3.30.13 \(\int \frac {x^4 \sqrt {b+a^2 x^2}}{x^2-\sqrt {a x-\sqrt {b+a^2 x^2}}} \, dx\)

Optimal. Leaf size=328 \[ \frac {2 \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^5 a^2-2 \text {$\#$1}^4 b+b^2\& ,\frac {\text {$\#$1}^3 a^6 \log \left (\sqrt {a x-\sqrt {a^2 x^2+b}}-\text {$\#$1}\right )+\text {$\#$1}^2 a^4 b \log \left (\sqrt {a x-\sqrt {a^2 x^2+b}}-\text {$\#$1}\right )}{-2 \text {$\#$1}^4+5 \text {$\#$1} a^2+2 b}\& \right ]}{a^3}+\frac {b^2}{6 a \left (a x-\sqrt {a^2 x^2+b}\right )^{3/2}}-\frac {b \sqrt {a x-\sqrt {a^2 x^2+b}}}{a}-\frac {\left (a x-\sqrt {a^2 x^2+b}\right )^{5/2}}{10 a}+a \sqrt {a^2 x^2+b}-a^2 x+\frac {b^4}{64 a^3 \left (\sqrt {a^2 x^2+b}-a x\right )^4}+\frac {b^2 \log \left (\sqrt {a^2 x^2+b}-a x\right )}{8 a^3}-\frac {\left (\sqrt {a^2 x^2+b}-a x\right )^4}{64 a^3} \]

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Rubi [F]  time = 6.45, 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^4 \sqrt {b+a^2 x^2}}{x^2-\sqrt {a x-\sqrt {b+a^2 x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-(a^2*x) + (b*x*Sqrt[b + a^2*x^2])/(8*a^2) + (x^3*Sqrt[b + a^2*x^2])/4 + b^2/(6*a*(a*x - Sqrt[b + a^2*x^2])^(3
/2)) - (b*Sqrt[a*x - Sqrt[b + a^2*x^2]])/a - (a*x - Sqrt[b + a^2*x^2])^(5/2)/(10*a) - (b^2*ArcTanh[(a*x)/Sqrt[
b + a^2*x^2]])/(8*a^3) + a^2*b*Defer[Int][(b + 2*a*x^5 - x^8)^(-1), x] + 2*a^3*Defer[Int][x^5/(b + 2*a*x^5 - x
^8), x] + b*Defer[Int][x^6/(b + 2*a*x^5 - x^8), x] - b*Defer[Int][(x^2*Sqrt[b + a^2*x^2])/(b + 2*a*x^5 - x^8),
x] - a*Defer[Int][(x^7*Sqrt[b + a^2*x^2])/(b + 2*a*x^5 - x^8), x] + b*Defer[Int][(x^4*Sqrt[a*x - Sqrt[b + a^2
*x^2]])/(b + 2*a*x^5 - x^8), x] + a^2*Defer[Int][(x^6*Sqrt[a*x - Sqrt[b + a^2*x^2]])/(b + 2*a*x^5 - x^8), x] -
b*Defer[Int][(Sqrt[b + a^2*x^2]*Sqrt[a*x - Sqrt[b + a^2*x^2]])/(b + 2*a*x^5 - x^8), x] - a*Defer[Int][(x^5*Sq
rt[b + a^2*x^2]*Sqrt[a*x - Sqrt[b + a^2*x^2]])/(b + 2*a*x^5 - x^8), x]

