Optimal. Leaf size=31 \[ -4 \sqrt [4]{1+x}+2 \sqrt {1+x}+4 \log \left (1+\sqrt [4]{1+x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2037, 1607,
272, 45} \begin {gather*} 2 \sqrt {x+1}-4 \sqrt [4]{x+1}+4 \log \left (\sqrt [4]{x+1}+1\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 272
Rule 1607
Rule 2037
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [4]{1+x}+\sqrt {1+x}} \, dx &=\text {Subst}\left (\int \frac {1}{\sqrt [4]{x}+\sqrt {x}} \, dx,x,1+x\right )\\ &=\text {Subst}\left (\int \frac {1}{\left (1+\sqrt [4]{x}\right ) \sqrt [4]{x}} \, dx,x,1+x\right )\\ &=4 \text {Subst}\left (\int \frac {x^2}{1+x} \, dx,x,\sqrt [4]{1+x}\right )\\ &=4 \text {Subst}\left (\int \left (-1+x+\frac {1}{1+x}\right ) \, dx,x,\sqrt [4]{1+x}\right )\\ &=-4 \sqrt [4]{1+x}+2 \sqrt {1+x}+4 \log \left (1+\sqrt [4]{1+x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 31, normalized size = 1.00 \begin {gather*} 2 \sqrt [4]{1+x} \left (-2+\sqrt [4]{1+x}\right )+4 \log \left (1+\sqrt [4]{1+x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.02, size = 26, normalized size = 0.84
method | result | size |
derivativedivides | \(-4 \left (1+x \right )^{\frac {1}{4}}+4 \ln \left (1+\left (1+x \right )^{\frac {1}{4}}\right )+2 \sqrt {1+x}\) | \(26\) |
default | \(-4 \left (1+x \right )^{\frac {1}{4}}+4 \ln \left (1+\left (1+x \right )^{\frac {1}{4}}\right )+2 \sqrt {1+x}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.37, size = 25, normalized size = 0.81 \begin {gather*} 2 \, \sqrt {x + 1} - 4 \, {\left (x + 1\right )}^{\frac {1}{4}} + 4 \, \log \left ({\left (x + 1\right )}^{\frac {1}{4}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.32, size = 25, normalized size = 0.81 \begin {gather*} 2 \, \sqrt {x + 1} - 4 \, {\left (x + 1\right )}^{\frac {1}{4}} + 4 \, \log \left ({\left (x + 1\right )}^{\frac {1}{4}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.10, size = 27, normalized size = 0.87 \begin {gather*} - 4 \sqrt [4]{x + 1} + 2 \sqrt {x + 1} + 4 \log {\left (\sqrt [4]{x + 1} + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 4.10, size = 25, normalized size = 0.81 \begin {gather*} 2 \, \sqrt {x + 1} - 4 \, {\left (x + 1\right )}^{\frac {1}{4}} + 4 \, \log \left ({\left (x + 1\right )}^{\frac {1}{4}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.07, size = 25, normalized size = 0.81 \begin {gather*} 4\,\ln \left ({\left (x+1\right )}^{1/4}+1\right )+2\,\sqrt {x+1}-4\,{\left (x+1\right )}^{1/4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________