3.28.5 \(\int \frac {\sqrt {b+a x} \sqrt {1+\sqrt {b+a x}}}{x^2 \sqrt {1+\sqrt {1+\sqrt {b+a x}}}} \, dx\) [2705]

Optimal. Leaf size=246 \[ -\frac {\sqrt {1+\sqrt {1+\sqrt {b+a x}}}}{x}-\frac {\sqrt {b+a x} \sqrt {1+\sqrt {1+\sqrt {b+a x}}}}{x}+\frac {\sqrt {1+\sqrt {b+a x}} \sqrt {1+\sqrt {1+\sqrt {b+a x}}}}{x}+\frac {1}{8} a \text {RootSum}\left [b-4 \text {$\#$1}^4+4 \text {$\#$1}^6-\text {$\#$1}^8\& ,\frac {2 \log \left (\sqrt {1+\sqrt {1+\sqrt {b+a x}}}-\text {$\#$1}\right )-9 \log \left (\sqrt {1+\sqrt {1+\sqrt {b+a x}}}-\text {$\#$1}\right ) \text {$\#$1}^2+5 \log \left (\sqrt {1+\sqrt {1+\sqrt {b+a x}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\& \right ] \]

[Out]

Unintegrable

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Rubi [F]
time = 2.23, 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 x} \sqrt {1+\sqrt {b+a x}}}{x^2 \sqrt {1+\sqrt {1+\sqrt {b+a x}}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(Sqrt[b + a*x]*Sqrt[1 + Sqrt[b + a*x]])/(x^2*Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]]),x]

[Out]

(-376*Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]])/(105*x) + (112*(1 + Sqrt[1 + Sqrt[b + a*x]])^(3/2))/(15*x) - (8*(1 +
Sqrt[1 + Sqrt[b + a*x]])^(5/2))/(3*x) - (376*a*b*Defer[Subst][Defer[Int][(b - 4*x^4 + 4*x^6 - x^8)^(-2), x], x
, Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]]])/105 + (112*a*b*Defer[Subst][Defer[Int][x^2/(b - 4*x^4 + 4*x^6 - x^8)^2,
x], x, Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]]])/5 - (8*a*(144 + 175*b)*Defer[Subst][Defer[Int][x^4/(b - 4*x^4 + 4*x
^6 - x^8)^2, x], x, Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]]])/105 + (192*a*Defer[Subst][Defer[Int][x^6/(-b + 4*x^4 -
4*x^6 + x^8)^2, x], x, Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]]])/35

