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

Optimal. Leaf size=409 \[ \frac{d x^{5/2} \left (17 a^2 d^2-39 a b c d+27 b^2 c^2\right )}{10 b^4}+\frac{\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} b^{21/4}}-\frac{\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} b^{21/4}}+\frac{\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} b^{21/4}}-\frac{\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} b^{21/4}}+\frac{\sqrt{x} (5 b c-17 a d) (b c-a d)^2}{2 b^5}+\frac{d^2 x^{9/2} (39 b c-17 a d)}{18 b^3}-\frac{x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{17 d^3 x^{13/2}}{26 b^2} \]

[Out]

((5*b*c - 17*a*d)*(b*c - a*d)^2*Sqrt[x])/(2*b^5) + (d*(27*b^2*c^2 - 39*a*b*c*d +
17*a^2*d^2)*x^(5/2))/(10*b^4) + (d^2*(39*b*c - 17*a*d)*x^(9/2))/(18*b^3) + (17*
d^3*x^(13/2))/(26*b^2) - (x^(5/2)*(c + d*x^2)^3)/(2*b*(a + b*x^2)) + (a^(1/4)*(5
*b*c - 17*a*d)*(b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*S
qrt[2]*b^(21/4)) - (a^(1/4)*(5*b*c - 17*a*d)*(b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*b
^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*b^(21/4)) + (a^(1/4)*(5*b*c - 17*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]*
b^(21/4)) - (a^(1/4)*(5*b*c - 17*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]*b^(21/4))

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Rubi [A]  time = 0.986217, antiderivative size = 409, 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 \[ \frac{d x^{5/2} \left (17 a^2 d^2-39 a b c d+27 b^2 c^2\right )}{10 b^4}+\frac{\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} b^{21/4}}-\frac{\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} b^{21/4}}+\frac{\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} b^{21/4}}-\frac{\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} b^{21/4}}+\frac{\sqrt{x} (5 b c-17 a d) (b c-a d)^2}{2 b^5}+\frac{d^2 x^{9/2} (39 b c-17 a d)}{18 b^3}-\frac{x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{17 d^3 x^{13/2}}{26 b^2} \]

Antiderivative was successfully verified.

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

[Out]

((5*b*c - 17*a*d)*(b*c - a*d)^2*Sqrt[x])/(2*b^5) + (d*(27*b^2*c^2 - 39*a*b*c*d +
17*a^2*d^2)*x^(5/2))/(10*b^4) + (d^2*(39*b*c - 17*a*d)*x^(9/2))/(18*b^3) + (17*
d^3*x^(13/2))/(26*b^2) - (x^(5/2)*(c + d*x^2)^3)/(2*b*(a + b*x^2)) + (a^(1/4)*(5
*b*c - 17*a*d)*(b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*S
qrt[2]*b^(21/4)) - (a^(1/4)*(5*b*c - 17*a*d)*(b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*b
^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*b^(21/4)) + (a^(1/4)*(5*b*c - 17*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]*
b^(21/4)) - (a^(1/4)*(5*b*c - 17*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]*b^(21/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\sqrt{2} \sqrt [4]{a} \left (a d - b c\right )^{2} \left (17 a d - 5 b c\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 b^{\frac{21}{4}}} + \frac{\sqrt{2} \sqrt [4]{a} \left (a d - b c\right )^{2} \left (17 a d - 5 b c\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 b^{\frac{21}{4}}} - \frac{\sqrt{2} \sqrt [4]{a} \left (a d - b c\right )^{2} \left (17 a d - 5 b c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 b^{\frac{21}{4}}} + \frac{\sqrt{2} \sqrt [4]{a} \left (a d - b c\right )^{2} \left (17 a d - 5 b c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 b^{\frac{21}{4}}} - \frac{x^{\frac{5}{2}} \left (c + d x^{2}\right )^{3}}{2 b \left (a + b x^{2}\right )} - \frac{\left (a d - b c\right )^{2} \left (17 a d - 5 b c\right ) \int ^{\sqrt{x}} \frac{1}{b^{4}}\, dx}{2 b} + \frac{17 d^{3} x^{\frac{13}{2}}}{26 b^{2}} - \frac{d^{2} x^{\frac{9}{2}} \left (17 a d - 39 b c\right )}{18 b^{3}} + \frac{d x^{\frac{5}{2}} \left (17 a^{2} d^{2} - 39 a b c d + 27 b^{2} c^{2}\right )}{10 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*a**(1/4)*(a*d - b*c)**2*(17*a*d - 5*b*c)*log(-sqrt(2)*a**(1/4)*b**(1/4)
*sqrt(x) + sqrt(a) + sqrt(b)*x)/(16*b**(21/4)) + sqrt(2)*a**(1/4)*(a*d - b*c)**2
*(17*a*d - 5*b*c)*log(sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(
16*b**(21/4)) - sqrt(2)*a**(1/4)*(a*d - b*c)**2*(17*a*d - 5*b*c)*atan(1 - sqrt(2
)*b**(1/4)*sqrt(x)/a**(1/4))/(8*b**(21/4)) + sqrt(2)*a**(1/4)*(a*d - b*c)**2*(17
*a*d - 5*b*c)*atan(1 + sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(8*b**(21/4)) - x**(5/
2)*(c + d*x**2)**3/(2*b*(a + b*x**2)) - (a*d - b*c)**2*(17*a*d - 5*b*c)*Integral
(b**(-4), (x, sqrt(x)))/(2*b) + 17*d**3*x**(13/2)/(26*b**2) - d**2*x**(9/2)*(17*
a*d - 39*b*c)/(18*b**3) + d*x**(5/2)*(17*a**2*d**2 - 39*a*b*c*d + 27*b**2*c**2)/
(10*b**4)

