3.460 \(\int \frac{\left (c+d x^2\right )^3}{x^{9/2} \left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=376 \[ -\frac{(b c-a d)^2 (a d+11 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{15/4} b^{5/4}}+\frac{(b c-a d)^2 (a d+11 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{15/4} b^{5/4}}-\frac{(b c-a d)^2 (a d+11 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{15/4} b^{5/4}}+\frac{(b c-a d)^2 (a d+11 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{15/4} b^{5/4}}-\frac{c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac{c \left (6 a^2 d^2-21 a b c d+11 b^2 c^2\right )}{6 a^3 b x^{3/2}}+\frac{\left (c+d x^2\right )^2 (b c-a d)}{2 a b x^{7/2} \left (a+b x^2\right )} \]

[Out]

-(c^2*(11*b*c - 7*a*d))/(14*a^2*b*x^(7/2)) + (c*(11*b^2*c^2 - 21*a*b*c*d + 6*a^2
*d^2))/(6*a^3*b*x^(3/2)) + ((b*c - a*d)*(c + d*x^2)^2)/(2*a*b*x^(7/2)*(a + b*x^2
)) - ((b*c - a*d)^2*(11*b*c + a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]
)/(4*Sqrt[2]*a^(15/4)*b^(5/4)) + ((b*c - a*d)^2*(11*b*c + a*d)*ArcTan[1 + (Sqrt[
2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(15/4)*b^(5/4)) - ((b*c - a*d)^2*(11*
b*c + a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2
]*a^(15/4)*b^(5/4)) + ((b*c - a*d)^2*(11*b*c + a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4
)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(15/4)*b^(5/4))

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Rubi [A]  time = 0.904467, 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 \[ -\frac{(b c-a d)^2 (a d+11 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{15/4} b^{5/4}}+\frac{(b c-a d)^2 (a d+11 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{15/4} b^{5/4}}-\frac{(b c-a d)^2 (a d+11 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{15/4} b^{5/4}}+\frac{(b c-a d)^2 (a d+11 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{15/4} b^{5/4}}-\frac{c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac{c \left (6 a^2 d^2-21 a b c d+11 b^2 c^2\right )}{6 a^3 b x^{3/2}}+\frac{\left (c+d x^2\right )^2 (b c-a d)}{2 a b x^{7/2} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)^3/(x^(9/2)*(a + b*x^2)^2),x]

[Out]

-(c^2*(11*b*c - 7*a*d))/(14*a^2*b*x^(7/2)) + (c*(11*b^2*c^2 - 21*a*b*c*d + 6*a^2
*d^2))/(6*a^3*b*x^(3/2)) + ((b*c - a*d)*(c + d*x^2)^2)/(2*a*b*x^(7/2)*(a + b*x^2
)) - ((b*c - a*d)^2*(11*b*c + a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]
)/(4*Sqrt[2]*a^(15/4)*b^(5/4)) + ((b*c - a*d)^2*(11*b*c + a*d)*ArcTan[1 + (Sqrt[
2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(15/4)*b^(5/4)) - ((b*c - a*d)^2*(11*
b*c + a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2
]*a^(15/4)*b^(5/4)) + ((b*c - a*d)^2*(11*b*c + a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4
)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(15/4)*b^(5/4))

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Rubi in Sympy [A]  time = 153.168, size = 347, normalized size = 0.92 \[ - \frac{\left (c + d x^{2}\right )^{2} \left (a d - b c\right )}{2 a b x^{\frac{7}{2}} \left (a + b x^{2}\right )} + \frac{c^{2} \left (7 a d - 11 b c\right )}{14 a^{2} b x^{\frac{7}{2}}} + \frac{c \left (6 a^{2} d^{2} - 21 a b c d + 11 b^{2} c^{2}\right )}{6 a^{3} b x^{\frac{3}{2}}} - \frac{\sqrt{2} \left (a d - b c\right )^{2} \left (a d + 11 b c\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{15}{4}} b^{\frac{5}{4}}} + \frac{\sqrt{2} \left (a d - b c\right )^{2} \left (a d + 11 b c\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{15}{4}} b^{\frac{5}{4}}} - \frac{\sqrt{2} \left (a d - b c\right )^{2} \left (a d + 11 b c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{15}{4}} b^{\frac{5}{4}}} + \frac{\sqrt{2} \left (a d - b c\right )^{2} \left (a d + 11 b c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{15}{4}} b^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)**3/x**(9/2)/(b*x**2+a)**2,x)