Rubi steps

\begin {align*} \int \frac {\sqrt {b+a x} \sqrt {1+\sqrt {b+a x}}}{x^2 \sqrt {1+\sqrt {1+\sqrt {b+a x}}}} \, dx &=(2 a) \text {Subst}\left (\int \frac {x^2 \sqrt {1+x}}{\left (b-x^2\right )^2 \sqrt {1+\sqrt {1+x}}} \, dx,x,\sqrt {b+a x}\right )\\ &=(4 a) \text {Subst}\left (\int \frac {x^2 \left (-1+x^2\right )^2}{\sqrt {1+x} \left (b-\left (-1+x^2\right )^2\right )^2} \, dx,x,\sqrt {1+\sqrt {b+a x}}\right )\\ &=(4 a) \text {Subst}\left (\int \frac {(-1+x)^2 x^2 (1+x)^{3/2}}{\left (b-\left (-1+x^2\right )^2\right )^2} \, dx,x,\sqrt {1+\sqrt {b+a x}}\right )\\ &=(8 a) \text {Subst}\left (\int \frac {x^4 \left (2-3 x^2+x^4\right )^2}{\left (b-x^4 \left (-2+x^2\right )^2\right )^2} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {b+a x}}}\right )\\ &=-\frac {8 \left (1+\sqrt {1+\sqrt {b+a x}}\right )^{5/2}}{3 x}+\frac {1}{3} (8 a) \text {Subst}\left (\int \frac {(12-5 b) x^4-36 x^6+43 x^8-14 x^{10}}{\left (b-x^4 \left (-2+x^2\right )^2\right )^2} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {b+a x}}}\right )\\ &=\frac {112 \left (1+\sqrt {1+\sqrt {b+a x}}\right )^{3/2}}{15 x}-\frac {8 \left (1+\sqrt {1+\sqrt {b+a x}}\right )^{5/2}}{3 x}+\frac {1}{15} (8 a) \text {Subst}\left (\int \frac {42 b x^2+5 (12-5 b) x^4-124 x^6+47 x^8}{\left (b-x^4 \left (-2+x^2\right )^2\right )^2} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {b+a x}}}\right )\\ &=-\frac {376 \sqrt {1+\sqrt {1+\sqrt {b+a x}}}}{105 x}+\frac {112 \left (1+\sqrt {1+\sqrt {b+a x}}\right )^{3/2}}{15 x}-\frac {8 \left (1+\sqrt {1+\sqrt {b+a x}}\right )^{5/2}}{3 x}+\frac {1}{105} (8 a) \text {Subst}\left (\int \frac {-47 b+294 b x^2-(144+175 b) x^4+72 x^6}{\left (b-x^4 \left (-2+x^2\right )^2\right )^2} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {b+a x}}}\right )\\ &=-\frac {376 \sqrt {1+\sqrt {1+\sqrt {b+a x}}}}{105 x}+\frac {112 \left (1+\sqrt {1+\sqrt {b+a x}}\right )^{3/2}}{15 x}-\frac {8 \left (1+\sqrt {1+\sqrt {b+a x}}\right )^{5/2}}{3 x}+\frac {1}{105} (8 a) \text {Subst}\left (\int \left (-\frac {47 b}{\left (b-4 x^4+4 x^6-x^8\right )^2}+\frac {294 b x^2}{\left (b-4 x^4+4 x^6-x^8\right )^2}-\frac {(144+175 b) x^4}{\left (b-4 x^4+4 x^6-x^8\right )^2}+\frac {72 x^6}{\left (-b+4 x^4-4 x^6+x^8\right )^2}\right ) \, dx,x,\sqrt {1+\sqrt {1+\sqrt {b+a x}}}\right )\\ &=-\frac {376 \sqrt {1+\sqrt {1+\sqrt {b+a x}}}}{105 x}+\frac {112 \left (1+\sqrt {1+\sqrt {b+a x}}\right )^{3/2}}{15 x}-\frac {8 \left (1+\sqrt {1+\sqrt {b+a x}}\right )^{5/2}}{3 x}+\frac {1}{35} (192 a) \text {Subst}\left (\int \frac {x^6}{\left (-b+4 x^4-4 x^6+x^8\right )^2} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {b+a x}}}\right )-\frac {1}{105} (376 a b) \text {Subst}\left (\int \frac {1}{\left (b-4 x^4+4 x^6-x^8\right )^2} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {b+a x}}}\right )+\frac {1}{5} (112 a b) \text {Subst}\left (\int \frac {x^2}{\left (b-4 x^4+4 x^6-x^8\right )^2} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {b+a x}}}\right )-\frac {1}{105} (8 a (144+175 b)) \text {Subst}\left (\int \frac {x^4}{\left (b-4 x^4+4 x^6-x^8\right )^2} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {b+a x}}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 299, normalized size = 1.22 \begin {gather*} \frac {\sqrt {1+\sqrt {1+\sqrt {b+a x}}} \left (-1-\sqrt {b+a x}+\sqrt {1+\sqrt {b+a x}}\right )}{x}+a \text {RootSum}\left [b-4 \text {$\#$1}^4+4 \text {$\#$1}^6-\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {1+\sqrt {1+\sqrt {b+a x}}}-\text {$\#$1}\right )+\log \left (\sqrt {1+\sqrt {1+\sqrt {b+a x}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-2 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]-\frac {1}{8} a \text {RootSum}\left [b-4 \text {$\#$1}^4+4 \text {$\#$1}^6-\text {$\#$1}^8\&,\frac {6 \log \left (\sqrt {1+\sqrt {1+\sqrt {b+a x}}}-\text {$\#$1}\right )-7 \log \left (\sqrt {1+\sqrt {1+\sqrt {b+a x}}}-\text {$\#$1}\right ) \text {$\#$1}^2+3 \log \left (\sqrt {1+\sqrt {1+\sqrt {b+a x}}}-\text {$\#$1}\right ) \text {$\#$1}^4}{2 \text {$\#$1}^3-3 \text {$\#$1}^5+\text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[b + a*x]*Sqrt[1 + Sqrt[b + a*x]])/(x^2*Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]]),x]

[Out]

(Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]]*(-1 - Sqrt[b + a*x] + Sqrt[1 + Sqrt[b + a*x]]))/x + a*RootSum[b - 4*#1^4 +
4*#1^6 - #1^8 & , (-Log[Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]] - #1] + Log[Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]] - #1]*
#1^2)/(-2*#1^3 + #1^5) & ] - (a*RootSum[b - 4*#1^4 + 4*#1^6 - #1^8 & , (6*Log[Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]
] - #1] - 7*Log[Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]] - #1]*#1^2 + 3*Log[Sqrt[1 + Sqrt[1 + Sqrt[b + a*x]]] - #1]*#
1^4)/(2*#1^3 - 3*#1^5 + #1^7) & ])/8

