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

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

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Rubi [F]  time = 0.44, 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}}}{d+c x^2} \, 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),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*} \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx &=\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*}

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Mathematica [F]  time = 0.10, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.00, size = 398, normalized size = 1.00 \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

[Out]

Timed out

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

[Out]

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

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

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),x)

[Out]

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

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

[Out]

integrate(sqrt(a*x^2 + 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}}{c\,x^2+d} \,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),x)

[Out]

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

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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}}}{c x^{2} + d}\, 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),x)

[Out]

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

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