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

Optimal. Leaf size=257 \[ \frac{b^{3/4} (b B-A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} c^{11/4}}-\frac{b^{3/4} (b B-A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} c^{11/4}}-\frac{b^{3/4} (b B-A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} c^{11/4}}+\frac{b^{3/4} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} c^{11/4}}-\frac{2 x^{3/2} (b B-A c)}{3 c^2}+\frac{2 B x^{7/2}}{7 c} \]

[Out]

(-2*(b*B - A*c)*x^(3/2))/(3*c^2) + (2*B*x^(7/2))/(7*c) - (b^(3/4)*(b*B - A*c)*Ar
cTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(Sqrt[2]*c^(11/4)) + (b^(3/4)*(b*B
- A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(Sqrt[2]*c^(11/4)) + (b^(3
/4)*(b*B - A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*S
qrt[2]*c^(11/4)) - (b^(3/4)*(b*B - A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sq
rt[x] + Sqrt[c]*x])/(2*Sqrt[2]*c^(11/4))

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Rubi [A]  time = 0.434281, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ \frac{b^{3/4} (b B-A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} c^{11/4}}-\frac{b^{3/4} (b B-A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} c^{11/4}}-\frac{b^{3/4} (b B-A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} c^{11/4}}+\frac{b^{3/4} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} c^{11/4}}-\frac{2 x^{3/2} (b B-A c)}{3 c^2}+\frac{2 B x^{7/2}}{7 c} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*B - A*c)*x^(3/2))/(3*c^2) + (2*B*x^(7/2))/(7*c) - (b^(3/4)*(b*B - A*c)*Ar
cTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(Sqrt[2]*c^(11/4)) + (b^(3/4)*(b*B
- A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(Sqrt[2]*c^(11/4)) + (b^(3
/4)*(b*B - A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*S
qrt[2]*c^(11/4)) - (b^(3/4)*(b*B - A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sq
rt[x] + Sqrt[c]*x])/(2*Sqrt[2]*c^(11/4))

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Rubi in Sympy [A]  time = 72.5311, size = 240, normalized size = 0.93 \[ \frac{2 B x^{\frac{7}{2}}}{7 c} - \frac{\sqrt{2} b^{\frac{3}{4}} \left (A c - B b\right ) \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 c^{\frac{11}{4}}} + \frac{\sqrt{2} b^{\frac{3}{4}} \left (A c - B b\right ) \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 c^{\frac{11}{4}}} + \frac{\sqrt{2} b^{\frac{3}{4}} \left (A c - B b\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 c^{\frac{11}{4}}} - \frac{\sqrt{2} b^{\frac{3}{4}} \left (A c - B b\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 c^{\frac{11}{4}}} + \frac{2 x^{\frac{3}{2}} \left (A c - B b\right )}{3 c^{2}} \]

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),x)

[Out]

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

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Mathematica [A]  time = 0.338133, size = 243, normalized size = 0.95 \[ \frac{21 \sqrt{2} b^{3/4} (b B-A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-21 \sqrt{2} b^{3/4} (b B-A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-42 \sqrt{2} b^{3/4} (b B-A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )+42 \sqrt{2} b^{3/4} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )+56 c^{3/4} x^{3/2} (A c-b B)+24 B c^{7/4} x^{7/2}}{84 c^{11/4}} \]

Antiderivative was successfully verified.

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

[Out]

(56*c^(3/4)*(-(b*B) + A*c)*x^(3/2) + 24*B*c^(7/4)*x^(7/2) - 42*Sqrt[2]*b^(3/4)*(
b*B - A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 42*Sqrt[2]*b^(3/4)*(b
*B - A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 21*Sqrt[2]*b^(3/4)*(b*
B - A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - 21*Sqrt[2]
*b^(3/4)*(b*B - A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])
/(84*c^(11/4))

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Maple [A]  time = 0.014, size = 308, normalized size = 1.2 \[{\frac{2\,B}{7\,c}{x}^{{\frac{7}{2}}}}+{\frac{2\,A}{3\,c}{x}^{{\frac{3}{2}}}}-{\frac{2\,Bb}{3\,{c}^{2}}{x}^{{\frac{3}{2}}}}-{\frac{b\sqrt{2}A}{2\,{c}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{b\sqrt{2}A}{2\,{c}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{b\sqrt{2}A}{4\,{c}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{{b}^{2}\sqrt{2}B}{2\,{c}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{{b}^{2}\sqrt{2}B}{2\,{c}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{{b}^{2}\sqrt{2}B}{4\,{c}^{3}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}} \]

