3.457 \(\int \frac{\left (c+d x^2\right )^3}{x^{3/2} \left (a+b x^2\right )^2} \, dx\)
Optimal. Leaf size=368 \[ -\frac{(b c-a d)^2 (7 a d+5 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{9/4} b^{11/4}}+\frac{(b c-a d)^2 (7 a d+5 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{9/4} b^{11/4}}+\frac{(b c-a d)^2 (7 a d+5 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{9/4} b^{11/4}}-\frac{(b c-a d)^2 (7 a d+5 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{9/4} b^{11/4}}-\frac{c^2 (5 b c-a d)}{2 a^2 b \sqrt{x}}-\frac{d^2 x^{3/2} (3 b c-7 a d)}{6 a b^2}+\frac{\left (c+d x^2\right )^2 (b c-a d)}{2 a b \sqrt{x} \left (a+b x^2\right )} \]
[Out]
-(c^2*(5*b*c - a*d))/(2*a^2*b*Sqrt[x]) - (d^2*(3*b*c - 7*a*d)*x^(3/2))/(6*a*b^2)
+ ((b*c - a*d)*(c + d*x^2)^2)/(2*a*b*Sqrt[x]*(a + b*x^2)) + ((b*c - a*d)^2*(5*b
*c + 7*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(9/4)*b^
(11/4)) - ((b*c - a*d)^2*(5*b*c + 7*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^
(1/4)])/(4*Sqrt[2]*a^(9/4)*b^(11/4)) - ((b*c - a*d)^2*(5*b*c + 7*a*d)*Log[Sqrt[a
] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4)*b^(11/4)) +
((b*c - a*d)^2*(5*b*c + 7*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] +
Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4)*b^(11/4))
_______________________________________________________________________________________
Rubi [A] time = 0.91854, antiderivative size = 368, 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
\[ -\frac{(b c-a d)^2 (7 a d+5 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{9/4} b^{11/4}}+\frac{(b c-a d)^2 (7 a d+5 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{9/4} b^{11/4}}+\frac{(b c-a d)^2 (7 a d+5 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{9/4} b^{11/4}}-\frac{(b c-a d)^2 (7 a d+5 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{9/4} b^{11/4}}-\frac{c^2 (5 b c-a d)}{2 a^2 b \sqrt{x}}-\frac{d^2 x^{3/2} (3 b c-7 a d)}{6 a b^2}+\frac{\left (c+d x^2\right )^2 (b c-a d)}{2 a b \sqrt{x} \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^2)^3/(x^(3/2)*(a + b*x^2)^2),x]
[Out]
-(c^2*(5*b*c - a*d))/(2*a^2*b*Sqrt[x]) - (d^2*(3*b*c - 7*a*d)*x^(3/2))/(6*a*b^2)
+ ((b*c - a*d)*(c + d*x^2)^2)/(2*a*b*Sqrt[x]*(a + b*x^2)) + ((b*c - a*d)^2*(5*b
*c + 7*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(9/4)*b^
(11/4)) - ((b*c - a*d)^2*(5*b*c + 7*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^
(1/4)])/(4*Sqrt[2]*a^(9/4)*b^(11/4)) - ((b*c - a*d)^2*(5*b*c + 7*a*d)*Log[Sqrt[a
] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4)*b^(11/4)) +
((b*c - a*d)^2*(5*b*c + 7*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] +
Sqrt[b]*x])/(8*Sqrt[2]*a^(9/4)*b^(11/4))
