3.455 \(\int \frac{\sqrt{x} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx\)
Optimal. Leaf size=376 \[ \frac{(b c-a d)^2 (11 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} b^{15/4}}-\frac{(b c-a d)^2 (11 a d+b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} b^{15/4}}-\frac{(b c-a d)^2 (11 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{5/4} b^{15/4}}+\frac{(b c-a d)^2 (11 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{5/4} b^{15/4}}-\frac{d x^{3/2} \left (11 a^2 d^2-21 a b c d+6 b^2 c^2\right )}{6 a b^3}-\frac{d^2 x^{7/2} (7 b c-11 a d)}{14 a b^2}+\frac{x^{3/2} \left (c+d x^2\right )^2 (b c-a d)}{2 a b \left (a+b x^2\right )} \]
[Out]
-(d*(6*b^2*c^2 - 21*a*b*c*d + 11*a^2*d^2)*x^(3/2))/(6*a*b^3) - (d^2*(7*b*c - 11*
a*d)*x^(7/2))/(14*a*b^2) + ((b*c - a*d)*x^(3/2)*(c + d*x^2)^2)/(2*a*b*(a + b*x^2
)) - ((b*c - a*d)^2*(b*c + 11*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]
)/(4*Sqrt[2]*a^(5/4)*b^(15/4)) + ((b*c - a*d)^2*(b*c + 11*a*d)*ArcTan[1 + (Sqrt[
2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(5/4)*b^(15/4)) + ((b*c - a*d)^2*(b*c
+ 11*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2
]*a^(5/4)*b^(15/4)) - ((b*c - a*d)^2*(b*c + 11*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4
)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(5/4)*b^(15/4))
_______________________________________________________________________________________
Rubi [A] time = 0.910527, 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
\[ \frac{(b c-a d)^2 (11 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} b^{15/4}}-\frac{(b c-a d)^2 (11 a d+b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{5/4} b^{15/4}}-\frac{(b c-a d)^2 (11 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{5/4} b^{15/4}}+\frac{(b c-a d)^2 (11 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{5/4} b^{15/4}}-\frac{d x^{3/2} \left (11 a^2 d^2-21 a b c d+6 b^2 c^2\right )}{6 a b^3}-\frac{d^2 x^{7/2} (7 b c-11 a d)}{14 a b^2}+\frac{x^{3/2} \left (c+d x^2\right )^2 (b c-a d)}{2 a b \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[x]*(c + d*x^2)^3)/(a + b*x^2)^2,x]
[Out]
-(d*(6*b^2*c^2 - 21*a*b*c*d + 11*a^2*d^2)*x^(3/2))/(6*a*b^3) - (d^2*(7*b*c - 11*
a*d)*x^(7/2))/(14*a*b^2) + ((b*c - a*d)*x^(3/2)*(c + d*x^2)^2)/(2*a*b*(a + b*x^2
)) - ((b*c - a*d)^2*(b*c + 11*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]
)/(4*Sqrt[2]*a^(5/4)*b^(15/4)) + ((b*c - a*d)^2*(b*c + 11*a*d)*ArcTan[1 + (Sqrt[
2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(5/4)*b^(15/4)) + ((b*c - a*d)^2*(b*c
+ 11*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2
]*a^(5/4)*b^(15/4)) - ((b*c - a*d)^2*(b*c + 11*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4
)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(5/4)*b^(15/4))
