3.459 \(\int \frac{\left (c+d x^2\right )^3}{x^{7/2} \left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=376 \[ \frac{3 (b c-a d)^2 (a d+3 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{13/4} b^{7/4}}-\frac{3 (b c-a d)^2 (a d+3 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{13/4} b^{7/4}}-\frac{3 (b c-a d)^2 (a d+3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{13/4} b^{7/4}}+\frac{3 (b c-a d)^2 (a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{13/4} b^{7/4}}-\frac{c^2 (9 b c-5 a d)}{10 a^2 b x^{5/2}}+\frac{c \left (2 a^2 d^2-15 a b c d+9 b^2 c^2\right )}{2 a^3 b \sqrt{x}}+\frac{\left (c+d x^2\right )^2 (b c-a d)}{2 a b x^{5/2} \left (a+b x^2\right )} \]

[Out]

-(c^2*(9*b*c - 5*a*d))/(10*a^2*b*x^(5/2)) + (c*(9*b^2*c^2 - 15*a*b*c*d + 2*a^2*d
^2))/(2*a^3*b*Sqrt[x]) + ((b*c - a*d)*(c + d*x^2)^2)/(2*a*b*x^(5/2)*(a + b*x^2))
- (3*(b*c - a*d)^2*(3*b*c + a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])
/(4*Sqrt[2]*a^(13/4)*b^(7/4)) + (3*(b*c - a*d)^2*(3*b*c + a*d)*ArcTan[1 + (Sqrt[
2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(13/4)*b^(7/4)) + (3*(b*c - a*d)^2*(3
*b*c + a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[
2]*a^(13/4)*b^(7/4)) - (3*(b*c - a*d)^2*(3*b*c + a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1
/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(13/4)*b^(7/4))

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Rubi [A]  time = 0.940374, antiderivative size = 376, 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 \[ \frac{3 (b c-a d)^2 (a d+3 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{13/4} b^{7/4}}-\frac{3 (b c-a d)^2 (a d+3 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{13/4} b^{7/4}}-\frac{3 (b c-a d)^2 (a d+3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{13/4} b^{7/4}}+\frac{3 (b c-a d)^2 (a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{13/4} b^{7/4}}-\frac{c^2 (9 b c-5 a d)}{10 a^2 b x^{5/2}}+\frac{c \left (2 a^2 d^2-15 a b c d+9 b^2 c^2\right )}{2 a^3 b \sqrt{x}}+\frac{\left (c+d x^2\right )^2 (b c-a d)}{2 a b x^{5/2} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)^3/(x^(7/2)*(a + b*x^2)^2),x]

[Out]

-(c^2*(9*b*c - 5*a*d))/(10*a^2*b*x^(5/2)) + (c*(9*b^2*c^2 - 15*a*b*c*d + 2*a^2*d
^2))/(2*a^3*b*Sqrt[x]) + ((b*c - a*d)*(c + d*x^2)^2)/(2*a*b*x^(5/2)*(a + b*x^2))
- (3*(b*c - a*d)^2*(3*b*c + a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])
/(4*Sqrt[2]*a^(13/4)*b^(7/4)) + (3*(b*c - a*d)^2*(3*b*c + a*d)*ArcTan[1 + (Sqrt[
2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(13/4)*b^(7/4)) + (3*(b*c - a*d)^2*(3
*b*c + a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[
2]*a^(13/4)*b^(7/4)) - (3*(b*c - a*d)^2*(3*b*c + a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1
/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(13/4)*b^(7/4))

