3.316 \(\int \frac{x^{13/2}}{b x^2+c x^4} \, dx\)

Optimal. Leaf size=217 \[ \frac{b^{7/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} c^{11/4}}-\frac{b^{7/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} c^{11/4}}-\frac{b^{7/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} c^{11/4}}+\frac{b^{7/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} c^{11/4}}-\frac{2 b x^{3/2}}{3 c^2}+\frac{2 x^{7/2}}{7 c} \]

[Out]

(-2*b*x^(3/2))/(3*c^2) + (2*x^(7/2))/(7*c) - (b^(7/4)*ArcTan[1 - (Sqrt[2]*c^(1/4
)*Sqrt[x])/b^(1/4)])/(Sqrt[2]*c^(11/4)) + (b^(7/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*S
qrt[x])/b^(1/4)])/(Sqrt[2]*c^(11/4)) + (b^(7/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^
(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*c^(11/4)) - (b^(7/4)*Log[Sqrt[b] + Sqrt[2
]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*c^(11/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.449725, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474 \[ \frac{b^{7/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} c^{11/4}}-\frac{b^{7/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} c^{11/4}}-\frac{b^{7/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} c^{11/4}}+\frac{b^{7/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} c^{11/4}}-\frac{2 b x^{3/2}}{3 c^2}+\frac{2 x^{7/2}}{7 c} \]

Antiderivative was successfully verified.

[In]  Int[x^(13/2)/(b*x^2 + c*x^4),x]

[Out]

(-2*b*x^(3/2))/(3*c^2) + (2*x^(7/2))/(7*c) - (b^(7/4)*ArcTan[1 - (Sqrt[2]*c^(1/4
)*Sqrt[x])/b^(1/4)])/(Sqrt[2]*c^(11/4)) + (b^(7/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*S
qrt[x])/b^(1/4)])/(Sqrt[2]*c^(11/4)) + (b^(7/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^
(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*c^(11/4)) - (b^(7/4)*Log[Sqrt[b] + Sqrt[2
]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*c^(11/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 71.3276, size = 206, normalized size = 0.95 \[ \frac{\sqrt{2} b^{\frac{7}{4}} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 c^{\frac{11}{4}}} - \frac{\sqrt{2} b^{\frac{7}{4}} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 c^{\frac{11}{4}}} - \frac{\sqrt{2} b^{\frac{7}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 c^{\frac{11}{4}}} + \frac{\sqrt{2} b^{\frac{7}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 c^{\frac{11}{4}}} - \frac{2 b x^{\frac{3}{2}}}{3 c^{2}} + \frac{2 x^{\frac{7}{2}}}{7 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(13/2)/(c*x**4+b*x**2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.118326, size = 203, normalized size = 0.94 \[ \frac{21 \sqrt{2} b^{7/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-21 \sqrt{2} b^{7/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-42 \sqrt{2} b^{7/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )+42 \sqrt{2} b^{7/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )-56 b c^{3/4} x^{3/2}+24 c^{7/4} x^{7/2}}{84 c^{11/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(13/2)/(b*x^2 + c*x^4),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.039, size = 158, normalized size = 0.7 \[{\frac{2}{7\,c}{x}^{{\frac{7}{2}}}}-{\frac{2\,b}{3\,{c}^{2}}{x}^{{\frac{3}{2}}}}+{\frac{{b}^{2}\sqrt{2}}{4\,{c}^{3}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{{b}^{2}\sqrt{2}}{2\,{c}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{{b}^{2}\sqrt{2}}{2\,{c}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(13/2)/(c*x^4+b*x^2),x)

[Out]

2/7*x^(7/2)/c-2/3*b*x^(3/2)/c^2+1/4*b^2/c^3/(b/c)^(1/4)*2^(1/2)*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)*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)*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(x^(13/2)/(c*x^4 + b*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.282712, size = 223, normalized size = 1.03 \[ \frac{84 \, c^{2} \left (-\frac{b^{7}}{c^{11}}\right )^{\frac{1}{4}} \arctan \left (\frac{c^{8} \left (-\frac{b^{7}}{c^{11}}\right )^{\frac{3}{4}}}{b^{5} \sqrt{x} + \sqrt{-b^{7} c^{5} \sqrt{-\frac{b^{7}}{c^{11}}} + b^{10} x}}\right ) + 21 \, c^{2} \left (-\frac{b^{7}}{c^{11}}\right )^{\frac{1}{4}} \log \left (c^{8} \left (-\frac{b^{7}}{c^{11}}\right )^{\frac{3}{4}} + b^{5} \sqrt{x}\right ) - 21 \, c^{2} \left (-\frac{b^{7}}{c^{11}}\right )^{\frac{1}{4}} \log \left (-c^{8} \left (-\frac{b^{7}}{c^{11}}\right )^{\frac{3}{4}} + b^{5} \sqrt{x}\right ) + 4 \,{\left (3 \, c x^{3} - 7 \, b x\right )} \sqrt{x}}{42 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(13/2)/(c*x^4 + b*x^2),x, algorithm="fricas")

[Out]

1/42*(84*c^2*(-b^7/c^11)^(1/4)*arctan(c^8*(-b^7/c^11)^(3/4)/(b^5*sqrt(x) + sqrt(
-b^7*c^5*sqrt(-b^7/c^11) + b^10*x))) + 21*c^2*(-b^7/c^11)^(1/4)*log(c^8*(-b^7/c^
11)^(3/4) + b^5*sqrt(x)) - 21*c^2*(-b^7/c^11)^(1/4)*log(-c^8*(-b^7/c^11)^(3/4) +
b^5*sqrt(x)) + 4*(3*c*x^3 - 7*b*x)*sqrt(x))/c^2

_______________________________________________________________________________________

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**(13/2)/(c*x**4+b*x**2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.278023, size = 266, normalized size = 1.23 \[ \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} b \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, c^{5}} + \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} b \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, c^{5}} - \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} b{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, c^{5}} + \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} b{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, c^{5}} + \frac{2 \,{\left (3 \, c^{6} x^{\frac{7}{2}} - 7 \, b c^{5} x^{\frac{3}{2}}\right )}}{21 \, c^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(13/2)/(c*x^4 + b*x^2),x, algorithm="giac")

[Out]

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