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

Optimal. Leaf size=220 \[ \frac {\sqrt {a^2 x^2-b x} \sqrt {x \left (\sqrt {a^2 x^2-b x}+a x\right )} \left (32 a^4 x^2-88 a^2 b x+115 b^2\right )}{40 b^2 x}+\sqrt {x \left (\sqrt {a^2 x^2-b x}+a x\right )} \left (\frac {9 \sqrt {b} \sqrt {\sqrt {a^2 x^2-b x}-a x} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {\sqrt {a^2 x^2-b x}-a x}}{\sqrt {b}}\right )}{8 \sqrt {2} \sqrt {a} x}+\frac {-32 a^5 x^2+104 a^3 b x-145 a b^2}{40 b^2}\right ) \]

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Rubi [F]  time = 4.13, 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 {\left (-b x+a^2 x^2\right )^{3/2}}{\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[(-(b*x) + a^2*x^2)^(3/2)/(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^4*(-b + a^2*x^2)^(3/2))/(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 {\left (-b x+a^2 x^2\right )^{3/2}}{\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 {x^{3/2} \left (-b+a^2 x\right )^{3/2}}{\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^4 \left (-b+a^2 x^2\right )^{3/2}}{\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 = 0.41, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-b x+a^2 x^2\right )^{3/2}}{\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[(-(b*x) + a^2*x^2)^(3/2)/(a*x^2 + x*Sqrt[-(b*x) + a^2*x^2])^(3/2),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 7.12, size = 220, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-b x+a^2 x^2} \left (115 b^2-88 a^2 b x+32 a^4 x^2\right ) \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{40 b^2 x}+\sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )} \left (\frac {-145 a b^2+104 a^3 b x-32 a^5 x^2}{40 b^2}+\frac {9 \sqrt {b} \sqrt {-a x+\sqrt {-b x+a^2 x^2}} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {-a x+\sqrt {-b x+a^2 x^2}}}{\sqrt {b}}\right )}{8 \sqrt {2} \sqrt {a} x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-(b*x) + a^2*x^2]*(115*b^2 - 88*a^2*b*x + 32*a^4*x^2)*Sqrt[x*(a*x + Sqrt[-(b*x) + a^2*x^2])])/(40*b^2*x)
+ Sqrt[x*(a*x + Sqrt[-(b*x) + a^2*x^2])]*((-145*a*b^2 + 104*a^3*b*x - 32*a^5*x^2)/(40*b^2) + (9*Sqrt[b]*Sqrt[
-(a*x) + Sqrt[-(b*x) + a^2*x^2]]*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[-(a*x) + Sqrt[-(b*x) + a^2*x^2]])/Sqrt[b]])/(8*S
qrt[2]*Sqrt[a]*x))

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fricas [A]  time = 0.65, size = 367, normalized size = 1.67 \begin {gather*} \left [\frac {45 \, \sqrt {2} \sqrt {a} b^{3} x \log \left (-\frac {4 \, a^{2} x^{2} + 4 \, \sqrt {a^{2} x^{2} - b x} a x - b x - 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x + \sqrt {2} \sqrt {a^{2} x^{2} - b x} \sqrt {a}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{x}\right ) - 4 \, {\left (32 \, a^{6} x^{3} - 104 \, a^{4} b x^{2} + 145 \, a^{2} b^{2} x - {\left (32 \, a^{5} x^{2} - 88 \, a^{3} b x + 115 \, a b^{2}\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{160 \, a b^{2} x}, \frac {45 \, \sqrt {2} \sqrt {-a} b^{3} x \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x} \sqrt {-a}}{2 \, a x}\right ) - 2 \, {\left (32 \, a^{6} x^{3} - 104 \, a^{4} b x^{2} + 145 \, a^{2} b^{2} x - {\left (32 \, a^{5} x^{2} - 88 \, a^{3} b x + 115 \, a b^{2}\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{80 \, a b^{2} x}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/160*(45*sqrt(2)*sqrt(a)*b^3*x*log(-(4*a^2*x^2 + 4*sqrt(a^2*x^2 - b*x)*a*x - b*x - 2*(sqrt(2)*a^(3/2)*x + sq
rt(2)*sqrt(a^2*x^2 - b*x)*sqrt(a))*sqrt(a*x^2 + sqrt(a^2*x^2 - b*x)*x))/x) - 4*(32*a^6*x^3 - 104*a^4*b*x^2 + 1
45*a^2*b^2*x - (32*a^5*x^2 - 88*a^3*b*x + 115*a*b^2)*sqrt(a^2*x^2 - b*x))*sqrt(a*x^2 + sqrt(a^2*x^2 - b*x)*x))
/(a*b^2*x), 1/80*(45*sqrt(2)*sqrt(-a)*b^3*x*arctan(1/2*sqrt(2)*sqrt(a*x^2 + sqrt(a^2*x^2 - b*x)*x)*sqrt(-a)/(a
*x)) - 2*(32*a^6*x^3 - 104*a^4*b*x^2 + 145*a^2*b^2*x - (32*a^5*x^2 - 88*a^3*b*x + 115*a*b^2)*sqrt(a^2*x^2 - b*
x))*sqrt(a*x^2 + sqrt(a^2*x^2 - b*x)*x))/(a*b^2*x)]

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

[Out]

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

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maple [F]  time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\left (a^{2} x^{2}-b x \right )^{\frac {3}{2}}}{\left (a \,x^{2}+x \sqrt {a^{2} x^{2}-b x}\right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a^2*x^2-b*x)^(3/2)/(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 {{\left (a^{2} x^{2} - b x\right )}^{\frac {3}{2}}}{{\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)^(3/2)/(a*x^2+x*(a^2*x^2-b*x)^(1/2))^(3/2),x, algorithm="maxima")

[Out]

integrate((a^2*x^2 - b*x)^(3/2)/(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 = -0.00 \begin {gather*} \int \frac {{\left (a^2\,x^2-b\,x\right )}^{3/2}}{{\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)^(3/2)/(a*x^2 + x*(a^2*x^2 - b*x)^(1/2))^(3/2),x)

[Out]

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

[Out]

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

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