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Rubi in Sympy [A]  time = 157.362, size = 354, normalized size = 0.94 \[ - \frac{\left (c + d x^{2}\right )^{2} \left (a d - b c\right )}{2 a b x^{\frac{5}{2}} \left (a + b x^{2}\right )} + \frac{c^{2} \left (5 a d - 9 b c\right )}{10 a^{2} b x^{\frac{5}{2}}} + \frac{c \left (2 a^{2} d^{2} - 15 a b c d + 9 b^{2} c^{2}\right )}{2 a^{3} b \sqrt{x}} + \frac{3 \sqrt{2} \left (a d - b c\right )^{2} \left (a d + 3 b c\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{13}{4}} b^{\frac{7}{4}}} - \frac{3 \sqrt{2} \left (a d - b c\right )^{2} \left (a d + 3 b c\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{13}{4}} b^{\frac{7}{4}}} - \frac{3 \sqrt{2} \left (a d - b c\right )^{2} \left (a d + 3 b c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{13}{4}} b^{\frac{7}{4}}} + \frac{3 \sqrt{2} \left (a d - b c\right )^{2} \left (a d + 3 b c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{13}{4}} b^{\frac{7}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)**3/x**(7/2)/(b*x**2+a)**2,x)

[Out]

-(c + d*x**2)**2*(a*d - b*c)/(2*a*b*x**(5/2)*(a + b*x**2)) + c**2*(5*a*d - 9*b*c
)/(10*a**2*b*x**(5/2)) + c*(2*a**2*d**2 - 15*a*b*c*d + 9*b**2*c**2)/(2*a**3*b*sq
rt(x)) + 3*sqrt(2)*(a*d - b*c)**2*(a*d + 3*b*c)*log(-sqrt(2)*a**(1/4)*b**(1/4)*s
qrt(x) + sqrt(a) + sqrt(b)*x)/(16*a**(13/4)*b**(7/4)) - 3*sqrt(2)*(a*d - b*c)**2
*(a*d + 3*b*c)*log(sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(16*
a**(13/4)*b**(7/4)) - 3*sqrt(2)*(a*d - b*c)**2*(a*d + 3*b*c)*atan(1 - sqrt(2)*b*
*(1/4)*sqrt(x)/a**(1/4))/(8*a**(13/4)*b**(7/4)) + 3*sqrt(2)*(a*d - b*c)**2*(a*d
+ 3*b*c)*atan(1 + sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(8*a**(13/4)*b**(7/4))

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Mathematica [A]  time = 0.416625, size = 323, normalized size = 0.86 \[ \frac{-\frac{32 a^{5/4} c^3}{x^{5/2}}+\frac{15 \sqrt{2} (b c-a d)^2 (a d+3 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{b^{7/4}}-\frac{15 \sqrt{2} (b c-a d)^2 (a d+3 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{b^{7/4}}-\frac{30 \sqrt{2} (b c-a d)^2 (a d+3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{b^{7/4}}+\frac{30 \sqrt{2} (b c-a d)^2 (a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{b^{7/4}}-\frac{160 \sqrt [4]{a} c^2 (3 a d-2 b c)}{\sqrt{x}}-\frac{40 \sqrt [4]{a} x^{3/2} (a d-b c)^3}{b \left (a+b x^2\right )}}{80 a^{13/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)^3/(x^(7/2)*(a + b*x^2)^2),x]

[Out]

((-32*a^(5/4)*c^3)/x^(5/2) - (160*a^(1/4)*c^2*(-2*b*c + 3*a*d))/Sqrt[x] - (40*a^
(1/4)*(-(b*c) + a*d)^3*x^(3/2))/(b*(a + b*x^2)) - (30*Sqrt[2]*(b*c - a*d)^2*(3*b
*c + a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(7/4) + (30*Sqrt[2]*(
b*c - a*d)^2*(3*b*c + a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(7/4
) + (15*Sqrt[2]*(b*c - a*d)^2*(3*b*c + a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4
)*Sqrt[x] + Sqrt[b]*x])/b^(7/4) - (15*Sqrt[2]*(b*c - a*d)^2*(3*b*c + a*d)*Log[Sq
rt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(7/4))/(80*a^(13/4))

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Maple [B]  time = 0.03, size = 697, normalized size = 1.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)^3/x^(7/2)/(b*x^2+a)^2,x)

[Out]