_______________________________________________________________________________________
Rubi in Sympy [A] time = 152.817, size = 347, normalized size = 0.92 \[ - \frac{x^{\frac{3}{2}} \left (c + d x^{2}\right )^{2} \left (a d - b c\right )}{2 a b \left (a + b x^{2}\right )} + \frac{d^{2} x^{\frac{7}{2}} \left (11 a d - 7 b c\right )}{14 a b^{2}} - \frac{d x^{\frac{3}{2}} \left (11 a^{2} d^{2} - 21 a b c d + 6 b^{2} c^{2}\right )}{6 a b^{3}} + \frac{\sqrt{2} \left (a d - b c\right )^{2} \left (11 a d + b c\right ) \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{5}{4}} b^{\frac{15}{4}}} - \frac{\sqrt{2} \left (a d - b c\right )^{2} \left (11 a d + b c\right ) \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 a^{\frac{5}{4}} b^{\frac{15}{4}}} - \frac{\sqrt{2} \left (a d - b c\right )^{2} \left (11 a d + b c\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{5}{4}} b^{\frac{15}{4}}} + \frac{\sqrt{2} \left (a d - b c\right )^{2} \left (11 a d + b c\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{5}{4}} b^{\frac{15}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**2+c)**3*x**(1/2)/(b*x**2+a)**2,x)
[Out]
-x**(3/2)*(c + d*x**2)**2*(a*d - b*c)/(2*a*b*(a + b*x**2)) + d**2*x**(7/2)*(11*a
*d - 7*b*c)/(14*a*b**2) - d*x**(3/2)*(11*a**2*d**2 - 21*a*b*c*d + 6*b**2*c**2)/(
6*a*b**3) + sqrt(2)*(a*d - b*c)**2*(11*a*d + b*c)*log(-sqrt(2)*a**(1/4)*b**(1/4)
*sqrt(x) + sqrt(a) + sqrt(b)*x)/(16*a**(5/4)*b**(15/4)) - sqrt(2)*(a*d - b*c)**2
*(11*a*d + b*c)*log(sqrt(2)*a**(1/4)*b**(1/4)*sqrt(x) + sqrt(a) + sqrt(b)*x)/(16
*a**(5/4)*b**(15/4)) - sqrt(2)*(a*d - b*c)**2*(11*a*d + b*c)*atan(1 - sqrt(2)*b*
*(1/4)*sqrt(x)/a**(1/4))/(8*a**(5/4)*b**(15/4)) + sqrt(2)*(a*d - b*c)**2*(11*a*d
+ b*c)*atan(1 + sqrt(2)*b**(1/4)*sqrt(x)/a**(1/4))/(8*a**(5/4)*b**(15/4))
_______________________________________________________________________________________
Mathematica [A] time = 0.410647, size = 323, normalized size = 0.86 \[ \frac{\frac{21 \sqrt{2} (b c-a d)^2 (11 a d+b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{5/4}}-\frac{21 \sqrt{2} (b c-a d)^2 (11 a d+b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{a^{5/4}}-\frac{42 \sqrt{2} (b c-a d)^2 (11 a d+b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{a^{5/4}}+\frac{42 \sqrt{2} (b c-a d)^2 (11 a d+b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{a^{5/4}}+224 b^{3/4} d^2 x^{3/2} (3 b c-2 a d)+\frac{168 b^{3/4} x^{3/2} (b c-a d)^3}{a \left (a+b x^2\right )}+96 b^{7/4} d^3 x^{7/2}}{336 b^{15/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[x]*(c + d*x^2)^3)/(a + b*x^2)^2,x]
[Out]
(224*b^(3/4)*d^2*(3*b*c - 2*a*d)*x^(3/2) + 96*b^(7/4)*d^3*x^(7/2) + (168*b^(3/4)
*(b*c - a*d)^3*x^(3/2))/(a*(a + b*x^2)) - (42*Sqrt[2]*(b*c - a*d)^2*(b*c + 11*a*
d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(5/4) + (42*Sqrt[2]*(b*c - a
