3.22.31 \(\int \frac {1}{x^2 \sqrt {-b x+a^2 x^2} (a x^2+x \sqrt {-b x+a^2 x^2})^{3/2}} \, dx\)

Optimal. Leaf size=154 \[ \frac {4 \left (8192 a^7 x^3+1024 a^5 b x^2+448 a^3 b^2 x+6699 a b^3\right ) \sqrt {x \left (\sqrt {a^2 x^2-b x}+a x\right )}}{45045 b^5 x^4}-\frac {4 \sqrt {a^2 x^2-b x} \left (-8192 a^6 x^3-5120 a^4 b x^2-4032 a^2 b^2 x+3003 b^3\right ) \sqrt {x \left (\sqrt {a^2 x^2-b x}+a x\right )}}{45045 b^5 x^5} \]

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Rubi [F]  time = 4.77, 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 {1}{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[1/(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][1/(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 {1}{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 {1}{x^{5/2} \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 {1}{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 = 1.77, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{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[1/(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[1/(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 = 5.55, size = 154, normalized size = 1.00 \begin {gather*} -\frac {4 \sqrt {-b x+a^2 x^2} \left (3003 b^3-4032 a^2 b^2 x-5120 a^4 b x^2-8192 a^6 x^3\right ) \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{45045 b^5 x^5}+\frac {4 \left (6699 a b^3+448 a^3 b^2 x+1024 a^5 b x^2+8192 a^7 x^3\right ) \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{45045 b^5 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(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*Sqrt[-(b*x) + a^2*x^2]*(3003*b^3 - 4032*a^2*b^2*x - 5120*a^4*b*x^2 - 8192*a^6*x^3)*Sqrt[x*(a*x + Sqrt[-(b*
x) + a^2*x^2])])/(45045*b^5*x^5) + (4*(6699*a*b^3 + 448*a^3*b^2*x + 1024*a^5*b*x^2 + 8192*a^7*x^3)*Sqrt[x*(a*x
+ Sqrt[-(b*x) + a^2*x^2])])/(45045*b^5*x^4)

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fricas [A]  time = 0.51, size = 115, normalized size = 0.75 \begin {gather*} \frac {4 \, {\left (8192 \, a^{7} x^{4} + 1024 \, a^{5} b x^{3} + 448 \, a^{3} b^{2} x^{2} + 6699 \, a b^{3} x + {\left (8192 \, a^{6} x^{3} + 5120 \, a^{4} b x^{2} + 4032 \, a^{2} b^{2} x - 3003 \, b^{3}\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{45045 \, b^{5} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/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]

4/45045*(8192*a^7*x^4 + 1024*a^5*b*x^3 + 448*a^3*b^2*x^2 + 6699*a*b^3*x + (8192*a^6*x^3 + 5120*a^4*b*x^2 + 403
2*a^2*b^2*x - 3003*b^3)*sqrt(a^2*x^2 - b*x))*sqrt(a*x^2 + sqrt(a^2*x^2 - b*x)*x)/(b^5*x^5)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a^{2} x^{2} - b x} {\left (a x^{2} + \sqrt {a^{2} x^{2} - b x} x\right )}^{\frac {3}{2}} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{2} \sqrt {a^{2} x^{2}-b x}\, \left (a \,x^{2}+x \sqrt {a^{2} x^{2}-b x}\right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/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(1/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 {1}{\sqrt {a^{2} x^{2} - b x} {\left (a x^{2} + \sqrt {a^{2} x^{2} - b x} x\right )}^{\frac {3}{2}} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{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(1/(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(1/(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 {1}{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(1/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(1/(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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