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Mathematica [A]  time = 0.395073, size = 378, normalized size = 0.92 \[ \frac{2080 b^{9/4} d^2 x^{9/2} (3 b c-2 a d)+11232 b^{5/4} d x^{5/2} (b c-a d)^2+\frac{4680 a \sqrt [4]{b} \sqrt{x} (b c-a d)^3}{a+b x^2}+18720 \sqrt [4]{b} \sqrt{x} (b c-4 a d) (b c-a d)^2-585 \sqrt{2} \sqrt [4]{a} (b c-a d)^2 (17 a d-5 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+585 \sqrt{2} \sqrt [4]{a} (b c-a d)^2 (17 a d-5 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-1170 \sqrt{2} \sqrt [4]{a} (b c-a d)^2 (17 a d-5 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )+1170 \sqrt{2} \sqrt [4]{a} (b c-a d)^2 (17 a d-5 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )+1440 b^{13/4} d^3 x^{13/2}}{9360 b^{21/4}} \]

Antiderivative was successfully verified.

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

[Out]

(18720*b^(1/4)*(b*c - 4*a*d)*(b*c - a*d)^2*Sqrt[x] + 11232*b^(5/4)*d*(b*c - a*d)
^2*x^(5/2) + 2080*b^(9/4)*d^2*(3*b*c - 2*a*d)*x^(9/2) + 1440*b^(13/4)*d^3*x^(13/
2) + (4680*a*b^(1/4)*(b*c - a*d)^3*Sqrt[x])/(a + b*x^2) - 1170*Sqrt[2]*a^(1/4)*(
b*c - a*d)^2*(-5*b*c + 17*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 1
170*Sqrt[2]*a^(1/4)*(b*c - a*d)^2*(-5*b*c + 17*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*
Sqrt[x])/a^(1/4)] - 585*Sqrt[2]*a^(1/4)*(b*c - a*d)^2*(-5*b*c + 17*a*d)*Log[Sqrt
[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 585*Sqrt[2]*a^(1/4)*(b*c -
a*d)^2*(-5*b*c + 17*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]
*x])/(9360*b^(21/4))

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Maple [B]  time = 0.027, size = 804, normalized size = 2. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/2*a^3/b^4*x^(1/2)/(b*x^2+a)*c*d^2-3/2*a^2/b^3*x^(1/2)/(b*x^2+a)*c^2*d+17/16*a^
3/b^5*(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+17/8*a^3/b^5*(a/b)^(1/4)*2^(1/2)*arcta
n(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*d^3+17/8*a^3/b^5*(a/b)^(1/4)*2^(1/2)*arctan(2^(
1/2)/(a/b)^(1/4)*x^(1/2)-1)*d^3+18/b^4*a^2*c*d^2*x^(1/2)-12/b^3*a*c^2*d*x^(1/2)-
1/2*a^4/b^5*x^(1/2)/(b*x^2+a)*d^3+1/2*a/b^2*x^(1/2)/(b*x^2+a)*c^3-5/16/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)))*c^3-5/8/b^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^
(1/4)*x^(1/2)+1)*c^3+2/13*d^3*x^(13/2)/b^2-39/16*a^2/b^4*(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+27/16*a/b^3*(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-39/8*a^2/b^4*(a
/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*c*d^2+27/8*a/b^3*(a/b)^(
1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*c^2*d-5/8/b^2*(a/b)^(1/4)*2^(
1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*c^3-12/5/b^3*x^(5/2)*a*c*d^2-4/9/b^3*
x^(9/2)*a*d^3+2/3/b^2*x^(9/2)*c*d^2+6/5/b^4*x^(5/2)*a^2*d^3+6/5/b^2*x^(5/2)*c^2*
d-8/b^5*a^3*d^3*x^(1/2)-39/8*a^2/b^4*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1
/4)*x^(1/2)-1)*c*d^2+27/8*a/b^3*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x
^(1/2)-1)*c^2*d+2/b^2*c^3*x^(1/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.275052, size = 2257, normalized size = 5.52 \[ \text{result too large to display} \]

