3.323 \(\int \frac{1}{\sqrt{x} \left (b x^2+c x^4\right )} \, dx\)

Optimal. Leaf size=204 \[ \frac{c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{7/4}}-\frac{c^{3/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{7/4}}+\frac{c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{7/4}}-\frac{c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} b^{7/4}}-\frac{2}{3 b x^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.340984, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474 \[ \frac{c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{7/4}}-\frac{c^{3/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{7/4}}+\frac{c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{7/4}}-\frac{c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} b^{7/4}}-\frac{2}{3 b x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 64.6819, size = 192, normalized size = 0.94 \[ - \frac{2}{3 b x^{\frac{3}{2}}} + \frac{\sqrt{2} c^{\frac{3}{4}} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 b^{\frac{7}{4}}} - \frac{\sqrt{2} c^{\frac{3}{4}} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 b^{\frac{7}{4}}} + \frac{\sqrt{2} c^{\frac{3}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 b^{\frac{7}{4}}} - \frac{\sqrt{2} c^{\frac{3}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 b^{\frac{7}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.116057, size = 190, normalized size = 0.93 \[ \frac{-\frac{8 b^{3/4}}{x^{3/2}}+3 \sqrt{2} c^{3/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-3 \sqrt{2} c^{3/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+6 \sqrt{2} c^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )-6 \sqrt{2} c^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{12 b^{7/4}} \]

Antiderivative was successfully verified.

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

[Out]

((-8*b^(3/4))/x^(3/2) + 6*Sqrt[2]*c^(3/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b
^(1/4)] - 6*Sqrt[2]*c^(3/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 3*Sq
rt[2]*c^(3/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - 3*Sqr
t[2]*c^(3/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(12*b^(
7/4))

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Maple [A]  time = 0.012, size = 143, normalized size = 0.7 \[ -{\frac{c\sqrt{2}}{4\,{b}^{2}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }-{\frac{c\sqrt{2}}{2\,{b}^{2}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }-{\frac{c\sqrt{2}}{2\,{b}^{2}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }-{\frac{2}{3\,b}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^4+b*x^2)/x^(1/2),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)-2/3/b/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)*sqrt(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.282166, size = 197, normalized size = 0.97 \[ \frac{12 \, b x^{\frac{3}{2}} \left (-\frac{c^{3}}{b^{7}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{2} \left (-\frac{c^{3}}{b^{7}}\right )^{\frac{1}{4}}}{c \sqrt{x} + \sqrt{b^{4} \sqrt{-\frac{c^{3}}{b^{7}}} + c^{2} x}}\right ) - 3 \, b x^{\frac{3}{2}} \left (-\frac{c^{3}}{b^{7}}\right )^{\frac{1}{4}} \log \left (b^{2} \left (-\frac{c^{3}}{b^{7}}\right )^{\frac{1}{4}} + c \sqrt{x}\right ) + 3 \, b x^{\frac{3}{2}} \left (-\frac{c^{3}}{b^{7}}\right )^{\frac{1}{4}} \log \left (-b^{2} \left (-\frac{c^{3}}{b^{7}}\right )^{\frac{1}{4}} + c \sqrt{x}\right ) - 4}{6 \, b x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 140.023, size = 184, normalized size = 0.9 \[ \begin{cases} \frac{\tilde{\infty }}{x^{\frac{7}{2}}} & \text{for}\: b = 0 \wedge c = 0 \\- \frac{2}{7 c x^{\frac{7}{2}}} & \text{for}\: b = 0 \\- \frac{2}{3 b x^{\frac{3}{2}}} & \text{for}\: c = 0 \\- \frac{2}{3 b x^{\frac{3}{2}}} - \frac{\sqrt [4]{-1} \log{\left (\sqrt [4]{-1} \sqrt [4]{b} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 b^{\frac{7}{4}} c^{3} \left (\frac{1}{c}\right )^{\frac{15}{4}}} + \frac{\sqrt [4]{-1} \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} \sqrt{x}}{\sqrt [4]{b} \sqrt [4]{\frac{1}{c}}} \right )}}{b^{\frac{7}{4}} c^{3} \left (\frac{1}{c}\right )^{\frac{15}{4}}} + \frac{\sqrt [4]{-1} \log{\left (- \sqrt [4]{-1} \sqrt [4]{b} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 b^{\frac{7}{4}} c^{41} \left (\frac{1}{c}\right )^{\frac{167}{4}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((zoo/x**(7/2), Eq(b, 0) & Eq(c, 0)), (-2/(7*c*x**(7/2)), Eq(b, 0)), (-
2/(3*b*x**(3/2)), Eq(c, 0)), (-2/(3*b*x**(3/2)) - (-1)**(1/4)*log((-1)**(1/4)*b*
*(1/4)*(1/c)**(1/4) + sqrt(x))/(2*b**(7/4)*c**3*(1/c)**(15/4)) + (-1)**(1/4)*ata
n((-1)**(3/4)*sqrt(x)/(b**(1/4)*(1/c)**(1/4)))/(b**(7/4)*c**3*(1/c)**(15/4)) + (
-1)**(1/4)*log(-(-1)**(1/4)*b**(1/4)*(1/c)**(1/4) + sqrt(x))/(2*b**(7/4)*c**41*(
1/c)**(167/4)), True))

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GIAC/XCAS [A]  time = 0.273638, size = 240, normalized size = 1.18 \[ -\frac{\sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, b^{2}} - \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, b^{2}} - \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{2}} + \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{2}} - \frac{2}{3 \, b x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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