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

Optimal. Leaf size=93 \[ -\frac{2 c \log \left (c \sqrt{a+b x^3}+d\right )}{3 \left (a c^2-d^2\right )}+\frac{2 d \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 \sqrt{a} \left (a c^2-d^2\right )}+\frac{c \log (x)}{a c^2-d^2} \]

[Out]

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

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Rubi [A]  time = 0.40595, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ -\frac{2 c \log \left (c \sqrt{a+b x^3}+d\right )}{3 \left (a c^2-d^2\right )}+\frac{2 d \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 \sqrt{a} \left (a c^2-d^2\right )}+\frac{c \log (x)}{a c^2-d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.122586, size = 0, normalized size = 0. \[ \int \frac{1}{x \left (a c+b c x^3+d \sqrt{a+b x^3}\right )} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [C]  time = 0.048, size = 636, normalized size = 6.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*ln(x)/(a*c^2-d^2)-1/3*a*c^3/(a*c^2-d^2)/d^2*ln(b*c^2*x^3+a*c^2-d^2)+1/3*c/d^2*
ln(b*c^2*x^3+a*c^2-d^2)-2/3*d/a/(a*c^2-d^2)*(b*x^3+a)^(1/2)+2/3*d*arctanh((b*x^3
+a)^(1/2)/a^(1/2))/(a*c^2-d^2)/a^(1/2)-2/3/a/d*(b*x^3+a)^(1/2)+2/3*c^2/(a*c^2-d^
2)/d*(b*x^3+a)^(1/2)+1/3*I/b^2*c^2/(a*c^2-d^2)/d*2^(1/2)*sum((-a*b^2)^(1/3)*(1/2
*I*b*(2*x+1/b*((-a*b^2)^(1/3)-I*3^(1/2)*(-a*b^2)^(1/3)))/(-a*b^2)^(1/3))^(1/2)*(
b*(x-1/b*(-a*b^2)^(1/3))/(-3*(-a*b^2)^(1/3)+I*3^(1/2)*(-a*b^2)^(1/3)))^(1/2)*(-1
/2*I*b*(2*x+1/b*((-a*b^2)^(1/3)+I*3^(1/2)*(-a*b^2)^(1/3)))/(-a*b^2)^(1/3))^(1/2)
/(b*x^3+a)^(1/2)*(I*(-a*b^2)^(1/3)*3^(1/2)*_alpha*b-I*(-a*b^2)^(2/3)*3^(1/2)+2*_
alpha^2*b^2-(-a*b^2)^(1/3)*_alpha*b-(-a*b^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x
+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^
(1/2),-1/2*c^2/b*(2*I*3^(1/2)*(-a*b^2)^(1/3)*_alpha^2*b-I*3^(1/2)*(-a*b^2)^(2/3)
*_alpha+I*3^(1/2)*a*b-3*(-a*b^2)^(2/3)*_alpha-3*a*b)/d^2,(I*3^(1/2)/b*(-a*b^2)^(
1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)),_alpha=RootO
f(_Z^3*b*c^2+a*c^2-d^2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b c x^{3} + a c + \sqrt{b x^{3} + a} d\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.338577, size = 1, normalized size = 0.01 \[ \left [-\frac{\sqrt{a} c \log \left (\sqrt{b x^{3} + a} c + d\right ) - \sqrt{a} c \log \left (\sqrt{b x^{3} + a} c - d\right ) - d \log \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x^{3} + a} a}{x^{3}}\right ) +{\left (c \log \left (b c^{2} x^{3} + a c^{2} - d^{2}\right ) - 3 \, c \log \left (x\right )\right )} \sqrt{a}}{3 \,{\left (a c^{2} - d^{2}\right )} \sqrt{a}}, -\frac{\sqrt{-a} c \log \left (\sqrt{b x^{3} + a} c + d\right ) - \sqrt{-a} c \log \left (\sqrt{b x^{3} + a} c - d\right ) + 2 \, d \arctan \left (\frac{a}{\sqrt{b x^{3} + a} \sqrt{-a}}\right ) +{\left (c \log \left (b c^{2} x^{3} + a c^{2} - d^{2}\right ) - 3 \, c \log \left (x\right )\right )} \sqrt{-a}}{3 \,{\left (a c^{2} - d^{2}\right )} \sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.27708, size = 127, normalized size = 1.37 \[ -\frac{2 \, c^{2}{\rm ln}\left ({\left | \sqrt{b x^{3} + a} c + d \right |}\right )}{3 \,{\left (a c^{3} - c d^{2}\right )}} + \frac{c{\rm ln}\left (b x^{3}\right )}{3 \,{\left (a c^{2} - d^{2}\right )}} - \frac{2 \, d \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{3 \,{\left (a c^{2} - d^{2}\right )} \sqrt{-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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