3.453 \(\int \frac{x^{5/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=374 \[ \frac{d x^{3/2} \left (5 a^2 d^2-11 a b c d+7 b^2 c^2\right )}{2 b^4}+\frac{3 (b c-5 a d) (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} b^{19/4}}-\frac{3 (b c-5 a d) (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} b^{19/4}}-\frac{3 (b c-5 a d) (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} \sqrt [4]{a} b^{19/4}}+\frac{3 (b c-5 a d) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} \sqrt [4]{a} b^{19/4}}+\frac{3 d^2 x^{7/2} (11 b c-5 a d)}{14 b^3}-\frac{x^{3/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{15 d^3 x^{11/2}}{22 b^2} \]

[Out]

(d*(7*b^2*c^2 - 11*a*b*c*d + 5*a^2*d^2)*x^(3/2))/(2*b^4) + (3*d^2*(11*b*c - 5*a*
d)*x^(7/2))/(14*b^3) + (15*d^3*x^(11/2))/(22*b^2) - (x^(3/2)*(c + d*x^2)^3)/(2*b
*(a + b*x^2)) - (3*(b*c - 5*a*d)*(b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[
x])/a^(1/4)])/(4*Sqrt[2]*a^(1/4)*b^(19/4)) + (3*(b*c - 5*a*d)*(b*c - a*d)^2*ArcT
an[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(1/4)*b^(19/4)) + (3*(b*
c - 5*a*d)*(b*c - a*d)^2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]
*x])/(8*Sqrt[2]*a^(1/4)*b^(19/4)) - (3*(b*c - 5*a*d)*(b*c - a*d)^2*Log[Sqrt[a] +
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(1/4)*b^(19/4))

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Rubi [A]  time = 0.9084, antiderivative size = 374, 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 \[ \frac{d x^{3/2} \left (5 a^2 d^2-11 a b c d+7 b^2 c^2\right )}{2 b^4}+\frac{3 (b c-5 a d) (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} b^{19/4}}-\frac{3 (b c-5 a d) (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} b^{19/4}}-\frac{3 (b c-5 a d) (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} \sqrt [4]{a} b^{19/4}}+\frac{3 (b c-5 a d) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} \sqrt [4]{a} b^{19/4}}+\frac{3 d^2 x^{7/2} (11 b c-5 a d)}{14 b^3}-\frac{x^{3/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{15 d^3 x^{11/2}}{22 b^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^(5/2)*(c + d*x^2)^3)/(a + b*x^2)^2,x]

[Out]

(d*(7*b^2*c^2 - 11*a*b*c*d + 5*a^2*d^2)*x^(3/2))/(2*b^4) + (3*d^2*(11*b*c - 5*a*
d)*x^(7/2))/(14*b^3) + (15*d^3*x^(11/2))/(22*b^2) - (x^(3/2)*(c + d*x^2)^3)/(2*b
*(a + b*x^2)) - (3*(b*c - 5*a*d)*(b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[
x])/a^(1/4)])/(4*Sqrt[2]*a^(1/4)*b^(19/4)) + (3*(b*c - 5*a*d)*(b*c - a*d)^2*ArcT
an[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(1/4)*b^(19/4)) + (3*(b*
c - 5*a*d)*(b*c - a*d)^2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]
*x])/(8*Sqrt[2]*a^(1/4)*b^(19/4)) - (3*(b*c - 5*a*d)*(b*c - a*d)^2*Log[Sqrt[a] +
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(1/4)*b^(19/4))