*d)^2*(b*c + 11*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(5/4) + (2
1*Sqrt[2]*(b*c - a*d)^2*(b*c + 11*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqr
t[x] + Sqrt[b]*x])/a^(5/4) - (21*Sqrt[2]*(b*c - a*d)^2*(b*c + 11*a*d)*Log[Sqrt[a
] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(5/4))/(336*b^(15/4))
_______________________________________________________________________________________
Maple [B] time = 0.024, 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^(1/2)/(b*x^2+a)^2,x)
[Out]
2/7*d^3/b^2*x^(7/2)-4/3*d^3/b^3*x^(3/2)*a+2*d^2/b^2*x^(3/2)*c-1/2/b^3*a^2*x^(3/2
)/(b*x^2+a)*d^3+3/2/b^2*a*x^(3/2)/(b*x^2+a)*c*d^2-3/2/b*x^(3/2)/(b*x^2+a)*c^2*d+
1/2/a*x^(3/2)/(b*x^2+a)*c^3+11/8/b^4*a^2/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b
)^(1/4)*x^(1/2)-1)*d^3-21/8/b^3*a/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)
*x^(1/2)-1)*c*d^2+9/8/b^2/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)
-1)*c^2*d+1/8/b/a/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*c^3+
11/16/b^4*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)))*d^3-21/16/b^3*a/(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+9/16/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^2*d+1/16/b/a/(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+11/8/b^4*a^2/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/
(a/b)^(1/4)*x^(1/2)+1)*d^3-21/8/b^3*a/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(
1/4)*x^(1/2)+1)*c*d^2+9/8/b^2/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(
1/2)+1)*c^2*d+1/8/b/a/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*
c^3
_______________________________________________________________________________________
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*sqrt(x)/(b*x^2 + a)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
_______________________________________________________________________________________
Fricas [A] time = 0.284317, size = 2955, normalized size = 7.86 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3*sqrt(x)/(b*x^2 + a)^2,x, algorithm="fricas")
[Out]
1/168*(84*(a*b^4*x^2 + a^2*b^3)*(-(b^12*c^12 + 36*a*b^11*c^11*d + 402*a^2*b^10*c
^10*d^2 + 692*a^3*b^9*c^9*d^3 - 10017*a^4*b^8*c^8*d^4 - 5688*a^5*b^7*c^7*d^5 + 1
60188*a^6*b^6*c^6*d^6 - 486648*a^7*b^5*c^5*d^7 + 746703*a^8*b^4*c^4*d^8 - 676588
*a^9*b^3*c^3*d^9 + 368082*a^10*b^2*c^2*d^10 - 111804*a^11*b*c*d^11 + 14641*a^12*
d^12)/(a^5*b^15))^(1/4)*arctan(a^4*b^11*(-(b^12*c^12 + 36*a*b^11*c^11*d + 402*a^
2*b^10*c^10*d^2 + 692*a^3*b^9*c^9*d^3 - 10017*a^4*b^8*c^8*d^4 - 5688*a^5*b^7*c^7
*d^5 + 160188*a^6*b^6*c^6*d^6 - 486648*a^7*b^5*c^5*d^7 + 746703*a^8*b^4*c^4*d^8
- 676588*a^9*b^3*c^3*d^9 + 368082*a^10*b^2*c^2*d^10 - 111804*a^11*b*c*d^11 + 146
41*a^12*d^12)/(a^5*b^15))^(3/4)/((b^9*c^9 + 27*a*b^8*c^8*d + 180*a^2*b^7*c^7*d^2
- 372*a^3*b^6*c^6*d^3 - 3186*a^4*b^5*c^5*d^4 + 13194*a^5*b^4*c^4*d^5 - 21372*a^
