3.208 \(\int \frac{x^{21/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=322 \[ -\frac{7 (11 b B-3 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} \sqrt [4]{b} c^{15/4}}+\frac{7 (11 b B-3 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} \sqrt [4]{b} c^{15/4}}+\frac{7 (11 b B-3 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} \sqrt [4]{b} c^{15/4}}-\frac{7 (11 b B-3 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} \sqrt [4]{b} c^{15/4}}+\frac{7 x^{3/2} (11 b B-3 A c)}{48 b c^3}-\frac{x^{7/2} (11 b B-3 A c)}{16 b c^2 \left (b+c x^2\right )}-\frac{x^{11/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2} \]

[Out]

(7*(11*b*B - 3*A*c)*x^(3/2))/(48*b*c^3) - ((b*B - A*c)*x^(11/2))/(4*b*c*(b + c*x
^2)^2) - ((11*b*B - 3*A*c)*x^(7/2))/(16*b*c^2*(b + c*x^2)) + (7*(11*b*B - 3*A*c)
*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*b^(1/4)*c^(15/4)) -
(7*(11*b*B - 3*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*b
^(1/4)*c^(15/4)) - (7*(11*b*B - 3*A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqr
t[x] + Sqrt[c]*x])/(64*Sqrt[2]*b^(1/4)*c^(15/4)) + (7*(11*b*B - 3*A*c)*Log[Sqrt[
b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*b^(1/4)*c^(15/4))

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Rubi [A]  time = 0.535153, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.423 \[ -\frac{7 (11 b B-3 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} \sqrt [4]{b} c^{15/4}}+\frac{7 (11 b B-3 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} \sqrt [4]{b} c^{15/4}}+\frac{7 (11 b B-3 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} \sqrt [4]{b} c^{15/4}}-\frac{7 (11 b B-3 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} \sqrt [4]{b} c^{15/4}}+\frac{7 x^{3/2} (11 b B-3 A c)}{48 b c^3}-\frac{x^{7/2} (11 b B-3 A c)}{16 b c^2 \left (b+c x^2\right )}-\frac{x^{11/2} (b B-A c)}{4 b c \left (b+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^(21/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]

[Out]

(7*(11*b*B - 3*A*c)*x^(3/2))/(48*b*c^3) - ((b*B - A*c)*x^(11/2))/(4*b*c*(b + c*x
^2)^2) - ((11*b*B - 3*A*c)*x^(7/2))/(16*b*c^2*(b + c*x^2)) + (7*(11*b*B - 3*A*c)
*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*b^(1/4)*c^(15/4)) -
(7*(11*b*B - 3*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*b
^(1/4)*c^(15/4)) - (7*(11*b*B - 3*A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqr
t[x] + Sqrt[c]*x])/(64*Sqrt[2]*b^(1/4)*c^(15/4)) + (7*(11*b*B - 3*A*c)*Log[Sqrt[
b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*b^(1/4)*c^(15/4))

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Rubi in Sympy [A]  time = 84.3215, size = 308, normalized size = 0.96 \[ \frac{x^{\frac{11}{2}} \left (A c - B b\right )}{4 b c \left (b + c x^{2}\right )^{2}} + \frac{x^{\frac{7}{2}} \left (3 A c - 11 B b\right )}{16 b c^{2} \left (b + c x^{2}\right )} - \frac{7 x^{\frac{3}{2}} \left (3 A c - 11 B b\right )}{48 b c^{3}} + \frac{7 \sqrt{2} \left (3 A c - 11 B b\right ) \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 \sqrt [4]{b} c^{\frac{15}{4}}} - \frac{7 \sqrt{2} \left (3 A c - 11 B b\right ) \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 \sqrt [4]{b} c^{\frac{15}{4}}} - \frac{7 \sqrt{2} \left (3 A c - 11 B b\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 \sqrt [4]{b} c^{\frac{15}{4}}} + \frac{7 \sqrt{2} \left (3 A c - 11 B b\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 \sqrt [4]{b} c^{\frac{15}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(21/2)*(B*x**2+A)/(c*x**4+b*x**2)**3,x)

[Out]

x**(11/2)*(A*c - B*b)/(4*b*c*(b + c*x**2)**2) + x**(7/2)*(3*A*c - 11*B*b)/(16*b*
c**2*(b + c*x**2)) - 7*x**(3/2)*(3*A*c - 11*B*b)/(48*b*c**3) + 7*sqrt(2)*(3*A*c
- 11*B*b)*log(-sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(128*b**
(1/4)*c**(15/4)) - 7*sqrt(2)*(3*A*c - 11*B*b)*log(sqrt(2)*b**(1/4)*c**(1/4)*sqrt
(x) + sqrt(b) + sqrt(c)*x)/(128*b**(1/4)*c**(15/4)) - 7*sqrt(2)*(3*A*c - 11*B*b)
*atan(1 - sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(64*b**(1/4)*c**(15/4)) + 7*sqrt(2)
*(3*A*c - 11*B*b)*atan(1 + sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(64*b**(1/4)*c**(1
5/4))