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Rubi in Sympy [A]  time = 153.945, size = 359, normalized size = 0.96 \[ - \frac{x^{\frac{3}{2}} \left (c + d x^{2}\right )^{3}}{2 b \left (a + b x^{2}\right )} + \frac{15 d^{3} x^{\frac{11}{2}}}{22 b^{2}} - \frac{3 d^{2} x^{\frac{7}{2}} \left (5 a d - 11 b c\right )}{14 b^{3}} + \frac{d x^{\frac{3}{2}} \left (5 a^{2} d^{2} - 11 a b c d + 7 b^{2} c^{2}\right )}{2 b^{4}} - \frac{3 \sqrt{2} \left (a d - b c\right )^{2} \left (5 a d - b c\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 \sqrt [4]{a} b^{\frac{19}{4}}} + \frac{3 \sqrt{2} \left (a d - b c\right )^{2} \left (5 a d - b c\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 \sqrt [4]{a} b^{\frac{19}{4}}} + \frac{3 \sqrt{2} \left (a d - b c\right )^{2} \left (5 a d - b c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 \sqrt [4]{a} b^{\frac{19}{4}}} - \frac{3 \sqrt{2} \left (a d - b c\right )^{2} \left (5 a d - b c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 \sqrt [4]{a} b^{\frac{19}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)*(d*x**2+c)**3/(b*x**2+a)**2,x)

[Out]

-x**(3/2)*(c + d*x**2)**3/(2*b*(a + b*x**2)) + 15*d**3*x**(11/2)/(22*b**2) - 3*d
**2*x**(7/2)*(5*a*d - 11*b*c)/(14*b**3) + d*x**(3/2)*(5*a**2*d**2 - 11*a*b*c*d +
7*b**2*c**2)/(2*b**4) - 3*sqrt(2)*(a*d - b*c)**2*(5*a*d - b*c)*log(-sqrt(2)*a**
(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(16*a**(1/4)*b**(19/4)) + 3*sqrt(2
)*(a*d - b*c)**2*(5*a*d - b*c)*log(sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) +
sqrt(b)*x)/(16*a**(1/4)*b**(19/4)) + 3*sqrt(2)*(a*d - b*c)**2*(5*a*d - b*c)*ata
n(1 - sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(8*a**(1/4)*b**(19/4)) - 3*sqrt(2)*(a*d
- b*c)**2*(5*a*d - b*c)*atan(1 + sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(8*a**(1/4)
*b**(19/4))

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Mathematica [A]  time = 0.409179, size = 345, normalized size = 0.92 \[ \frac{352 b^{7/4} d^2 x^{7/2} (3 b c-2 a d)+2464 b^{3/4} d x^{3/2} (b c-a d)^2-\frac{616 b^{3/4} x^{3/2} (b c-a d)^3}{a+b x^2}+\frac{231 \sqrt{2} (b c-5 a d) (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{a}}+\frac{231 \sqrt{2} (b c-a d)^2 (5 a d-b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{\sqrt [4]{a}}+\frac{462 \sqrt{2} (b c-a d)^2 (5 a d-b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a}}+\frac{462 \sqrt{2} (b c-5 a d) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{a}}+224 b^{11/4} d^3 x^{11/2}}{1232 b^{19/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(5/2)*(c + d*x^2)^3)/(a + b*x^2)^2,x]

[Out]

(2464*b^(3/4)*d*(b*c - a*d)^2*x^(3/2) + 352*b^(7/4)*d^2*(3*b*c - 2*a*d)*x^(7/2)
+ 224*b^(11/4)*d^3*x^(11/2) - (616*b^(3/4)*(b*c - a*d)^3*x^(3/2))/(a + b*x^2) +
(462*Sqrt[2]*(b*c - a*d)^2*(-(b*c) + 5*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])
/a^(1/4)])/a^(1/4) + (462*Sqrt[2]*(b*c - 5*a*d)*(b*c - a*d)^2*ArcTan[1 + (Sqrt[2
]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(1/4) + (231*Sqrt[2]*(b*c - 5*a*d)*(b*c - a*d)^2*
Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(1/4) + (231*Sqrt[
2]*(b*c - a*d)^2*(-(b*c) + 5*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x]
+ Sqrt[b]*x])/a^(1/4))/(1232*b^(19/4))

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Maple [B]  time = 0.026, size = 748, normalized size = 2. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(d*x^2+c)^3/(b*x^2+a)^2,x)

[Out]

