3.325 \(\int \frac{1}{x^{5/2} \left (b x^2+c x^4\right )} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.387007, 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 \[ -\frac{c^{7/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{11/4}}+\frac{c^{7/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{11/4}}-\frac{c^{7/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{11/4}}+\frac{c^{7/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} b^{11/4}}+\frac{2 c}{3 b^2 x^{3/2}}-\frac{2}{7 b x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(5/2)*(b*x^2 + c*x^4)),x]

[Out]

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

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Rubi in Sympy [A]  time = 71.9281, size = 206, normalized size = 0.95 \[ - \frac{2}{7 b x^{\frac{7}{2}}} + \frac{2 c}{3 b^{2} x^{\frac{3}{2}}} - \frac{\sqrt{2} c^{\frac{7}{4}} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 b^{\frac{11}{4}}} + \frac{\sqrt{2} c^{\frac{7}{4}} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 b^{\frac{11}{4}}} - \frac{\sqrt{2} c^{\frac{7}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 b^{\frac{11}{4}}} + \frac{\sqrt{2} c^{\frac{7}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 b^{\frac{11}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(5/2)/(c*x**4+b*x**2),x)

[Out]

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

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Mathematica [A]  time = 0.0965798, size = 221, normalized size = 1.02 \[ \frac{56 b^{3/4} c x^2-24 b^{7/4}-21 \sqrt{2} c^{7/4} x^{7/2} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+21 \sqrt{2} c^{7/4} x^{7/2} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-42 \sqrt{2} c^{7/4} x^{7/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )+42 \sqrt{2} c^{7/4} x^{7/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{84 b^{11/4} x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(5/2)*(b*x^2 + c*x^4)),x]

[Out]

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

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Maple [A]  time = 0.014, size = 158, normalized size = 0.7 \[{\frac{{c}^{2}\sqrt{2}}{4\,{b}^{3}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }+{\frac{{c}^{2}\sqrt{2}}{2\,{b}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }+{\frac{{c}^{2}\sqrt{2}}{2\,{b}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }-{\frac{2}{7\,b}{x}^{-{\frac{7}{2}}}}+{\frac{2\,c}{3\,{b}^{2}}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(5/2)/(c*x^4+b*x^2),x)

[Out]

1/4*c^2/b^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*c^2/b^3*(b/c)^(1/4)*2^(1/2)*arct
an(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)+1/2*c^2/b^3*(b/c)^(1/4)*2^(1/2)*arctan(2^(1/2)
/(b/c)^(1/4)*x^(1/2)-1)-2/7/b/x^(7/2)+2/3*c/b^2/x^(3/2)

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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(1/((c*x^4 + b*x^2)*x^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.293846, size = 224, normalized size = 1.03 \[ -\frac{84 \, b^{2} x^{\frac{7}{2}} \left (-\frac{c^{7}}{b^{11}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{3} \left (-\frac{c^{7}}{b^{11}}\right )^{\frac{1}{4}}}{c^{2} \sqrt{x} + \sqrt{b^{6} \sqrt{-\frac{c^{7}}{b^{11}}} + c^{4} x}}\right ) - 21 \, b^{2} x^{\frac{7}{2}} \left (-\frac{c^{7}}{b^{11}}\right )^{\frac{1}{4}} \log \left (b^{3} \left (-\frac{c^{7}}{b^{11}}\right )^{\frac{1}{4}} + c^{2} \sqrt{x}\right ) + 21 \, b^{2} x^{\frac{7}{2}} \left (-\frac{c^{7}}{b^{11}}\right )^{\frac{1}{4}} \log \left (-b^{3} \left (-\frac{c^{7}}{b^{11}}\right )^{\frac{1}{4}} + c^{2} \sqrt{x}\right ) - 28 \, c x^{2} + 12 \, b}{42 \, b^{2} x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2)*x^(5/2)),x, algorithm="fricas")

[Out]

-1/42*(84*b^2*x^(7/2)*(-c^7/b^11)^(1/4)*arctan(b^3*(-c^7/b^11)^(1/4)/(c^2*sqrt(x
) + sqrt(b^6*sqrt(-c^7/b^11) + c^4*x))) - 21*b^2*x^(7/2)*(-c^7/b^11)^(1/4)*log(b
^3*(-c^7/b^11)^(1/4) + c^2*sqrt(x)) + 21*b^2*x^(7/2)*(-c^7/b^11)^(1/4)*log(-b^3*
(-c^7/b^11)^(1/4) + c^2*sqrt(x)) - 28*c*x^2 + 12*b)/(b^2*x^(7/2))

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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(1/x**(5/2)/(c*x**4+b*x**2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.274811, size = 259, normalized size = 1.19 \[ \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} c \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, b^{3}} + \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} c \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, b^{3}} + \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} c{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{3}} - \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} c{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{3}} + \frac{2 \,{\left (7 \, c x^{2} - 3 \, b\right )}}{21 \, b^{2} x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2)*x^(5/2)),x, algorithm="giac")

[Out]

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