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Mathematica [A]  time = 0.5296, size = 287, normalized size = 0.89 \[ \frac{-\frac{24 c^{3/4} x^{3/2} (11 A c-19 b B)}{b+c x^2}+\frac{96 b c^{3/4} x^{3/2} (A c-b B)}{\left (b+c x^2\right )^2}-\frac{21 \sqrt{2} (11 b B-3 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{\sqrt [4]{b}}+\frac{21 \sqrt{2} (11 b B-3 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{\sqrt [4]{b}}+\frac{42 \sqrt{2} (11 b B-3 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt [4]{b}}-\frac{42 \sqrt{2} (11 b B-3 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt [4]{b}}+256 B c^{3/4} x^{3/2}}{384 c^{15/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(21/2)*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]

[Out]

(256*B*c^(3/4)*x^(3/2) + (96*b*c^(3/4)*(-(b*B) + A*c)*x^(3/2))/(b + c*x^2)^2 - (
24*c^(3/4)*(-19*b*B + 11*A*c)*x^(3/2))/(b + c*x^2) + (42*Sqrt[2]*(11*b*B - 3*A*c
)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/b^(1/4) - (42*Sqrt[2]*(11*b*B -
3*A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/b^(1/4) - (21*Sqrt[2]*(11
*b*B - 3*A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/b^(1/4
) + (21*Sqrt[2]*(11*b*B - 3*A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] +
Sqrt[c]*x])/b^(1/4))/(384*c^(15/4))

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Maple [A]  time = 0.027, size = 357, normalized size = 1.1 \[{\frac{2\,B}{3\,{c}^{3}}{x}^{{\frac{3}{2}}}}-{\frac{11\,A}{16\,c \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{7}{2}}}}+{\frac{19\,Bb}{16\,{c}^{2} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{7}{2}}}}-{\frac{7\,Ab}{16\,{c}^{2} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{15\,{b}^{2}B}{16\,{c}^{3} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{21\,\sqrt{2}A}{128\,{c}^{3}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{21\,\sqrt{2}A}{64\,{c}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{21\,\sqrt{2}A}{64\,{c}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{77\,\sqrt{2}Bb}{128\,{c}^{4}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{77\,\sqrt{2}Bb}{64\,{c}^{4}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{77\,\sqrt{2}Bb}{64\,{c}^{4}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(21/2)*(B*x^2+A)/(c*x^4+b*x^2)^3,x)

[Out]

2/3/c^3*B*x^(3/2)-11/16/c/(c*x^2+b)^2*A*x^(7/2)+19/16/c^2/(c*x^2+b)^2*B*x^(7/2)*
b-7/16/c^2/(c*x^2+b)^2*x^(3/2)*A*b+15/16/c^3/(c*x^2+b)^2*x^(3/2)*b^2*B+21/128/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)))+21/64/c^3/(b/c)^(1/4)*2^(1/2)*A*arctan(2^(1/
2)/(b/c)^(1/4)*x^(1/2)+1)+21/64/c^3/(b/c)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(b/c)^(
1/4)*x^(1/2)-1)-77/128/c^4/(b/c)^(1/4)*2^(1/2)*B*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)))-77/64/c^4/(b/c)^(
1/4)*2^(1/2)*B*b*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)-77/64/c^4/(b/c)^(1/4)*2^(
1/2)*B*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)*x^(21/2)/(c*x^4 + b*x^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.250983, size = 1187, normalized size = 3.69 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(21/2)/(c*x^4 + b*x^2)^3,x, algorithm="fricas")

[Out]

