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

Optimal. Leaf size=1356 \[ \frac {a^3 \tan ^{-1}\left (\frac {c \sqrt {a x^2+\sqrt {a^2 x^4+b}}}{\sqrt {-a d^2-\sqrt {b c^4+a^2 d^4}}}\right ) d^5}{\left (b c^4+a^2 d^4\right )^{3/2} \sqrt {-a d^2-\sqrt {b c^4+a^2 d^4}}}-\frac {a^3 \tan ^{-1}\left (\frac {c \sqrt {a x^2+\sqrt {a^2 x^4+b}}}{\sqrt {\sqrt {b c^4+a^2 d^4}-a d^2}}\right ) d^5}{\left (b c^4+a^2 d^4\right )^{3/2} \sqrt {\sqrt {b c^4+a^2 d^4}-a d^2}}+\frac {a^2 \tan ^{-1}\left (\frac {c \sqrt {a x^2+\sqrt {a^2 x^4+b}}}{\sqrt {-a d^2-\sqrt {b c^4+a^2 d^4}}}\right ) d^3}{\left (b c^4+a^2 d^4\right ) \sqrt {-a d^2-\sqrt {b c^4+a^2 d^4}}}+\frac {a^2 \tan ^{-1}\left (\frac {c \sqrt {a x^2+\sqrt {a^2 x^4+b}}}{\sqrt {\sqrt {b c^4+a^2 d^4}-a d^2}}\right ) d^3}{\left (b c^4+a^2 d^4\right ) \sqrt {\sqrt {b c^4+a^2 d^4}-a d^2}}+\frac {a c \sqrt {a x^2+\sqrt {a^2 x^4+b}} d^3}{\left (b c^4+a^2 d^4\right ) \left (c^2 x^2-d^2\right )}+\frac {a^{3/2} \text {RootSum}\left [c^2 \text {$\#$1}^4-4 i a d^2 \text {$\#$1}^3+2 b c^2 \text {$\#$1}^2+4 i a b d^2 \text {$\#$1}+b^2 c^2\& ,\frac {2 i b \log \left (i a x^2+i \sqrt {2} \sqrt {a} \sqrt {a x^2+\sqrt {a^2 x^4+b}} x-\text {$\#$1}+i \sqrt {a^2 x^4+b}\right ) \text {$\#$1} c^2-a d^2 \log \left (i a x^2+i \sqrt {2} \sqrt {a} \sqrt {a x^2+\sqrt {a^2 x^4+b}} x-\text {$\#$1}+i \sqrt {a^2 x^4+b}\right ) \text {$\#$1}^2+a b d^2 \log \left (i a x^2+i \sqrt {2} \sqrt {a} \sqrt {a x^2+\sqrt {a^2 x^4+b}} x-\text {$\#$1}+i \sqrt {a^2 x^4+b}\right )}{-i b \text {$\#$1}^3 c^6-i b^2 \text {$\#$1} c^6+a b^2 d^2 c^4-3 a b d^2 \text {$\#$1}^2 c^4-i a^2 d^4 \text {$\#$1}^3 c^2-i a^2 b d^4 \text {$\#$1} c^2+a^3 b d^6-3 a^3 d^6 \text {$\#$1}^2}\& \right ] d^2}{\sqrt {2}}-\frac {a b c^4 \tan ^{-1}\left (\frac {c \sqrt {a x^2+\sqrt {a^2 x^4+b}}}{\sqrt {-a d^2-\sqrt {b c^4+a^2 d^4}}}\right ) d}{\left (b c^4+a^2 d^4\right )^{3/2} \sqrt {-a d^2-\sqrt {b c^4+a^2 d^4}}}+\frac {a b c^4 \tan ^{-1}\left (\frac {c \sqrt {a x^2+\sqrt {a^2 x^4+b}}}{\sqrt {\sqrt {b c^4+a^2 d^4}-a d^2}}\right ) d}{\left (b c^4+a^2 d^4\right )^{3/2} \sqrt {\sqrt {b c^4+a^2 d^4}-a d^2}}+\frac {b c^3 d}{\left (b c^4+a^2 d^4\right ) \left (c^2 x^2-d^2\right ) \sqrt {a x^2+\sqrt {a^2 x^4+b}}}+\frac {i \sqrt {a} x \left (\frac {i a^{3/2} c^2 x^2 d^2}{b c^4+a^2 d^4}+\frac {i \sqrt {a} c^2 \sqrt {a^2 x^4+b} d^2}{b c^4+a^2 d^4}\right )}{\left (c^2 x^2-d^2\right ) \sqrt {a x^2+\sqrt {a^2 x^4+b}}}-\frac {b c^4 x}{\left (b c^4+a^2 d^4\right ) \left (c^2 x^2-d^2\right ) \sqrt {a x^2+\sqrt {a^2 x^4+b}}} \]

