3.195 \(\int \frac{x^{19/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=332 \[ \frac{b^{5/4} (13 b B-9 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{17/4}}-\frac{b^{5/4} (13 b B-9 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{17/4}}+\frac{b^{5/4} (13 b B-9 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} c^{17/4}}-\frac{b^{5/4} (13 b B-9 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} c^{17/4}}+\frac{b \sqrt{x} (13 b B-9 A c)}{2 c^4}-\frac{x^{5/2} (13 b B-9 A c)}{10 c^3}+\frac{x^{9/2} (13 b B-9 A c)}{18 b c^2}-\frac{x^{13/2} (b B-A c)}{2 b c \left (b+c x^2\right )} \]

[Out]

(b*(13*b*B - 9*A*c)*Sqrt[x])/(2*c^4) - ((13*b*B - 9*A*c)*x^(5/2))/(10*c^3) + ((1
3*b*B - 9*A*c)*x^(9/2))/(18*b*c^2) - ((b*B - A*c)*x^(13/2))/(2*b*c*(b + c*x^2))
+ (b^(5/4)*(13*b*B - 9*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sq
rt[2]*c^(17/4)) - (b^(5/4)*(13*b*B - 9*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])
/b^(1/4)])/(4*Sqrt[2]*c^(17/4)) + (b^(5/4)*(13*b*B - 9*A*c)*Log[Sqrt[b] - Sqrt[2
]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*c^(17/4)) - (b^(5/4)*(13*b*B
- 9*A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*
c^(17/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.572476, antiderivative size = 332, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ \frac{b^{5/4} (13 b B-9 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{17/4}}-\frac{b^{5/4} (13 b B-9 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{17/4}}+\frac{b^{5/4} (13 b B-9 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} c^{17/4}}-\frac{b^{5/4} (13 b B-9 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} c^{17/4}}+\frac{b \sqrt{x} (13 b B-9 A c)}{2 c^4}-\frac{x^{5/2} (13 b B-9 A c)}{10 c^3}+\frac{x^{9/2} (13 b B-9 A c)}{18 b c^2}-\frac{x^{13/2} (b B-A c)}{2 b c \left (b+c x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(x^(19/2)*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]

[Out]

(b*(13*b*B - 9*A*c)*Sqrt[x])/(2*c^4) - ((13*b*B - 9*A*c)*x^(5/2))/(10*c^3) + ((1
3*b*B - 9*A*c)*x^(9/2))/(18*b*c^2) - ((b*B - A*c)*x^(13/2))/(2*b*c*(b + c*x^2))
+ (b^(5/4)*(13*b*B - 9*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sq
rt[2]*c^(17/4)) - (b^(5/4)*(13*b*B - 9*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])
/b^(1/4)])/(4*Sqrt[2]*c^(17/4)) + (b^(5/4)*(13*b*B - 9*A*c)*Log[Sqrt[b] - Sqrt[2
]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*c^(17/4)) - (b^(5/4)*(13*b*B
- 9*A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*
c^(17/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 91.5936, size = 311, normalized size = 0.94 \[ - \frac{\sqrt{2} b^{\frac{5}{4}} \left (9 A c - 13 B b\right ) \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 c^{\frac{17}{4}}} + \frac{\sqrt{2} b^{\frac{5}{4}} \left (9 A c - 13 B b\right ) \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 c^{\frac{17}{4}}} - \frac{\sqrt{2} b^{\frac{5}{4}} \left (9 A c - 13 B b\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 c^{\frac{17}{4}}} + \frac{\sqrt{2} b^{\frac{5}{4}} \left (9 A c - 13 B b\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 c^{\frac{17}{4}}} - \frac{b \sqrt{x} \left (9 A c - 13 B b\right )}{2 c^{4}} + \frac{x^{\frac{5}{2}} \left (9 A c - 13 B b\right )}{10 c^{3}} + \frac{x^{\frac{13}{2}} \left (A c - B b\right )}{2 b c \left (b + c x^{2}\right )} - \frac{x^{\frac{9}{2}} \left (9 A c - 13 B b\right )}{18 b c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(19/2)*(B*x**2+A)/(c*x**4+b*x**2)**2,x)

[Out]

-sqrt(2)*b**(5/4)*(9*A*c - 13*B*b)*log(-sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt
(b) + sqrt(c)*x)/(16*c**(17/4)) + sqrt(2)*b**(5/4)*(9*A*c - 13*B*b)*log(sqrt(2)*
b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(16*c**(17/4)) - sqrt(2)*b**(5/
4)*(9*A*c - 13*B*b)*atan(1 - sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(8*c**(17/4)) +
sqrt(2)*b**(5/4)*(9*A*c - 13*B*b)*atan(1 + sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(8
*c**(17/4)) - b*sqrt(x)*(9*A*c - 13*B*b)/(2*c**4) + x**(5/2)*(9*A*c - 13*B*b)/(1
0*c**3) + x**(13/2)*(A*c - B*b)/(2*b*c*(b + c*x**2)) - x**(9/2)*(9*A*c - 13*B*b)
/(18*b*c**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.448331, size = 301, normalized size = 0.91 \[ \frac{45 \sqrt{2} b^{5/4} (13 b B-9 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-45 \sqrt{2} b^{5/4} (13 b B-9 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+90 \sqrt{2} b^{5/4} (13 b B-9 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )-90 \sqrt{2} b^{5/4} (13 b B-9 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )+\frac{360 b^2 \sqrt [4]{c} \sqrt{x} (b B-A c)}{b+c x^2}+288 c^{5/4} x^{5/2} (A c-2 b B)+1440 b \sqrt [4]{c} \sqrt{x} (3 b B-2 A c)+160 B c^{9/4} x^{9/2}}{720 c^{17/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(19/2)*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]