1/192*(84*(c^5*x^4 + 2*b*c^4*x^2 + b^2*c^3)*(-(14641*B^4*b^4 - 15972*A*B^3*b^3*c
+ 6534*A^2*B^2*b^2*c^2 - 1188*A^3*B*b*c^3 + 81*A^4*c^4)/(b*c^15))^(1/4)*arctan(
-b*c^11*(-(14641*B^4*b^4 - 15972*A*B^3*b^3*c + 6534*A^2*B^2*b^2*c^2 - 1188*A^3*B
*b*c^3 + 81*A^4*c^4)/(b*c^15))^(3/4)/((1331*B^3*b^3 - 1089*A*B^2*b^2*c + 297*A^2
*B*b*c^2 - 27*A^3*c^3)*sqrt(x) - sqrt((1771561*B^6*b^6 - 2898918*A*B^5*b^5*c + 1
976535*A^2*B^4*b^4*c^2 - 718740*A^3*B^3*b^3*c^3 + 147015*A^4*B^2*b^2*c^4 - 16038
*A^5*B*b*c^5 + 729*A^6*c^6)*x - (14641*B^4*b^5*c^7 - 15972*A*B^3*b^4*c^8 + 6534*
A^2*B^2*b^3*c^9 - 1188*A^3*B*b^2*c^10 + 81*A^4*b*c^11)*sqrt(-(14641*B^4*b^4 - 15
972*A*B^3*b^3*c + 6534*A^2*B^2*b^2*c^2 - 1188*A^3*B*b*c^3 + 81*A^4*c^4)/(b*c^15)
)))) + 21*(c^5*x^4 + 2*b*c^4*x^2 + b^2*c^3)*(-(14641*B^4*b^4 - 15972*A*B^3*b^3*c
+ 6534*A^2*B^2*b^2*c^2 - 1188*A^3*B*b*c^3 + 81*A^4*c^4)/(b*c^15))^(1/4)*log(343
*b*c^11*(-(14641*B^4*b^4 - 15972*A*B^3*b^3*c + 6534*A^2*B^2*b^2*c^2 - 1188*A^3*B
*b*c^3 + 81*A^4*c^4)/(b*c^15))^(3/4) - 343*(1331*B^3*b^3 - 1089*A*B^2*b^2*c + 29
7*A^2*B*b*c^2 - 27*A^3*c^3)*sqrt(x)) - 21*(c^5*x^4 + 2*b*c^4*x^2 + b^2*c^3)*(-(1
4641*B^4*b^4 - 15972*A*B^3*b^3*c + 6534*A^2*B^2*b^2*c^2 - 1188*A^3*B*b*c^3 + 81*
A^4*c^4)/(b*c^15))^(1/4)*log(-343*b*c^11*(-(14641*B^4*b^4 - 15972*A*B^3*b^3*c +
6534*A^2*B^2*b^2*c^2 - 1188*A^3*B*b*c^3 + 81*A^4*c^4)/(b*c^15))^(3/4) - 343*(133
1*B^3*b^3 - 1089*A*B^2*b^2*c + 297*A^2*B*b*c^2 - 27*A^3*c^3)*sqrt(x)) + 4*(32*B*
c^2*x^5 + 11*(11*B*b*c - 3*A*c^2)*x^3 + 7*(11*B*b^2 - 3*A*b*c)*x)*sqrt(x))/(c^5*
x^4 + 2*b*c^4*x^2 + b^2*c^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(x**(21/2)*(B*x**2+A)/(c*x**4+b*x**2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.229003, size = 410, normalized size = 1.27 \[ \frac{2 \, B x^{\frac{3}{2}}}{3 \, c^{3}} + \frac{19 \, B b c x^{\frac{7}{2}} - 11 \, A c^{2} x^{\frac{7}{2}} + 15 \, B b^{2} x^{\frac{3}{2}} - 7 \, A b c x^{\frac{3}{2}}}{16 \,{\left (c x^{2} + b\right )}^{2} c^{3}} - \frac{7 \, \sqrt{2}{\left (11 \, \left (b c^{3}\right )^{\frac{3}{4}} B b - 3 \, \left (b c^{3}\right )^{\frac{3}{4}} A c\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b c^{6}} - \frac{7 \, \sqrt{2}{\left (11 \, \left (b c^{3}\right )^{\frac{3}{4}} B b - 3 \, \left (b c^{3}\right )^{\frac{3}{4}} A c\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b c^{6}} + \frac{7 \, \sqrt{2}{\left (11 \, \left (b c^{3}\right )^{\frac{3}{4}} B b - 3 \, \left (b c^{3}\right )^{\frac{3}{4}} A c\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b c^{6}} - \frac{7 \, \sqrt{2}{\left (11 \, \left (b c^{3}\right )^{\frac{3}{4}} B b - 3 \, \left (b c^{3}\right )^{\frac{3}{4}} A c\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(21/2)/(c*x^4 + b*x^2)^3,x, algorithm="giac")

[Out]

2/3*B*x^(3/2)/c^3 + 1/16*(19*B*b*c*x^(7/2) - 11*A*c^2*x^(7/2) + 15*B*b^2*x^(3/2)
- 7*A*b*c*x^(3/2))/((c*x^2 + b)^2*c^3) - 7/64*sqrt(2)*(11*(b*c^3)^(3/4)*B*b - 3
*(b*c^3)^(3/4)*A*c)*arctan(1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) + 2*sqrt(x))/(b/c)^(
1/4))/(b*c^6) - 7/64*sqrt(2)*(11*(b*c^3)^(3/4)*B*b - 3*(b*c^3)^(3/4)*A*c)*arctan
(-1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) - 2*sqrt(x))/(b/c)^(1/4))/(b*c^6) + 7/128*sqr
t(2)*(11*(b*c^3)^(3/4)*B*b - 3*(b*c^3)^(3/4)*A*c)*ln(sqrt(2)*sqrt(x)*(b/c)^(1/4)
+ x + sqrt(b/c))/(b*c^6) - 7/128*sqrt(2)*(11*(b*c^3)^(3/4)*B*b - 3*(b*c^3)^(3/4
)*A*c)*ln(-sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(b*c^6)