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

Optimal. Leaf size=83 \[ \frac {4 \sqrt {a^2 x^2-b x} \sqrt {x \left (\sqrt {a^2 x^2-b x}+a x\right )}}{b^2 x}-\frac {4 a \sqrt {x \left (\sqrt {a^2 x^2-b x}+a x\right )}}{b^2} \]

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Rubi [F]  time = 5.25, 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 {x^2}{\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[x^2/(Sqrt[-(b*x) + a^2*x^2]*(a*x^2 + x*Sqrt[-(b*x) + a^2*x^2])^(3/2)),x]

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.75, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2}{\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[x^2/(Sqrt[-(b*x) + a^2*x^2]*(a*x^2 + x*Sqrt[-(b*x) + a^2*x^2])^(3/2)),x]

[Out]

Integrate[x^2/(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 = 4.06, size = 83, normalized size = 1.00 \begin {gather*} -\frac {4 a \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{b^2}+\frac {4 \sqrt {-b x+a^2 x^2} \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{b^2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.54, size = 52, normalized size = 0.63 \begin {gather*} -\frac {4 \, \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x} {\left (a x - \sqrt {a^{2} x^{2} - b x}\right )}}{b^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(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(x^2/(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 {x^{2}}{\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(x^2/(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(x^2/(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 {x^2}{\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(x^2/((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(x^2/((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 {x^{2}}{\left (x \left (a x + \sqrt {a^{2} x^{2} - b x}\right )\right )^{\frac {3}{2}} \sqrt {x \left (a^{2} x - b\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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