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 1.
time = 0.31, size = 188, normalized size = 0.76

method	result	size
derivativedivides	\(2 a \left (\frac {\frac {\left (1+\sqrt {1+\sqrt {a x +b}}\right )^{\frac {5}{2}}}{2}-\frac {3 \left (1+\sqrt {1+\sqrt {a x +b}}\right )^{\frac {3}{2}}}{2}+\sqrt {1+\sqrt {1+\sqrt {a x +b}}}}{-\left (1+\sqrt {1+\sqrt {a x +b}}\right )^{4}+4 \left (1+\sqrt {1+\sqrt {a x +b}}\right )^{3}-4 \left (1+\sqrt {1+\sqrt {a x +b}}\right )^{2}+b}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+4 \textit {\_Z}^{4}-b \right )}{\sum }\frac {\left (-5 \textit {\_R}^{4}+9 \textit {\_R}^{2}-2\right ) \ln \left (\sqrt {1+\sqrt {1+\sqrt {a x +b}}}-\textit {\_R} \right )}{-\textit {\_R}^{7}+3 \textit {\_R}^{5}-2 \textit {\_R}^{3}}\right )}{16}\right )\)	\(188\)
default	\(2 a \left (\frac {\frac {\left (1+\sqrt {1+\sqrt {a x +b}}\right )^{\frac {5}{2}}}{2}-\frac {3 \left (1+\sqrt {1+\sqrt {a x +b}}\right )^{\frac {3}{2}}}{2}+\sqrt {1+\sqrt {1+\sqrt {a x +b}}}}{-\left (1+\sqrt {1+\sqrt {a x +b}}\right )^{4}+4 \left (1+\sqrt {1+\sqrt {a x +b}}\right )^{3}-4 \left (1+\sqrt {1+\sqrt {a x +b}}\right )^{2}+b}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+4 \textit {\_Z}^{4}-b \right )}{\sum }\frac {\left (-5 \textit {\_R}^{4}+9 \textit {\_R}^{2}-2\right ) \ln \left (\sqrt {1+\sqrt {1+\sqrt {a x +b}}}-\textit {\_R} \right )}{-\textit {\_R}^{7}+3 \textit {\_R}^{5}-2 \textit {\_R}^{3}}\right )}{16}\right )\)	\(188\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+b)^(1/2)*(1+(a*x+b)^(1/2))^(1/2)/x^2/(1+(1+(a*x+b)^(1/2))^(1/2))^(1/2),x,method=_RETURNVERBOSE)

[Out]

2*a*(4*(1/8*(1+(1+(a*x+b)^(1/2))^(1/2))^(5/2)-3/8*(1+(1+(a*x+b)^(1/2))^(1/2))^(3/2)+1/4*(1+(1+(a*x+b)^(1/2))^(
1/2))^(1/2))/(-(1+(1+(a*x+b)^(1/2))^(1/2))^4+4*(1+(1+(a*x+b)^(1/2))^(1/2))^3-4*(1+(1+(a*x+b)^(1/2))^(1/2))^2+b
)+1/16*sum((-5*_R^4+9*_R^2-2)/(-_R^7+3*_R^5-2*_R^3)*ln((1+(1+(a*x+b)^(1/2))^(1/2))^(1/2)-_R),_R=RootOf(_Z^8-4*
_Z^6+4*_Z^4-b)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+b)^(1/2)*(1+(a*x+b)^(1/2))^(1/2)/x^2/(1+(1+(a*x+b)^(1/2))^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a*x + b)*sqrt(sqrt(a*x + b) + 1)/(x^2*sqrt(sqrt(sqrt(a*x + b) + 1) + 1)), x)

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Fricas [F(-1)] Timed out
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+b)^(1/2)*(1+(a*x+b)^(1/2))^(1/2)/x^2/(1+(1+(a*x+b)^(1/2))^(1/2))^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)] Timed out
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+b)**(1/2)*(1+(a*x+b)**(1/2))**(1/2)/x**2/(1+(1+(a*x+b)**(1/2))**(1/2))**(1/2),x)

[Out]

Timed out

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Giac [F(-1)] Timed out
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+b)^(1/2)*(1+(a*x+b)^(1/2))^(1/2)/x^2/(1+(1+(a*x+b)^(1/2))^(1/2))^(1/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {\sqrt {b+a\,x}+1}\,\sqrt {b+a\,x}}{x^2\,\sqrt {\sqrt {\sqrt {b+a\,x}+1}+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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