-2/5*c^3/a^2/x^(5/2)-6*c^2/a^2/x^(1/2)*d+4*c^3/a^3/x^(1/2)*b-1/2/b*x^(3/2)/(b*x^
2+a)*d^3+3/2/a*x^(3/2)/(b*x^2+a)*c*d^2-3/2/a^2*b*x^(3/2)/(b*x^2+a)*c^2*d+1/2/a^3
*b^2*x^(3/2)/(b*x^2+a)*c^3+3/8/b^2/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4
)*x^(1/2)+1)*d^3+3/8/a/b/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+
1)*c*d^2-15/8/a^2/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*c^2*
d+9/8/a^3*b/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*c^3+3/8/b^
2/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*d^3+3/8/a/b/(a/b)^(1
/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*c*d^2-15/8/a^2/(a/b)^(1/4)*2^(
1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*c^2*d+9/8/a^3*b/(a/b)^(1/4)*2^(1/2)*a
rctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*c^3+3/16/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))
)*d^3+3/16/a/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*d^2-15/16/a^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+9/16/a^3*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^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)^2*x^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.286593, size = 2982, normalized size = 7.93 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^3/((b*x^2 + a)^2*x^(7/2)),x, algorithm="fricas")

[Out]

-1/40*(16*a^2*b*c^3 - 20*(9*b^3*c^3 - 15*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*
x^4 - 48*(3*a*b^2*c^3 - 5*a^2*b*c^2*d)*x^2 - 60*(a^3*b^2*x^4 + a^4*b*x^2)*sqrt(x
)*(-(81*b^12*c^12 - 540*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 1932*a^3*b^9*c^
9*d^3 + 1039*a^4*b^8*c^8*d^4 + 328*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 136*a
^7*b^5*c^5*d^7 + 127*a^8*b^4*c^4*d^8 - 44*a^9*b^3*c^3*d^9 - 14*a^10*b^2*c^2*d^10
+ 4*a^11*b*c*d^11 + a^12*d^12)/(a^13*b^7))^(1/4)*arctan(a^10*b^5*(-(81*b^12*c^1
2 - 540*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 1932*a^3*b^9*c^9*d^3 + 1039*a^4
*b^8*c^8*d^4 + 328*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 136*a^7*b^5*c^5*d^7 +
127*a^8*b^4*c^4*d^8 - 44*a^9*b^3*c^3*d^9 - 14*a^10*b^2*c^2*d^10 + 4*a^11*b*c*d^
11 + a^12*d^12)/(a^13*b^7))^(3/4)/((27*b^9*c^9 - 135*a*b^8*c^8*d + 252*a^2*b^7*c
^7*d^2 - 188*a^3*b^6*c^6*d^3 - 6*a^4*b^5*c^5*d^4 + 78*a^5*b^4*c^4*d^5 - 20*a^6*b
^3*c^3*d^6 - 12*a^7*b^2*c^2*d^7 + 3*a^8*b*c*d^8 + a^9*d^9)*sqrt(x) + sqrt((729*b
^18*c^18 - 7290*a*b^17*c^17*d + 31833*a^2*b^16*c^16*d^2 - 78192*a^3*b^15*c^15*d^
3 + 113940*a^4*b^14*c^14*d^4 - 88920*a^5*b^13*c^13*d^5 + 10180*a^6*b^12*c^12*d^6
+ 46320*a^7*b^11*c^11*d^7 - 35970*a^8*b^10*c^10*d^8 - 220*a^9*b^9*c^9*d^9 + 120
78*a^10*b^8*c^8*d^10 - 3600*a^11*b^7*c^7*d^11 - 1884*a^12*b^6*c^6*d^12 + 936*a^1
3*b^5*c^5*d^13 + 180*a^14*b^4*c^4*d^14 - 112*a^15*b^3*c^3*d^15 - 15*a^16*b^2*c^2
*d^16 + 6*a^17*b*c*d^17 + a^18*d^18)*x - (81*a^7*b^15*c^12 - 540*a^8*b^14*c^11*d
+ 1458*a^9*b^13*c^10*d^2 - 1932*a^10*b^12*c^9*d^3 + 1039*a^11*b^11*c^8*d^4 + 32
8*a^12*b^10*c^7*d^5 - 644*a^13*b^9*c^6*d^6 + 136*a^14*b^8*c^5*d^7 + 127*a^15*b^7
*c^4*d^8 - 44*a^16*b^6*c^3*d^9 - 14*a^17*b^5*c^2*d^10 + 4*a^18*b^4*c*d^11 + a^19
*b^3*d^12)*sqrt(-(81*b^12*c^12 - 540*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 19
32*a^3*b^9*c^9*d^3 + 1039*a^4*b^8*c^8*d^4 + 328*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^
6*d^6 + 136*a^7*b^5*c^5*d^7 + 127*a^8*b^4*c^4*d^8 - 44*a^9*b^3*c^3*d^9 - 14*a^10
*b^2*c^2*d^10 + 4*a^11*b*c*d^11 + a^12*d^12)/(a^13*b^7))))) - 15*(a^3*b^2*x^4 +
a^4*b*x^2)*sqrt(x)*(-(81*b^12*c^12 - 540*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2