[Out]

-(c + d*x**2)**2*(a*d - b*c)/(2*a*b*x**(7/2)*(a + b*x**2)) + c**2*(7*a*d - 11*b*
c)/(14*a**2*b*x**(7/2)) + c*(6*a**2*d**2 - 21*a*b*c*d + 11*b**2*c**2)/(6*a**3*b*
x**(3/2)) - sqrt(2)*(a*d - b*c)**2*(a*d + 11*b*c)*log(-sqrt(2)*a**(1/4)*b**(1/4)
*sqrt(x) + sqrt(a) + sqrt(b)*x)/(16*a**(15/4)*b**(5/4)) + sqrt(2)*(a*d - b*c)**2
*(a*d + 11*b*c)*log(sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(16
*a**(15/4)*b**(5/4)) - sqrt(2)*(a*d - b*c)**2*(a*d + 11*b*c)*atan(1 - sqrt(2)*b*
*(1/4)*sqrt(x)/a**(1/4))/(8*a**(15/4)*b**(5/4)) + sqrt(2)*(a*d - b*c)**2*(a*d +
11*b*c)*atan(1 + sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(8*a**(15/4)*b**(5/4))

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Mathematica [A]  time = 0.413445, size = 323, normalized size = 0.86 \[ \frac{-\frac{224 a^{3/4} c^2 (3 a d-2 b c)}{x^{3/2}}-\frac{168 a^{3/4} \sqrt{x} (a d-b c)^3}{b \left (a+b x^2\right )}-\frac{96 a^{7/4} c^3}{x^{7/2}}-\frac{21 \sqrt{2} (b c-a d)^2 (a d+11 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{b^{5/4}}+\frac{21 \sqrt{2} (b c-a d)^2 (a d+11 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{b^{5/4}}-\frac{42 \sqrt{2} (b c-a d)^2 (a d+11 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{b^{5/4}}+\frac{42 \sqrt{2} (b c-a d)^2 (a d+11 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{b^{5/4}}}{336 a^{15/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)^3/(x^(9/2)*(a + b*x^2)^2),x]

[Out]

((-96*a^(7/4)*c^3)/x^(7/2) - (224*a^(3/4)*c^2*(-2*b*c + 3*a*d))/x^(3/2) - (168*a
^(3/4)*(-(b*c) + a*d)^3*Sqrt[x])/(b*(a + b*x^2)) - (42*Sqrt[2]*(b*c - a*d)^2*(11
*b*c + a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(5/4) + (42*Sqrt[2]
*(b*c - a*d)^2*(11*b*c + a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/b^(
5/4) - (21*Sqrt[2]*(b*c - a*d)^2*(11*b*c + a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^
(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(5/4) + (21*Sqrt[2]*(b*c - a*d)^2*(11*b*c + a*d)*L
og[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/b^(5/4))/(336*a^(15/4
))

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Maple [B]  time = 0.03, size = 706, 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^(9/2)/(b*x^2+a)^2,x)

[Out]

-2/7*c^3/a^2/x^(7/2)-2*c^2/a^2/x^(3/2)*d+4/3*c^3/a^3/x^(3/2)*b-1/2/b*x^(1/2)/(b*
x^2+a)*d^3+3/2/a*x^(1/2)/(b*x^2+a)*c*d^2-3/2/a^2*b*x^(1/2)/(b*x^2+a)*c^2*d+1/2/a
^3*b^2*x^(1/2)/(b*x^2+a)*c^3+1/8/a/b*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1
/4)*x^(1/2)+1)*d^3+9/8/a^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2
)+1)*c*d^2-21/8/a^3*b*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*
c^2*d+11/8/a^4*b^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*c^3
+1/8/a/b*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*d^3+9/8/a^2*(
a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*c*d^2-21/8/a^3*b*(a/b)^
(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*c^2*d+11/8/a^4*b^2*(a/b)^(1/
4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*c^3+1/16/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)))*d^3+9/16/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*d^2-21/16/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^2*d+11/16/a^4*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)))
*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^(9/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.278786, size = 2178, normalized size = 5.79 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^3/((b*x^2 + a)^2*x^(9/2)),x, algorithm="fricas")

