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

Optimal. Leaf size=463 \[ \frac {\sqrt {2} \sqrt {a} d \text {RootSum}\left [\text {$\#$1}^4 c+4 \text {$\#$1}^3 a d-2 \text {$\#$1}^2 b c+4 \text {$\#$1} a b d+b^2 c\& ,\frac {\text {$\#$1}^2 a d \log \left (-\text {$\#$1}+\sqrt {a^2 x^4+b}+\sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+a x^2\right )-2 \text {$\#$1} b c \log \left (-\text {$\#$1}+\sqrt {a^2 x^4+b}+\sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+a x^2\right )+a b d \log \left (-\text {$\#$1}+\sqrt {a^2 x^4+b}+\sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+a x^2\right )}{\text {$\#$1}^3 c+3 \text {$\#$1}^2 a d-\text {$\#$1} b c+a b d}\& \right ]}{c}-\frac {\sqrt {2} \sqrt {a} d \log \left (\sqrt {a^2 x^4+b}+\sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+a x^2\right )}{c}+\frac {1}{2} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a^2 x^4+b}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}}{\sqrt {b}}+\frac {a x^2}{\sqrt {b}}\right )}{\sqrt {2} \sqrt {a}} \]

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Rubi [F]  time = 0.73, 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 {\left (-d+c x^2\right ) \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[((-d + c*x^2)*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]], x] - Sqrt[d]*Defer[Int][Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/(Sqrt[d] -
Sqrt[-c]*x), x] - Sqrt[d]*Defer[Int][Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/(Sqrt[d] + Sqrt[-c]*x), x]

Rubi steps

\begin {align*} \int \frac {\left (-d+c x^2\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx &=\int \left (\sqrt {a x^2+\sqrt {b+a^2 x^4}}-\frac {2 d \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2}\right ) \, dx\\ &=-\left ((2 d) \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx\right )+\int \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx\\ &=-\left ((2 d) \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\right )+\int \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx\\ &=-\left (\sqrt {d} \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {d}-\sqrt {-c} x} \, dx\right )-\sqrt {d} \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {d}+\sqrt {-c} x} \, dx+\int \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.16, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-d+c x^2\right ) \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[((-d + c*x^2)*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/(d + c*x^2),x]

[Out]

Integrate[((-d + c*x^2)*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 = 463, normalized size = 1.00 \begin {gather*} \frac {1}{2} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}-\frac {\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} \sqrt {a}}+\frac {\sqrt {2} \sqrt {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 )}{c}-\frac {\sqrt {2} \sqrt {a} d \text {RootSum}\left [b^2 c-4 a b d \text {$\#$1}-2 b c \text {$\#$1}^2-4 a d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {a b d \log \left (a x^2+\sqrt {b+a^2 x^4}-\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+\text {$\#$1}\right )+2 b c \log \left (a x^2+\sqrt {b+a^2 x^4}-\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+\text {$\#$1}\right ) \text {$\#$1}+a d \log \left (a x^2+\sqrt {b+a^2 x^4}-\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+\text {$\#$1}\right ) \text {$\#$1}^2}{a b d+b c \text {$\#$1}+3 a d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/2 - (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]*Sqrt[a]) + (Sqrt[2]*Sqrt[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]]])/c - (Sqrt[2]*Sqrt[a]*d*RootSum[b^2*c - 4*a*
b*d*#1 - 2*b*c*#1^2 - 4*a*d*#1^3 + c*#1^4 & , (a*b*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]] + #1] + 2*b*c*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 + 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]] +
#1]*#1^2)/(a*b*d + b*c*#1 + 3*a*d*#1^2 - c*#1^3) & ])/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((c*x^2-d)*(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}} {\left (c x^{2} - d\right )}}{c x^{2} + d}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2-d)*(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)/(c*x^2 + d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((c*x^2-d)*(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}} {\left (c x^{2} - d\right )}}{c x^{2} + d}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2-d)*(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)/(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 (d-c\,x^2\right )}{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))/(d + c*x^2),x)

[Out]

int(-(((b + a^2*x^4)^(1/2) + a*x^2)^(1/2)*(d - c*x^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}} \left (c x^{2} - d\right )}{c x^{2} + d}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2-d)*(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)/(c*x**2 + d), x)

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