_______________________________________________________________________________________
Rubi in Sympy [A] time = 153.543, size = 338, normalized size = 0.92 \[ - \frac{\left (c + d x^{2}\right )^{2} \left (a d - b c\right )}{2 a b \sqrt{x} \left (a + b x^{2}\right )} + \frac{d^{2} x^{\frac{3}{2}} \left (7 a d - 3 b c\right )}{6 a b^{2}} + \frac{c^{2} \left (a d - 5 b c\right )}{2 a^{2} b \sqrt{x}} - \frac{\sqrt{2} \left (a d - b c\right )^{2} \left (7 a d + 5 b c\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{9}{4}} b^{\frac{11}{4}}} + \frac{\sqrt{2} \left (a d - b c\right )^{2} \left (7 a d + 5 b c\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{9}{4}} b^{\frac{11}{4}}} + \frac{\sqrt{2} \left (a d - b c\right )^{2} \left (7 a d + 5 b c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{9}{4}} b^{\frac{11}{4}}} - \frac{\sqrt{2} \left (a d - b c\right )^{2} \left (7 a d + 5 b c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{9}{4}} b^{\frac{11}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**2+c)**3/x**(3/2)/(b*x**2+a)**2,x)
[Out]
-(c + d*x**2)**2*(a*d - b*c)/(2*a*b*sqrt(x)*(a + b*x**2)) + d**2*x**(3/2)*(7*a*d
- 3*b*c)/(6*a*b**2) + c**2*(a*d - 5*b*c)/(2*a**2*b*sqrt(x)) - sqrt(2)*(a*d - b*
c)**2*(7*a*d + 5*b*c)*log(-sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)
*x)/(16*a**(9/4)*b**(11/4)) + sqrt(2)*(a*d - b*c)**2*(7*a*d + 5*b*c)*log(sqrt(2)
*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(16*a**(9/4)*b**(11/4)) + sqrt
(2)*(a*d - b*c)**2*(7*a*d + 5*b*c)*atan(1 - sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(
8*a**(9/4)*b**(11/4)) - sqrt(2)*(a*d - b*c)**2*(7*a*d + 5*b*c)*atan(1 + sqrt(2)*
b**(1/4)*sqrt(x)/a**(1/4))/(8*a**(9/4)*b**(11/4))
_______________________________________________________________________________________
Mathematica [A] time = 0.413562, size = 327, normalized size = 0.89 \[ \frac{1}{48} \left (-\frac{3 \sqrt{2} (b c-a d)^2 (7 a d+5 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{9/4} b^{11/4}}+\frac{3 \sqrt{2} (b c-a d)^2 (7 a d+5 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{9/4} b^{11/4}}+\frac{6 \sqrt{2} (b c-a d)^2 (7 a d+5 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{9/4} b^{11/4}}-\frac{6 \sqrt{2} (b c-a d)^2 (7 a d+5 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{9/4} b^{11/4}}+\frac{24 x^{3/2} (a d-b c)^3}{a^2 b^2 \left (a+b x^2\right )}-\frac{96 c^3}{a^2 \sqrt{x}}+\frac{32 d^3 x^{3/2}}{b^2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^2)^3/(x^(3/2)*(a + b*x^2)^2),x]
[Out]
((-96*c^3)/(a^2*Sqrt[x]) + (32*d^3*x^(3/2))/b^2 + (24*(-(b*c) + a*d)^3*x^(3/2))/
(a^2*b^2*(a + b*x^2)) + (6*Sqrt[2]*(b*c - a*d)^2*(5*b*c + 7*a*d)*ArcTan[1 - (Sqr
t[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(9/4)*b^(11/4)) - (6*Sqrt[2]*(b*c - a*d)^2*(5
*b*c + 7*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(9/4)*b^(11/4))
- (3*Sqrt[2]*(b*c - a*d)^2*(5*b*c + 7*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)
*Sqrt[x] + Sqrt[b]*x])/(a^(9/4)*b^(11/4)) + (3*Sqrt[2]*(b*c - a*d)^2*(5*b*c + 7*