Rubi steps

\begin {align*} \int \frac {x^4 \sqrt {b+a^2 x^2}}{x^2-\sqrt {a x-\sqrt {b+a^2 x^2}}} \, dx &=\int \left (x^2 \sqrt {b+a^2 x^2}+\frac {x^6 \left (b+a^2 x^2\right )}{b+2 a x^5-x^8}+\frac {x^2 \sqrt {b+a^2 x^2} \left (b+a x^5\right )}{-b-2 a x^5+x^8}+\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}-\frac {b \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8}-\frac {a x^5 \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8}+\frac {x^4 \left (b+a^2 x^2\right ) \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8}\right ) \, dx\\ &=-\left (a \int \frac {x^5 \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx\right )-b \int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx+\int x^2 \sqrt {b+a^2 x^2} \, dx+\int \frac {x^6 \left (b+a^2 x^2\right )}{b+2 a x^5-x^8} \, dx+\int \frac {x^2 \sqrt {b+a^2 x^2} \left (b+a x^5\right )}{-b-2 a x^5+x^8} \, dx+\int \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}} \, dx+\int \frac {x^4 \left (b+a^2 x^2\right ) \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx\\ &=\frac {1}{4} x^3 \sqrt {b+a^2 x^2}-\frac {\operatorname {Subst}\left (\int \frac {\left (b+x^2\right )^2}{x^{5/2}} \, dx,x,a x-\sqrt {b+a^2 x^2}\right )}{4 a}-a \int \frac {x^5 \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx+\frac {1}{4} b \int \frac {x^2}{\sqrt {b+a^2 x^2}} \, dx-b \int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx+\int \left (-\frac {b x^2 \sqrt {b+a^2 x^2}}{b+2 a x^5-x^8}-\frac {a x^7 \sqrt {b+a^2 x^2}}{b+2 a x^5-x^8}\right ) \, dx+\int \left (-a^2+\frac {a^2 b+2 a^3 x^5+b x^6}{b+2 a x^5-x^8}\right ) \, dx+\int \left (\frac {b x^4 \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8}+\frac {a^2 x^6 \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8}\right ) \, dx\\ &=-a^2 x+\frac {b x \sqrt {b+a^2 x^2}}{8 a^2}+\frac {1}{4} x^3 \sqrt {b+a^2 x^2}-\frac {\operatorname {Subst}\left (\int \left (\frac {b^2}{x^{5/2}}+\frac {2 b}{\sqrt {x}}+x^{3/2}\right ) \, dx,x,a x-\sqrt {b+a^2 x^2}\right )}{4 a}-a \int \frac {x^7 \sqrt {b+a^2 x^2}}{b+2 a x^5-x^8} \, dx-a \int \frac {x^5 \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx+a^2 \int \frac {x^6 \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx-b \int \frac {x^2 \sqrt {b+a^2 x^2}}{b+2 a x^5-x^8} \, dx+b \int \frac {x^4 \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx-b \int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx-\frac {b^2 \int \frac {1}{\sqrt {b+a^2 x^2}} \, dx}{8 a^2}+\int \frac {a^2 b+2 a^3 x^5+b x^6}{b+2 a x^5-x^8} \, dx\\ &=-a^2 x+\frac {b x \sqrt {b+a^2 x^2}}{8 a^2}+\frac {1}{4} x^3 \sqrt {b+a^2 x^2}+\frac {b^2}{6 a \left (a x-\sqrt {b+a^2 x^2}\right )^{3/2}}-\frac {b \sqrt {a x-\sqrt {b+a^2 x^2}}}{a}-\frac {\left (a x-\sqrt {b+a^2 x^2}\right )^{5/2}}{10 a}-a \int \frac {x^7 \sqrt {b+a^2 x^2}}{b+2 a x^5-x^8} \, dx-a \int \frac {x^5 \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx+a^2 \int \frac {x^6 \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx-b \int \frac {x^2 \sqrt {b+a^2 x^2}}{b+2 a x^5-x^8} \, dx+b \int \frac {x^4 \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx-b \int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx-\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{1-a^2 x^2} \, dx,x,\frac {x}{\sqrt {b+a^2 x^2}}\right )}{8 a^2}+\int \left (\frac {a^2 b}{b+2 a x^5-x^8}+\frac {2 a^3 x^5}{b+2 a x^5-x^8}+\frac {b x^6}{b+2 a x^5-x^8}\right ) \, dx\\ &=-a^2 x+\frac {b x \sqrt {b+a^2 x^2}}{8 a^2}+\frac {1}{4} x^3 \sqrt {b+a^2 x^2}+\frac {b^2}{6 a \left (a x-\sqrt {b+a^2 x^2}\right )^{3/2}}-\frac {b \sqrt {a x-\sqrt {b+a^2 x^2}}}{a}-\frac {\left (a x-\sqrt {b+a^2 x^2}\right )^{5/2}}{10 a}-\frac {b^2 \tanh ^{-1}\left (\frac {a x}{\sqrt {b+a^2 x^2}}\right )}{8 a^3}-a \int \frac {x^7 \sqrt {b+a^2 x^2}}{b+2 a x^5-x^8} \, dx-a \int \frac {x^5 \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx+a^2 \int \frac {x^6 \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx+\left (2 a^3\right ) \int \frac {x^5}{b+2 a x^5-x^8} \, dx+b \int \frac {x^6}{b+2 a x^5-x^8} \, dx-b \int \frac {x^2 \sqrt {b+a^2 x^2}}{b+2 a x^5-x^8} \, dx+b \int \frac {x^4 \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx-b \int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+2 a x^5-x^8} \, dx+\left (a^2 b\right ) \int \frac {1}{b+2 a x^5-x^8} \, dx\\ \end {align*}

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Mathematica [F]  time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

