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

Optimal. Leaf size=151 \[ \frac {2 a^{5/2} x^6+2 a^{3/2} x^4 \sqrt {a^2 x^4+b}+3 \sqrt {a} b x^2}{8 \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}}-\frac {11 b \log \left (\sqrt {a^2 x^4+b}+\sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+a x^2\right )}{8 \sqrt {2} \sqrt {a}} \]

________________________________________________________________________________________

Rubi [F]  time = 1.24, 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^4\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [F]  time = 0.29, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-b+a^2 x^4\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.84, size = 151, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt {a} b x^2+2 a^{5/2} x^6+2 a^{3/2} x^4 \sqrt {b+a^2 x^4}}{8 \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}}-\frac {11 b \log \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{8 \sqrt {2} \sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*Sqrt[a]*b*x^2 + 2*a^(5/2)*x^6 + 2*a^(3/2)*x^4*Sqrt[b + a^2*x^4])/(8*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4
]]) - (11*b*Log[a*x^2 + Sqrt[b + a^2*x^4] + Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]])/(8*Sqrt[2]*Sqr
t[a])

________________________________________________________________________________________

fricas [A]  time = 2.40, size = 236, normalized size = 1.56 \begin {gather*} \left [\frac {11 \, \sqrt {2} \sqrt {a} b \log \left (4 \, a^{2} x^{4} + 4 \, \sqrt {a^{2} x^{4} + b} a x^{2} - 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x^{3} + \sqrt {2} \sqrt {a^{2} x^{4} + b} \sqrt {a} x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} + b\right ) - 4 \, {\left (a^{2} x^{3} - 3 \, \sqrt {a^{2} x^{4} + b} a x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{32 \, a}, \frac {11 \, \sqrt {2} \sqrt {-a} b \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} \sqrt {-a}}{2 \, a x}\right ) - 2 \, {\left (a^{2} x^{3} - 3 \, \sqrt {a^{2} x^{4} + b} a x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{16 \, a}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/32*(11*sqrt(2)*sqrt(a)*b*log(4*a^2*x^4 + 4*sqrt(a^2*x^4 + b)*a*x^2 - 2*(sqrt(2)*a^(3/2)*x^3 + sqrt(2)*sqrt(
a^2*x^4 + b)*sqrt(a)*x)*sqrt(a*x^2 + sqrt(a^2*x^4 + b)) + b) - 4*(a^2*x^3 - 3*sqrt(a^2*x^4 + b)*a*x)*sqrt(a*x^
2 + sqrt(a^2*x^4 + b)))/a, 1/16*(11*sqrt(2)*sqrt(-a)*b*arctan(1/2*sqrt(2)*sqrt(a*x^2 + sqrt(a^2*x^4 + b))*sqrt
(-a)/(a*x)) - 2*(a^2*x^3 - 3*sqrt(a^2*x^4 + b)*a*x)*sqrt(a*x^2 + sqrt(a^2*x^4 + b)))/a]

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a^{2} x^{4} - b\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{\sqrt {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)*(a*x^2+(a^2*x^4+b)^(1/2))^(1/2)/(a^2*x^4+b)^(1/2),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a^{2} x^{4} - b\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{\sqrt {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)*(a*x^2+(a^2*x^4+b)^(1/2))^(1/2)/(a^2*x^4+b)^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\sqrt {\sqrt {a^2\,x^4+b}+a\,x^2}\,\left (b-a^2\,x^4\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^4))/(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^4))/(b + a^2*x^4)^(1/2), x)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________