[Out]

-1/168*(48*a^2*b*c^3 - 28*(11*b^3*c^3 - 21*a*b^2*c^2*d + 9*a^2*b*c*d^2 - 3*a^3*d
^3)*x^4 - 16*(11*a*b^2*c^3 - 21*a^2*b*c^2*d)*x^2 + 84*(a^3*b^2*x^5 + a^4*b*x^3)*
sqrt(x)*(-(14641*b^12*c^12 - 111804*a*b^11*c^11*d + 368082*a^2*b^10*c^10*d^2 - 6
76588*a^3*b^9*c^9*d^3 + 746703*a^4*b^8*c^8*d^4 - 486648*a^5*b^7*c^7*d^5 + 160188
*a^6*b^6*c^6*d^6 - 5688*a^7*b^5*c^5*d^7 - 10017*a^8*b^4*c^4*d^8 + 692*a^9*b^3*c^
3*d^9 + 402*a^10*b^2*c^2*d^10 + 36*a^11*b*c*d^11 + a^12*d^12)/(a^15*b^5))^(1/4)*
arctan(a^4*b*(-(14641*b^12*c^12 - 111804*a*b^11*c^11*d + 368082*a^2*b^10*c^10*d^
2 - 676588*a^3*b^9*c^9*d^3 + 746703*a^4*b^8*c^8*d^4 - 486648*a^5*b^7*c^7*d^5 + 1
60188*a^6*b^6*c^6*d^6 - 5688*a^7*b^5*c^5*d^7 - 10017*a^8*b^4*c^4*d^8 + 692*a^9*b
^3*c^3*d^9 + 402*a^10*b^2*c^2*d^10 + 36*a^11*b*c*d^11 + a^12*d^12)/(a^15*b^5))^(
1/4)/((11*b^3*c^3 - 21*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*sqrt(x) + sqrt(a^8
*b^2*sqrt(-(14641*b^12*c^12 - 111804*a*b^11*c^11*d + 368082*a^2*b^10*c^10*d^2 -
676588*a^3*b^9*c^9*d^3 + 746703*a^4*b^8*c^8*d^4 - 486648*a^5*b^7*c^7*d^5 + 16018
8*a^6*b^6*c^6*d^6 - 5688*a^7*b^5*c^5*d^7 - 10017*a^8*b^4*c^4*d^8 + 692*a^9*b^3*c
^3*d^9 + 402*a^10*b^2*c^2*d^10 + 36*a^11*b*c*d^11 + a^12*d^12)/(a^15*b^5)) + (12
1*b^6*c^6 - 462*a*b^5*c^5*d + 639*a^2*b^4*c^4*d^2 - 356*a^3*b^3*c^3*d^3 + 39*a^4
*b^2*c^2*d^4 + 18*a^5*b*c*d^5 + a^6*d^6)*x))) - 21*(a^3*b^2*x^5 + a^4*b*x^3)*sqr
t(x)*(-(14641*b^12*c^12 - 111804*a*b^11*c^11*d + 368082*a^2*b^10*c^10*d^2 - 6765
88*a^3*b^9*c^9*d^3 + 746703*a^4*b^8*c^8*d^4 - 486648*a^5*b^7*c^7*d^5 + 160188*a^
6*b^6*c^6*d^6 - 5688*a^7*b^5*c^5*d^7 - 10017*a^8*b^4*c^4*d^8 + 692*a^9*b^3*c^3*d
^9 + 402*a^10*b^2*c^2*d^10 + 36*a^11*b*c*d^11 + a^12*d^12)/(a^15*b^5))^(1/4)*log
(a^4*b*(-(14641*b^12*c^12 - 111804*a*b^11*c^11*d + 368082*a^2*b^10*c^10*d^2 - 67
6588*a^3*b^9*c^9*d^3 + 746703*a^4*b^8*c^8*d^4 - 486648*a^5*b^7*c^7*d^5 + 160188*
a^6*b^6*c^6*d^6 - 5688*a^7*b^5*c^5*d^7 - 10017*a^8*b^4*c^4*d^8 + 692*a^9*b^3*c^3
*d^9 + 402*a^10*b^2*c^2*d^10 + 36*a^11*b*c*d^11 + a^12*d^12)/(a^15*b^5))^(1/4) +
(11*b^3*c^3 - 21*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*sqrt(x)) + 21*(a^3*b^2*
x^5 + a^4*b*x^3)*sqrt(x)*(-(14641*b^12*c^12 - 111804*a*b^11*c^11*d + 368082*a^2*
b^10*c^10*d^2 - 676588*a^3*b^9*c^9*d^3 + 746703*a^4*b^8*c^8*d^4 - 486648*a^5*b^7
*c^7*d^5 + 160188*a^6*b^6*c^6*d^6 - 5688*a^7*b^5*c^5*d^7 - 10017*a^8*b^4*c^4*d^8
+ 692*a^9*b^3*c^3*d^9 + 402*a^10*b^2*c^2*d^10 + 36*a^11*b*c*d^11 + a^12*d^12)/(
a^15*b^5))^(1/4)*log(-a^4*b*(-(14641*b^12*c^12 - 111804*a*b^11*c^11*d + 368082*a
^2*b^10*c^10*d^2 - 676588*a^3*b^9*c^9*d^3 + 746703*a^4*b^8*c^8*d^4 - 486648*a^5*
b^7*c^7*d^5 + 160188*a^6*b^6*c^6*d^6 - 5688*a^7*b^5*c^5*d^7 - 10017*a^8*b^4*c^4*
d^8 + 692*a^9*b^3*c^3*d^9 + 402*a^10*b^2*c^2*d^10 + 36*a^11*b*c*d^11 + a^12*d^12
)/(a^15*b^5))^(1/4) + (11*b^3*c^3 - 21*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*sq
rt(x)))/((a^3*b^2*x^5 + a^4*b*x^3)*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**(9/2)/(b*x**2+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.270616, size = 687, normalized size = 1.83 \[ \frac{\sqrt{2}{\left (11 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 21 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 9 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} + \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{4} b^{2}} + \frac{\sqrt{2}{\left (11 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 21 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 9 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} + \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{4} b^{2}} + \frac{\sqrt{2}{\left (11 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 21 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 9 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} + \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{4} b^{2}} - \frac{\sqrt{2}{\left (11 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 21 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 9 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} + \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{4} b^{2}} + \frac{b^{3} c^{3} \sqrt{x} - 3 \, a b^{2} c^{2} d \sqrt{x} + 3 \, a^{2} b c d^{2} \sqrt{x} - a^{3} d^{3} \sqrt{x}}{2 \,{\left (b x^{2} + a\right )} a^{3} b} + \frac{2 \,{\left (14 \, b c^{3} x^{2} - 21 \, a c^{2} d x^{2} - 3 \, a c^{3}\right )}}{21 \, a^{3} x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^3/((b*x^2 + a)^2*x^(9/2)),x, algorithm="giac")

