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

Optimal. Leaf size=117 \[ \frac {4 \sqrt {a x^2-b x} \left (20 a x+15 b+4 c^2 x\right ) \sqrt {-x \left (\sqrt {a x^2-b x}-c x\right )}}{105 b^2 x^3}-\frac {4 \left (32 a c x+3 b c-8 c^3 x\right ) \sqrt {-x \left (\sqrt {a x^2-b x}-c x\right )}}{105 b^2 x^2} \]

________________________________________________________________________________________

Rubi [F]  time = 3.92, 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 \sqrt {-b x+a x^2}} \, 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*Sqrt[-(b*x) + a*x^2]),x]

[Out]

(2*Sqrt[x]*Sqrt[-b + a*x]*Defer[Subst][Defer[Int][Sqrt[c*x^4 - x^2*Sqrt[-(b*x^2) + a*x^4]]/(x^6*Sqrt[-b + a*x^
2]), x], x, Sqrt[x]])/Sqrt[-(b*x) + a*x^2]

Rubi steps

\begin {align*} \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3 \sqrt {-b x+a x^2}} \, dx &=\frac {\left (\sqrt {x} \sqrt {-b+a x}\right ) \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^{7/2} \sqrt {-b+a x}} \, dx}{\sqrt {-b x+a x^2}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-b+a x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {c x^4-x^2 \sqrt {-b x^2+a x^4}}}{x^6 \sqrt {-b+a x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b x+a x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 1.08, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3 \sqrt {-b x+a x^2}} \, 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*Sqrt[-(b*x) + a*x^2]),x]

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 180.02, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

$Aborted

________________________________________________________________________________________

fricas [A]  time = 0.46, size = 78, normalized size = 0.67 \begin {gather*} -\frac {4 \, {\left (3 \, b c x - 8 \, {\left (c^{3} - 4 \, a c\right )} x^{2} - \sqrt {a x^{2} - b x} {\left (4 \, {\left (c^{2} + 5 \, a\right )} x + 15 \, b\right )}\right )} \sqrt {c x^{2} - \sqrt {a x^{2} - b x} x}}{105 \, b^{2} x^{3}} \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/(a*x^2-b*x)^(1/2),x, algorithm="fricas")

[Out]

-4/105*(3*b*c*x - 8*(c^3 - 4*a*c)*x^2 - sqrt(a*x^2 - b*x)*(4*(c^2 + 5*a)*x + 15*b))*sqrt(c*x^2 - sqrt(a*x^2 -
b*x)*x)/(b^2*x^3)

________________________________________________________________________________________

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}}{\sqrt {a x^{2} - b 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/(a*x^2-b*x)^(1/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {c \,x^{2}-x \sqrt {a \,x^{2}-b x}}}{x^{3} \sqrt {a \,x^{2}-b x}}\, dx\]

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

[Out]

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

________________________________________________________________________________________

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}}{\sqrt {a x^{2} - b 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/(a*x^2-b*x)^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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\,\sqrt {a\,x^2-b\,x}} \,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*(a*x^2 - b*x)^(1/2)),x)

[Out]

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

________________________________________________________________________________________

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} \sqrt {x \left (a x - b\right )}}\, 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/(a*x**2-b*x)**(1/2),x)

[Out]

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

________________________________________________________________________________________