Optimal. Leaf size=198 \[ \frac {8 \left (35 a x+35 b-28 c^2+36 c d^2-16 d^4\right ) \sqrt {\sqrt {\sqrt {a x+b}+c}+d}}{315 a}-\frac {64 \left (2 c d-d^3\right ) \sqrt {\sqrt {a x+b}+c} \sqrt {\sqrt {\sqrt {a x+b}+c}+d}}{315 a}+\sqrt {a x+b} \left (\frac {8 \left (7 c-6 d^2\right ) \sqrt {\sqrt {\sqrt {a x+b}+c}+d}}{315 a}+\frac {8 d \sqrt {\sqrt {a x+b}+c} \sqrt {\sqrt {\sqrt {a x+b}+c}+d}}{63 a}\right ) \]
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Rubi [A] time = 0.15, antiderivative size = 129, normalized size of antiderivative = 0.65, 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 )^{5/2}}{5 a}+\frac {8 d \left (c-d^2\right ) \left (\sqrt {\sqrt {a x+b}+c}+d\right )^{3/2}}{3 a}+\frac {8 \left (\sqrt {\sqrt {a x+b}+c}+d\right )^{9/2}}{9 a}-\frac {24 d \left (\sqrt {\sqrt {a x+b}+c}+d\right )^{7/2}}{7 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 772
Rule 1398
Rubi steps
\begin {align*} \int \sqrt {d+\sqrt {c+\sqrt {b+a x}}} \, dx &=\frac {2 \operatorname {Subst}\left (\int x \sqrt {d+\sqrt {c+x}} \, dx,x,\sqrt {b+a x}\right )}{a}\\ &=\frac {2 \operatorname {Subst}\left (\int \sqrt {d+\sqrt {x}} (-c+x) \, dx,x,c+\sqrt {b+a x}\right )}{a}\\ &=\frac {4 \operatorname {Subst}\left (\int x \sqrt {d+x} \left (-c+x^2\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=\frac {4 \operatorname {Subst}\left (\int \left (-d \left (-c+d^2\right ) \sqrt {d+x}+\left (-c+3 d^2\right ) (d+x)^{3/2}-3 d (d+x)^{5/2}+(d+x)^{7/2}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a}\\ &=\frac {8 d \left (c-d^2\right ) \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{3/2}}{3 a}-\frac {8 \left (c-3 d^2\right ) \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{5/2}}{5 a}-\frac {24 d \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{7/2}}{7 a}+\frac {8 \left (d+\sqrt {c+\sqrt {b+a x}}\right )^{9/2}}{9 a}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 114, normalized size = 0.58 \begin {gather*} \frac {8 \left (\sqrt {\sqrt {a x+b}+c}+d\right )^{3/2} \left (24 d^2 \sqrt {\sqrt {a x+b}+c}-28 c \sqrt {\sqrt {a x+b}+c}+35 \sqrt {a x+b} \sqrt {\sqrt {a x+b}+c}-30 d \sqrt {a x+b}+12 c d-16 d^3\right )}{315 a} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 144, normalized size = 0.73 \begin {gather*} -\frac {8 \sqrt {c+\sqrt {b+a x}} \left (16 c d-8 d^3-5 d \sqrt {b+a x}\right ) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{315 a}-\frac {8 \left (28 c^2-36 c d^2+16 d^4-7 c \sqrt {b+a x}+6 d^2 \sqrt {b+a x}-35 (b+a x)\right ) \sqrt {d+\sqrt {c+\sqrt {b+a x}}}}{315 a} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 94, normalized size = 0.47 \begin {gather*} -\frac {8 \, {\left (16 \, d^{4} - 36 \, c d^{2} + 28 \, c^{2} - 35 \, a x + {\left (6 \, d^{2} - 7 \, c\right )} \sqrt {a x + b} - {\left (8 \, d^{3} - 16 \, c d + 5 \, \sqrt {a x + b} d\right )} \sqrt {c + \sqrt {a x + b}} - 35 \, b\right )} \sqrt {d + \sqrt {c + \sqrt {a x + b}}}}{315 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.66, size = 444, normalized size = 2.24 \begin {gather*} \frac {8 \, {\left (35 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {9}{2}} - 180 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {7}{2}} d + 378 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {d + \sqrt {c + \sqrt {a x + b}}} d^{4} - 126 \, c {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {5}{2}} + 420 \, c {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {3}{2}} d - 630 \, c \sqrt {d + \sqrt {c + \sqrt {a x + b}}} d^{2} + 315 \, c^{2} \sqrt {d + \sqrt {c + \sqrt {a x + b}}} + 21 \, {\left (3 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {5}{2}} - 10 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {3}{2}} d + 15 \, \sqrt {d + \sqrt {c + \sqrt {a x + b}}} d^{2} - 15 \, c \sqrt {d + \sqrt {c + \sqrt {a x + b}}}\right )} c + 3 \, {\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 )} d\right )}}{315 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 93, normalized size = 0.47
method | result | size |
derivativedivides | \(\frac {\frac {8 \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {9}{2}}}{9}-\frac {24 d \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {7}{2}}}{7}+\frac {8 \left (3 d^{2}-c \right ) \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {5}{2}}}{5}-\frac {8 \left (d^{2}-c \right ) d \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {3}{2}}}{3}}{a}\) | \(93\) |
default | \(\frac {\frac {8 \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {9}{2}}}{9}-\frac {24 d \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {7}{2}}}{7}+\frac {8 \left (3 d^{2}-c \right ) \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {5}{2}}}{5}-\frac {8 \left (d^{2}-c \right ) d \left (d +\sqrt {c +\sqrt {a x +b}}\right )^{\frac {3}{2}}}{3}}{a}\) | \(93\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 92, normalized size = 0.46 \begin {gather*} \frac {8 \, {\left (35 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {9}{2}} - 135 \, {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {7}{2}} d + 63 \, {\left (3 \, d^{2} - c\right )} {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {5}{2}} - 105 \, {\left (d^{3} - c d\right )} {\left (d + \sqrt {c + \sqrt {a x + b}}\right )}^{\frac {3}{2}}\right )}}{315 \, 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 \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 \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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