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),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.257279, size = 1054, normalized size = 4.1 \[ -\frac{84 \, c^{2} \left (-\frac{B^{4} b^{7} - 4 \, A B^{3} b^{6} c + 6 \, A^{2} B^{2} b^{5} c^{2} - 4 \, A^{3} B b^{4} c^{3} + A^{4} b^{3} c^{4}}{c^{11}}\right )^{\frac{1}{4}} \arctan \left (-\frac{c^{8} \left (-\frac{B^{4} b^{7} - 4 \, A B^{3} b^{6} c + 6 \, A^{2} B^{2} b^{5} c^{2} - 4 \, A^{3} B b^{4} c^{3} + A^{4} b^{3} c^{4}}{c^{11}}\right )^{\frac{3}{4}}}{{\left (B^{3} b^{5} - 3 \, A B^{2} b^{4} c + 3 \, A^{2} B b^{3} c^{2} - A^{3} b^{2} c^{3}\right )} \sqrt{x} - \sqrt{{\left (B^{6} b^{10} - 6 \, A B^{5} b^{9} c + 15 \, A^{2} B^{4} b^{8} c^{2} - 20 \, A^{3} B^{3} b^{7} c^{3} + 15 \, A^{4} B^{2} b^{6} c^{4} - 6 \, A^{5} B b^{5} c^{5} + A^{6} b^{4} c^{6}\right )} x -{\left (B^{4} b^{7} c^{5} - 4 \, A B^{3} b^{6} c^{6} + 6 \, A^{2} B^{2} b^{5} c^{7} - 4 \, A^{3} B b^{4} c^{8} + A^{4} b^{3} c^{9}\right )} \sqrt{-\frac{B^{4} b^{7} - 4 \, A B^{3} b^{6} c + 6 \, A^{2} B^{2} b^{5} c^{2} - 4 \, A^{3} B b^{4} c^{3} + A^{4} b^{3} c^{4}}{c^{11}}}}}\right ) + 21 \, c^{2} \left (-\frac{B^{4} b^{7} - 4 \, A B^{3} b^{6} c + 6 \, A^{2} B^{2} b^{5} c^{2} - 4 \, A^{3} B b^{4} c^{3} + A^{4} b^{3} c^{4}}{c^{11}}\right )^{\frac{1}{4}} \log \left (c^{8} \left (-\frac{B^{4} b^{7} - 4 \, A B^{3} b^{6} c + 6 \, A^{2} B^{2} b^{5} c^{2} - 4 \, A^{3} B b^{4} c^{3} + A^{4} b^{3} c^{4}}{c^{11}}\right )^{\frac{3}{4}} -{\left (B^{3} b^{5} - 3 \, A B^{2} b^{4} c + 3 \, A^{2} B b^{3} c^{2} - A^{3} b^{2} c^{3}\right )} \sqrt{x}\right ) - 21 \, c^{2} \left (-\frac{B^{4} b^{7} - 4 \, A B^{3} b^{6} c + 6 \, A^{2} B^{2} b^{5} c^{2} - 4 \, A^{3} B b^{4} c^{3} + A^{4} b^{3} c^{4}}{c^{11}}\right )^{\frac{1}{4}} \log \left (-c^{8} \left (-\frac{B^{4} b^{7} - 4 \, A B^{3} b^{6} c + 6 \, A^{2} B^{2} b^{5} c^{2} - 4 \, A^{3} B b^{4} c^{3} + A^{4} b^{3} c^{4}}{c^{11}}\right )^{\frac{3}{4}} -{\left (B^{3} b^{5} - 3 \, A B^{2} b^{4} c + 3 \, A^{2} B b^{3} c^{2} - A^{3} b^{2} c^{3}\right )} \sqrt{x}\right ) - 4 \,{\left (3 \, B c x^{3} - 7 \,{\left (B b - A c\right )} x\right )} \sqrt{x}}{42 \, c^{2}} \]

