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

Optimal. Leaf size=278 \[ -\frac{c^{7/4} (b B-A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{15/4}}+\frac{c^{7/4} (b B-A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{15/4}}-\frac{c^{7/4} (b B-A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{15/4}}+\frac{c^{7/4} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} b^{15/4}}+\frac{2 c (b B-A c)}{3 b^3 x^{3/2}}-\frac{2 (b B-A c)}{7 b^2 x^{7/2}}-\frac{2 A}{11 b x^{11/2}} \]

[Out]

(-2*A)/(11*b*x^(11/2)) - (2*(b*B - A*c))/(7*b^2*x^(7/2)) + (2*c*(b*B - A*c))/(3*
b^3*x^(3/2)) - (c^(7/4)*(b*B - A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)
])/(Sqrt[2]*b^(15/4)) + (c^(7/4)*(b*B - A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x]
)/b^(1/4)])/(Sqrt[2]*b^(15/4)) - (c^(7/4)*(b*B - A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1
/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*b^(15/4)) + (c^(7/4)*(b*B - A*c)*Lo
g[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*b^(15/4))

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Rubi [A]  time = 0.499605, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ -\frac{c^{7/4} (b B-A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{15/4}}+\frac{c^{7/4} (b B-A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{15/4}}-\frac{c^{7/4} (b B-A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{15/4}}+\frac{c^{7/4} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} b^{15/4}}+\frac{2 c (b B-A c)}{3 b^3 x^{3/2}}-\frac{2 (b B-A c)}{7 b^2 x^{7/2}}-\frac{2 A}{11 b x^{11/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*A)/(11*b*x^(11/2)) - (2*(b*B - A*c))/(7*b^2*x^(7/2)) + (2*c*(b*B - A*c))/(3*
b^3*x^(3/2)) - (c^(7/4)*(b*B - A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)
])/(Sqrt[2]*b^(15/4)) + (c^(7/4)*(b*B - A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x]
)/b^(1/4)])/(Sqrt[2]*b^(15/4)) - (c^(7/4)*(b*B - A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1
/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*b^(15/4)) + (c^(7/4)*(b*B - A*c)*Lo
g[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*b^(15/4))

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Rubi in Sympy [A]  time = 79.2934, size = 262, normalized size = 0.94 \[ - \frac{2 A}{11 b x^{\frac{11}{2}}} + \frac{2 \left (A c - B b\right )}{7 b^{2} x^{\frac{7}{2}}} - \frac{2 c \left (A c - B b\right )}{3 b^{3} x^{\frac{3}{2}}} + \frac{\sqrt{2} c^{\frac{7}{4}} \left (A c - B b\right ) \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 b^{\frac{15}{4}}} - \frac{\sqrt{2} c^{\frac{7}{4}} \left (A c - B b\right ) \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 b^{\frac{15}{4}}} + \frac{\sqrt{2} c^{\frac{7}{4}} \left (A c - B b\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 b^{\frac{15}{4}}} - \frac{\sqrt{2} c^{\frac{7}{4}} \left (A c - B b\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 b^{\frac{15}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.618897, size = 264, normalized size = 0.95 \[ \frac{\frac{264 b^{7/4} (A c-b B)}{x^{7/2}}+\frac{616 b^{3/4} c (b B-A c)}{x^{3/2}}-\frac{168 A b^{11/4}}{x^{11/2}}+231 \sqrt{2} c^{7/4} (A c-b B) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+231 \sqrt{2} c^{7/4} (b B-A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+462 \sqrt{2} c^{7/4} (A c-b B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )+462 \sqrt{2} c^{7/4} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{924 b^{15/4}} \]

Antiderivative was successfully verified.

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

[Out]

((-168*A*b^(11/4))/x^(11/2) + (264*b^(7/4)*(-(b*B) + A*c))/x^(7/2) + (616*b^(3/4
)*c*(b*B - A*c))/x^(3/2) + 462*Sqrt[2]*c^(7/4)*(-(b*B) + A*c)*ArcTan[1 - (Sqrt[2
]*c^(1/4)*Sqrt[x])/b^(1/4)] + 462*Sqrt[2]*c^(7/4)*(b*B - A*c)*ArcTan[1 + (Sqrt[2
]*c^(1/4)*Sqrt[x])/b^(1/4)] + 231*Sqrt[2]*c^(7/4)*(-(b*B) + A*c)*Log[Sqrt[b] - S
qrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 231*Sqrt[2]*c^(7/4)*(b*B - A*c)*Lo
g[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(924*b^(15/4))

