Optimal. Leaf size=76 \[ \frac {4 \left (1+2 \sqrt {x}\right )}{15 \left (1+\sqrt {x}+x\right )^{5/2}}+\frac {64 \left (1+2 \sqrt {x}\right )}{135 \left (1+\sqrt {x}+x\right )^{3/2}}+\frac {512 \left (1+2 \sqrt {x}\right )}{405 \sqrt {1+\sqrt {x}+x}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1366, 628, 627}
\begin {gather*} \frac {512 \left (2 \sqrt {x}+1\right )}{405 \sqrt {x+\sqrt {x}+1}}+\frac {64 \left (2 \sqrt {x}+1\right )}{135 \left (x+\sqrt {x}+1\right )^{3/2}}+\frac {4 \left (2 \sqrt {x}+1\right )}{15 \left (x+\sqrt {x}+1\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 627
Rule 628
Rule 1366
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x} \left (1+\sqrt {x}+x\right )^{7/2}} \, dx &=2 \text {Subst}\left (\int \frac {1}{\left (1+x+x^2\right )^{7/2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {4 \left (1+2 \sqrt {x}\right )}{15 \left (1+\sqrt {x}+x\right )^{5/2}}+\frac {32}{15} \text {Subst}\left (\int \frac {1}{\left (1+x+x^2\right )^{5/2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {4 \left (1+2 \sqrt {x}\right )}{15 \left (1+\sqrt {x}+x\right )^{5/2}}+\frac {64 \left (1+2 \sqrt {x}\right )}{135 \left (1+\sqrt {x}+x\right )^{3/2}}+\frac {256}{135} \text {Subst}\left (\int \frac {1}{\left (1+x+x^2\right )^{3/2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {4 \left (1+2 \sqrt {x}\right )}{15 \left (1+\sqrt {x}+x\right )^{5/2}}+\frac {64 \left (1+2 \sqrt {x}\right )}{135 \left (1+\sqrt {x}+x\right )^{3/2}}+\frac {512 \left (1+2 \sqrt {x}\right )}{405 \sqrt {1+\sqrt {x}+x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.23, size = 49, normalized size = 0.64 \begin {gather*} \frac {4 \left (1+2 \sqrt {x}\right ) \left (203+304 \sqrt {x}+432 x+256 x^{3/2}+128 x^2\right )}{405 \left (1+\sqrt {x}+x\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.01, size = 53, normalized size = 0.70
method | result | size |
derivativedivides | \(\frac {\frac {4}{15}+\frac {8 \sqrt {x}}{15}}{\left (1+x +\sqrt {x}\right )^{\frac {5}{2}}}+\frac {\frac {64}{135}+\frac {128 \sqrt {x}}{135}}{\left (1+x +\sqrt {x}\right )^{\frac {3}{2}}}+\frac {\frac {512}{405}+\frac {1024 \sqrt {x}}{405}}{\sqrt {1+x +\sqrt {x}}}\) | \(53\) |
default | \(\frac {\frac {4}{15}+\frac {8 \sqrt {x}}{15}}{\left (1+x +\sqrt {x}\right )^{\frac {5}{2}}}+\frac {\frac {64}{135}+\frac {128 \sqrt {x}}{135}}{\left (1+x +\sqrt {x}\right )^{\frac {3}{2}}}+\frac {\frac {512}{405}+\frac {1024 \sqrt {x}}{405}}{\sqrt {1+x +\sqrt {x}}}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 95, normalized size = 1.25 \begin {gather*} -\frac {4 \, {\left (128 \, x^{5} + 272 \, x^{4} + 455 \, x^{3} + 232 \, x^{2} - {\left (256 \, x^{5} + 736 \, x^{4} + 1366 \, x^{3} + 1427 \, x^{2} + 839 \, x + 101\right )} \sqrt {x} - 128 \, x - 203\right )} \sqrt {x + \sqrt {x} + 1}}{405 \, {\left (x^{6} + 3 \, x^{5} + 6 \, x^{4} + 7 \, x^{3} + 6 \, x^{2} + 3 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {x} \left (\sqrt {x} + x + 1\right )^{\frac {7}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 3.28, size = 45, normalized size = 0.59 \begin {gather*} \frac {4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, \sqrt {x} {\left (2 \, \sqrt {x} + 5\right )} + 35\right )} \sqrt {x} + 65\right )} \sqrt {x} + 355\right )} \sqrt {x} + 203\right )}}{405 \, {\left (x + \sqrt {x} + 1\right )}^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {x}\,{\left (x+\sqrt {x}+1\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________