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

Optimal. Leaf size=472 \[ -\frac {\sqrt {2} \text {RootSum}\left [\text {$\#$1}^4 c+4 \text {$\#$1}^3 a d-2 \text {$\#$1}^2 b c+4 \text {$\#$1} a b d+b^2 c\& ,\frac {\text {$\#$1}^2 a c d \log \left (-\text {$\#$1}+\sqrt {a^2 x^4+b}+\sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+a x^2\right )+a b c d \log \left (-\text {$\#$1}+\sqrt {a^2 x^4+b}+\sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+a x^2\right )+2 \text {$\#$1} a^2 d^2 \log \left (-\text {$\#$1}+\sqrt {a^2 x^4+b}+\sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+a x^2\right )}{\text {$\#$1}^3 c+3 \text {$\#$1}^2 a d-\text {$\#$1} b c+a b d}\& \right ]}{\sqrt {a} c}+\frac {2 \sqrt {2} \sqrt {a} d \tan ^{-1}\left (\frac {\sqrt {a^2 x^4+b}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}}{\sqrt {b}}+\frac {a x^2}{\sqrt {b}}\right )}{\sqrt {b} c}+\frac {x}{2 \sqrt {\sqrt {a^2 x^4+b}+a x^2}}+\frac {\log \left (\sqrt {a^2 x^4+b}+\sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+a x^2\right )}{2 \sqrt {2} \sqrt {a}} \]

________________________________________________________________________________________

Rubi [F]  time = 0.80, 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 {-d+c x^2}{\left (d+c x^2\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[Int][1/Sqrt[a*x^2 + Sqrt[b + a^2*x^4]], x] - Sqrt[d]*Defer[Int][1/((Sqrt[d] - Sqrt[-c]*x)*Sqrt[a*x^2 + S
qrt[b + a^2*x^4]]), x] - Sqrt[d]*Defer[Int][1/((Sqrt[d] + Sqrt[-c]*x)*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]), x]

Rubi steps

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

________________________________________________________________________________________

Mathematica [F]  time = 0.16, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-d+c x^2}{\left (d+c x^2\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.00, size = 463, normalized size = 0.98 \begin {gather*} \frac {x}{2 \sqrt {a x^2+\sqrt {b+a^2 x^4}}}-\frac {2 \sqrt {2} \sqrt {a} d \tan ^{-1}\left (\frac {a x^2}{\sqrt {b}}+\frac {\sqrt {b+a^2 x^4}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b}}\right )}{\sqrt {b} c}-\frac {\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 )}{2 \sqrt {2} \sqrt {a}}+\frac {\sqrt {2} \sqrt {a} d \text {RootSum}\left [b^2 c-4 a b d \text {$\#$1}-2 b c \text {$\#$1}^2-4 a d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {b c \log \left (a x^2+\sqrt {b+a^2 x^4}-\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+\text {$\#$1}\right )-2 a d \log \left (a x^2+\sqrt {b+a^2 x^4}-\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+\text {$\#$1}\right ) \text {$\#$1}+c \log \left (a x^2+\sqrt {b+a^2 x^4}-\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+\text {$\#$1}\right ) \text {$\#$1}^2}{a b d+b c \text {$\#$1}+3 a d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x/(2*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]) - (2*Sqrt[2]*Sqrt[a]*d*ArcTan[(a*x^2)/Sqrt[b] + Sqrt[b + a^2*x^4]/Sqrt[b
] - (Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[b]])/(Sqrt[b]*c) - 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]]]/(2*Sqrt[2]*Sqrt[a]) + (Sqrt[2]*Sqrt[a]*d*RootSum[b^2*c -
4*a*b*d*#1 - 2*b*c*#1^2 - 4*a*d*#1^3 + c*#1^4 & , (b*c*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]] + #1] - 2*a*d*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]] + #1]*#1 + c*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]] +
#1]*#1^2)/(a*b*d + b*c*#1 + 3*a*d*#1^2 - c*#1^3) & ])/c

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________