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

Optimal. Leaf size=421 \[ \frac {\log \left (i a^{3/2} x^2+i \sqrt {a} \sqrt {a^2 x^4+b}+i \sqrt {2} a x \sqrt {\sqrt {a^2 x^4+b}+a x^2}\right )}{\sqrt {2} \sqrt {a}}-2 \sqrt {2} \sqrt {a} b \text {RootSum}\left [\text {$\#$1}^8+20 \text {$\#$1}^6 a b-26 \text {$\#$1}^4 a^2 b^2+20 \text {$\#$1}^2 a^3 b^3+a^4 b^4\& ,\frac {\text {$\#$1}^4 \log \left (-\text {$\#$1}+i a^{3/2} x^2+i \sqrt {a} \sqrt {a^2 x^4+b}+i \sqrt {2} a x \sqrt {\sqrt {a^2 x^4+b}+a x^2}\right )-2 \text {$\#$1}^2 a b \log \left (-\text {$\#$1}+i a^{3/2} x^2+i \sqrt {a} \sqrt {a^2 x^4+b}+i \sqrt {2} a x \sqrt {\sqrt {a^2 x^4+b}+a x^2}\right )+a^2 b^2 \log \left (-\text {$\#$1}+i a^{3/2} x^2+i \sqrt {a} \sqrt {a^2 x^4+b}+i \sqrt {2} a x \sqrt {\sqrt {a^2 x^4+b}+a x^2}\right )}{\text {$\#$1}^6+15 \text {$\#$1}^4 a b-13 \text {$\#$1}^2 a^2 b^2+5 a^3 b^3}\& \right ] \]

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Rubi [F]  time = 3.16, 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 {\sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{-b+a^2 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-1/4*Defer[Int][(Sqrt[b + a^2*x^4]*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/(b^(1/4) - Sqrt[-a]*x), x]/b^(3/4) - Defer
[Int][(Sqrt[b + a^2*x^4]*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/(b^(1/4) + Sqrt[-a]*x), x]/(4*b^(3/4)) - Defer[Int][
(Sqrt[b + a^2*x^4]*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/(b^(1/4) - Sqrt[a]*x), x]/(4*b^(3/4)) - Defer[Int][(Sqrt[b
+ a^2*x^4]*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/(b^(1/4) + Sqrt[a]*x), x]/(4*b^(3/4))

Rubi steps

\begin {align*} \int \frac {\sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{-b+a^2 x^4} \, dx &=\int \left (-\frac {\sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{2 \sqrt {b} \left (\sqrt {b}-a x^2\right )}-\frac {\sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{2 \sqrt {b} \left (\sqrt {b}+a x^2\right )}\right ) \, dx\\ &=-\frac {\int \frac {\sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b}-a x^2} \, dx}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b}+a x^2} \, dx}{2 \sqrt {b}}\\ &=-\frac {\int \left (\frac {\sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{2 \sqrt [4]{b} \left (\sqrt [4]{b}-\sqrt {-a} x\right )}+\frac {\sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{2 \sqrt [4]{b} \left (\sqrt [4]{b}+\sqrt {-a} x\right )}\right ) \, dx}{2 \sqrt {b}}-\frac {\int \left (\frac {\sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{2 \sqrt [4]{b} \left (\sqrt [4]{b}-\sqrt {a} x\right )}+\frac {\sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{2 \sqrt [4]{b} \left (\sqrt [4]{b}+\sqrt {a} x\right )}\right ) \, dx}{2 \sqrt {b}}\\ &=-\frac {\int \frac {\sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt [4]{b}-\sqrt {-a} x} \, dx}{4 b^{3/4}}-\frac {\int \frac {\sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt [4]{b}+\sqrt {-a} x} \, dx}{4 b^{3/4}}-\frac {\int \frac {\sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt [4]{b}-\sqrt {a} x} \, dx}{4 b^{3/4}}-\frac {\int \frac {\sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt [4]{b}+\sqrt {a} x} \, dx}{4 b^{3/4}}\\ \end {align*}

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Mathematica [F]  time = 0.27, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{-b+a^2 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.00, size = 421, normalized size = 1.00 \begin {gather*} \frac {\log \left (i a^{3/2} x^2+i \sqrt {a} \sqrt {b+a^2 x^4}+i \sqrt {2} a x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{\sqrt {2} \sqrt {a}}-2 \sqrt {2} \sqrt {a} b \text {RootSum}\left [a^4 b^4+20 a^3 b^3 \text {$\#$1}^2-26 a^2 b^2 \text {$\#$1}^4+20 a b \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {a^2 b^2 \log \left (i a^{3/2} x^2+i \sqrt {a} \sqrt {b+a^2 x^4}+i \sqrt {2} a x \sqrt {a x^2+\sqrt {b+a^2 x^4}}-\text {$\#$1}\right )-2 a b \log \left (i a^{3/2} x^2+i \sqrt {a} \sqrt {b+a^2 x^4}+i \sqrt {2} a x \sqrt {a x^2+\sqrt {b+a^2 x^4}}-\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (i a^{3/2} x^2+i \sqrt {a} \sqrt {b+a^2 x^4}+i \sqrt {2} a x \sqrt {a x^2+\sqrt {b+a^2 x^4}}-\text {$\#$1}\right ) \text {$\#$1}^4}{5 a^3 b^3-13 a^2 b^2 \text {$\#$1}^2+15 a b \text {$\#$1}^4+\text {$\#$1}^6}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[I*a^(3/2)*x^2 + I*Sqrt[a]*Sqrt[b + a^2*x^4] + I*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]]/(Sqrt[2]*Sqrt
[a]) - 2*Sqrt[2]*Sqrt[a]*b*RootSum[a^4*b^4 + 20*a^3*b^3*#1^2 - 26*a^2*b^2*#1^4 + 20*a*b*#1^6 + #1^8 & , (a^2*b
^2*Log[I*a^(3/2)*x^2 + I*Sqrt[a]*Sqrt[b + a^2*x^4] + I*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1] - 2*a
*b*Log[I*a^(3/2)*x^2 + I*Sqrt[a]*Sqrt[b + a^2*x^4] + I*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1]*#1^2
+ Log[I*a^(3/2)*x^2 + I*Sqrt[a]*Sqrt[b + a^2*x^4] + I*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1]*#1^4)/
(5*a^3*b^3 - 13*a^2*b^2*#1^2 + 15*a*b*#1^4 + #1^6) & ]

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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((a^2*x^4+b)^(1/2)*(a*x^2+(a^2*x^4+b)^(1/2))^(1/2)/(a^2*x^4-b),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^{4} + b} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{a^{2} x^{4} - b}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a^{2} x^{4}+b}\, \sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}}{a^{2} x^{4}-b}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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