3.15.55 \(\int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3} \, dx\)

Optimal. Leaf size=102 \[ -\frac {4 \left (-3 a x+3 b+2 c^2 x\right ) \sqrt {-x \left (\sqrt {a x^2-b x}-c x\right )}}{15 b x^2}-\frac {4 c \sqrt {a x^2-b x} \sqrt {-x \left (\sqrt {a x^2-b x}-c x\right )}}{15 b x^2} \]

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Rubi [F]  time = 0.34, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[Int][Sqrt[c*x^2 - x*Sqrt[-(b*x) + a*x^2]]/x^3, x]

Rubi steps

\begin {align*} \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3} \, dx &=\int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.31, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 7.07, size = 102, normalized size = 1.00 \begin {gather*} -\frac {4 \left (3 b-3 a x+2 c^2 x\right ) \sqrt {-x \left (-c x+\sqrt {-b x+a x^2}\right )}}{15 b x^2}-\frac {4 c \sqrt {-b x+a x^2} \sqrt {-x \left (-c x+\sqrt {-b x+a x^2}\right )}}{15 b x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(3*b - 3*a*x + 2*c^2*x)*Sqrt[-(x*(-(c*x) + Sqrt[-(b*x) + a*x^2]))])/(15*b*x^2) - (4*c*Sqrt[-(b*x) + a*x^2]
*Sqrt[-(x*(-(c*x) + Sqrt[-(b*x) + a*x^2]))])/(15*b*x^2)

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fricas [A]  time = 0.85, size = 60, normalized size = 0.59 \begin {gather*} -\frac {4 \, \sqrt {c x^{2} - \sqrt {a x^{2} - b x} x} {\left ({\left (2 \, c^{2} - 3 \, a\right )} x + \sqrt {a x^{2} - b x} c + 3 \, b\right )}}{15 \, b x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2-x*(a*x^2-b*x)^(1/2))^(1/2)/x^3,x, algorithm="fricas")

[Out]

-4/15*sqrt(c*x^2 - sqrt(a*x^2 - b*x)*x)*((2*c^2 - 3*a)*x + sqrt(a*x^2 - b*x)*c + 3*b)/(b*x^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c x^{2} - \sqrt {a x^{2} - b x} x}}{x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2-x*(a*x^2-b*x)^(1/2))^(1/2)/x^3,x, algorithm="giac")

[Out]

integrate(sqrt(c*x^2 - sqrt(a*x^2 - b*x)*x)/x^3, x)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {c \,x^{2}-x \sqrt {a \,x^{2}-b x}}}{x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c x^{2} - \sqrt {a x^{2} - b x} x}}{x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2-x*(a*x^2-b*x)^(1/2))^(1/2)/x^3,x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^2 - sqrt(a*x^2 - b*x)*x)/x^3, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {c\,x^2-x\,\sqrt {a\,x^2-b\,x}}}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x \left (c x - \sqrt {a x^{2} - b x}\right )}}{x^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2-x*(a*x**2-b*x)**(1/2))**(1/2)/x**3,x)

[Out]

Integral(sqrt(x*(c*x - sqrt(a*x**2 - b*x)))/x**3, x)

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