3.2956 \(\int \frac{\sqrt{a+b \sqrt{c x^3}}}{x^4} \, dx\)

Optimal. Leaf size=97 \[ \frac{b^2 c \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^3}}}{\sqrt{a}}\right )}{6 a^{3/2}}-\frac{b c \sqrt{a+b \sqrt{c x^3}}}{6 a \sqrt{c x^3}}-\frac{\sqrt{a+b \sqrt{c x^3}}}{3 x^3} \]

[Out]

-Sqrt[a + b*Sqrt[c*x^3]]/(3*x^3) - (b*c*Sqrt[a + b*Sqrt[c*x^3]])/(6*a*Sqrt[c*x^3
]) + (b^2*c*ArcTanh[Sqrt[a + b*Sqrt[c*x^3]]/Sqrt[a]])/(6*a^(3/2))

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Rubi [A]  time = 0.155149, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{b^2 c \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c x^3}}}{\sqrt{a}}\right )}{6 a^{3/2}}-\frac{b c \sqrt{a+b \sqrt{c x^3}}}{6 a \sqrt{c x^3}}-\frac{\sqrt{a+b \sqrt{c x^3}}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*Sqrt[c*x^3]]/x^4,x]

[Out]

-Sqrt[a + b*Sqrt[c*x^3]]/(3*x^3) - (b*c*Sqrt[a + b*Sqrt[c*x^3]])/(6*a*Sqrt[c*x^3
]) + (b^2*c*ArcTanh[Sqrt[a + b*Sqrt[c*x^3]]/Sqrt[a]])/(6*a^(3/2))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \sqrt{c x^{3}}}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(a + b*sqrt(c*x**3))/x**4, x)

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Mathematica [A]  time = 0.0334027, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \sqrt{c x^3}}}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[Sqrt[a + b*Sqrt[c*x^3]]/x^4,x]

[Out]

Integrate[Sqrt[a + b*Sqrt[c*x^3]]/x^4, x]

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Maple [A]  time = 0.181, size = 81, normalized size = 0.8 \[ -{\frac{1}{6\,{x}^{3}} \left ( -{b}^{2}{\it Artanh} \left ({1\sqrt{a+b\sqrt{c{x}^{3}}}{\frac{1}{\sqrt{a}}}} \right ) c{x}^{3}a+\sqrt{c{x}^{3}}b\sqrt{a+b\sqrt{c{x}^{3}}}{a}^{{\frac{3}{2}}}+2\,\sqrt{a+b\sqrt{c{x}^{3}}}{a}^{5/2} \right ){a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(-b^2*arctanh((a+b*(c*x^3)^(1/2))^(1/2)/a^(1/2))*c*x^3*a+(c*x^3)^(1/2)*b*(a
+b*(c*x^3)^(1/2))^(1/2)*a^(3/2)+2*(a+b*(c*x^3)^(1/2))^(1/2)*a^(5/2))/x^3/a^(5/2)

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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(sqrt(sqrt(c*x^3)*b + a)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \sqrt{c x^{3}}}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(a + b*sqrt(c*x**3))/x**4, x)

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GIAC/XCAS [A]  time = 0.242669, size = 127, normalized size = 1.31 \[ -\frac{1}{6} \, b^{2} c^{\frac{3}{2}}{\left (\frac{\arctan \left (\frac{\sqrt{\sqrt{c x} b c x + a c}}{\sqrt{-a c}}\right )}{\sqrt{-a c} a c} + \frac{\sqrt{\sqrt{c x} b c x + a c} a c +{\left (\sqrt{c x} b c x + a c\right )}^{\frac{3}{2}}}{a b^{2} c^{4} x^{3}}\right )}{\left | c \right |} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*b^2*c^(3/2)*(arctan(sqrt(sqrt(c*x)*b*c*x + a*c)/sqrt(-a*c))/(sqrt(-a*c)*a*c
) + (sqrt(sqrt(c*x)*b*c*x + a*c)*a*c + (sqrt(c*x)*b*c*x + a*c)^(3/2))/(a*b^2*c^4
*x^3))*abs(c)