[Out]

1/8*sqrt(2)*(11*(a*b^3)^(1/4)*b^3*c^3 - 21*(a*b^3)^(1/4)*a*b^2*c^2*d + 9*(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^4*b^2) + 1/8*sqrt(2)*(11*(a*b^3)^(1/4)*b^3*c^3
- 21*(a*b^3)^(1/4)*a*b^2*c^2*d + 9*(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^4*b^
2) + 1/16*sqrt(2)*(11*(a*b^3)^(1/4)*b^3*c^3 - 21*(a*b^3)^(1/4)*a*b^2*c^2*d + 9*(
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^4*b^2) - 1/16*sqrt(2)*(11*(a*b^3)^(1/4)*b^3*c^3 - 21*(a*b^3
)^(1/4)*a*b^2*c^2*d + 9*(a*b^3)^(1/4)*a^2*b*c*d^2 + (a*b^3)^(1/4)*a^3*d^3)*ln(-s
qrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^4*b^2) + 1/2*(b^3*c^3*sqrt(x) - 3
*a*b^2*c^2*d*sqrt(x) + 3*a^2*b*c*d^2*sqrt(x) - a^3*d^3*sqrt(x))/((b*x^2 + a)*a^3
*b) + 2/21*(14*b*c^3*x^2 - 21*a*c^2*d*x^2 - 3*a*c^3)/(a^3*x^(7/2))