$Aborted

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IntegrateAlgebraic [A]  time = 2.31, size = 323, normalized size = 0.98 \begin {gather*} -a^2 x+a \sqrt {b+a^2 x^2}+\frac {b^2}{6 a \left (a x-\sqrt {b+a^2 x^2}\right )^{3/2}}-\frac {b \sqrt {a x-\sqrt {b+a^2 x^2}}}{a}-\frac {\left (a x-\sqrt {b+a^2 x^2}\right )^{5/2}}{10 a}+\frac {b^4}{64 a^3 \left (-a x+\sqrt {b+a^2 x^2}\right )^4}-\frac {\left (-a x+\sqrt {b+a^2 x^2}\right )^4}{64 a^3}+\frac {b^2 \log \left (-a x+\sqrt {b+a^2 x^2}\right )}{8 a^3}+2 a \text {RootSum}\left [b^2-2 b \text {$\#$1}^4-4 a^2 \text {$\#$1}^5+\text {$\#$1}^8\&,\frac {b \log \left (\sqrt {a x-\sqrt {b+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^2+a^2 \log \left (\sqrt {a x-\sqrt {b+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^3}{2 b+5 a^2 \text {$\#$1}-2 \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^2*x) + a*Sqrt[b + a^2*x^2] + b^2/(6*a*(a*x - Sqrt[b + a^2*x^2])^(3/2)) - (b*Sqrt[a*x - Sqrt[b + a^2*x^2]])
/a - (a*x - Sqrt[b + a^2*x^2])^(5/2)/(10*a) + b^4/(64*a^3*(-(a*x) + Sqrt[b + a^2*x^2])^4) - (-(a*x) + Sqrt[b +
a^2*x^2])^4/(64*a^3) + (b^2*Log[-(a*x) + Sqrt[b + a^2*x^2]])/(8*a^3) + 2*a*RootSum[b^2 - 2*b*#1^4 - 4*a^2*#1^
5 + #1^8 & , (b*Log[Sqrt[a*x - Sqrt[b + a^2*x^2]] - #1]*#1^2 + a^2*Log[Sqrt[a*x - Sqrt[b + a^2*x^2]] - #1]*#1^
3)/(2*b + 5*a^2*#1 - 2*#1^4) & ]

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a^2*x^2+b)^(1/2)/(x^2-(a*x-(a^2*x^2+b)^(1/2))^(1/2)),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a^{2} x^{2} + b} x^{4}}{x^{2} - \sqrt {a x - \sqrt {a^{2} x^{2} + b}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a^2*x^2+b)^(1/2)/(x^2-(a*x-(a^2*x^2+b)^(1/2))^(1/2)),x, algorithm="giac")

[Out]

integrate(sqrt(a^2*x^2 + b)*x^4/(x^2 - sqrt(a*x - sqrt(a^2*x^2 + b))), x)

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maple [F]  time = 0.16, size = 0, normalized size = 0.00 \[\int \frac {x^{4} \sqrt {a^{2} x^{2}+b}}{x^{2}-\sqrt {a x -\sqrt {a^{2} x^{2}+b}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a^2*x^2+b)^(1/2)/(x^2-(a*x-(a^2*x^2+b)^(1/2))^(1/2)),x)

[Out]

int(x^4*(a^2*x^2+b)^(1/2)/(x^2-(a*x-(a^2*x^2+b)^(1/2))^(1/2)),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a^{2} x^{2} + b} x^{4}}{x^{2} - \sqrt {a x - \sqrt {a^{2} x^{2} + b}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a^2*x^2+b)^(1/2)/(x^2-(a*x-(a^2*x^2+b)^(1/2))^(1/2)),x, algorithm="maxima")

[Out]

integrate(sqrt(a^2*x^2 + b)*x^4/(x^2 - sqrt(a*x - sqrt(a^2*x^2 + b))), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {x^4\,\sqrt {a^2\,x^2+b}}{\sqrt {a\,x-\sqrt {a^2\,x^2+b}}-x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^4*(b + a^2*x^2)^(1/2))/((a*x - (b + a^2*x^2)^(1/2))^(1/2) - x^2),x)

[Out]

-int((x^4*(b + a^2*x^2)^(1/2))/((a*x - (b + a^2*x^2)^(1/2))^(1/2) - x^2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} \sqrt {a^{2} x^{2} + b}}{x^{2} - \sqrt {a x - \sqrt {a^{2} x^{2} + b}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a**2*x**2+b)**(1/2)/(x**2-(a*x-(a**2*x**2+b)**(1/2))**(1/2)),x)

[Out]

Integral(x**4*sqrt(a**2*x**2 + b)/(x**2 - sqrt(a*x - sqrt(a**2*x**2 + b))), x)

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