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, algorithm="fricas")

[Out]

-1/4680*(2340*(b^6*x^2 + a*b^5)*(-(625*a*b^12*c^12 - 13500*a^2*b^11*c^11*d + 128
850*a^3*b^10*c^10*d^2 - 718060*a^4*b^9*c^9*d^3 + 2603151*a^5*b^8*c^8*d^4 - 64770
48*a^6*b^7*c^7*d^5 + 11369148*a^7*b^6*c^6*d^6 - 14225976*a^8*b^5*c^5*d^7 + 12631
455*a^9*b^4*c^4*d^8 - 7783756*a^10*b^3*c^3*d^9 + 3168018*a^11*b^2*c^2*d^10 - 766
428*a^12*b*c*d^11 + 83521*a^13*d^12)/b^21)^(1/4)*arctan(-b^5*(-(625*a*b^12*c^12
- 13500*a^2*b^11*c^11*d + 128850*a^3*b^10*c^10*d^2 - 718060*a^4*b^9*c^9*d^3 + 26
03151*a^5*b^8*c^8*d^4 - 6477048*a^6*b^7*c^7*d^5 + 11369148*a^7*b^6*c^6*d^6 - 142
25976*a^8*b^5*c^5*d^7 + 12631455*a^9*b^4*c^4*d^8 - 7783756*a^10*b^3*c^3*d^9 + 31
68018*a^11*b^2*c^2*d^10 - 766428*a^12*b*c*d^11 + 83521*a^13*d^12)/b^21)^(1/4)/((
5*b^3*c^3 - 27*a*b^2*c^2*d + 39*a^2*b*c*d^2 - 17*a^3*d^3)*sqrt(x) - sqrt(b^10*sq
rt(-(625*a*b^12*c^12 - 13500*a^2*b^11*c^11*d + 128850*a^3*b^10*c^10*d^2 - 718060
*a^4*b^9*c^9*d^3 + 2603151*a^5*b^8*c^8*d^4 - 6477048*a^6*b^7*c^7*d^5 + 11369148*
a^7*b^6*c^6*d^6 - 14225976*a^8*b^5*c^5*d^7 + 12631455*a^9*b^4*c^4*d^8 - 7783756*
a^10*b^3*c^3*d^9 + 3168018*a^11*b^2*c^2*d^10 - 766428*a^12*b*c*d^11 + 83521*a^13
*d^12)/b^21) + (25*b^6*c^6 - 270*a*b^5*c^5*d + 1119*a^2*b^4*c^4*d^2 - 2276*a^3*b
^3*c^3*d^3 + 2439*a^4*b^2*c^2*d^4 - 1326*a^5*b*c*d^5 + 289*a^6*d^6)*x))) - 585*(
b^6*x^2 + a*b^5)*(-(625*a*b^12*c^12 - 13500*a^2*b^11*c^11*d + 128850*a^3*b^10*c^
10*d^2 - 718060*a^4*b^9*c^9*d^3 + 2603151*a^5*b^8*c^8*d^4 - 6477048*a^6*b^7*c^7*
d^5 + 11369148*a^7*b^6*c^6*d^6 - 14225976*a^8*b^5*c^5*d^7 + 12631455*a^9*b^4*c^4
*d^8 - 7783756*a^10*b^3*c^3*d^9 + 3168018*a^11*b^2*c^2*d^10 - 766428*a^12*b*c*d^
11 + 83521*a^13*d^12)/b^21)^(1/4)*log(b^5*(-(625*a*b^12*c^12 - 13500*a^2*b^11*c^
11*d + 128850*a^3*b^10*c^10*d^2 - 718060*a^4*b^9*c^9*d^3 + 2603151*a^5*b^8*c^8*d
^4 - 6477048*a^6*b^7*c^7*d^5 + 11369148*a^7*b^6*c^6*d^6 - 14225976*a^8*b^5*c^5*d
^7 + 12631455*a^9*b^4*c^4*d^8 - 7783756*a^10*b^3*c^3*d^9 + 3168018*a^11*b^2*c^2*
d^10 - 766428*a^12*b*c*d^11 + 83521*a^13*d^12)/b^21)^(1/4) - (5*b^3*c^3 - 27*a*b
^2*c^2*d + 39*a^2*b*c*d^2 - 17*a^3*d^3)*sqrt(x)) + 585*(b^6*x^2 + a*b^5)*(-(625*
a*b^12*c^12 - 13500*a^2*b^11*c^11*d + 128850*a^3*b^10*c^10*d^2 - 718060*a^4*b^9*
c^9*d^3 + 2603151*a^5*b^8*c^8*d^4 - 6477048*a^6*b^7*c^7*d^5 + 11369148*a^7*b^6*c
^6*d^6 - 14225976*a^8*b^5*c^5*d^7 + 12631455*a^9*b^4*c^4*d^8 - 7783756*a^10*b^3*
c^3*d^9 + 3168018*a^11*b^2*c^2*d^10 - 766428*a^12*b*c*d^11 + 83521*a^13*d^12)/b^
21)^(1/4)*log(-b^5*(-(625*a*b^12*c^12 - 13500*a^2*b^11*c^11*d + 128850*a^3*b^10*
c^10*d^2 - 718060*a^4*b^9*c^9*d^3 + 2603151*a^5*b^8*c^8*d^4 - 6477048*a^6*b^7*c^
7*d^5 + 11369148*a^7*b^6*c^6*d^6 - 14225976*a^8*b^5*c^5*d^7 + 12631455*a^9*b^4*c
^4*d^8 - 7783756*a^10*b^3*c^3*d^9 + 3168018*a^11*b^2*c^2*d^10 - 766428*a^12*b*c*
d^11 + 83521*a^13*d^12)/b^21)^(1/4) - (5*b^3*c^3 - 27*a*b^2*c^2*d + 39*a^2*b*c*d
^2 - 17*a^3*d^3)*sqrt(x)) - 4*(180*b^4*d^3*x^8 + 2925*a*b^3*c^3 - 15795*a^2*b^2*
c^2*d + 22815*a^3*b*c*d^2 - 9945*a^4*d^3 + 20*(39*b^4*c*d^2 - 17*a*b^3*d^3)*x^6
+ 52*(27*b^4*c^2*d - 39*a*b^3*c*d^2 + 17*a^2*b^2*d^3)*x^4 + 468*(5*b^4*c^3 - 27*
a*b^3*c^2*d + 39*a^2*b^2*c*d^2 - 17*a^3*b*d^3)*x^2)*sqrt(x))/(b^6*x^2 + a*b^5)

