3.27.44 \(\int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{(d+c x^2) \sqrt {b+a^2 x^4}} \, dx\) [2644]

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

[Out]

Unintegrable

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Rubi [F]
time = 1.36, 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 {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\left (d+c x^2\right ) \sqrt {b+a^2 x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[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*} \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 \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*}

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Mathematica [A]
time = 8.76, size = 340, normalized size = 1.45 \begin {gather*} \frac {\sqrt {-\sqrt {b} c-\sqrt {b c^2+a^2 d^2}} \left (-\sqrt {b} c-a d+\sqrt {b c^2+a^2 d^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {-\sqrt {b} c-\sqrt {b c^2+a^2 d^2}} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {d} \left (\sqrt {b}+a x^2+\sqrt {b+a^2 x^4}\right )}\right )+\sqrt {-\sqrt {b} c+\sqrt {b c^2+a^2 d^2}} \left (\sqrt {b} c+a d+\sqrt {b c^2+a^2 d^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {-\sqrt {b} c+\sqrt {b c^2+a^2 d^2}} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {d} \left (\sqrt {b}+a x^2+\sqrt {b+a^2 x^4}\right )}\right )}{\sqrt {2} a d^{3/2} \sqrt {b c^2+a^2 d^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-(Sqrt[b]*c) - Sqrt[b*c^2 + a^2*d^2]]*(-(Sqrt[b]*c) - a*d + Sqrt[b*c^2 + a^2*d^2])*ArcTanh[(Sqrt[2]*Sqrt
[-(Sqrt[b]*c) - Sqrt[b*c^2 + a^2*d^2]]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/(Sqrt[d]*(Sqrt[b] + a*x^2 + Sqrt[b +
a^2*x^4]))] + Sqrt[-(Sqrt[b]*c) + Sqrt[b*c^2 + a^2*d^2]]*(Sqrt[b]*c + a*d + Sqrt[b*c^2 + a^2*d^2])*ArcTanh[(S
qrt[2]*Sqrt[-(Sqrt[b]*c) + Sqrt[b*c^2 + a^2*d^2]]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/(Sqrt[d]*(Sqrt[b] + a*x^2
+ Sqrt[b + a^2*x^4]))])/(Sqrt[2]*a*d^(3/2)*Sqrt[b*c^2 + a^2*d^2])

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F(-1)] Timed out
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((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

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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}}}{\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((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)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \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)/((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)

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