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Maple [A]  time = 0.018, size = 336, normalized size = 1.2 \[ -{\frac{2\,A}{11\,b}{x}^{-{\frac{11}{2}}}}+{\frac{2\,Ac}{7\,{b}^{2}}{x}^{-{\frac{7}{2}}}}-{\frac{2\,B}{7\,b}{x}^{-{\frac{7}{2}}}}-{\frac{2\,A{c}^{2}}{3\,{b}^{3}}{x}^{-{\frac{3}{2}}}}+{\frac{2\,Bc}{3\,{b}^{2}}{x}^{-{\frac{3}{2}}}}-{\frac{{c}^{3}\sqrt{2}A}{2\,{b}^{4}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }-{\frac{{c}^{3}\sqrt{2}A}{2\,{b}^{4}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }-{\frac{{c}^{3}\sqrt{2}A}{4\,{b}^{4}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }+{\frac{{c}^{2}\sqrt{2}B}{2\,{b}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }+{\frac{{c}^{2}\sqrt{2}B}{2\,{b}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }+{\frac{{c}^{2}\sqrt{2}B}{4\,{b}^{3}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/11*A/b/x^(11/2)+2/7/b^2/x^(7/2)*A*c-2/7/b/x^(7/2)*B-2/3/b^3*c^2/x^(3/2)*A+2/3
/b^2*c/x^(3/2)*B-1/2*c^3/b^4*(b/c)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(b/c)^(1/4)*x^
(1/2)+1)-1/2*c^3/b^4*(b/c)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)-1)
-1/4*c^3/b^4*(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)))+1/2*c^2/b^3*(b/c)^(1/4)*2^(1/2)*B
*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)+1/2*c^2/b^3*(b/c)^(1/4)*2^(1/2)*B*arctan(
2^(1/2)/(b/c)^(1/4)*x^(1/2)-1)+1/4*c^2/b^3*(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)/((c*x^4 + b*x^2)*x^(9/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242681, size = 824, normalized size = 2.96 \[ \frac{924 \, b^{3} x^{\frac{11}{2}} \left (-\frac{B^{4} b^{4} c^{7} - 4 \, A B^{3} b^{3} c^{8} + 6 \, A^{2} B^{2} b^{2} c^{9} - 4 \, A^{3} B b c^{10} + A^{4} c^{11}}{b^{15}}\right )^{\frac{1}{4}} \arctan \left (-\frac{b^{4} \left (-\frac{B^{4} b^{4} c^{7} - 4 \, A B^{3} b^{3} c^{8} + 6 \, A^{2} B^{2} b^{2} c^{9} - 4 \, A^{3} B b c^{10} + A^{4} c^{11}}{b^{15}}\right )^{\frac{1}{4}}}{{\left (B b c^{2} - A c^{3}\right )} \sqrt{x} - \sqrt{b^{8} \sqrt{-\frac{B^{4} b^{4} c^{7} - 4 \, A B^{3} b^{3} c^{8} + 6 \, A^{2} B^{2} b^{2} c^{9} - 4 \, A^{3} B b c^{10} + A^{4} c^{11}}{b^{15}}} +{\left (B^{2} b^{2} c^{4} - 2 \, A B b c^{5} + A^{2} c^{6}\right )} x}}\right ) - 231 \, b^{3} x^{\frac{11}{2}} \left (-\frac{B^{4} b^{4} c^{7} - 4 \, A B^{3} b^{3} c^{8} + 6 \, A^{2} B^{2} b^{2} c^{9} - 4 \, A^{3} B b c^{10} + A^{4} c^{11}}{b^{15}}\right )^{\frac{1}{4}} \log \left (b^{4} \left (-\frac{B^{4} b^{4} c^{7} - 4 \, A B^{3} b^{3} c^{8} + 6 \, A^{2} B^{2} b^{2} c^{9} - 4 \, A^{3} B b c^{10} + A^{4} c^{11}}{b^{15}}\right )^{\frac{1}{4}} -{\left (B b c^{2} - A c^{3}\right )} \sqrt{x}\right ) + 231 \, b^{3} x^{\frac{11}{2}} \left (-\frac{B^{4} b^{4} c^{7} - 4 \, A B^{3} b^{3} c^{8} + 6 \, A^{2} B^{2} b^{2} c^{9} - 4 \, A^{3} B b c^{10} + A^{4} c^{11}}{b^{15}}\right )^{\frac{1}{4}} \log \left (-b^{4} \left (-\frac{B^{4} b^{4} c^{7} - 4 \, A B^{3} b^{3} c^{8} + 6 \, A^{2} B^{2} b^{2} c^{9} - 4 \, A^{3} B b c^{10} + A^{4} c^{11}}{b^{15}}\right )^{\frac{1}{4}} -{\left (B b c^{2} - A c^{3}\right )} \sqrt{x}\right ) + 308 \,{\left (B b c - A c^{2}\right )} x^{4} - 84 \, A b^{2} - 132 \,{\left (B b^{2} - A b c\right )} x^{2}}{462 \, b^{3} x^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/462*(924*b^3*x^(11/2)*(-(B^4*b^4*c^7 - 4*A*B^3*b^3*c^8 + 6*A^2*B^2*b^2*c^9 - 4
*A^3*B*b*c^10 + A^4*c^11)/b^15)^(1/4)*arctan(-b^4*(-(B^4*b^4*c^7 - 4*A*B^3*b^3*c
^8 + 6*A^2*B^2*b^2*c^9 - 4*A^3*B*b*c^10 + A^4*c^11)/b^15)^(1/4)/((B*b*c^2 - A*c^
3)*sqrt(x) - sqrt(b^8*sqrt(-(B^4*b^4*c^7 - 4*A*B^3*b^3*c^8 + 6*A^2*B^2*b^2*c^9 -
4*A^3*B*b*c^10 + A^4*c^11)/b^15) + (B^2*b^2*c^4 - 2*A*B*b*c^5 + A^2*c^6)*x))) -
231*b^3*x^(11/2)*(-(B^4*b^4*c^7 - 4*A*B^3*b^3*c^8 + 6*A^2*B^2*b^2*c^9 - 4*A^3*B
*b*c^10 + A^4*c^11)/b^15)^(1/4)*log(b^4*(-(B^4*b^4*c^7 - 4*A*B^3*b^3*c^8 + 6*A^2
*B^2*b^2*c^9 - 4*A^3*B*b*c^10 + A^4*c^11)/b^15)^(1/4) - (B*b*c^2 - A*c^3)*sqrt(x
)) + 231*b^3*x^(11/2)*(-(B^4*b^4*c^7 - 4*A*B^3*b^3*c^8 + 6*A^2*B^2*b^2*c^9 - 4*A
^3*B*b*c^10 + A^4*c^11)/b^15)^(1/4)*log(-b^4*(-(B^4*b^4*c^7 - 4*A*B^3*b^3*c^8 +
6*A^2*B^2*b^2*c^9 - 4*A^3*B*b*c^10 + A^4*c^11)/b^15)^(1/4) - (B*b*c^2 - A*c^3)*s
qrt(x)) + 308*(B*b*c - A*c^2)*x^4 - 84*A*b^2 - 132*(B*b^2 - A*b*c)*x^2)/(b^3*x^(
11/2))