2/11*d^3*x^(11/2)/b^2-4/7*d^3/b^3*x^(7/2)*a+6/7*d^2/b^2*x^(7/2)*c+2*d^3/b^4*x^(3
/2)*a^2-4*d^2/b^3*x^(3/2)*a*c+2*d/b^2*x^(3/2)*c^2+1/2/b^4*x^(3/2)/(b*x^2+a)*a^3*
d^3-3/2/b^3*x^(3/2)/(b*x^2+a)*a^2*c*d^2+3/2/b^2*x^(3/2)/(b*x^2+a)*a*c^2*d-1/2/b*
x^(3/2)/(b*x^2+a)*c^3-15/16/b^5/(a/b)^(1/4)*2^(1/2)*a^3*d^3*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)))-15/8/b^5
/(a/b)^(1/4)*2^(1/2)*a^3*d^3*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)-15/8/b^5/(a/b
)^(1/4)*2^(1/2)*a^3*d^3*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)+33/16/b^4/(a/b)^(1
/4)*2^(1/2)*a^2*c*d^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)))+33/8/b^4/(a/b)^(1/4)*2^(1/2)*a^2*c*d^2*arctan(
2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+33/8/b^4/(a/b)^(1/4)*2^(1/2)*a^2*c*d^2*arctan(2^(
1/2)/(a/b)^(1/4)*x^(1/2)-1)-21/16/b^3/(a/b)^(1/4)*2^(1/2)*a*c^2*d*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)))-21
/8/b^3/(a/b)^(1/4)*2^(1/2)*a*c^2*d*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)-21/8/b^
3/(a/b)^(1/4)*2^(1/2)*a*c^2*d*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)+3/16/b^2/(a/
b)^(1/4)*2^(1/2)*c^3*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)))+3/8/b^2/(a/b)^(1/4)*2^(1/2)*c^3*arctan(2^(1/2)/
(a/b)^(1/4)*x^(1/2)+1)+3/8/b^2/(a/b)^(1/4)*2^(1/2)*c^3*arctan(2^(1/2)/(a/b)^(1/4
)*x^(1/2)-1)

_______________________________________________________________________________________

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*x^(5/2)/(b*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.286594, size = 2978, normalized size = 7.96 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^3*x^(5/2)/(b*x^2 + a)^2,x, algorithm="fricas")

[Out]

