3.218 \(\int \frac{\sqrt{x} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=365 \[ \frac{13 c^{5/4} (9 b B-17 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{21/4}}-\frac{13 c^{5/4} (9 b B-17 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{21/4}}-\frac{13 c^{5/4} (9 b B-17 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{21/4}}+\frac{13 c^{5/4} (9 b B-17 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{21/4}}+\frac{13 c (9 b B-17 A c)}{16 b^5 \sqrt{x}}-\frac{13 (9 b B-17 A c)}{80 b^4 x^{5/2}}+\frac{13 (9 b B-17 A c)}{144 b^3 c x^{9/2}}-\frac{9 b B-17 A c}{16 b^2 c x^{9/2} \left (b+c x^2\right )}-\frac{b B-A c}{4 b c x^{9/2} \left (b+c x^2\right )^2} \]

[Out]

(13*(9*b*B - 17*A*c))/(144*b^3*c*x^(9/2)) - (13*(9*b*B - 17*A*c))/(80*b^4*x^(5/2
)) + (13*c*(9*b*B - 17*A*c))/(16*b^5*Sqrt[x]) - (b*B - A*c)/(4*b*c*x^(9/2)*(b +
c*x^2)^2) - (9*b*B - 17*A*c)/(16*b^2*c*x^(9/2)*(b + c*x^2)) - (13*c^(5/4)*(9*b*B
- 17*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*b^(21/4))
+ (13*c^(5/4)*(9*b*B - 17*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(3
2*Sqrt[2]*b^(21/4)) + (13*c^(5/4)*(9*b*B - 17*A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)
*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*b^(21/4)) - (13*c^(5/4)*(9*b*B - 17*A
*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*b^(2
1/4))

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Rubi [A]  time = 0.669857, antiderivative size = 365, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.423 \[ \frac{13 c^{5/4} (9 b B-17 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{21/4}}-\frac{13 c^{5/4} (9 b B-17 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{21/4}}-\frac{13 c^{5/4} (9 b B-17 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{21/4}}+\frac{13 c^{5/4} (9 b B-17 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{21/4}}+\frac{13 c (9 b B-17 A c)}{16 b^5 \sqrt{x}}-\frac{13 (9 b B-17 A c)}{80 b^4 x^{5/2}}+\frac{13 (9 b B-17 A c)}{144 b^3 c x^{9/2}}-\frac{9 b B-17 A c}{16 b^2 c x^{9/2} \left (b+c x^2\right )}-\frac{b B-A c}{4 b c x^{9/2} \left (b+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[x]*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]

[Out]

(13*(9*b*B - 17*A*c))/(144*b^3*c*x^(9/2)) - (13*(9*b*B - 17*A*c))/(80*b^4*x^(5/2
)) + (13*c*(9*b*B - 17*A*c))/(16*b^5*Sqrt[x]) - (b*B - A*c)/(4*b*c*x^(9/2)*(b +
c*x^2)^2) - (9*b*B - 17*A*c)/(16*b^2*c*x^(9/2)*(b + c*x^2)) - (13*c^(5/4)*(9*b*B
- 17*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*b^(21/4))
+ (13*c^(5/4)*(9*b*B - 17*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(3
2*Sqrt[2]*b^(21/4)) + (13*c^(5/4)*(9*b*B - 17*A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)
*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*b^(21/4)) - (13*c^(5/4)*(9*b*B - 17*A
*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*b^(2
1/4))

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Rubi in Sympy [A]  time = 102.433, size = 350, normalized size = 0.96 \[ \frac{A c - B b}{4 b c x^{\frac{9}{2}} \left (b + c x^{2}\right )^{2}} + \frac{17 A c - 9 B b}{16 b^{2} c x^{\frac{9}{2}} \left (b + c x^{2}\right )} - \frac{13 \left (17 A c - 9 B b\right )}{144 b^{3} c x^{\frac{9}{2}}} + \frac{13 \left (17 A c - 9 B b\right )}{80 b^{4} x^{\frac{5}{2}}} - \frac{13 c \left (17 A c - 9 B b\right )}{16 b^{5} \sqrt{x}} - \frac{13 \sqrt{2} c^{\frac{5}{4}} \left (17 A c - 9 B b\right ) \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 b^{\frac{21}{4}}} + \frac{13 \sqrt{2} c^{\frac{5}{4}} \left (17 A c - 9 B b\right ) \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 b^{\frac{21}{4}}} + \frac{13 \sqrt{2} c^{\frac{5}{4}} \left (17 A c - 9 B b\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 b^{\frac{21}{4}}} - \frac{13 \sqrt{2} c^{\frac{5}{4}} \left (17 A c - 9 B b\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 b^{\frac{21}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)*x**(1/2)/(c*x**4+b*x**2)**3,x)

