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

Optimal. Leaf size=549 \[ -\frac {\sqrt {a} \text {RootSum}\left [\text {$\#$1}^4 c+4 \text {$\#$1}^3 a d-2 \text {$\#$1}^2 b c+4 \text {$\#$1} a b d+b^2 c\& ,\frac {\text {$\#$1}^2 b c^2 \log \left (-\text {$\#$1}+\sqrt {a^2 x^4+b}+\sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+a x^2\right )+\text {$\#$1}^2 a^2 d^2 \log \left (-\text {$\#$1}+\sqrt {a^2 x^4+b}+\sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+a x^2\right )+b^2 c^2 \log \left (-\text {$\#$1}+\sqrt {a^2 x^4+b}+\sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+a x^2\right )+a^2 b d^2 \log \left (-\text {$\#$1}+\sqrt {a^2 x^4+b}+\sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+a x^2\right )}{\text {$\#$1}^3 (-c)-3 \text {$\#$1}^2 a d+\text {$\#$1} b c-a b d}\& \right ]}{\sqrt {2} c^2}+\frac {a x \sqrt {\sqrt {a^2 x^4+b}+a x^2}}{2 c}-\frac {\sqrt {a} \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a^2 x^4+b}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}}{\sqrt {b}}+\frac {a x^2}{\sqrt {b}}\right )}{\sqrt {2} c}-\frac {a^{3/2} d \log \left (\sqrt {a^2 x^4+b}+\sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+a x^2\right )}{\sqrt {2} c^2} \]

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Rubi [F]  time = 1.29, 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 {b+a^2 x^4} \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[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*} \int \frac {\sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx &=\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*}

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Mathematica [F]  time = 0.11, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {b+a^2 x^4} \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[b + a^2*x^4]*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/(d + c*x^2),x]

[Out]

Integrate[(Sqrt[b + a^2*x^4]*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 = 544, normalized size = 0.99 \begin {gather*} \frac {a x \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{2 c}+\frac {\sqrt {a} \sqrt {b} \tan ^{-1}\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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^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

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^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)

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

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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}\,\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]

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)

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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}}{c x^{2} + d}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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