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

Optimal. Leaf size=351 \[ \frac {\log \left (i a^{3/2} x^2+i \sqrt {a} \sqrt {a^2 x^4+b}+i \sqrt {2} a x \sqrt {\sqrt {a^2 x^4+b}+a x^2}\right )}{\sqrt {2} \sqrt {a}}-\frac {i \sqrt {2} \text {RootSum}\left [\text {$\#$1}^4 \sqrt {a}-4 i \text {$\#$1}^3 b+2 \text {$\#$1}^2 a^{3/2} b+4 i \text {$\#$1} a b^2+a^{5/2} b^2\& ,\frac {a b^2 \log \left (-\text {$\#$1}+i a^{3/2} x^2+i \sqrt {a} \sqrt {a^2 x^4+b}+i \sqrt {2} a x \sqrt {\sqrt {a^2 x^4+b}+a x^2}\right )-\text {$\#$1}^2 b \log \left (-\text {$\#$1}+i a^{3/2} x^2+i \sqrt {a} \sqrt {a^2 x^4+b}+i \sqrt {2} a x \sqrt {\sqrt {a^2 x^4+b}+a x^2}\right )}{\text {$\#$1}^3 \sqrt {a}-3 i \text {$\#$1}^2 b+\text {$\#$1} a^{3/2} b+i a b^2}\& \right ]}{\sqrt {a}} \]

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Rubi [F]  time = 2.62, 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+a^2 x^2\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\left (-b+a^2 x^2\right ) \sqrt {b+a^2 x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.63, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b+a^2 x^2\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\left (-b+a^2 x^2\right ) \sqrt {b+a^2 x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [B]  time = 36.02, size = 1255, normalized size = 3.58 \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} a^{3/2} b^4 \text {RootSum}\left [a^5 b^4+4 a^4 b^3 \text {$\#$1}^2+16 a^2 b^4 \text {$\#$1}^2+6 a^3 b^2 \text {$\#$1}^4-32 a b^3 \text {$\#$1}^4+4 a^2 b \text {$\#$1}^6+16 b^2 \text {$\#$1}^6+a \text {$\#$1}^8\&,\frac {\log \left (i a^{3/2} x^2+i \sqrt {a} \sqrt {b+a^2 x^4}-i \sqrt {2} a x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+\text {$\#$1}\right )}{a^4 b^3+4 a^2 b^4+3 a^3 b^2 \text {$\#$1}^2-16 a b^3 \text {$\#$1}^2+3 a^2 b \text {$\#$1}^4+12 b^2 \text {$\#$1}^4+a \text {$\#$1}^6}\&\right ]-4 \sqrt {2} \sqrt {a} b^3 \text {RootSum}\left [a^5 b^4+4 a^4 b^3 \text {$\#$1}^2+16 a^2 b^4 \text {$\#$1}^2+6 a^3 b^2 \text {$\#$1}^4-32 a b^3 \text {$\#$1}^4+4 a^2 b \text {$\#$1}^6+16 b^2 \text {$\#$1}^6+a \text {$\#$1}^8\&,\frac {\log \left (i a^{3/2} x^2+i \sqrt {a} \sqrt {b+a^2 x^4}-i \sqrt {2} a x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+\text {$\#$1}\right ) \text {$\#$1}^2}{a^4 b^3+4 a^2 b^4+3 a^3 b^2 \text {$\#$1}^2-16 a b^3 \text {$\#$1}^2+3 a^2 b \text {$\#$1}^4+12 b^2 \text {$\#$1}^4+a \text {$\#$1}^6}\&\right ]+\frac {2 \sqrt {2} b^2 \text {RootSum}\left [a^5 b^4+4 a^4 b^3 \text {$\#$1}^2+16 a^2 b^4 \text {$\#$1}^2+6 a^3 b^2 \text {$\#$1}^4-32 a b^3 \text {$\#$1}^4+4 a^2 b \text {$\#$1}^6+16 b^2 \text {$\#$1}^6+a \text {$\#$1}^8\&,\frac {\log \left (i a^{3/2} x^2+i \sqrt {a} \sqrt {b+a^2 x^4}-i \sqrt {2} a x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+\text {$\#$1}\right ) \text {$\#$1}^4}{a^4 b^3+4 a^2 b^4+3 a^3 b^2 \text {$\#$1}^2-16 a b^3 \text {$\#$1}^2+3 a^2 b \text {$\#$1}^4+12 b^2 \text {$\#$1}^4+a \text {$\#$1}^6}\&\right ]}{\sqrt {a}}+\frac {i b \text {RootSum}\left [a^5 b^4+4 a^4 b^3 \text {$\#$1}^2+16 a^2 b^4 \text {$\#$1}^2+6 a^3 b^2 \text {$\#$1}^4-32 a b^3 \text {$\#$1}^4+4 a^2 b \text {$\#$1}^6+16 b^2 \text {$\#$1}^6+a \text {$\#$1}^8\&,\frac {-a^3 b^3 \log \left (-i \sqrt {2} a x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+i \sqrt {a} \left (a x^2+\sqrt {b+a^2 x^4}\right )+\text {$\#$1}\right )-a^2 b^2 \log \left (-i \sqrt {2} a x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+i \sqrt {a} \left (a x^2+\sqrt {b+a^2 x^4}\right )+\text {$\#$1}\right ) \text {$\#$1}^2+a b \log \left (-i \sqrt {2} a x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+i \sqrt {a} \left (a x^2+\sqrt {b+a^2 x^4}\right )+\text {$\#$1}\right ) \text {$\#$1}^4+\log \left (-i \sqrt {2} a x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+i \sqrt {a} \left (a x^2+\sqrt {b+a^2 x^4}\right )+\text {$\#$1}\right ) \text {$\#$1}^6}{a^4 b^3 \text {$\#$1}+4 a^2 b^4 \text {$\#$1}+3 a^3 b^2 \text {$\#$1}^3-16 a b^3 \text {$\#$1}^3+3 a^2 b \text {$\#$1}^5+12 b^2 \text {$\#$1}^5+a \text {$\#$1}^7}\&\right ]}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((b + a^2*x^2)*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/((-b + a^2*x^2)*Sqrt[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]*Sq
rt[a])) + 2*Sqrt[2]*a^(3/2)*b^4*RootSum[a^5*b^4 + 4*a^4*b^3*#1^2 + 16*a^2*b^4*#1^2 + 6*a^3*b^2*#1^4 - 32*a*b^3
*#1^4 + 4*a^2*b*#1^6 + 16*b^2*#1^6 + a*#1^8 & , 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]/(a^4*b^3 + 4*a^2*b^4 + 3*a^3*b^2*#1^2 - 16*a*b^3*#1^2 + 3*a^2*b*#1^4 +
12*b^2*#1^4 + a*#1^6) & ] - 4*Sqrt[2]*Sqrt[a]*b^3*RootSum[a^5*b^4 + 4*a^4*b^3*#1^2 + 16*a^2*b^4*#1^2 + 6*a^3*
b^2*#1^4 - 32*a*b^3*#1^4 + 4*a^2*b*#1^6 + 16*b^2*#1^6 + a*#1^8 & , (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)/(a^4*b^3 + 4*a^2*b^4 + 3*a^3*b^2*#1^2 - 16*a
*b^3*#1^2 + 3*a^2*b*#1^4 + 12*b^2*#1^4 + a*#1^6) & ] + (2*Sqrt[2]*b^2*RootSum[a^5*b^4 + 4*a^4*b^3*#1^2 + 16*a^
2*b^4*#1^2 + 6*a^3*b^2*#1^4 - 32*a*b^3*#1^4 + 4*a^2*b*#1^6 + 16*b^2*#1^6 + a*#1^8 & , (Log[I*a^(3/2)*x^2 + I*S
qrt[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)/(a^4*b^3 + 4*a^2*b^4 + 3*
a^3*b^2*#1^2 - 16*a*b^3*#1^2 + 3*a^2*b*#1^4 + 12*b^2*#1^4 + a*#1^6) & ])/Sqrt[a] + (I*b*RootSum[a^5*b^4 + 4*a^
4*b^3*#1^2 + 16*a^2*b^4*#1^2 + 6*a^3*b^2*#1^4 - 32*a*b^3*#1^4 + 4*a^2*b*#1^6 + 16*b^2*#1^6 + a*#1^8 & , (-(a^3
*b^3*Log[(-I)*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + I*Sqrt[a]*(a*x^2 + Sqrt[b + a^2*x^4]) + #1]) - a^2
*b^2*Log[(-I)*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + I*Sqrt[a]*(a*x^2 + Sqrt[b + a^2*x^4]) + #1]*#1^2 +
a*b*Log[(-I)*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + I*Sqrt[a]*(a*x^2 + Sqrt[b + a^2*x^4]) + #1]*#1^4 +
Log[(-I)*Sqrt[2]*a*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + I*Sqrt[a]*(a*x^2 + Sqrt[b + a^2*x^4]) + #1]*#1^6)/(a^4
*b^3*#1 + 4*a^2*b^4*#1 + 3*a^3*b^2*#1^3 - 16*a*b^3*#1^3 + 3*a^2*b*#1^5 + 12*b^2*#1^5 + a*#1^7) & ])/Sqrt[2]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a^2*x^2 + b)*sqrt(a*x^2 + sqrt(a^2*x^4 + b))/(sqrt(a^2*x^4 + b)*(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 {\sqrt {\sqrt {a^2\,x^4+b}+a\,x^2}\,\left (a^2\,x^2+b\right )}{\left (b-a^2\,x^2\right )\,\sqrt {a^2\,x^4+b}} \,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^2))/((b - a^2*x^2)*(b + a^2*x^4)^(1/2)),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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