a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(9/4)*b^(11/
4)))/48
_______________________________________________________________________________________
Maple [B] time = 0.03, size = 682, 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^(3/2)/(b*x^2+a)^2,x)
[Out]
2/3*d^3*x^(3/2)/b^2-2*c^3/x^(1/2)/a^2+1/2*a/b^2*x^(3/2)/(b*x^2+a)*d^3-3/2/b*x^(3
/2)/(b*x^2+a)*c*d^2+3/2/a*x^(3/2)/(b*x^2+a)*c^2*d-1/2/a^2*b*x^(3/2)/(b*x^2+a)*c^
3-7/16*a/b^3/(a/b)^(1/4)*2^(1/2)*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)))-7/8*a/b^3/(a/b)^(1/4)*2^(1/2)*d
^3*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)-7/8*a/b^3/(a/b)^(1/4)*2^(1/2)*d^3*arcta
n(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)+3/16/a/b/(a/b)^(1/4)*2^(1/2)*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)))+
3/8/a/b/(a/b)^(1/4)*2^(1/2)*c^2*d*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+3/8/a/b/
(a/b)^(1/4)*2^(1/2)*c^2*d*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)-5/16/a^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)))-5/8/a^2/(a/b)^(1/4)*2^(1/2)*c^3*arctan(2^(1/2)/(a/b
)^(1/4)*x^(1/2)+1)-5/8/a^2/(a/b)^(1/4)*2^(1/2)*c^3*arctan(2^(1/2)/(a/b)^(1/4)*x^
(1/2)-1)+9/16/b^2/(a/b)^(1/4)*2^(1/2)*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)))+9/8/b^2/(a/b)^(1/4)*2^(1
/2)*c*d^2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+9/8/b^2/(a/b)^(1/4)*2^(1/2)*c*d^
2*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/((b*x^2 + a)^2*x^(3/2)),x, algorithm="maxima")
[Out]
Exception raised: ValueError
_______________________________________________________________________________________
Fricas [A] time = 0.28554, size = 2971, normalized size = 8.07 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3/((b*x^2 + a)^2*x^(3/2)),x, algorithm="fricas")
[Out]
1/24*(16*a^2*b*d^3*x^4 - 48*a*b^2*c^3 - 4*(15*b^3*c^3 - 9*a*b^2*c^2*d + 9*a^2*b*
c*d^2 - 7*a^3*d^3)*x^2 - 12*(a^2*b^3*x^2 + a^3*b^2)*sqrt(x)*(-(625*b^12*c^12 - 1
500*a*b^11*c^11*d - 3150*a^2*b^10*c^10*d^2 + 11060*a^3*b^9*c^9*d^3 + 1071*a^4*b^
8*c^8*d^4 - 28728*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 + 27144*a^7*b^5*c^5*d^
7 - 37665*a^8*b^4*c^4*d^8 + 2324*a^9*b^3*c^3*d^9 + 19698*a^10*b^2*c^2*d^10 - 123
48*a^11*b*c*d^11 + 2401*a^12*d^12)/(a^9*b^11))^(1/4)*arctan(a^7*b^8*(-(625*b^12*
c^12 - 1500*a*b^11*c^11*d - 3150*a^2*b^10*c^10*d^2 + 11060*a^3*b^9*c^9*d^3 + 107
1*a^4*b^8*c^8*d^4 - 28728*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 + 27144*a^7*b^
5*c^5*d^7 - 37665*a^8*b^4*c^4*d^8 + 2324*a^9*b^3*c^3*d^9 + 19698*a^10*b^2*c^2*d^
10 - 12348*a^11*b*c*d^11 + 2401*a^12*d^12)/(a^9*b^11))^(3/4)/((125*b^9*c^9 - 225
*a*b^8*c^8*d - 540*a^2*b^7*c^7*d^2 + 1308*a^3*b^6*c^6*d^3 + 342*a^4*b^5*c^5*d^4
- 2430*a^5*b^4*c^4*d^5 + 1140*a^6*b^3*c^3*d^6 + 1260*a^7*b^2*c^2*d^7 - 1323*a^8*
b*c*d^8 + 343*a^9*d^9)*sqrt(x) + sqrt((15625*b^18*c^18 - 56250*a*b^17*c^17*d - 8