[Out]

(1440*b*c^(1/4)*(3*b*B - 2*A*c)*Sqrt[x] + 288*c^(5/4)*(-2*b*B + A*c)*x^(5/2) + 1
60*B*c^(9/4)*x^(9/2) + (360*b^2*c^(1/4)*(b*B - A*c)*Sqrt[x])/(b + c*x^2) + 90*Sq
rt[2]*b^(5/4)*(13*b*B - 9*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - 9
0*Sqrt[2]*b^(5/4)*(13*b*B - 9*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)]
+ 45*Sqrt[2]*b^(5/4)*(13*b*B - 9*A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqr
t[x] + Sqrt[c]*x] - 45*Sqrt[2]*b^(5/4)*(13*b*B - 9*A*c)*Log[Sqrt[b] + Sqrt[2]*b^
(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(720*c^(17/4))

_______________________________________________________________________________________

Maple [A]  time = 0.023, size = 372, normalized size = 1.1 \[{\frac{2\,B}{9\,{c}^{2}}{x}^{{\frac{9}{2}}}}+{\frac{2\,A}{5\,{c}^{2}}{x}^{{\frac{5}{2}}}}-{\frac{4\,Bb}{5\,{c}^{3}}{x}^{{\frac{5}{2}}}}-4\,{\frac{Ab\sqrt{x}}{{c}^{3}}}+6\,{\frac{\sqrt{x}B{b}^{2}}{{c}^{4}}}-{\frac{{b}^{2}A}{2\,{c}^{3} \left ( c{x}^{2}+b \right ) }\sqrt{x}}+{\frac{B{b}^{3}}{2\,{c}^{4} \left ( c{x}^{2}+b \right ) }\sqrt{x}}+{\frac{9\,b\sqrt{2}A}{8\,{c}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }+{\frac{9\,b\sqrt{2}A}{16\,{c}^{3}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }+{\frac{9\,b\sqrt{2}A}{8\,{c}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }-{\frac{13\,{b}^{2}\sqrt{2}B}{8\,{c}^{4}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }-{\frac{13\,{b}^{2}\sqrt{2}B}{16\,{c}^{4}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }-{\frac{13\,{b}^{2}\sqrt{2}B}{8\,{c}^{4}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(19/2)*(B*x^2+A)/(c*x^4+b*x^2)^2,x)

[Out]

2/9/c^2*B*x^(9/2)+2/5/c^2*A*x^(5/2)-4/5/c^3*B*x^(5/2)*b-4/c^3*A*x^(1/2)*b+6/c^4*
x^(1/2)*B*b^2-1/2*b^2/c^3*x^(1/2)/(c*x^2+b)*A+1/2*b^3/c^4*x^(1/2)/(c*x^2+b)*B+9/
8*b/c^3*(b/c)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)-1)+9/16*b/c^3*(
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)))+9/8*b/c^3*(b/c)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/
(b/c)^(1/4)*x^(1/2)+1)-13/8*b^2/c^4*(b/c)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(b/c)^(
1/4)*x^(1/2)-1)-13/16*b^2/c^4*(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)))-13/8*b^2/c^4*(b/
c)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(b/c)^(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((B*x^2 + A)*x^(19/2)/(c*x^4 + b*x^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.240446, size = 921, normalized size = 2.77 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(19/2)/(c*x^4 + b*x^2)^2,x, algorithm="fricas")

[Out]

