3.324 \(\int \frac{1}{x^{3/2} \left (b x^2+c x^4\right )} \, dx\)

Optimal. Leaf size=215 \[ \frac{c^{5/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{9/4}}-\frac{c^{5/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{9/4}}-\frac{c^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{9/4}}+\frac{c^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} b^{9/4}}+\frac{2 c}{b^2 \sqrt{x}}-\frac{2}{5 b x^{5/2}} \]

[Out]

-2/(5*b*x^(5/2)) + (2*c)/(b^2*Sqrt[x]) - (c^(5/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sq
rt[x])/b^(1/4)])/(Sqrt[2]*b^(9/4)) + (c^(5/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x
])/b^(1/4)])/(Sqrt[2]*b^(9/4)) + (c^(5/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*
Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*b^(9/4)) - (c^(5/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/
4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*b^(9/4))

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Rubi [A]  time = 0.392114, antiderivative size = 215, 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 \[ \frac{c^{5/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{9/4}}-\frac{c^{5/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{9/4}}-\frac{c^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{9/4}}+\frac{c^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} b^{9/4}}+\frac{2 c}{b^2 \sqrt{x}}-\frac{2}{5 b x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(3/2)*(b*x^2 + c*x^4)),x]

[Out]

-2/(5*b*x^(5/2)) + (2*c)/(b^2*Sqrt[x]) - (c^(5/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sq
rt[x])/b^(1/4)])/(Sqrt[2]*b^(9/4)) + (c^(5/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x
])/b^(1/4)])/(Sqrt[2]*b^(9/4)) + (c^(5/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*
Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*b^(9/4)) - (c^(5/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/
4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*b^(9/4))

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Rubi in Sympy [A]  time = 71.7935, size = 204, normalized size = 0.95 \[ - \frac{2}{5 b x^{\frac{5}{2}}} + \frac{2 c}{b^{2} \sqrt{x}} + \frac{\sqrt{2} c^{\frac{5}{4}} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 b^{\frac{9}{4}}} - \frac{\sqrt{2} c^{\frac{5}{4}} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 b^{\frac{9}{4}}} - \frac{\sqrt{2} c^{\frac{5}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 b^{\frac{9}{4}}} + \frac{\sqrt{2} c^{\frac{5}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 b^{\frac{9}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**(3/2)/(c*x**4+b*x**2),x)

[Out]

-2/(5*b*x**(5/2)) + 2*c/(b**2*sqrt(x)) + sqrt(2)*c**(5/4)*log(-sqrt(2)*b**(1/4)*
c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(4*b**(9/4)) - sqrt(2)*c**(5/4)*log(sqrt
(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(4*b**(9/4)) - sqrt(2)*c**(
5/4)*atan(1 - sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(2*b**(9/4)) + sqrt(2)*c**(5/4)
*atan(1 + sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(2*b**(9/4))

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Mathematica [A]  time = 0.173897, size = 203, normalized size = 0.94 \[ \frac{-\frac{8 b^{5/4}}{x^{5/2}}+5 \sqrt{2} c^{5/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-5 \sqrt{2} c^{5/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-10 \sqrt{2} c^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )+10 \sqrt{2} c^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )+\frac{40 \sqrt [4]{b} c}{\sqrt{x}}}{20 b^{9/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(3/2)*(b*x^2 + c*x^4)),x]

[Out]

((-8*b^(5/4))/x^(5/2) + (40*b^(1/4)*c)/Sqrt[x] - 10*Sqrt[2]*c^(5/4)*ArcTan[1 - (
Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 10*Sqrt[2]*c^(5/4)*ArcTan[1 + (Sqrt[2]*c^(1/
4)*Sqrt[x])/b^(1/4)] + 5*Sqrt[2]*c^(5/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*S
qrt[x] + Sqrt[c]*x] - 5*Sqrt[2]*c^(5/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sq
rt[x] + Sqrt[c]*x])/(20*b^(9/4))

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Maple [A]  time = 0.015, size = 152, normalized size = 0.7 \[{\frac{c\sqrt{2}}{4\,{b}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{c\sqrt{2}}{2\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{c\sqrt{2}}{2\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{2}{5\,b}{x}^{-{\frac{5}{2}}}}+2\,{\frac{c}{{b}^{2}\sqrt{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(3/2)/(c*x^4+b*x^2),x)

[Out]

1/4*c/b^2/(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/b^2/(b/c)^(1/4)*2^(1/2)*arctan(2
^(1/2)/(b/c)^(1/4)*x^(1/2)+1)+1/2*c/b^2/(b/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(b/c)
^(1/4)*x^(1/2)-1)-2/5/b/x^(5/2)+2*c/b^2/x^(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(1/((c*x^4 + b*x^2)*x^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.285196, size = 230, normalized size = 1.07 \[ \frac{20 \, b^{2} x^{\frac{5}{2}} \left (-\frac{c^{5}}{b^{9}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{7} \left (-\frac{c^{5}}{b^{9}}\right )^{\frac{3}{4}}}{c^{4} \sqrt{x} + \sqrt{-b^{5} c^{5} \sqrt{-\frac{c^{5}}{b^{9}}} + c^{8} x}}\right ) + 5 \, b^{2} x^{\frac{5}{2}} \left (-\frac{c^{5}}{b^{9}}\right )^{\frac{1}{4}} \log \left (b^{7} \left (-\frac{c^{5}}{b^{9}}\right )^{\frac{3}{4}} + c^{4} \sqrt{x}\right ) - 5 \, b^{2} x^{\frac{5}{2}} \left (-\frac{c^{5}}{b^{9}}\right )^{\frac{1}{4}} \log \left (-b^{7} \left (-\frac{c^{5}}{b^{9}}\right )^{\frac{3}{4}} + c^{4} \sqrt{x}\right ) + 20 \, c x^{2} - 4 \, b}{10 \, b^{2} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2)*x^(3/2)),x, algorithm="fricas")

[Out]

1/10*(20*b^2*x^(5/2)*(-c^5/b^9)^(1/4)*arctan(b^7*(-c^5/b^9)^(3/4)/(c^4*sqrt(x) +
sqrt(-b^5*c^5*sqrt(-c^5/b^9) + c^8*x))) + 5*b^2*x^(5/2)*(-c^5/b^9)^(1/4)*log(b^
7*(-c^5/b^9)^(3/4) + c^4*sqrt(x)) - 5*b^2*x^(5/2)*(-c^5/b^9)^(1/4)*log(-b^7*(-c^
5/b^9)^(3/4) + c^4*sqrt(x)) + 20*c*x^2 - 4*b)/(b^2*x^(5/2))

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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**(3/2)/(c*x**4+b*x**2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.275866, size = 270, normalized size = 1.26 \[ \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, b^{3} c} + \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, b^{3} c} - \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{3} c} + \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{3} c} + \frac{2 \,{\left (5 \, c x^{2} - b\right )}}{5 \, b^{2} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2)*x^(3/2)),x, algorithm="giac")

[Out]

1/2*sqrt(2)*(b*c^3)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) + 2*sqrt(x))/(
b/c)^(1/4))/(b^3*c) + 1/2*sqrt(2)*(b*c^3)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(b/
c)^(1/4) - 2*sqrt(x))/(b/c)^(1/4))/(b^3*c) - 1/4*sqrt(2)*(b*c^3)^(3/4)*ln(sqrt(2
)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(b^3*c) + 1/4*sqrt(2)*(b*c^3)^(3/4)*ln(-s
qrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(b^3*c) + 2/5*(5*c*x^2 - b)/(b^2*x^(
5/2))