4375*a^2*b^16*c^16*d^2 + 570000*a^3*b^15*c^15*d^3 - 211500*a^4*b^14*c^14*d^4 - 2
174040*a^5*b^13*c^13*d^5 + 2720004*a^6*b^12*c^12*d^6 + 3321072*a^7*b^11*c^11*d^7
- 8368866*a^8*b^10*c^10*d^8 + 640420*a^9*b^9*c^9*d^9 + 11255310*a^10*b^8*c^8*d^
10 - 8509968*a^11*b^7*c^7*d^11 - 4831644*a^12*b^6*c^6*d^12 + 9537192*a^13*b^5*c^
5*d^13 - 3095820*a^14*b^4*c^4*d^14 - 2551920*a^15*b^3*c^3*d^15 + 2614689*a^16*b^
2*c^2*d^16 - 907578*a^17*b*c*d^17 + 117649*a^18*d^18)*x - (625*a^5*b^17*c^12 - 1
500*a^6*b^16*c^11*d - 3150*a^7*b^15*c^10*d^2 + 11060*a^8*b^14*c^9*d^3 + 1071*a^9
*b^13*c^8*d^4 - 28728*a^10*b^12*c^7*d^5 + 19068*a^11*b^11*c^6*d^6 + 27144*a^12*b
^10*c^5*d^7 - 37665*a^13*b^9*c^4*d^8 + 2324*a^14*b^8*c^3*d^9 + 19698*a^15*b^7*c^
2*d^10 - 12348*a^16*b^6*c*d^11 + 2401*a^17*b^5*d^12)*sqrt(-(625*b^12*c^12 - 1500
*a*b^11*c^11*d - 3150*a^2*b^10*c^10*d^2 + 11060*a^3*b^9*c^9*d^3 + 1071*a^4*b^8*c
^8*d^4 - 28728*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 + 27144*a^7*b^5*c^5*d^7 -
37665*a^8*b^4*c^4*d^8 + 2324*a^9*b^3*c^3*d^9 + 19698*a^10*b^2*c^2*d^10 - 12348*
a^11*b*c*d^11 + 2401*a^12*d^12)/(a^9*b^11))))) - 3*(a^2*b^3*x^2 + a^3*b^2)*sqrt(
x)*(-(625*b^12*c^12 - 1500*a*b^11*c^11*d - 3150*a^2*b^10*c^10*d^2 + 11060*a^3*b^
9*c^9*d^3 + 1071*a^4*b^8*c^8*d^4 - 28728*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6
+ 27144*a^7*b^5*c^5*d^7 - 37665*a^8*b^4*c^4*d^8 + 2324*a^9*b^3*c^3*d^9 + 19698*
a^10*b^2*c^2*d^10 - 12348*a^11*b*c*d^11 + 2401*a^12*d^12)/(a^9*b^11))^(1/4)*log(
a^7*b^8*(-(625*b^12*c^12 - 1500*a*b^11*c^11*d - 3150*a^2*b^10*c^10*d^2 + 11060*a
^3*b^9*c^9*d^3 + 1071*a^4*b^8*c^8*d^4 - 28728*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^
6*d^6 + 27144*a^7*b^5*c^5*d^7 - 37665*a^8*b^4*c^4*d^8 + 2324*a^9*b^3*c^3*d^9 + 1
9698*a^10*b^2*c^2*d^10 - 12348*a^11*b*c*d^11 + 2401*a^12*d^12)/(a^9*b^11))^(3/4)
+ (125*b^9*c^9 - 225*a*b^8*c^8*d - 540*a^2*b^7*c^7*d^2 + 1308*a^3*b^6*c^6*d^3 +
342*a^4*b^5*c^5*d^4 - 2430*a^5*b^4*c^4*d^5 + 1140*a^6*b^3*c^3*d^6 + 1260*a^7*b^
2*c^2*d^7 - 1323*a^8*b*c*d^8 + 343*a^9*d^9)*sqrt(x)) + 3*(a^2*b^3*x^2 + a^3*b^2)
*sqrt(x)*(-(625*b^12*c^12 - 1500*a*b^11*c^11*d - 3150*a^2*b^10*c^10*d^2 + 11060*
a^3*b^9*c^9*d^3 + 1071*a^4*b^8*c^8*d^4 - 28728*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c
^6*d^6 + 27144*a^7*b^5*c^5*d^7 - 37665*a^8*b^4*c^4*d^8 + 2324*a^9*b^3*c^3*d^9 +
19698*a^10*b^2*c^2*d^10 - 12348*a^11*b*c*d^11 + 2401*a^12*d^12)/(a^9*b^11))^(1/4
)*log(-a^7*b^8*(-(625*b^12*c^12 - 1500*a*b^11*c^11*d - 3150*a^2*b^10*c^10*d^2 +
11060*a^3*b^9*c^9*d^3 + 1071*a^4*b^8*c^8*d^4 - 28728*a^5*b^7*c^7*d^5 + 19068*a^6
*b^6*c^6*d^6 + 27144*a^7*b^5*c^5*d^7 - 37665*a^8*b^4*c^4*d^8 + 2324*a^9*b^3*c^3*
d^9 + 19698*a^10*b^2*c^2*d^10 - 12348*a^11*b*c*d^11 + 2401*a^12*d^12)/(a^9*b^11)
)^(3/4) + (125*b^9*c^9 - 225*a*b^8*c^8*d - 540*a^2*b^7*c^7*d^2 + 1308*a^3*b^6*c^
6*d^3 + 342*a^4*b^5*c^5*d^4 - 2430*a^5*b^4*c^4*d^5 + 1140*a^6*b^3*c^3*d^6 + 1260
