3.26.72 \(\int \frac {(-b x+a^2 x^2)^{3/2}}{(a x^2+x \sqrt {-b x+a^2 x^2})^{3/2}} \, dx\) [2572]

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

[Out]

1/40*(a^2*x^2-b*x)^(1/2)*(32*a^4*x^2-88*a^2*b*x+115*b^2)*(x*(a*x+(a^2*x^2-b*x)^(1/2)))^(1/2)/b^2/x+(x*(a*x+(a^
2*x^2-b*x)^(1/2)))^(1/2)*(1/40*(-32*a^5*x^2+104*a^3*b*x-145*a*b^2)/b^2+9/16*b^(1/2)*(-a*x+(a^2*x^2-b*x)^(1/2))
^(1/2)*arctan(2^(1/2)*a^(1/2)*(-a*x+(a^2*x^2-b*x)^(1/2))^(1/2)/b^(1/2))*2^(1/2)/a^(1/2)/x)

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Rubi [F]
time = 2.78, 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 ) \text {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 [A]
time = 4.95, size = 247, normalized size = 1.12 \begin {gather*} \frac {\sqrt {x \left (a x+\sqrt {x \left (-b+a^2 x\right )}\right )} \left (-2 \sqrt {a} x \left (115 b^3+8 a^3 b x \left (15 a x-13 \sqrt {x \left (-b+a^2 x\right )}\right )+32 a^5 x^2 \left (-a x+\sqrt {x \left (-b+a^2 x\right )}\right )+29 a b^2 \left (-7 a x+5 \sqrt {x \left (-b+a^2 x\right )}\right )\right )+45 \sqrt {2} b^{5/2} \sqrt {x \left (-b+a^2 x\right )} \sqrt {-a x+\sqrt {x \left (-b+a^2 x\right )}} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {-a x+\sqrt {x \left (-b+a^2 x\right )}}}{\sqrt {b}}\right )\right )}{80 \sqrt {a} b^2 x \sqrt {x \left (-b+a^2 x\right )}} \end {gather*}

Antiderivative was successfully verified.

[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]

(Sqrt[x*(a*x + Sqrt[x*(-b + a^2*x)])]*(-2*Sqrt[a]*x*(115*b^3 + 8*a^3*b*x*(15*a*x - 13*Sqrt[x*(-b + a^2*x)]) +
32*a^5*x^2*(-(a*x) + Sqrt[x*(-b + a^2*x)]) + 29*a*b^2*(-7*a*x + 5*Sqrt[x*(-b + a^2*x)])) + 45*Sqrt[2]*b^(5/2)*
Sqrt[x*(-b + a^2*x)]*Sqrt[-(a*x) + Sqrt[x*(-b + a^2*x)]]*ArcTan[(Sqrt[2]*Sqrt[a]*Sqrt[-(a*x) + Sqrt[x*(-b + a^
2*x)]])/Sqrt[b]]))/(80*Sqrt[a]*b^2*x*Sqrt[x*(-b + a^2*x)])

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Maple [F]
time = 0.06, 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*} \text {Failed to integrate} \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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Fricas [A]
time = 0.41, 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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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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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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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