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

Optimal. Leaf size=163 \[ \frac {5}{12} x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}} \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}+\frac {\left (-2 a x^2-9\right ) \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{24 b}+\frac {3 \tanh ^{-1}\left (\sqrt {2} \sqrt {b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}\right )}{8 \sqrt {2} b} \]

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Rubi [F]  time = 0.43, 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 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx &=\int \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx\\ \end {align*}

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Mathematica [B]  time = 4.60, size = 714, normalized size = 4.38 \begin {gather*} \frac {\left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^6 \left (b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x^2-1\right ) \left (2 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+2 a x^2-1\right )^2 \left (12 \sqrt {a} x \left (4 a^2 x^3+a x \left (4 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}-3\right )-b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}\right ) \tanh ^{-1}\left (\frac {\sqrt {\left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )^2+a}}{\sqrt {a}}\right )+\sqrt {2} x \sqrt {a x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )} \left (32 a^3 x^5+16 a^2 x^3 \left (2 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}-5\right )-a x \left (64 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}-37\right )+9 b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}\right )-3 \sqrt {x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )} \sqrt {a x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )} \left (2 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+2 a x^2-1\right ) \tanh ^{-1}\left (\sqrt {2} \sqrt {x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )}\right )\right )}{24 \sqrt {2} a^2 b^2 \sqrt {\frac {a \left (a x^2-1\right )}{b^2}} \sqrt {x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )} \sqrt {a x \left (b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}+a x\right )} \left (4096 a^7 x^{13}+1024 a^6 x^{11} \left (4 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}-13\right )-256 a^5 x^9 \left (44 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}-65\right )+768 a^4 x^7 \left (15 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}-13\right )-224 a^3 x^5 \left (24 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}-13\right )+28 a^2 x^3 \left (40 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}-13\right )-a x \left (84 b x \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}-13\right )+b \sqrt {\frac {a \left (a x^2-1\right )}{b^2}}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])^6*(-1 + a*x^2 + b*x*Sqrt[(a*(-1 + a*x^2))/b^2])*(-1 + 2*a*x^2 + 2*b*x*Sq
rt[(a*(-1 + a*x^2))/b^2])^2*(Sqrt[2]*x*Sqrt[a*x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])]*(32*a^3*x^5 + 9*b*Sqrt[(
a*(-1 + a*x^2))/b^2] + 16*a^2*x^3*(-5 + 2*b*x*Sqrt[(a*(-1 + a*x^2))/b^2]) - a*x*(-37 + 64*b*x*Sqrt[(a*(-1 + a*
x^2))/b^2])) - 3*Sqrt[x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])]*Sqrt[a*x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])]*(
-1 + 2*a*x^2 + 2*b*x*Sqrt[(a*(-1 + a*x^2))/b^2])*ArcTanh[Sqrt[2]*Sqrt[x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])]]
+ 12*Sqrt[a]*x*(4*a^2*x^3 - b*Sqrt[(a*(-1 + a*x^2))/b^2] + a*x*(-3 + 4*b*x*Sqrt[(a*(-1 + a*x^2))/b^2]))*ArcTa
nh[Sqrt[a + (a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])^2]/Sqrt[a]]))/(24*Sqrt[2]*a^2*b^2*Sqrt[(a*(-1 + a*x^2))/b^2]*
Sqrt[x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])]*Sqrt[a*x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])]*(4096*a^7*x^13 + b
*Sqrt[(a*(-1 + a*x^2))/b^2] + 1024*a^6*x^11*(-13 + 4*b*x*Sqrt[(a*(-1 + a*x^2))/b^2]) + 768*a^4*x^7*(-13 + 15*b
*x*Sqrt[(a*(-1 + a*x^2))/b^2]) - 224*a^3*x^5*(-13 + 24*b*x*Sqrt[(a*(-1 + a*x^2))/b^2]) + 28*a^2*x^3*(-13 + 40*
b*x*Sqrt[(a*(-1 + a*x^2))/b^2]) - 256*a^5*x^9*(-65 + 44*b*x*Sqrt[(a*(-1 + a*x^2))/b^2]) - a*x*(-13 + 84*b*x*Sq
rt[(a*(-1 + a*x^2))/b^2])))

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IntegrateAlgebraic [A]  time = 3.27, size = 223, normalized size = 1.37 \begin {gather*} \frac {\left (-9-2 a x^2\right ) \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{24 b}+\frac {5}{12} x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}+\frac {3 \log \left (1+\sqrt {2} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{16 \sqrt {2} b}-\frac {3 \log \left (-b+\sqrt {2} b \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{16 \sqrt {2} b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-9 - 2*a*x^2)*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]])/(24*b) + (5*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]
*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]])/12 + (3*Log[1 + Sqrt[2]*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a
^2*x^2)/b^2]]])/(16*Sqrt[2]*b) - (3*Log[-b + Sqrt[2]*b*Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]]])/(16*
Sqrt[2]*b)

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fricas [A]  time = 21.29, size = 168, normalized size = 1.03 \begin {gather*} -\frac {4 \, {\left (2 \, a x^{2} - 10 \, b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} + 9\right )} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} - 9 \, \sqrt {2} \log \left (4 \, a x^{2} - 4 \, b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} - 2 \, \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} {\left (2 \, \sqrt {2} b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} - \sqrt {2} {\left (2 \, a x^{2} - 1\right )}\right )} - 1\right )}{96 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/96*(4*(2*a*x^2 - 10*b*x*sqrt((a^2*x^2 - a)/b^2) + 9)*sqrt(a*x^2 + b*x*sqrt((a^2*x^2 - a)/b^2)) - 9*sqrt(2)*
log(4*a*x^2 - 4*b*x*sqrt((a^2*x^2 - a)/b^2) - 2*sqrt(a*x^2 + b*x*sqrt((a^2*x^2 - a)/b^2))*(2*sqrt(2)*b*x*sqrt(
(a^2*x^2 - a)/b^2) - sqrt(2)*(2*a*x^2 - 1)) - 1))/b

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}\, \sqrt {a \,x^{2}+b x \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x*(a*x + b*sqrt(a**2*x**2/b**2 - a/b**2)))*sqrt(a*(a*x**2 - 1)/b**2), x)

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