3.200 \(\int \frac{x^{9/2} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=261 \[ \frac{(A c+3 b B) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{5/4} c^{7/4}}-\frac{(A c+3 b B) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{5/4} c^{7/4}}-\frac{(A c+3 b B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{5/4} c^{7/4}}+\frac{(A c+3 b B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{5/4} c^{7/4}}-\frac{x^{3/2} (b B-A c)}{2 b c \left (b+c x^2\right )} \]

[Out]

-((b*B - A*c)*x^(3/2))/(2*b*c*(b + c*x^2)) - ((3*b*B + A*c)*ArcTan[1 - (Sqrt[2]*
c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(5/4)*c^(7/4)) + ((3*b*B + A*c)*ArcTan[1
+ (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(5/4)*c^(7/4)) + ((3*b*B + A
*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*b^(5/
4)*c^(7/4)) - ((3*b*B + A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqr
t[c]*x])/(8*Sqrt[2]*b^(5/4)*c^(7/4))

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Rubi [A]  time = 0.409988, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346 \[ \frac{(A c+3 b B) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{5/4} c^{7/4}}-\frac{(A c+3 b B) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{5/4} c^{7/4}}-\frac{(A c+3 b B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{5/4} c^{7/4}}+\frac{(A c+3 b B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{5/4} c^{7/4}}-\frac{x^{3/2} (b B-A c)}{2 b c \left (b+c x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(x^(9/2)*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]

[Out]

-((b*B - A*c)*x^(3/2))/(2*b*c*(b + c*x^2)) - ((3*b*B + A*c)*ArcTan[1 - (Sqrt[2]*
c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(5/4)*c^(7/4)) + ((3*b*B + A*c)*ArcTan[1
+ (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(5/4)*c^(7/4)) + ((3*b*B + A
*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*b^(5/
4)*c^(7/4)) - ((3*b*B + A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqr
t[c]*x])/(8*Sqrt[2]*b^(5/4)*c^(7/4))

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Rubi in Sympy [A]  time = 68.5861, size = 240, normalized size = 0.92 \[ \frac{x^{\frac{3}{2}} \left (A c - B b\right )}{2 b c \left (b + c x^{2}\right )} + \frac{\sqrt{2} \left (A c + 3 B b\right ) \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 b^{\frac{5}{4}} c^{\frac{7}{4}}} - \frac{\sqrt{2} \left (A c + 3 B b\right ) \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 b^{\frac{5}{4}} c^{\frac{7}{4}}} - \frac{\sqrt{2} \left (A c + 3 B b\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 b^{\frac{5}{4}} c^{\frac{7}{4}}} + \frac{\sqrt{2} \left (A c + 3 B b\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 b^{\frac{5}{4}} c^{\frac{7}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(9/2)*(B*x**2+A)/(c*x**4+b*x**2)**2,x)

[Out]

x**(3/2)*(A*c - B*b)/(2*b*c*(b + c*x**2)) + sqrt(2)*(A*c + 3*B*b)*log(-sqrt(2)*b
**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(16*b**(5/4)*c**(7/4)) - sqrt(2)
*(A*c + 3*B*b)*log(sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(16*
b**(5/4)*c**(7/4)) - sqrt(2)*(A*c + 3*B*b)*atan(1 - sqrt(2)*c**(1/4)*sqrt(x)/b**
(1/4))/(8*b**(5/4)*c**(7/4)) + sqrt(2)*(A*c + 3*B*b)*atan(1 + sqrt(2)*c**(1/4)*s
qrt(x)/b**(1/4))/(8*b**(5/4)*c**(7/4))

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Mathematica [A]  time = 0.325744, size = 228, normalized size = 0.87 \[ \frac{-\frac{8 \sqrt [4]{b} c^{3/4} x^{3/2} (b B-A c)}{b+c x^2}+\sqrt{2} (A c+3 b B) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-\sqrt{2} (A c+3 b B) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-2 \sqrt{2} (A c+3 b B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )+2 \sqrt{2} (A c+3 b B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{16 b^{5/4} c^{7/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(9/2)*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]

[Out]

((-8*b^(1/4)*c^(3/4)*(b*B - A*c)*x^(3/2))/(b + c*x^2) - 2*Sqrt[2]*(3*b*B + A*c)*
ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 2*Sqrt[2]*(3*b*B + A*c)*ArcTan[1
+ (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + Sqrt[2]*(3*b*B + A*c)*Log[Sqrt[b] - Sqrt
[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - Sqrt[2]*(3*b*B + A*c)*Log[Sqrt[b] + S
qrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(16*b^(5/4)*c^(7/4))

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Maple [A]  time = 0.02, size = 305, normalized size = 1.2 \[{\frac{Ac-Bb}{2\,bc \left ( c{x}^{2}+b \right ) }{x}^{{\frac{3}{2}}}}+{\frac{\sqrt{2}A}{8\,bc}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{\sqrt{2}A}{8\,bc}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{\sqrt{2}A}{16\,bc}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{3\,\sqrt{2}B}{8\,{c}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{3\,\sqrt{2}B}{8\,{c}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{3\,\sqrt{2}B}{16\,{c}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(9/2)*(B*x^2+A)/(c*x^4+b*x^2)^2,x)

[Out]

1/2*(A*c-B*b)/b/c*x^(3/2)/(c*x^2+b)+1/8/b/c/(b/c)^(1/4)*2^(1/2)*A*arctan(2^(1/2)
/(b/c)^(1/4)*x^(1/2)+1)+1/8/b/c/(b/c)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(b/c)^(1/4)
*x^(1/2)-1)+1/16/b/c/(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)))+3/8/c^2/(b/c)^(1/4)*2^(1/
2)*B*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)+3/8/c^2/(b/c)^(1/4)*2^(1/2)*B*arctan(
2^(1/2)/(b/c)^(1/4)*x^(1/2)-1)+3/16/c^2/(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)))