- 1932*a^3*b^9*c^9*d^3 + 1039*a^4*b^8*c^8*d^4 + 328*a^5*b^7*c^7*d^5 - 644*a^6*b^
6*c^6*d^6 + 136*a^7*b^5*c^5*d^7 + 127*a^8*b^4*c^4*d^8 - 44*a^9*b^3*c^3*d^9 - 14*
a^10*b^2*c^2*d^10 + 4*a^11*b*c*d^11 + a^12*d^12)/(a^13*b^7))^(1/4)*log(27*a^10*b
^5*(-(81*b^12*c^12 - 540*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 1932*a^3*b^9*c
^9*d^3 + 1039*a^4*b^8*c^8*d^4 + 328*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 136*
a^7*b^5*c^5*d^7 + 127*a^8*b^4*c^4*d^8 - 44*a^9*b^3*c^3*d^9 - 14*a^10*b^2*c^2*d^1
0 + 4*a^11*b*c*d^11 + a^12*d^12)/(a^13*b^7))^(3/4) + 27*(27*b^9*c^9 - 135*a*b^8*
c^8*d + 252*a^2*b^7*c^7*d^2 - 188*a^3*b^6*c^6*d^3 - 6*a^4*b^5*c^5*d^4 + 78*a^5*b
^4*c^4*d^5 - 20*a^6*b^3*c^3*d^6 - 12*a^7*b^2*c^2*d^7 + 3*a^8*b*c*d^8 + a^9*d^9)*
sqrt(x)) + 15*(a^3*b^2*x^4 + a^4*b*x^2)*sqrt(x)*(-(81*b^12*c^12 - 540*a*b^11*c^1
1*d + 1458*a^2*b^10*c^10*d^2 - 1932*a^3*b^9*c^9*d^3 + 1039*a^4*b^8*c^8*d^4 + 328
*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 136*a^7*b^5*c^5*d^7 + 127*a^8*b^4*c^4*d
^8 - 44*a^9*b^3*c^3*d^9 - 14*a^10*b^2*c^2*d^10 + 4*a^11*b*c*d^11 + a^12*d^12)/(a
^13*b^7))^(1/4)*log(-27*a^10*b^5*(-(81*b^12*c^12 - 540*a*b^11*c^11*d + 1458*a^2*
b^10*c^10*d^2 - 1932*a^3*b^9*c^9*d^3 + 1039*a^4*b^8*c^8*d^4 + 328*a^5*b^7*c^7*d^
5 - 644*a^6*b^6*c^6*d^6 + 136*a^7*b^5*c^5*d^7 + 127*a^8*b^4*c^4*d^8 - 44*a^9*b^3
*c^3*d^9 - 14*a^10*b^2*c^2*d^10 + 4*a^11*b*c*d^11 + a^12*d^12)/(a^13*b^7))^(3/4)
+ 27*(27*b^9*c^9 - 135*a*b^8*c^8*d + 252*a^2*b^7*c^7*d^2 - 188*a^3*b^6*c^6*d^3
- 6*a^4*b^5*c^5*d^4 + 78*a^5*b^4*c^4*d^5 - 20*a^6*b^3*c^3*d^6 - 12*a^7*b^2*c^2*d
^7 + 3*a^8*b*c*d^8 + a^9*d^9)*sqrt(x)))/((a^3*b^2*x^4 + a^4*b*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((d*x**2+c)**3/x**(7/2)/(b*x**2+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.305752, size = 682, normalized size = 1.81 \[ \frac{b^{3} c^{3} x^{\frac{3}{2}} - 3 \, a b^{2} c^{2} d x^{\frac{3}{2}} + 3 \, a^{2} b c d^{2} x^{\frac{3}{2}} - a^{3} d^{3} x^{\frac{3}{2}}}{2 \,{\left (b x^{2} + a\right )} a^{3} b} + \frac{2 \,{\left (10 \, b c^{3} x^{2} - 15 \, a c^{2} d x^{2} - a c^{3}\right )}}{5 \, a^{3} x^{\frac{5}{2}}} + \frac{3 \, \sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{3}{4}} b^{3} c^{3} - 5 \, \left (a b^{3}\right )^{\frac{3}{4}} a b^{2} c^{2} d + \left (a b^{3}\right )^{\frac{3}{4}} a^{2} b c d^{2} + \left (a b^{3}\right )^{\frac{3}{4}} a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{4} b^{4}} + \frac{3 \, \sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{3}{4}} b^{3} c^{3} - 5 \, \left (a b^{3}\right )^{\frac{3}{4}} a b^{2} c^{2} d + \left (a b^{3}\right )^{\frac{3}{4}} a^{2} b c d^{2} + \left (a b^{3}\right )^{\frac{3}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{4} b^{4}} - \frac{3 \, \sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{3}{4}} b^{3} c^{3} - 5 \, \left (a b^{3}\right )^{\frac{3}{4}} a b^{2} c^{2} d + \left (a b^{3}\right )^{\frac{3}{4}} a^{2} b c d^{2} + \left (a b^{3}\right )^{\frac{3}{4}} a^{3} d^{3}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{4} b^{4}} + \frac{3 \, \sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{3}{4}} b^{3} c^{3} - 5 \, \left (a b^{3}\right )^{\frac{3}{4}} a b^{2} c^{2} d + \left (a b^{3}\right )^{\frac{3}{4}} a^{2} b c d^{2} + \left (a b^{3}\right )^{\frac{3}{4}} a^{3} d^{3}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{4} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^3/((b*x^2 + a)^2*x^(7/2)),x, algorithm="giac")