[Out]

(A*c - B*b)/(4*b*c*x**(9/2)*(b + c*x**2)**2) + (17*A*c - 9*B*b)/(16*b**2*c*x**(9
/2)*(b + c*x**2)) - 13*(17*A*c - 9*B*b)/(144*b**3*c*x**(9/2)) + 13*(17*A*c - 9*B
*b)/(80*b**4*x**(5/2)) - 13*c*(17*A*c - 9*B*b)/(16*b**5*sqrt(x)) - 13*sqrt(2)*c*
*(5/4)*(17*A*c - 9*B*b)*log(-sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(
c)*x)/(128*b**(21/4)) + 13*sqrt(2)*c**(5/4)*(17*A*c - 9*B*b)*log(sqrt(2)*b**(1/4
)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(128*b**(21/4)) + 13*sqrt(2)*c**(5/4)*
(17*A*c - 9*B*b)*atan(1 - sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(64*b**(21/4)) - 13
*sqrt(2)*c**(5/4)*(17*A*c - 9*B*b)*atan(1 + sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(
64*b**(21/4))

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Mathematica [A]  time = 0.629817, size = 333, normalized size = 0.91 \[ \frac{\frac{1440 b^{5/4} c^2 x^{3/2} (b B-A c)}{\left (b+c x^2\right )^2}-\frac{2304 b^{5/4} (b B-3 A c)}{x^{5/2}}-\frac{1280 A b^{9/4}}{x^{9/2}}+585 \sqrt{2} c^{5/4} (9 b B-17 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+585 \sqrt{2} c^{5/4} (17 A c-9 b B) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+1170 \sqrt{2} c^{5/4} (17 A c-9 b B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )+1170 \sqrt{2} c^{5/4} (9 b B-17 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )+\frac{360 \sqrt [4]{b} c^2 x^{3/2} (21 b B-29 A c)}{b+c x^2}+\frac{34560 \sqrt [4]{b} c (b B-2 A c)}{\sqrt{x}}}{5760 b^{21/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[x]*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]

[Out]

((-1280*A*b^(9/4))/x^(9/2) - (2304*b^(5/4)*(b*B - 3*A*c))/x^(5/2) + (34560*b^(1/
4)*c*(b*B - 2*A*c))/Sqrt[x] + (1440*b^(5/4)*c^2*(b*B - A*c)*x^(3/2))/(b + c*x^2)
^2 + (360*b^(1/4)*c^2*(21*b*B - 29*A*c)*x^(3/2))/(b + c*x^2) + 1170*Sqrt[2]*c^(5
/4)*(-9*b*B + 17*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 1170*Sqrt[
2]*c^(5/4)*(9*b*B - 17*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 585*
Sqrt[2]*c^(5/4)*(9*b*B - 17*A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] +
Sqrt[c]*x] + 585*Sqrt[2]*c^(5/4)*(-9*b*B + 17*A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4
)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(5760*b^(21/4))

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Maple [A]  time = 0.039, size = 414, normalized size = 1.1 \[ -{\frac{2\,A}{9\,{b}^{3}}{x}^{-{\frac{9}{2}}}}+{\frac{6\,Ac}{5\,{b}^{4}}{x}^{-{\frac{5}{2}}}}-{\frac{2\,B}{5\,{b}^{3}}{x}^{-{\frac{5}{2}}}}-12\,{\frac{A{c}^{2}}{{b}^{5}\sqrt{x}}}+6\,{\frac{Bc}{{b}^{4}\sqrt{x}}}-{\frac{29\,{c}^{4}A}{16\,{b}^{5} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{7}{2}}}}+{\frac{21\,B{c}^{3}}{16\,{b}^{4} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{7}{2}}}}-{\frac{33\,A{c}^{3}}{16\,{b}^{4} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{25\,B{c}^{2}}{16\,{b}^{3} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{221\,{c}^{2}\sqrt{2}A}{128\,{b}^{5}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{221\,{c}^{2}\sqrt{2}A}{64\,{b}^{5}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{221\,{c}^{2}\sqrt{2}A}{64\,{b}^{5}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{117\,c\sqrt{2}B}{128\,{b}^{4}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{117\,c\sqrt{2}B}{64\,{b}^{4}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{117\,c\sqrt{2}B}{64\,{b}^{4}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)*x^(1/2)/(c*x^4+b*x^2)^3,x)