6*b^3*c^3*d^6 + 17820*a^7*b^2*c^2*d^7 - 7623*a^8*b*c*d^8 + 1331*a^9*d^9)*sqrt(x)
+ sqrt((b^18*c^18 + 54*a*b^17*c^17*d + 1089*a^2*b^16*c^16*d^2 + 8976*a^3*b^15*c
^15*d^3 + 5940*a^4*b^14*c^14*d^4 - 279576*a^5*b^13*c^13*d^5 - 338844*a^6*b^12*c^
12*d^6 + 6001776*a^7*b^11*c^11*d^7 - 6412626*a^8*b^10*c^10*d^8 - 62165180*a^9*b^
9*c^9*d^9 + 294333534*a^10*b^8*c^8*d^10 - 671362704*a^11*b^7*c^7*d^11 + 97458003
6*a^12*b^6*c^6*d^12 - 971334936*a^13*b^5*c^5*d^13 + 678512340*a^14*b^4*c^4*d^14
- 328575984*a^15*b^3*c^3*d^15 + 105546969*a^16*b^2*c^2*d^16 - 20292426*a^17*b*c*
d^17 + 1771561*a^18*d^18)*x - (a^3*b^19*c^12 + 36*a^4*b^18*c^11*d + 402*a^5*b^17
*c^10*d^2 + 692*a^6*b^16*c^9*d^3 - 10017*a^7*b^15*c^8*d^4 - 5688*a^8*b^14*c^7*d^
5 + 160188*a^9*b^13*c^6*d^6 - 486648*a^10*b^12*c^5*d^7 + 746703*a^11*b^11*c^4*d^
8 - 676588*a^12*b^10*c^3*d^9 + 368082*a^13*b^9*c^2*d^10 - 111804*a^14*b^8*c*d^11
+ 14641*a^15*b^7*d^12)*sqrt(-(b^12*c^12 + 36*a*b^11*c^11*d + 402*a^2*b^10*c^10*
d^2 + 692*a^3*b^9*c^9*d^3 - 10017*a^4*b^8*c^8*d^4 - 5688*a^5*b^7*c^7*d^5 + 16018
8*a^6*b^6*c^6*d^6 - 486648*a^7*b^5*c^5*d^7 + 746703*a^8*b^4*c^4*d^8 - 676588*a^9
*b^3*c^3*d^9 + 368082*a^10*b^2*c^2*d^10 - 111804*a^11*b*c*d^11 + 14641*a^12*d^12
)/(a^5*b^15))))) + 21*(a*b^4*x^2 + a^2*b^3)*(-(b^12*c^12 + 36*a*b^11*c^11*d + 40
2*a^2*b^10*c^10*d^2 + 692*a^3*b^9*c^9*d^3 - 10017*a^4*b^8*c^8*d^4 - 5688*a^5*b^7
*c^7*d^5 + 160188*a^6*b^6*c^6*d^6 - 486648*a^7*b^5*c^5*d^7 + 746703*a^8*b^4*c^4*
d^8 - 676588*a^9*b^3*c^3*d^9 + 368082*a^10*b^2*c^2*d^10 - 111804*a^11*b*c*d^11 +
14641*a^12*d^12)/(a^5*b^15))^(1/4)*log(a^4*b^11*(-(b^12*c^12 + 36*a*b^11*c^11*d
+ 402*a^2*b^10*c^10*d^2 + 692*a^3*b^9*c^9*d^3 - 10017*a^4*b^8*c^8*d^4 - 5688*a^
5*b^7*c^7*d^5 + 160188*a^6*b^6*c^6*d^6 - 486648*a^7*b^5*c^5*d^7 + 746703*a^8*b^4
*c^4*d^8 - 676588*a^9*b^3*c^3*d^9 + 368082*a^10*b^2*c^2*d^10 - 111804*a^11*b*c*d
^11 + 14641*a^12*d^12)/(a^5*b^15))^(3/4) + (b^9*c^9 + 27*a*b^8*c^8*d + 180*a^2*b
^7*c^7*d^2 - 372*a^3*b^6*c^6*d^3 - 3186*a^4*b^5*c^5*d^4 + 13194*a^5*b^4*c^4*d^5
- 21372*a^6*b^3*c^3*d^6 + 17820*a^7*b^2*c^2*d^7 - 7623*a^8*b*c*d^8 + 1331*a^9*d^
9)*sqrt(x)) - 21*(a*b^4*x^2 + a^2*b^3)*(-(b^12*c^12 + 36*a*b^11*c^11*d + 402*a^2
*b^10*c^10*d^2 + 692*a^3*b^9*c^9*d^3 - 10017*a^4*b^8*c^8*d^4 - 5688*a^5*b^7*c^7*
d^5 + 160188*a^6*b^6*c^6*d^6 - 486648*a^7*b^5*c^5*d^7 + 746703*a^8*b^4*c^4*d^8 -
676588*a^9*b^3*c^3*d^9 + 368082*a^10*b^2*c^2*d^10 - 111804*a^11*b*c*d^11 + 1464
1*a^12*d^12)/(a^5*b^15))^(1/4)*log(-a^4*b^11*(-(b^12*c^12 + 36*a*b^11*c^11*d + 4
02*a^2*b^10*c^10*d^2 + 692*a^3*b^9*c^9*d^3 - 10017*a^4*b^8*c^8*d^4 - 5688*a^5*b^
7*c^7*d^5 + 160188*a^6*b^6*c^6*d^6 - 486648*a^7*b^5*c^5*d^7 + 746703*a^8*b^4*c^4
*d^8 - 676588*a^9*b^3*c^3*d^9 + 368082*a^10*b^2*c^2*d^10 - 111804*a^11*b*c*d^11
+ 14641*a^12*d^12)/(a^5*b^15))^(3/4) + (b^9*c^9 + 27*a*b^8*c^8*d + 180*a^2*b^7*c