-1/360*(180*(c^5*x^2 + b*c^4)*(-(28561*B^4*b^9 - 79092*A*B^3*b^8*c + 82134*A^2*B
^2*b^7*c^2 - 37908*A^3*B*b^6*c^3 + 6561*A^4*b^5*c^4)/c^17)^(1/4)*arctan(-c^4*(-(
28561*B^4*b^9 - 79092*A*B^3*b^8*c + 82134*A^2*B^2*b^7*c^2 - 37908*A^3*B*b^6*c^3
+ 6561*A^4*b^5*c^4)/c^17)^(1/4)/((13*B*b^2 - 9*A*b*c)*sqrt(x) - sqrt(c^8*sqrt(-(
28561*B^4*b^9 - 79092*A*B^3*b^8*c + 82134*A^2*B^2*b^7*c^2 - 37908*A^3*B*b^6*c^3
+ 6561*A^4*b^5*c^4)/c^17) + (169*B^2*b^4 - 234*A*B*b^3*c + 81*A^2*b^2*c^2)*x)))
- 45*(c^5*x^2 + b*c^4)*(-(28561*B^4*b^9 - 79092*A*B^3*b^8*c + 82134*A^2*B^2*b^7*
c^2 - 37908*A^3*B*b^6*c^3 + 6561*A^4*b^5*c^4)/c^17)^(1/4)*log(c^4*(-(28561*B^4*b
^9 - 79092*A*B^3*b^8*c + 82134*A^2*B^2*b^7*c^2 - 37908*A^3*B*b^6*c^3 + 6561*A^4*
b^5*c^4)/c^17)^(1/4) - (13*B*b^2 - 9*A*b*c)*sqrt(x)) + 45*(c^5*x^2 + b*c^4)*(-(2
8561*B^4*b^9 - 79092*A*B^3*b^8*c + 82134*A^2*B^2*b^7*c^2 - 37908*A^3*B*b^6*c^3 +
6561*A^4*b^5*c^4)/c^17)^(1/4)*log(-c^4*(-(28561*B^4*b^9 - 79092*A*B^3*b^8*c + 8
2134*A^2*B^2*b^7*c^2 - 37908*A^3*B*b^6*c^3 + 6561*A^4*b^5*c^4)/c^17)^(1/4) - (13
*B*b^2 - 9*A*b*c)*sqrt(x)) - 4*(20*B*c^3*x^6 - 4*(13*B*b*c^2 - 9*A*c^3)*x^4 + 58
5*B*b^3 - 405*A*b^2*c + 36*(13*B*b^2*c - 9*A*b*c^2)*x^2)*sqrt(x))/(c^5*x^2 + b*c
^4)

_______________________________________________________________________________________

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**(19/2)*(B*x**2+A)/(c*x**4+b*x**2)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.221654, size = 452, normalized size = 1.36 \[ -\frac{\sqrt{2}{\left (13 \, \left (b c^{3}\right )^{\frac{1}{4}} B b^{2} - 9 \, \left (b c^{3}\right )^{\frac{1}{4}} A b c\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, c^{5}} - \frac{\sqrt{2}{\left (13 \, \left (b c^{3}\right )^{\frac{1}{4}} B b^{2} - 9 \, \left (b c^{3}\right )^{\frac{1}{4}} A b c\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, c^{5}} - \frac{\sqrt{2}{\left (13 \, \left (b c^{3}\right )^{\frac{1}{4}} B b^{2} - 9 \, \left (b c^{3}\right )^{\frac{1}{4}} A b c\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, c^{5}} + \frac{\sqrt{2}{\left (13 \, \left (b c^{3}\right )^{\frac{1}{4}} B b^{2} - 9 \, \left (b c^{3}\right )^{\frac{1}{4}} A b c\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, c^{5}} + \frac{B b^{3} \sqrt{x} - A b^{2} c \sqrt{x}}{2 \,{\left (c x^{2} + b\right )} c^{4}} + \frac{2 \,{\left (5 \, B c^{16} x^{\frac{9}{2}} - 18 \, B b c^{15} x^{\frac{5}{2}} + 9 \, A c^{16} x^{\frac{5}{2}} + 135 \, B b^{2} c^{14} \sqrt{x} - 90 \, A b c^{15} \sqrt{x}\right )}}{45 \, c^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(19/2)/(c*x^4 + b*x^2)^2,x, algorithm="giac")

[Out]

-1/8*sqrt(2)*(13*(b*c^3)^(1/4)*B*b^2 - 9*(b*c^3)^(1/4)*A*b*c)*arctan(1/2*sqrt(2)
*(sqrt(2)*(b/c)^(1/4) + 2*sqrt(x))/(b/c)^(1/4))/c^5 - 1/8*sqrt(2)*(13*(b*c^3)^(1
/4)*B*b^2 - 9*(b*c^3)^(1/4)*A*b*c)*arctan(-1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) - 2*
sqrt(x))/(b/c)^(1/4))/c^5 - 1/16*sqrt(2)*(13*(b*c^3)^(1/4)*B*b^2 - 9*(b*c^3)^(1/
4)*A*b*c)*ln(sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/c^5 + 1/16*sqrt(2)*(13
*(b*c^3)^(1/4)*B*b^2 - 9*(b*c^3)^(1/4)*A*b*c)*ln(-sqrt(2)*sqrt(x)*(b/c)^(1/4) +
x + sqrt(b/c))/c^5 + 1/2*(B*b^3*sqrt(x) - A*b^2*c*sqrt(x))/((c*x^2 + b)*c^4) + 2
/45*(5*B*c^16*x^(9/2) - 18*B*b*c^15*x^(5/2) + 9*A*c^16*x^(5/2) + 135*B*b^2*c^14*
sqrt(x) - 90*A*b*c^15*sqrt(x))/c^18