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Rubi [F]  time = 0.63, 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 \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]]/((d + c*x)^2*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)^2 \sqrt {b+a^2 x^4}} \, dx &=\int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{(d+c x)^2 \sqrt {b+a^2 x^4}} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.19, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{(d+c x)^2 \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)^2*Sqrt[b + a^2*x^4]),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 8.88, size = 2109, normalized size = 1.56 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*c^3*d)/((b*c^4 + a^2*d^4)*(-d^2 + c^2*x^2)*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]) - (b*c^4*x)/((b*c^4 + a^2*d^4)*
(-d^2 + c^2*x^2)*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]) + (a*c*d^3*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/((b*c^4 + a^2*d^
4)*(-d^2 + c^2*x^2)) + (I*Sqrt[a]*x*((I*a^(3/2)*c^2*d^2*x^2)/(b*c^4 + a^2*d^4) + (I*Sqrt[a]*c^2*d^2*Sqrt[b + a
^2*x^4])/(b*c^4 + a^2*d^4)))/((-d^2 + c^2*x^2)*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]) - (a*b*c^4*d*ArcTan[(c*Sqrt[a*
x^2 + Sqrt[b + a^2*x^4]])/Sqrt[-(a*d^2) - Sqrt[b*c^4 + a^2*d^4]]])/((b*c^4 + a^2*d^4)^(3/2)*Sqrt[-(a*d^2) - Sq
rt[b*c^4 + a^2*d^4]]) + (a^3*d^5*ArcTan[(c*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[-(a*d^2) - Sqrt[b*c^4 + a^2*d
^4]]])/((b*c^4 + a^2*d^4)^(3/2)*Sqrt[-(a*d^2) - Sqrt[b*c^4 + a^2*d^4]]) + (a^2*d^3*ArcTan[(c*Sqrt[a*x^2 + Sqrt
[b + a^2*x^4]])/Sqrt[-(a*d^2) - Sqrt[b*c^4 + a^2*d^4]]])/((b*c^4 + a^2*d^4)*Sqrt[-(a*d^2) - Sqrt[b*c^4 + a^2*d
^4]]) + (a*b*c^4*d*ArcTan[(c*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[-(a*d^2) + Sqrt[b*c^4 + a^2*d^4]]])/((b*c^4
+ a^2*d^4)^(3/2)*Sqrt[-(a*d^2) + Sqrt[b*c^4 + a^2*d^4]]) - (a^3*d^5*ArcTan[(c*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]
)/Sqrt[-(a*d^2) + Sqrt[b*c^4 + a^2*d^4]]])/((b*c^4 + a^2*d^4)^(3/2)*Sqrt[-(a*d^2) + Sqrt[b*c^4 + a^2*d^4]]) +
(a^2*d^3*ArcTan[(c*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[-(a*d^2) + Sqrt[b*c^4 + a^2*d^4]]])/((b*c^4 + a^2*d^4
)*Sqrt[-(a*d^2) + Sqrt[b*c^4 + a^2*d^4]]) - (I*Sqrt[a]*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*c^4*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] + 32*a^2*d^4*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] - (8*I)*a*c^2*d^2*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 - c^4*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]*c^4) + (I*Sqrt[a]*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^2*c^8*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] + 34*a^2*b*c^4*d^4*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] + 32*a^4*d^8*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] - (6*I)*a*b*c^6*d^2*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 - (8*I)*a^3*c^2*d^6*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 - b*c^8*Log[I*a*x^2 + I*Sq
rt[b + a^2*x^4] + I*Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] - #1]*#1^2 - 2*a^2*c^4*d^4*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^2*c^4*d^2 + I*a^
3*b*d^6 + b^2*c^6*#1 + a^2*b*c^2*d^4*#1 - (3*I)*a*b*c^4*d^2*#1^2 - (3*I)*a^3*d^6*#1^2 + b*c^6*#1^3 + a^2*c^2*d
^4*#1^3) & ])/(Sqrt[2]*c^4)

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

[Out]

Timed out

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giac [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)^2/(a^2*x^4+b)^(1/2),x, algorithm="giac")

[Out]

Timed out

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

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

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