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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^(9/2)/(c*x^4 + b*x^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247992, size = 1072, normalized size = 4.11 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(9/2)/(c*x^4 + b*x^2)^2,x, algorithm="fricas")

[Out]

-1/8*(4*(B*b - A*c)*x^(3/2) - 4*(b*c^2*x^2 + b^2*c)*(-(81*B^4*b^4 + 108*A*B^3*b^
3*c + 54*A^2*B^2*b^2*c^2 + 12*A^3*B*b*c^3 + A^4*c^4)/(b^5*c^7))^(1/4)*arctan(b^4
*c^5*(-(81*B^4*b^4 + 108*A*B^3*b^3*c + 54*A^2*B^2*b^2*c^2 + 12*A^3*B*b*c^3 + A^4
*c^4)/(b^5*c^7))^(3/4)/((27*B^3*b^3 + 27*A*B^2*b^2*c + 9*A^2*B*b*c^2 + A^3*c^3)*
sqrt(x) + sqrt((729*B^6*b^6 + 1458*A*B^5*b^5*c + 1215*A^2*B^4*b^4*c^2 + 540*A^3*
B^3*b^3*c^3 + 135*A^4*B^2*b^2*c^4 + 18*A^5*B*b*c^5 + A^6*c^6)*x - (81*B^4*b^7*c^
3 + 108*A*B^3*b^6*c^4 + 54*A^2*B^2*b^5*c^5 + 12*A^3*B*b^4*c^6 + A^4*b^3*c^7)*sqr
t(-(81*B^4*b^4 + 108*A*B^3*b^3*c + 54*A^2*B^2*b^2*c^2 + 12*A^3*B*b*c^3 + A^4*c^4
)/(b^5*c^7))))) - (b*c^2*x^2 + b^2*c)*(-(81*B^4*b^4 + 108*A*B^3*b^3*c + 54*A^2*B
^2*b^2*c^2 + 12*A^3*B*b*c^3 + A^4*c^4)/(b^5*c^7))^(1/4)*log(b^4*c^5*(-(81*B^4*b^
4 + 108*A*B^3*b^3*c + 54*A^2*B^2*b^2*c^2 + 12*A^3*B*b*c^3 + A^4*c^4)/(b^5*c^7))^
(3/4) + (27*B^3*b^3 + 27*A*B^2*b^2*c + 9*A^2*B*b*c^2 + A^3*c^3)*sqrt(x)) + (b*c^
2*x^2 + b^2*c)*(-(81*B^4*b^4 + 108*A*B^3*b^3*c + 54*A^2*B^2*b^2*c^2 + 12*A^3*B*b
*c^3 + A^4*c^4)/(b^5*c^7))^(1/4)*log(-b^4*c^5*(-(81*B^4*b^4 + 108*A*B^3*b^3*c +
54*A^2*B^2*b^2*c^2 + 12*A^3*B*b*c^3 + A^4*c^4)/(b^5*c^7))^(3/4) + (27*B^3*b^3 +
27*A*B^2*b^2*c + 9*A^2*B*b*c^2 + A^3*c^3)*sqrt(x)))/(b*c^2*x^2 + b^2*c)

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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**(9/2)*(B*x**2+A)/(c*x**4+b*x**2)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.224154, size = 369, normalized size = 1.41 \[ -\frac{B b x^{\frac{3}{2}} - A c x^{\frac{3}{2}}}{2 \,{\left (c x^{2} + b\right )} b c} + \frac{\sqrt{2}{\left (3 \, \left (b c^{3}\right )^{\frac{3}{4}} B b + \left (b c^{3}\right )^{\frac{3}{4}} A c\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, b^{2} c^{4}} + \frac{\sqrt{2}{\left (3 \, \left (b c^{3}\right )^{\frac{3}{4}} B b + \left (b c^{3}\right )^{\frac{3}{4}} A c\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, b^{2} c^{4}} - \frac{\sqrt{2}{\left (3 \, \left (b c^{3}\right )^{\frac{3}{4}} B b + \left (b c^{3}\right )^{\frac{3}{4}} A c\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, b^{2} c^{4}} + \frac{\sqrt{2}{\left (3 \, \left (b c^{3}\right )^{\frac{3}{4}} B b + \left (b c^{3}\right )^{\frac{3}{4}} A c\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, b^{2} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^(9/2)/(c*x^4 + b*x^2)^2,x, algorithm="giac")

[Out]

-1/2*(B*b*x^(3/2) - A*c*x^(3/2))/((c*x^2 + b)*b*c) + 1/8*sqrt(2)*(3*(b*c^3)^(3/4
)*B*b + (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^2*c^4) + 1/8*sqrt(2)*(3*(b*c^3)^(3/4)*B*b + (b*c^3)^(3/4)*A*c)*a
rctan(-1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) - 2*sqrt(x))/(b/c)^(1/4))/(b^2*c^4) - 1/
16*sqrt(2)*(3*(b*c^3)^(3/4)*B*b + (b*c^3)^(3/4)*A*c)*ln(sqrt(2)*sqrt(x)*(b/c)^(1
/4) + x + sqrt(b/c))/(b^2*c^4) + 1/16*sqrt(2)*(3*(b*c^3)^(3/4)*B*b + (b*c^3)^(3/
4)*A*c)*ln(-sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(b^2*c^4)