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

Optimal. Leaf size=108 \[ \frac {4 \left (3 a^2 x+b\right ) \sqrt {a^2 x^2-b x} \sqrt {x \left (\sqrt {a^2 x^2-b x}+a x\right )}}{3 b^2 x^2}-\frac {4 \left (3 a^3 x+5 a b\right ) \sqrt {x \left (\sqrt {a^2 x^2-b x}+a x\right )}}{3 b^2 x} \]

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Rubi [F]  time = 3.59, 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 {-b x+a^2 x^2}}{\left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(2*Sqrt[-(b*x) + a^2*x^2]*Defer[Subst][Defer[Int][(x^2*Sqrt[-b + a^2*x^2])/(a*x^4 + x^2*Sqrt[-(b*x^2) + a^2*x^
4])^(3/2), x], x, Sqrt[x]])/(Sqrt[x]*Sqrt[-b + a^2*x])

Rubi steps

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

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Mathematica [F]  time = 1.77, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-b x+a^2 x^2}}{\left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 4.92, size = 108, normalized size = 1.00 \begin {gather*} -\frac {4 \left (5 a b+3 a^3 x\right ) \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{3 b^2 x}+\frac {4 \left (b+3 a^2 x\right ) \sqrt {-b x+a^2 x^2} \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{3 b^2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(5*a*b + 3*a^3*x)*Sqrt[x*(a*x + Sqrt[-(b*x) + a^2*x^2])])/(3*b^2*x) + (4*(b + 3*a^2*x)*Sqrt[-(b*x) + a^2*x
^2]*Sqrt[x*(a*x + Sqrt[-(b*x) + a^2*x^2])])/(3*b^2*x^2)

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fricas [A]  time = 0.46, size = 70, normalized size = 0.65 \begin {gather*} -\frac {4 \, {\left (3 \, a^{3} x^{2} + 5 \, a b x - \sqrt {a^{2} x^{2} - b x} {\left (3 \, a^{2} x + b\right )}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{3 \, b^{2} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*(3*a^3*x^2 + 5*a*b*x - sqrt(a^2*x^2 - b*x)*(3*a^2*x + b))*sqrt(a*x^2 + sqrt(a^2*x^2 - b*x)*x)/(b^2*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a^{2} x^{2}-b x}}{\left (a \,x^{2}+x \sqrt {a^{2} x^{2}-b x}\right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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