\(\int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{(d+c x^2) \sqrt {b+a^2 x^4}} \, dx\) [2645]

Optimal result

Integrand size = 46, antiderivative size = 235 \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\left (d+c x^2\right ) \sqrt {b+a^2 x^4}} \, dx=-\frac {i \sqrt {a} \text {RootSum}\left [b^2 c-4 i a b d \text {$\#$1}+2 b c \text {$\#$1}^2+4 i a d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {b \log \left (i a x^2+i \sqrt {b+a^2 x^4}+i \sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}-\text {$\#$1}\right )-\log \left (i a x^2+i \sqrt {b+a^2 x^4}+i \sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-i a b d+b c \text {$\#$1}+3 i a d \text {$\#$1}^2+c \text {$\#$1}^3}\&\right ]}{\sqrt {2}} \]

[Out]

Unintegrable

Rubi [F]

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

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\left (d+c x^2\right ) \sqrt {b+a^2 x^4}} \, dx=-\frac {i \sqrt {a} \text {RootSum}\left [b^2 c-4 i a b d \text {$\#$1}+2 b c \text {$\#$1}^2+4 i a d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {b \log \left (i \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+i \text {$\#$1}\right )\right )-\log \left (i \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+i \text {$\#$1}\right )\right ) \text {$\#$1}^2}{-i a b d+b c \text {$\#$1}+3 i a d \text {$\#$1}^2+c \text {$\#$1}^3}\&\right ]}{\sqrt {2}} \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.00 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.17

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [N/A]

Not integrable

Time = 1.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.17 \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\left (d+c x^2\right ) \sqrt {b+a^2 x^4}} \, dx=\int \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{\sqrt {a^{2} x^{4} + b} \left (c x^{2} + d\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\left (d+c x^2\right ) \sqrt {b+a^2 x^4}} \, dx=\int { \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{\sqrt {a^{2} x^{4} + b} {\left (c x^{2} + d\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 1.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\left (d+c x^2\right ) \sqrt {b+a^2 x^4}} \, dx=\int { \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{\sqrt {a^{2} x^{4} + b} {\left (c x^{2} + d\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\left (d+c x^2\right ) \sqrt {b+a^2 x^4}} \, dx=\int \frac {\sqrt {\sqrt {a^2\,x^4+b}+a\,x^2}}{\left (c\,x^2+d\right )\,\sqrt {a^2\,x^4+b}} \,d x \]

[In]

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

[Out]

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