-1/616*(924*(b^5*x^2 + a*b^4)*(-(b^12*c^12 - 28*a*b^11*c^11*d + 338*a^2*b^10*c^1
0*d^2 - 2316*a^3*b^9*c^9*d^3 + 10015*a^4*b^8*c^8*d^4 - 28856*a^5*b^7*c^7*d^5 + 5
7148*a^6*b^6*c^6*d^6 - 78968*a^7*b^5*c^5*d^7 + 76111*a^8*b^4*c^4*d^8 - 50220*a^9
*b^3*c^3*d^9 + 21650*a^10*b^2*c^2*d^10 - 5500*a^11*b*c*d^11 + 625*a^12*d^12)/(a*
b^19))^(1/4)*arctan(-a*b^14*(-(b^12*c^12 - 28*a*b^11*c^11*d + 338*a^2*b^10*c^10*
d^2 - 2316*a^3*b^9*c^9*d^3 + 10015*a^4*b^8*c^8*d^4 - 28856*a^5*b^7*c^7*d^5 + 571
48*a^6*b^6*c^6*d^6 - 78968*a^7*b^5*c^5*d^7 + 76111*a^8*b^4*c^4*d^8 - 50220*a^9*b
^3*c^3*d^9 + 21650*a^10*b^2*c^2*d^10 - 5500*a^11*b*c*d^11 + 625*a^12*d^12)/(a*b^
19))^(3/4)/((b^9*c^9 - 21*a*b^8*c^8*d + 180*a^2*b^7*c^7*d^2 - 820*a^3*b^6*c^6*d^
3 + 2190*a^4*b^5*c^5*d^4 - 3606*a^5*b^4*c^4*d^5 + 3716*a^6*b^3*c^3*d^6 - 2340*a^
7*b^2*c^2*d^7 + 825*a^8*b*c*d^8 - 125*a^9*d^9)*sqrt(x) - sqrt((b^18*c^18 - 42*a*
b^17*c^17*d + 801*a^2*b^16*c^16*d^2 - 9200*a^3*b^15*c^15*d^3 + 71220*a^4*b^14*c^
14*d^4 - 394392*a^5*b^13*c^13*d^5 + 1619684*a^6*b^12*c^12*d^6 - 5050512*a^7*b^11
*c^11*d^7 + 12147630*a^8*b^10*c^10*d^8 - 22765820*a^9*b^9*c^9*d^9 + 33419166*a^1
0*b^8*c^8*d^10 - 38446992*a^11*b^7*c^7*d^11 + 34503236*a^12*b^6*c^6*d^12 - 23888
280*a^13*b^5*c^5*d^13 + 12508500*a^14*b^4*c^4*d^14 - 4790000*a^15*b^3*c^3*d^15 +
1265625*a^16*b^2*c^2*d^16 - 206250*a^17*b*c*d^17 + 15625*a^18*d^18)*x - (a*b^21
*c^12 - 28*a^2*b^20*c^11*d + 338*a^3*b^19*c^10*d^2 - 2316*a^4*b^18*c^9*d^3 + 100
15*a^5*b^17*c^8*d^4 - 28856*a^6*b^16*c^7*d^5 + 57148*a^7*b^15*c^6*d^6 - 78968*a^
8*b^14*c^5*d^7 + 76111*a^9*b^13*c^4*d^8 - 50220*a^10*b^12*c^3*d^9 + 21650*a^11*b
^11*c^2*d^10 - 5500*a^12*b^10*c*d^11 + 625*a^13*b^9*d^12)*sqrt(-(b^12*c^12 - 28*
a*b^11*c^11*d + 338*a^2*b^10*c^10*d^2 - 2316*a^3*b^9*c^9*d^3 + 10015*a^4*b^8*c^8
*d^4 - 28856*a^5*b^7*c^7*d^5 + 57148*a^6*b^6*c^6*d^6 - 78968*a^7*b^5*c^5*d^7 + 7
6111*a^8*b^4*c^4*d^8 - 50220*a^9*b^3*c^3*d^9 + 21650*a^10*b^2*c^2*d^10 - 5500*a^
11*b*c*d^11 + 625*a^12*d^12)/(a*b^19))))) + 231*(b^5*x^2 + a*b^4)*(-(b^12*c^12 -
28*a*b^11*c^11*d + 338*a^2*b^10*c^10*d^2 - 2316*a^3*b^9*c^9*d^3 + 10015*a^4*b^8
*c^8*d^4 - 28856*a^5*b^7*c^7*d^5 + 57148*a^6*b^6*c^6*d^6 - 78968*a^7*b^5*c^5*d^7
+ 76111*a^8*b^4*c^4*d^8 - 50220*a^9*b^3*c^3*d^9 + 21650*a^10*b^2*c^2*d^10 - 550
0*a^11*b*c*d^11 + 625*a^12*d^12)/(a*b^19))^(1/4)*log(27*a*b^14*(-(b^12*c^12 - 28
*a*b^11*c^11*d + 338*a^2*b^10*c^10*d^2 - 2316*a^3*b^9*c^9*d^3 + 10015*a^4*b^8*c^
8*d^4 - 28856*a^5*b^7*c^7*d^5 + 57148*a^6*b^6*c^6*d^6 - 78968*a^7*b^5*c^5*d^7 +
76111*a^8*b^4*c^4*d^8 - 50220*a^9*b^3*c^3*d^9 + 21650*a^10*b^2*c^2*d^10 - 5500*a
^11*b*c*d^11 + 625*a^12*d^12)/(a*b^19))^(3/4) - 27*(b^9*c^9 - 21*a*b^8*c^8*d + 1
80*a^2*b^7*c^7*d^2 - 820*a^3*b^6*c^6*d^3 + 2190*a^4*b^5*c^5*d^4 - 3606*a^5*b^4*c
^4*d^5 + 3716*a^6*b^3*c^3*d^6 - 2340*a^7*b^2*c^2*d^7 + 825*a^8*b*c*d^8 - 125*a^9
*d^9)*sqrt(x)) - 231*(b^5*x^2 + a*b^4)*(-(b^12*c^12 - 28*a*b^11*c^11*d + 338*a^2
*b^10*c^10*d^2 - 2316*a^3*b^9*c^9*d^3 + 10015*a^4*b^8*c^8*d^4 - 28856*a^5*b^7*c^
7*d^5 + 57148*a^6*b^6*c^6*d^6 - 78968*a^7*b^5*c^5*d^7 + 76111*a^8*b^4*c^4*d^8 -
50220*a^9*b^3*c^3*d^9 + 21650*a^10*b^2*c^2*d^10 - 5500*a^11*b*c*d^11 + 625*a^12*
d^12)/(a*b^19))^(1/4)*log(-27*a*b^14*(-(b^12*c^12 - 28*a*b^11*c^11*d + 338*a^2*b
^10*c^10*d^2 - 2316*a^3*b^9*c^9*d^3 + 10015*a^4*b^8*c^8*d^4 - 28856*a^5*b^7*c^7*
d^5 + 57148*a^6*b^6*c^6*d^6 - 78968*a^7*b^5*c^5*d^7 + 76111*a^8*b^4*c^4*d^8 - 50
220*a^9*b^3*c^3*d^9 + 21650*a^10*b^2*c^2*d^10 - 5500*a^11*b*c*d^11 + 625*a^12*d^
12)/(a*b^19))^(3/4) - 27*(b^9*c^9 - 21*a*b^8*c^8*d + 180*a^2*b^7*c^7*d^2 - 820*a
^3*b^6*c^6*d^3 + 2190*a^4*b^5*c^5*d^4 - 3606*a^5*b^4*c^4*d^5 + 3716*a^6*b^3*c^3*
d^6 - 2340*a^7*b^2*c^2*d^7 + 825*a^8*b*c*d^8 - 125*a^9*d^9)*sqrt(x)) - 4*(28*b^3
*d^3*x^7 + 12*(11*b^3*c*d^2 - 5*a*b^2*d^3)*x^5 + 44*(7*b^3*c^2*d - 11*a*b^2*c*d^
2 + 5*a^2*b*d^3)*x^3 - 77*(b^3*c^3 - 7*a*b^2*c^2*d + 11*a^2*b*c*d^2 - 5*a^3*d^3)
*x)*sqrt(x))/(b^5*x^2 + a*b^4)