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, algorithm="fricas")

[Out]

-1/42*(84*c^2*(-(B^4*b^7 - 4*A*B^3*b^6*c + 6*A^2*B^2*b^5*c^2 - 4*A^3*B*b^4*c^3 +
A^4*b^3*c^4)/c^11)^(1/4)*arctan(-c^8*(-(B^4*b^7 - 4*A*B^3*b^6*c + 6*A^2*B^2*b^5
*c^2 - 4*A^3*B*b^4*c^3 + A^4*b^3*c^4)/c^11)^(3/4)/((B^3*b^5 - 3*A*B^2*b^4*c + 3*
A^2*B*b^3*c^2 - A^3*b^2*c^3)*sqrt(x) - sqrt((B^6*b^10 - 6*A*B^5*b^9*c + 15*A^2*B
^4*b^8*c^2 - 20*A^3*B^3*b^7*c^3 + 15*A^4*B^2*b^6*c^4 - 6*A^5*B*b^5*c^5 + A^6*b^4
*c^6)*x - (B^4*b^7*c^5 - 4*A*B^3*b^6*c^6 + 6*A^2*B^2*b^5*c^7 - 4*A^3*B*b^4*c^8 +
A^4*b^3*c^9)*sqrt(-(B^4*b^7 - 4*A*B^3*b^6*c + 6*A^2*B^2*b^5*c^2 - 4*A^3*B*b^4*c
^3 + A^4*b^3*c^4)/c^11)))) + 21*c^2*(-(B^4*b^7 - 4*A*B^3*b^6*c + 6*A^2*B^2*b^5*c
^2 - 4*A^3*B*b^4*c^3 + A^4*b^3*c^4)/c^11)^(1/4)*log(c^8*(-(B^4*b^7 - 4*A*B^3*b^6
*c + 6*A^2*B^2*b^5*c^2 - 4*A^3*B*b^4*c^3 + A^4*b^3*c^4)/c^11)^(3/4) - (B^3*b^5 -
3*A*B^2*b^4*c + 3*A^2*B*b^3*c^2 - A^3*b^2*c^3)*sqrt(x)) - 21*c^2*(-(B^4*b^7 - 4
*A*B^3*b^6*c + 6*A^2*B^2*b^5*c^2 - 4*A^3*B*b^4*c^3 + A^4*b^3*c^4)/c^11)^(1/4)*lo
g(-c^8*(-(B^4*b^7 - 4*A*B^3*b^6*c + 6*A^2*B^2*b^5*c^2 - 4*A^3*B*b^4*c^3 + A^4*b^
3*c^4)/c^11)^(3/4) - (B^3*b^5 - 3*A*B^2*b^4*c + 3*A^2*B*b^3*c^2 - A^3*b^2*c^3)*s
qrt(x)) - 4*(3*B*c*x^3 - 7*(B*b - A*c)*x)*sqrt(x))/c^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(x**(9/2)*(B*x**2+A)/(c*x**4+b*x**2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.223348, size = 356, normalized size = 1.39 \[ \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{3}{4}} B b - \left (b c^{3}\right )^{\frac{3}{4}} A c\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, c^{5}} + \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{3}{4}} B b - \left (b c^{3}\right )^{\frac{3}{4}} A c\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, c^{5}} - \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{3}{4}} B b - \left (b c^{3}\right )^{\frac{3}{4}} A c\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, c^{5}} + \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{3}{4}} B b - \left (b c^{3}\right )^{\frac{3}{4}} A c\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, c^{5}} + \frac{2 \,{\left (3 \, B c^{6} x^{\frac{7}{2}} - 7 \, B b c^{5} x^{\frac{3}{2}} + 7 \, A c^{6} x^{\frac{3}{2}}\right )}}{21 \, c^{7}} \]

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, algorithm="giac")

[Out]

1/2*sqrt(2)*((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))/c^5 + 1/2*sqrt(2)*((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
))/c^5 - 1/4*sqrt(2)*((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))/c^5 + 1/4*sqrt(2)*((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))/c^5 + 2/21*(3*B*c^6*x^(7
/2) - 7*B*b*c^5*x^(3/2) + 7*A*c^6*x^(3/2))/c^7