[Out]

-2/9*A/b^3/x^(9/2)+6/5/x^(5/2)/b^4*A*c-2/5/x^(5/2)/b^3*B-12*c^2/b^5/x^(1/2)*A+6*
c/b^4/x^(1/2)*B-29/16/b^5*c^4/(c*x^2+b)^2*A*x^(7/2)+21/16/b^4*c^3/(c*x^2+b)^2*B*
x^(7/2)-33/16/b^4*c^3/(c*x^2+b)^2*x^(3/2)*A+25/16/b^3*c^2/(c*x^2+b)^2*x^(3/2)*B-
221/128/b^5*c^2/(b/c)^(1/4)*2^(1/2)*A*ln((x-(b/c)^(1/4)*x^(1/2)*2^(1/2)+(b/c)^(1
/2))/(x+(b/c)^(1/4)*x^(1/2)*2^(1/2)+(b/c)^(1/2)))-221/64/b^5*c^2/(b/c)^(1/4)*2^(
1/2)*A*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)-221/64/b^5*c^2/(b/c)^(1/4)*2^(1/2)*
A*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)-1)+117/128/b^4*c/(b/c)^(1/4)*2^(1/2)*B*ln((
x-(b/c)^(1/4)*x^(1/2)*2^(1/2)+(b/c)^(1/2))/(x+(b/c)^(1/4)*x^(1/2)*2^(1/2)+(b/c)^
(1/2)))+117/64/b^4*c/(b/c)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)
+117/64/b^4*c/(b/c)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)-1)

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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((B*x^2 + A)*sqrt(x)/(c*x^4 + b*x^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.25817, size = 1319, normalized size = 3.61 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*sqrt(x)/(c*x^4 + b*x^2)^3,x, algorithm="fricas")

[Out]