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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(x**(5/2)*(d*x**2+c)**3/(b*x**2+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.317961, size = 745, normalized size = 1.99 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^3*x^(5/2)/(b*x^2 + a)^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)*b^4) + 3/8*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 7*(a*b^3)^(3/4
)*a*b^2*c^2*d + 11*(a*b^3)^(3/4)*a^2*b*c*d^2 - 5*(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*b^7) + 3/8*sqrt(2)*
((a*b^3)^(3/4)*b^3*c^3 - 7*(a*b^3)^(3/4)*a*b^2*c^2*d + 11*(a*b^3)^(3/4)*a^2*b*c*
d^2 - 5*(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*b^7) - 3/16*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 7*(a*b^3)^(3/4
)*a*b^2*c^2*d + 11*(a*b^3)^(3/4)*a^2*b*c*d^2 - 5*(a*b^3)^(3/4)*a^3*d^3)*ln(sqrt(
2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a*b^7) + 3/16*sqrt(2)*((a*b^3)^(3/4)*b^
3*c^3 - 7*(a*b^3)^(3/4)*a*b^2*c^2*d + 11*(a*b^3)^(3/4)*a^2*b*c*d^2 - 5*(a*b^3)^(
3/4)*a^3*d^3)*ln(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a*b^7) + 2/77*(7
*b^20*d^3*x^(11/2) + 33*b^20*c*d^2*x^(7/2) - 22*a*b^19*d^3*x^(7/2) + 77*b^20*c^2
*d*x^(3/2) - 154*a*b^19*c*d^2*x^(3/2) + 77*a^2*b^18*d^3*x^(3/2))/b^22