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220634, size = 393, normalized size = 1.41 \[ \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{1}{4}} B b c - \left (b c^{3}\right )^{\frac{1}{4}} A c^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, b^{4}} + \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{1}{4}} B b c - \left (b c^{3}\right )^{\frac{1}{4}} A c^{2}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, b^{4}} + \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{1}{4}} B b c - \left (b c^{3}\right )^{\frac{1}{4}} A c^{2}\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{4}} - \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{1}{4}} B b c - \left (b c^{3}\right )^{\frac{1}{4}} A c^{2}\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{4}} + \frac{2 \,{\left (77 \, B b c x^{4} - 77 \, A c^{2} x^{4} - 33 \, B b^{2} x^{2} + 33 \, A b c x^{2} - 21 \, A b^{2}\right )}}{231 \, b^{3} x^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(2)*((b*c^3)^(1/4)*B*b*c - (b*c^3)^(1/4)*A*c^2)*arctan(1/2*sqrt(2)*(sqrt
(2)*(b/c)^(1/4) + 2*sqrt(x))/(b/c)^(1/4))/b^4 + 1/2*sqrt(2)*((b*c^3)^(1/4)*B*b*c
- (b*c^3)^(1/4)*A*c^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) - 2*sqrt(x))/(b
/c)^(1/4))/b^4 + 1/4*sqrt(2)*((b*c^3)^(1/4)*B*b*c - (b*c^3)^(1/4)*A*c^2)*ln(sqrt
(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/b^4 - 1/4*sqrt(2)*((b*c^3)^(1/4)*B*b*c
- (b*c^3)^(1/4)*A*c^2)*ln(-sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/b^4 + 2/
231*(77*B*b*c*x^4 - 77*A*c^2*x^4 - 33*B*b^2*x^2 + 33*A*b*c*x^2 - 21*A*b^2)/(b^3*
x^(11/2))