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

Optimal. Leaf size=460 \[ \frac {i \sqrt {a} d \text {RootSum}\left [\text {$\#$1}^4 c^2-4 i \text {$\#$1}^3 a d^2+2 \text {$\#$1}^2 b c^2+4 i \text {$\#$1} a b d^2+b^2 c^2\& ,\frac {b \log \left (-\text {$\#$1}+i \sqrt {a^2 x^4+b}+i \sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+i a x^2\right )-\text {$\#$1}^2 \log \left (-\text {$\#$1}+i \sqrt {a^2 x^4+b}+i \sqrt {2} \sqrt {a} x \sqrt {\sqrt {a^2 x^4+b}+a x^2}+i a x^2\right )}{\text {$\#$1}^3 c^2-3 i \text {$\#$1}^2 a d^2+\text {$\#$1} b c^2+i a b d^2}\& \right ]}{\sqrt {2}}-\frac {\sqrt {-\sqrt {a^2 d^4+b c^4}-a d^2} \tan ^{-1}\left (\frac {c \sqrt {\sqrt {a^2 x^4+b}+a x^2}}{\sqrt {-\sqrt {a^2 d^4+b c^4}-a d^2}}\right )}{\sqrt {a^2 d^4+b c^4}}+\frac {\sqrt {\sqrt {a^2 d^4+b c^4}-a d^2} \tan ^{-1}\left (\frac {c \sqrt {\sqrt {a^2 x^4+b}+a x^2}}{\sqrt {\sqrt {a^2 d^4+b c^4}-a d^2}}\right )}{\sqrt {a^2 d^4+b c^4}} \]

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Rubi [F]  time = 0.41, 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) \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)*Sqrt[b + a^2*x^4]),x]

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.09, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{(d+c x) \sqrt {b+a^2 x^4}} \, 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)*Sqrt[b + a^2*x^4]),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.00, size = 460, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {-a d^2-\sqrt {b c^4+a^2 d^4}} \tan ^{-1}\left (\frac {c \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {-a d^2-\sqrt {b c^4+a^2 d^4}}}\right )}{\sqrt {b c^4+a^2 d^4}}+\frac {\sqrt {-a d^2+\sqrt {b c^4+a^2 d^4}} \tan ^{-1}\left (\frac {c \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {-a d^2+\sqrt {b c^4+a^2 d^4}}}\right )}{\sqrt {b c^4+a^2 d^4}}+\frac {i \sqrt {a} d \text {RootSum}\left [b^2 c^2+4 i a b d^2 \text {$\#$1}+2 b c^2 \text {$\#$1}^2-4 i a d^2 \text {$\#$1}^3+c^2 \text {$\#$1}^4\&,\frac {b \log \left (i a x^2+i \sqrt {b+a^2 x^4}+i \sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}-\text {$\#$1}\right )-\log \left (i a x^2+i \sqrt {b+a^2 x^4}+i \sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}-\text {$\#$1}\right ) \text {$\#$1}^2}{i a b d^2+b c^2 \text {$\#$1}-3 i a d^2 \text {$\#$1}^2+c^2 \text {$\#$1}^3}\&\right ]}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[-(a*d^2) - Sqrt[b*c^4 + a^2*d^4]]*ArcTan[(c*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[-(a*d^2) - Sqrt[b*c^
4 + a^2*d^4]]])/Sqrt[b*c^4 + a^2*d^4]) + (Sqrt[-(a*d^2) + Sqrt[b*c^4 + a^2*d^4]]*ArcTan[(c*Sqrt[a*x^2 + Sqrt[b
+ a^2*x^4]])/Sqrt[-(a*d^2) + Sqrt[b*c^4 + a^2*d^4]]])/Sqrt[b*c^4 + a^2*d^4] + (I*Sqrt[a]*d*RootSum[b^2*c^2 +
(4*I)*a*b*d^2*#1 + 2*b*c^2*#1^2 - (4*I)*a*d^2*#1^3 + c^2*#1^4 & , (b*Log[I*a*x^2 + I*Sqrt[b + a^2*x^4] + I*Sqr
t[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1] - 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)/(I*a*b*d^2 + b*c^2*#1 - (3*I)*a*d^2*#1^2 + c^2*#1^3) & ])/Sqrt[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*x^2+(a^2*x^4+b)^(1/2))^(1/2)/(c*x+d)/(a^2*x^4+b)^(1/2),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}}}{\sqrt {a^{2} x^{4} + b} {\left (c x + d\right )}}\,{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+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 + d)), x)

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

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+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+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*} \int \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{\sqrt {a^{2} x^{4} + b} {\left (c x + d\right )}}\,{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+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 + 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}\,\left (d+c\,x\right )} \,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)),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)), 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} \left (c x + 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+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 + d)), x)

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