9*(c*d^3)^(3/4)*a^2*d^2)*ln(-sqr
t(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c^4*d^3)