\(\int \frac {\sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx\) [3098]

Optimal result

Integrand size = 46, antiderivative size = 549 \[ \int \frac {\sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx=\frac {a x \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{2 c}-\frac {\sqrt {a} \sqrt {b} \arctan \left (\frac {a x^2}{\sqrt {b}}+\frac {\sqrt {b+a^2 x^4}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b}}\right )}{\sqrt {2} c}-\frac {a^{3/2} d \log \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{\sqrt {2} c^2}-\frac {\sqrt {a} \text {RootSum}\left [b^2 c+4 a b d \text {$\#$1}-2 b c \text {$\#$1}^2+4 a d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {b^2 c^2 \log \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}-\text {$\#$1}\right )+a^2 b d^2 \log \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}-\text {$\#$1}\right )+b c^2 \log \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}-\text {$\#$1}\right ) \text {$\#$1}^2+a^2 d^2 \log \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-a b d+b c \text {$\#$1}-3 a d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]}{\sqrt {2} c^2} \]

[Out]

Unintegrable

Rubi [F]

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

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{2 \sqrt {d} \left (\sqrt {d}-\sqrt {-c} x\right )}+\frac {\sqrt {b+a^2 x^4} \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 {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {d}-\sqrt {-c} x} \, dx}{2 \sqrt {d}}+\frac {\int \frac {\sqrt {b+a^2 x^4} \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 = 0.00 (sec) , antiderivative size = 392, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx=\frac {\sqrt {a} \left (\sqrt {a} c x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+\sqrt {2} \sqrt {b} c \arctan \left (\frac {a x^2+\sqrt {b+a^2 x^4}-\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b}}\right )+\sqrt {2} a d \log \left (a x^2+\sqrt {b+a^2 x^4}-\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )-\sqrt {2} \left (b c^2+a^2 d^2\right ) \text {RootSum}\left [b^2 c-4 a b d \text {$\#$1}-2 b c \text {$\#$1}^2-4 a d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {b \log \left (a x^2+\sqrt {b+a^2 x^4}-\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+\text {$\#$1}\right )+\log \left (a x^2+\sqrt {b+a^2 x^4}-\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+\text {$\#$1}\right ) \text {$\#$1}^2}{a b d+b c \text {$\#$1}+3 a d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]\right )}{2 c^2} \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.00 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.07

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

integrate((a^2*x^4+b)^(1/2)*(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 = 0.63 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.07 \[ \int \frac {\sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx=\int \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} \sqrt {a^{2} x^{4} + b}}{c x^{2} + d}\, dx \]

[In]

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

Maxima [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.08 \[ \int \frac {\sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx=\int { \frac {\sqrt {a^{2} x^{4} + b} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{c x^{2} + d} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.95 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.08 \[ \int \frac {\sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx=\int { \frac {\sqrt {a^{2} x^{4} + b} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{c x^{2} + d} \,d x } \]

[In]

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

[Out]

integrate(sqrt(a^2*x^4 + b)*sqrt(a*x^2 + 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 = 0.08 \[ \int \frac {\sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx=\int \frac {\sqrt {\sqrt {a^2\,x^4+b}+a\,x^2}\,\sqrt {a^2\,x^4+b}}{c\,x^2+d} \,d x \]

[In]

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

[Out]

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