Optimal. Leaf size=174 \[ \frac {32 \left (13 c d-12 d^3\right ) \sqrt {\sqrt {\sqrt {a x+b}+c}+d}}{105 a}-\frac {32 \left (5 c-6 d^2\right ) \sqrt {\sqrt {a x+b}+c} \sqrt {\sqrt {\sqrt {a x+b}+c}+d}}{105 a}+\sqrt {a x+b} \left (\frac {8 \sqrt {\sqrt {a x+b}+c} \sqrt {\sqrt {\sqrt {a x+b}+c}+d}}{7 a}-\frac {48 d \sqrt {\sqrt {\sqrt {a x+b}+c}+d}}{35 a}\right ) \]
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Rubi [A] time = 0.16, antiderivative size = 127, normalized size of antiderivative = 0.73, number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {371, 1398, 772} \begin {gather*} -\frac {8 \left (c-3 d^2\right ) \left (\sqrt {\sqrt {a x+b}+c}+d\right )^{3/2}}{3 a}+\frac {8 d \left (c-d^2\right ) \sqrt {\sqrt {\sqrt {a x+b}+c}+d}}{a}+\frac {8 \left (\sqrt {\sqrt {a x+b}+c}+d\right )^{7/2}}{7 a}-\frac {24 d \left (\sqrt {\sqrt {a x+b}+c}+d\right )^{5/2}}{5 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 772
Rule 1398
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {d+\sqrt {c+\sqrt {b+a x}}}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {x}{\sqrt {d+\sqrt {c+x}}} \, dx,x,\sqrt {b+a x}\right )}{a}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {-c+x}{\sqrt {d+\sqrt {x}}} \, dx,x,c+\sqrt {b+a x}\right )}{a}\\ &=\frac {4 \operatorname {Subst}\left (\int \frac {x \left (-c+x^2\right )}{\sqrt {d+x}} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=\frac {4 \operatorname {Subst}\left (\int \left (-\frac {d \left (-c+d^2\right )}{\sqrt {d+x}}+\left (-c+3 d^2\right ) \sqrt {d+x}-3 d (d+x)^{3/2}+(d+x)^{5/2}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=\frac {8 d \left (c-d^2\right ) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{a}-\frac {8 \left (c-3 d^2\right ) \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{3/2}}{3 a}-\frac {24 d \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{5/2}}{5 a}+\frac {8 \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{7/2}}{7 a}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 114, normalized size = 0.66 \begin {gather*} \frac {8 \sqrt {\sqrt {\sqrt {a x+b}+c}+d} \left (24 d^2 \sqrt {\sqrt {a x+b}+c}-20 c \sqrt {\sqrt {a x+b}+c}+15 \sqrt {a x+b} \sqrt {\sqrt {a x+b}+c}-18 d \sqrt {a x+b}+52 c d-48 d^3\right )}{105 a} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 114, normalized size = 0.66 \begin {gather*} -\frac {8 \left (20 c-24 d^2-15 \sqrt {b+a x}\right ) \sqrt {c+\sqrt {b+a x}} \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{105 a}+\frac {16 \left (26 c d-24 d^3-9 d \sqrt {b+a x}\right ) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{105 a} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 71, normalized size = 0.41 \begin {gather*} -\frac {8 \, {\left (48 \, d^{3} - 52 \, c d - {\left (24 \, d^{2} - 20 \, c + 15 \, \sqrt {a x + b}\right )} \sqrt {c + \sqrt {a x + b}} + 18 \, \sqrt {a x + b} d\right )} \sqrt {d + \sqrt {c + \sqrt {a x + b}}}}{105 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 4.50, size = 190, normalized size = 1.09 \begin {gather*} \frac {8 \, {\left (15 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {7}{2}} \mathrm {sgn}\left (\sqrt {c + \sqrt {a x + b}}\right ) - 63 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {5}{2}} d \mathrm {sgn}\left (\sqrt {c + \sqrt {a x + b}}\right ) + 105 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {3}{2}} d^{2} \mathrm {sgn}\left (\sqrt {c + \sqrt {a x + b}}\right ) - 105 \, \sqrt {d + \sqrt {c + \sqrt {a x + b}}} d^{3} \mathrm {sgn}\left (\sqrt {c + \sqrt {a x + b}}\right ) - 35 \, c {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\sqrt {c + \sqrt {a x + b}}\right ) + 105 \, c \sqrt {d + \sqrt {c + \sqrt {a x + b}}} d \mathrm {sgn}\left (\sqrt {c + \sqrt {a x + b}}\right )\right )}}{105 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 93, normalized size = 0.53
method | result | size |
derivativedivides | \(\frac {\frac {8 \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {7}{2}}}{7}-\frac {24 d \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {5}{2}}}{5}+\frac {8 \left (3 d^{2}-c \right ) \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {3}{2}}}{3}-8 \left (d^{2}-c \right ) d \sqrt {d +\sqrt {c +\sqrt {a x +b}}}}{a}\) | \(93\) |
default | \(\frac {\frac {8 \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {7}{2}}}{7}-\frac {24 d \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {5}{2}}}{5}+\frac {8 \left (3 d^{2}-c \right ) \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {3}{2}}}{3}-8 \left (d^{2}-c \right ) d \sqrt {d +\sqrt {c +\sqrt {a x +b}}}}{a}\) | \(93\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 92, normalized size = 0.53 \begin {gather*} \frac {8 \, {\left (15 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {7}{2}} - 63 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {5}{2}} d + 35 \, {\left (3 \, d^{2} - c\right )} {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {3}{2}} - 105 \, {\left (d^{3} - c d\right )} \sqrt {d + \sqrt {c + \sqrt {a x + b}}}\right )}}{105 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {d+\sqrt {c+\sqrt {b+a\,x}}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {d + \sqrt {c + \sqrt {a x + b}}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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