^7*d^2 - 372*a^3*b^6*c^6*d^3 - 3186*a^4*b^5*c^5*d^4 + 13194*a^5*b^4*c^4*d^5 - 21
372*a^6*b^3*c^3*d^6 + 17820*a^7*b^2*c^2*d^7 - 7623*a^8*b*c*d^8 + 1331*a^9*d^9)*s
qrt(x)) + 4*(12*a*b^2*d^3*x^5 + 4*(21*a*b^2*c*d^2 - 11*a^2*b*d^3)*x^3 + 7*(3*b^3
*c^3 - 9*a*b^2*c^2*d + 21*a^2*b*c*d^2 - 11*a^3*d^3)*x)*sqrt(x))/(a*b^4*x^2 + a^2
*b^3)
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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**(1/2)/(b*x**2+a)**2,x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.308015, size = 697, normalized size = 1.85 \[ \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 b^{3}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} b^{3} c^{3} + 9 \, \left (a b^{3}\right )^{\frac{3}{4}} a b^{2} c^{2} d - 21 \, \left (a b^{3}\right )^{\frac{3}{4}} a^{2} b c d^{2} + 11 \, \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^{2} b^{6}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} b^{3} c^{3} + 9 \, \left (a b^{3}\right )^{\frac{3}{4}} a b^{2} c^{2} d - 21 \, \left (a b^{3}\right )^{\frac{3}{4}} a^{2} b c d^{2} + 11 \, \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^{2} b^{6}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} b^{3} c^{3} + 9 \, \left (a b^{3}\right )^{\frac{3}{4}} a b^{2} c^{2} d - 21 \, \left (a b^{3}\right )^{\frac{3}{4}} a^{2} b c d^{2} + 11 \, \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^{2} b^{6}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} b^{3} c^{3} + 9 \, \left (a b^{3}\right )^{\frac{3}{4}} a b^{2} c^{2} d - 21 \, \left (a b^{3}\right )^{\frac{3}{4}} a^{2} b c d^{2} + 11 \, \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^{2} b^{6}} + \frac{2 \,{\left (3 \, b^{12} d^{3} x^{\frac{7}{2}} + 21 \, b^{12} c d^{2} x^{\frac{3}{2}} - 14 \, a b^{11} d^{3} x^{\frac{3}{2}}\right )}}{21 \, b^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^3*sqrt(x)/(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)*a*b^3) + 1/8*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 + 9*(a*b^3)^(3/
4)*a*b^2*c^2*d - 21*(a*b^3)^(3/4)*a^2*b*c*d^2 + 11*(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^2*b^6) + 1/8*sqrt
(2)*((a*b^3)^(3/4)*b^3*c^3 + 9*(a*b^3)^(3/4)*a*b^2*c^2*d - 21*(a*b^3)^(3/4)*a^2*
b*c*d^2 + 11*(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^2*b^6) - 1/16*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 + 9*(a*b^
3)^(3/4)*a*b^2*c^2*d - 21*(a*b^3)^(3/4)*a^2*b*c*d^2 + 11*(a*b^3)^(3/4)*a^3*d^3)*
ln(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^2*b^6) + 1/16*sqrt(2)*((a*b^3
)^(3/4)*b^3*c^3 + 9*(a*b^3)^(3/4)*a*b^2*c^2*d - 21*(a*b^3)^(3/4)*a^2*b*c*d^2 + 1
1*(a*b^3)^(3/4)*a^3*d^3)*ln(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^2*b
^6) + 2/21*(3*b^12*d^3*x^(7/2) + 21*b^12*c*d^2*x^(3/2) - 14*a*b^11*d^3*x^(3/2))/
b^14