3.508 \(\int \frac{1}{x \sqrt{a+b x^3} \sqrt{c+d x^3}} \, dx\)

Optimal. Leaf size=48 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^3}}{\sqrt{a} \sqrt{c+d x^3}}\right )}{3 \sqrt{a} \sqrt{c}} \]

[Out]

(-2*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^3])/(Sqrt[a]*Sqrt[c + d*x^3])])/(3*Sqrt[a]*Sqr
t[c])

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Rubi [A]  time = 0.177712, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x^3}}{\sqrt{a} \sqrt{c+d x^3}}\right )}{3 \sqrt{a} \sqrt{c}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]

[Out]

(-2*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^3])/(Sqrt[a]*Sqrt[c + d*x^3])])/(3*Sqrt[a]*Sqr
t[c])

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Rubi in Sympy [A]  time = 13.7643, size = 46, normalized size = 0.96 \[ - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x^{3}}}{\sqrt{a} \sqrt{c + d x^{3}}} \right )}}{3 \sqrt{a} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*atanh(sqrt(c)*sqrt(a + b*x**3)/(sqrt(a)*sqrt(c + d*x**3)))/(3*sqrt(a)*sqrt(c)
)

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Mathematica [C]  time = 0.404513, size = 155, normalized size = 3.23 \[ \frac{4 b d x^3 F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{a}{b x^3},-\frac{c}{d x^3}\right )}{3 \sqrt{a+b x^3} \sqrt{c+d x^3} \left (-4 b d x^3 F_1\left (1;\frac{1}{2},\frac{1}{2};2;-\frac{a}{b x^3},-\frac{c}{d x^3}\right )+b c F_1\left (2;\frac{1}{2},\frac{3}{2};3;-\frac{a}{b x^3},-\frac{c}{d x^3}\right )+a d F_1\left (2;\frac{3}{2},\frac{1}{2};3;-\frac{a}{b x^3},-\frac{c}{d x^3}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(x*Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]

[Out]

(4*b*d*x^3*AppellF1[1, 1/2, 1/2, 2, -(a/(b*x^3)), -(c/(d*x^3))])/(3*Sqrt[a + b*x
^3]*Sqrt[c + d*x^3]*(-4*b*d*x^3*AppellF1[1, 1/2, 1/2, 2, -(a/(b*x^3)), -(c/(d*x^
3))] + b*c*AppellF1[2, 1/2, 3/2, 3, -(a/(b*x^3)), -(c/(d*x^3))] + a*d*AppellF1[2
, 3/2, 1/2, 3, -(a/(b*x^3)), -(c/(d*x^3))]))

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Maple [F]  time = 0.082, size = 0, normalized size = 0. \[ \int{\frac{1}{x}{\frac{1}{\sqrt{b{x}^{3}+a}}}{\frac{1}{\sqrt{d{x}^{3}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^3+a)^(1/2)/(d*x^3+c)^(1/2),x)

[Out]

int(1/x/(b*x^3+a)^(1/2)/(d*x^3+c)^(1/2),x)

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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/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.252921, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (-\frac{4 \,{\left (2 \, a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x^{3}\right )} \sqrt{b x^{3} + a} \sqrt{d x^{3} + c} -{\left ({\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{6} + 8 \, a^{2} c^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x^{3}\right )} \sqrt{a c}}{x^{6}}\right )}{6 \, \sqrt{a c}}, -\frac{\arctan \left (\frac{{\left ({\left (b c + a d\right )} x^{3} + 2 \, a c\right )} \sqrt{-a c}}{2 \, \sqrt{b x^{3} + a} \sqrt{d x^{3} + c} a c}\right )}{3 \, \sqrt{-a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*log(-(4*(2*a^2*c^2 + (a*b*c^2 + a^2*c*d)*x^3)*sqrt(b*x^3 + a)*sqrt(d*x^3 +
c) - ((b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^6 + 8*a^2*c^2 + 8*(a*b*c^2 + a^2*c*d)*x^
3)*sqrt(a*c))/x^6)/sqrt(a*c), -1/3*arctan(1/2*((b*c + a*d)*x^3 + 2*a*c)*sqrt(-a*
c)/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)*a*c))/sqrt(-a*c)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \sqrt{a + b x^{3}} \sqrt{c + d x^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b*x**3+a)**(1/2)/(d*x**3+c)**(1/2),x)

[Out]

Integral(1/(x*sqrt(a + b*x**3)*sqrt(c + d*x**3)), x)

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GIAC/XCAS [A]  time = 0.217419, size = 120, normalized size = 2.5 \[ -\frac{2 \, \sqrt{b d} b \arctan \left (-\frac{b^{2} c + a b d -{\left (\sqrt{b x^{3} + a} \sqrt{b d} - \sqrt{b^{2} c +{\left (b x^{3} + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt{-a b c d} b}\right )}{3 \, \sqrt{-a b c d}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*sqrt(b*d)*b*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*x^3 + a)*sqrt(b*d) - sqrt(
b^2*c + (b*x^3 + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*abs(b))