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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**(7/2)*(d*x**2+c)**3/(b*x**2+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.27865, size = 810, normalized size = 1.98 \[ \text{result too large to display} \]

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, algorithm="giac")

[Out]

-1/8*sqrt(2)*(5*(a*b^3)^(1/4)*b^3*c^3 - 27*(a*b^3)^(1/4)*a*b^2*c^2*d + 39*(a*b^3
)^(1/4)*a^2*b*c*d^2 - 17*(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))/b^6 - 1/8*sqrt(2)*(5*(a*b^3)^(1/4)*b^3*c^3 - 2
7*(a*b^3)^(1/4)*a*b^2*c^2*d + 39*(a*b^3)^(1/4)*a^2*b*c*d^2 - 17*(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))/b^6 -
1/16*sqrt(2)*(5*(a*b^3)^(1/4)*b^3*c^3 - 27*(a*b^3)^(1/4)*a*b^2*c^2*d + 39*(a*b^3
)^(1/4)*a^2*b*c*d^2 - 17*(a*b^3)^(1/4)*a^3*d^3)*ln(sqrt(2)*sqrt(x)*(a/b)^(1/4) +
x + sqrt(a/b))/b^6 + 1/16*sqrt(2)*(5*(a*b^3)^(1/4)*b^3*c^3 - 27*(a*b^3)^(1/4)*a
*b^2*c^2*d + 39*(a*b^3)^(1/4)*a^2*b*c*d^2 - 17*(a*b^3)^(1/4)*a^3*d^3)*ln(-sqrt(2
)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/b^6 + 1/2*(a*b^3*c^3*sqrt(x) - 3*a^2*b^2*
c^2*d*sqrt(x) + 3*a^3*b*c*d^2*sqrt(x) - a^4*d^3*sqrt(x))/((b*x^2 + a)*b^5) + 2/5
85*(45*b^24*d^3*x^(13/2) + 195*b^24*c*d^2*x^(9/2) - 130*a*b^23*d^3*x^(9/2) + 351
*b^24*c^2*d*x^(5/2) - 702*a*b^23*c*d^2*x^(5/2) + 351*a^2*b^22*d^3*x^(5/2) + 585*
b^24*c^3*sqrt(x) - 3510*a*b^23*c^2*d*sqrt(x) + 5265*a^2*b^22*c*d^2*sqrt(x) - 234
0*a^3*b^21*d^3*sqrt(x))/b^26