1/2880*(2340*(9*B*b*c^3 - 17*A*c^4)*x^8 + 4212*(9*B*b^2*c^2 - 17*A*b*c^3)*x^6 -
640*A*b^4 + 1664*(9*B*b^3*c - 17*A*b^2*c^2)*x^4 - 128*(9*B*b^4 - 17*A*b^3*c)*x^2
- 2340*(b^5*c^2*x^8 + 2*b^6*c*x^6 + b^7*x^4)*sqrt(x)*(-(6561*B^4*b^4*c^5 - 4957
2*A*B^3*b^3*c^6 + 140454*A^2*B^2*b^2*c^7 - 176868*A^3*B*b*c^8 + 83521*A^4*c^9)/b
^21)^(1/4)*arctan(-b^16*(-(6561*B^4*b^4*c^5 - 49572*A*B^3*b^3*c^6 + 140454*A^2*B
^2*b^2*c^7 - 176868*A^3*B*b*c^8 + 83521*A^4*c^9)/b^21)^(3/4)/((729*B^3*b^3*c^4 -
4131*A*B^2*b^2*c^5 + 7803*A^2*B*b*c^6 - 4913*A^3*c^7)*sqrt(x) - sqrt((531441*B^
6*b^6*c^8 - 6022998*A*B^5*b^5*c^9 + 28441935*A^2*B^4*b^4*c^10 - 71631540*A^3*B^3
*b^3*c^11 + 101478015*A^4*B^2*b^2*c^12 - 76672278*A^5*B*b*c^13 + 24137569*A^6*c^
14)*x - (6561*B^4*b^15*c^5 - 49572*A*B^3*b^14*c^6 + 140454*A^2*B^2*b^13*c^7 - 17
6868*A^3*B*b^12*c^8 + 83521*A^4*b^11*c^9)*sqrt(-(6561*B^4*b^4*c^5 - 49572*A*B^3*
b^3*c^6 + 140454*A^2*B^2*b^2*c^7 - 176868*A^3*B*b*c^8 + 83521*A^4*c^9)/b^21))))
- 585*(b^5*c^2*x^8 + 2*b^6*c*x^6 + b^7*x^4)*sqrt(x)*(-(6561*B^4*b^4*c^5 - 49572*
A*B^3*b^3*c^6 + 140454*A^2*B^2*b^2*c^7 - 176868*A^3*B*b*c^8 + 83521*A^4*c^9)/b^2
1)^(1/4)*log(2197*b^16*(-(6561*B^4*b^4*c^5 - 49572*A*B^3*b^3*c^6 + 140454*A^2*B^
2*b^2*c^7 - 176868*A^3*B*b*c^8 + 83521*A^4*c^9)/b^21)^(3/4) - 2197*(729*B^3*b^3*
c^4 - 4131*A*B^2*b^2*c^5 + 7803*A^2*B*b*c^6 - 4913*A^3*c^7)*sqrt(x)) + 585*(b^5*
c^2*x^8 + 2*b^6*c*x^6 + b^7*x^4)*sqrt(x)*(-(6561*B^4*b^4*c^5 - 49572*A*B^3*b^3*c
^6 + 140454*A^2*B^2*b^2*c^7 - 176868*A^3*B*b*c^8 + 83521*A^4*c^9)/b^21)^(1/4)*lo
g(-2197*b^16*(-(6561*B^4*b^4*c^5 - 49572*A*B^3*b^3*c^6 + 140454*A^2*B^2*b^2*c^7
- 176868*A^3*B*b*c^8 + 83521*A^4*c^9)/b^21)^(3/4) - 2197*(729*B^3*b^3*c^4 - 4131
*A*B^2*b^2*c^5 + 7803*A^2*B*b*c^6 - 4913*A^3*c^7)*sqrt(x)))/((b^5*c^2*x^8 + 2*b^
6*c*x^6 + b^7*x^4)*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((B*x**2+A)*x**(1/2)/(c*x**4+b*x**2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.229192, size = 474, normalized size = 1.3 \[ \frac{13 \, \sqrt{2}{\left (9 \, \left (b c^{3}\right )^{\frac{3}{4}} B b - 17 \, \left (b c^{3}\right )^{\frac{3}{4}} A c\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{6} c} + \frac{13 \, \sqrt{2}{\left (9 \, \left (b c^{3}\right )^{\frac{3}{4}} B b - 17 \, \left (b c^{3}\right )^{\frac{3}{4}} A c\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{6} c} - \frac{13 \, \sqrt{2}{\left (9 \, \left (b c^{3}\right )^{\frac{3}{4}} B b - 17 \, \left (b c^{3}\right )^{\frac{3}{4}} A c\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{6} c} + \frac{13 \, \sqrt{2}{\left (9 \, \left (b c^{3}\right )^{\frac{3}{4}} B b - 17 \, \left (b c^{3}\right )^{\frac{3}{4}} A c\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{6} c} + \frac{21 \, B b c^{3} x^{\frac{7}{2}} - 29 \, A c^{4} x^{\frac{7}{2}} + 25 \, B b^{2} c^{2} x^{\frac{3}{2}} - 33 \, A b c^{3} x^{\frac{3}{2}}}{16 \,{\left (c x^{2} + b\right )}^{2} b^{5}} + \frac{2 \,{\left (135 \, B b c x^{4} - 270 \, A c^{2} x^{4} - 9 \, B b^{2} x^{2} + 27 \, A b c x^{2} - 5 \, A b^{2}\right )}}{45 \, b^{5} x^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*sqrt(x)/(c*x^4 + b*x^2)^3,x, algorithm="giac")

[Out]

13/64*sqrt(2)*(9*(b*c^3)^(3/4)*B*b - 17*(b*c^3)^(3/4)*A*c)*arctan(1/2*sqrt(2)*(s
qrt(2)*(b/c)^(1/4) + 2*sqrt(x))/(b/c)^(1/4))/(b^6*c) + 13/64*sqrt(2)*(9*(b*c^3)^
(3/4)*B*b - 17*(b*c^3)^(3/4)*A*c)*arctan(-1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) - 2*s
qrt(x))/(b/c)^(1/4))/(b^6*c) - 13/128*sqrt(2)*(9*(b*c^3)^(3/4)*B*b - 17*(b*c^3)^
(3/4)*A*c)*ln(sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(b^6*c) + 13/128*sqrt
(2)*(9*(b*c^3)^(3/4)*B*b - 17*(b*c^3)^(3/4)*A*c)*ln(-sqrt(2)*sqrt(x)*(b/c)^(1/4)
+ x + sqrt(b/c))/(b^6*c) + 1/16*(21*B*b*c^3*x^(7/2) - 29*A*c^4*x^(7/2) + 25*B*b
^2*c^2*x^(3/2) - 33*A*b*c^3*x^(3/2))/((c*x^2 + b)^2*b^5) + 2/45*(135*B*b*c*x^4 -
270*A*c^2*x^4 - 9*B*b^2*x^2 + 27*A*b*c*x^2 - 5*A*b^2)/(b^5*x^(9/2))