[Out]

1/2*(b^3*c^3*x^(3/2) - 3*a*b^2*c^2*d*x^(3/2) + 3*a^2*b*c*d^2*x^(3/2) - a^3*d^3*x
^(3/2))/((b*x^2 + a)*a^3*b) + 2/5*(10*b*c^3*x^2 - 15*a*c^2*d*x^2 - a*c^3)/(a^3*x
^(5/2)) + 3/8*sqrt(2)*(3*(a*b^3)^(3/4)*b^3*c^3 - 5*(a*b^3)^(3/4)*a*b^2*c^2*d + (
a*b^3)^(3/4)*a^2*b*c*d^2 + (a*b^3)^(3/4)*a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*(a
/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a^4*b^4) + 3/8*sqrt(2)*(3*(a*b^3)^(3/4)*b^3
*c^3 - 5*(a*b^3)^(3/4)*a*b^2*c^2*d + (a*b^3)^(3/4)*a^2*b*c*d^2 + (a*b^3)^(3/4)*a
^3*d^3)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a^4*
b^4) - 3/16*sqrt(2)*(3*(a*b^3)^(3/4)*b^3*c^3 - 5*(a*b^3)^(3/4)*a*b^2*c^2*d + (a*
b^3)^(3/4)*a^2*b*c*d^2 + (a*b^3)^(3/4)*a^3*d^3)*ln(sqrt(2)*sqrt(x)*(a/b)^(1/4) +
x + sqrt(a/b))/(a^4*b^4) + 3/16*sqrt(2)*(3*(a*b^3)^(3/4)*b^3*c^3 - 5*(a*b^3)^(3
/4)*a*b^2*c^2*d + (a*b^3)^(3/4)*a^2*b*c*d^2 + (a*b^3)^(3/4)*a^3*d^3)*ln(-sqrt(2)
*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^4*b^4)