*a^7*b^2*c^2*d^7 - 1323*a^8*b*c*d^8 + 343*a^9*d^9)*sqrt(x)))/((a^2*b^3*x^2 + a^3
*b^2)*sqrt(x))
_______________________________________________________________________________________
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**(3/2)/(b*x**2+a)**2,x)
[Out]
Timed out
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.265592, size = 680, normalized size = 1.85 \[ \frac{2 \, d^{3} x^{\frac{3}{2}}}{3 \, b^{2}} - \frac{5 \, b^{3} c^{3} x^{2} - 3 \, a b^{2} c^{2} d x^{2} + 3 \, a^{2} b c d^{2} x^{2} - a^{3} d^{3} x^{2} + 4 \, a b^{2} c^{3}}{2 \,{\left (b x^{\frac{5}{2}} + a \sqrt{x}\right )} a^{2} b^{2}} - \frac{\sqrt{2}{\left (5 \, \left (a b^{3}\right )^{\frac{3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{3}{4}} a b^{2} c^{2} d - 9 \, \left (a b^{3}\right )^{\frac{3}{4}} a^{2} b c d^{2} + 7 \, \left (a b^{3}\right )^{\frac{3}{4}} a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{3} b^{5}} - \frac{\sqrt{2}{\left (5 \, \left (a b^{3}\right )^{\frac{3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{3}{4}} a b^{2} c^{2} d - 9 \, \left (a b^{3}\right )^{\frac{3}{4}} a^{2} b c d^{2} + 7 \, \left (a b^{3}\right )^{\frac{3}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{3} b^{5}} + \frac{\sqrt{2}{\left (5 \, \left (a b^{3}\right )^{\frac{3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{3}{4}} a b^{2} c^{2} d - 9 \, \left (a b^{3}\right )^{\frac{3}{4}} a^{2} b c d^{2} + 7 \, \left (a b^{3}\right )^{\frac{3}{4}} a^{3} d^{3}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{3} b^{5}} - \frac{\sqrt{2}{\left (5 \, \left (a b^{3}\right )^{\frac{3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{3}{4}} a b^{2} c^{2} d - 9 \, \left (a b^{3}\right )^{\frac{3}{4}} a^{2} b c d^{2} + 7 \, \left (a b^{3}\right )^{\frac{3}{4}} a^{3} d^{3}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{3} b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3/((b*x^2 + a)^2*x^(3/2)),x, algorithm="giac")
[Out]
2/3*d^3*x^(3/2)/b^2 - 1/2*(5*b^3*c^3*x^2 - 3*a*b^2*c^2*d*x^2 + 3*a^2*b*c*d^2*x^2
- a^3*d^3*x^2 + 4*a*b^2*c^3)/((b*x^(5/2) + a*sqrt(x))*a^2*b^2) - 1/8*sqrt(2)*(5
*(a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d - 9*(a*b^3)^(3/4)*a^2*b*c*d
^2 + 7*(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^3*b^5) - 1/8*sqrt(2)*(5*(a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/
4)*a*b^2*c^2*d - 9*(a*b^3)^(3/4)*a^2*b*c*d^2 + 7*(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^3*b^5) + 1/16*sqrt
(2)*(5*(a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d - 9*(a*b^3)^(3/4)*a^2
*b*c*d^2 + 7*(a*b^3)^(3/4)*a^3*d^3)*ln(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/
b))/(a^3*b^5) - 1/16*sqrt(2)*(5*(a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^
2*d - 9*(a*b^3)^(3/4)*a^2*b*c*d^2 + 7*(a*b^3)^(3/4)*a^3*d^3)*ln(-sqrt(2)*sqrt(x)
*(a/b)^(1/4) + x + sqrt(a/b))/(a^3*b^5)