\(\int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx\) [2995]

Optimal result

Integrand size = 33, antiderivative size = 398 \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx=\frac {\sqrt {a} \log \left (i a x^2+i \sqrt {b+a^2 x^4}+i \sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{\sqrt {2} c}-\frac {\sqrt {a} \text {RootSum}\left [b^2 c-4 i a b d \text {$\#$1}+2 b c \text {$\#$1}^2+4 i a d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {a b d \log \left (i a x^2+i \sqrt {b+a^2 x^4}+i \sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}-\text {$\#$1}\right )+2 i b c \log \left (i a x^2+i \sqrt {b+a^2 x^4}+i \sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}-\text {$\#$1}\right ) \text {$\#$1}-a d \log \left (i a x^2+i \sqrt {b+a^2 x^4}+i \sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}-\text {$\#$1}\right ) \text {$\#$1}^2}{a b d+i b c \text {$\#$1}-3 a d \text {$\#$1}^2+i c \text {$\#$1}^3}\&\right ]}{\sqrt {2} c} \]

[Out]

Unintegrable

Rubi [F]

\[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx=\int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 2.09 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx=\frac {\sqrt {a} \left (\log \left (i \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )\right )-\text {RootSum}\left [b^2 c-4 i a b d \text {$\#$1}+2 b c \text {$\#$1}^2+4 i a d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {a b d \log \left (i \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+i \text {$\#$1}\right )\right )+2 i b c \log \left (i \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+i \text {$\#$1}\right )\right ) \text {$\#$1}-a d \log \left (i \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+i \text {$\#$1}\right )\right ) \text {$\#$1}^2}{a b d+i b c \text {$\#$1}-3 a d \text {$\#$1}^2+i c \text {$\#$1}^3}\&\right ]\right )}{\sqrt {2} c} \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.07

\[\int \frac {\sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}}{c \,x^{2}+d}d x\]

[In]

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

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [N/A]

Not integrable

Time = 0.53 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.07 \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx=\int \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{c x^{2} + d}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.08 \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx=\int { \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{c x^{2} + d} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.52 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.08 \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx=\int { \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{c x^{2} + d} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 7.32 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.08 \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx=\int \frac {\sqrt {\sqrt {a^2\,x^4+b}+a\,x^2}}{c\,x^2+d} \,d x \]

[In]

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

[Out]

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