3.802 \(\int \frac{1}{\sqrt [4]{1+x}+\sqrt{1+x}} \, dx\)

Optimal. Leaf size=31 \[ 2 \sqrt{x+1}-4 \sqrt [4]{x+1}+4 \log \left (\sqrt [4]{x+1}+1\right ) \]

[Out]

-4*(1 + x)^(1/4) + 2*Sqrt[1 + x] + 4*Log[1 + (1 + x)^(1/4)]

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Rubi [A]  time = 0.034163, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ 2 \sqrt{x+1}-4 \sqrt [4]{x+1}+4 \log \left (\sqrt [4]{x+1}+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[((1 + x)^(1/4) + Sqrt[1 + x])^(-1),x]

[Out]

-4*(1 + x)^(1/4) + 2*Sqrt[1 + x] + 4*Log[1 + (1 + x)^(1/4)]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - 4 \sqrt [4]{x + 1} + 4 \log{\left (\sqrt [4]{x + 1} + 1 \right )} + 4 \int ^{\sqrt [4]{x + 1}} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/((1+x)**(1/4)+(1+x)**(1/2)),x)

[Out]

-4*(x + 1)**(1/4) + 4*log((x + 1)**(1/4) + 1) + 4*Integral(x, (x, (x + 1)**(1/4)
))

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Mathematica [A]  time = 0.0143116, size = 31, normalized size = 1. \[ 2 \sqrt{x+1}-4 \sqrt [4]{x+1}+4 \log \left (\sqrt [4]{x+1}+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((1 + x)^(1/4) + Sqrt[1 + x])^(-1),x]

[Out]

-4*(1 + x)^(1/4) + 2*Sqrt[1 + x] + 4*Log[1 + (1 + x)^(1/4)]

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Maple [A]  time = 0.011, size = 26, normalized size = 0.8 \[ -4\,\sqrt [4]{1+x}+4\,\ln \left ( 1+\sqrt [4]{1+x} \right ) +2\,\sqrt{1+x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/((1+x)^(1/4)+(1+x)^(1/2)),x)

[Out]

-4*(1+x)^(1/4)+4*ln(1+(1+x)^(1/4))+2*(1+x)^(1/2)

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Maxima [A]  time = 0.704884, size = 34, normalized size = 1.1 \[ 2 \, \sqrt{x + 1} - 4 \,{\left (x + 1\right )}^{\frac{1}{4}} + 4 \, \log \left ({\left (x + 1\right )}^{\frac{1}{4}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x + 1) + (x + 1)^(1/4)),x, algorithm="maxima")

[Out]

2*sqrt(x + 1) - 4*(x + 1)^(1/4) + 4*log((x + 1)^(1/4) + 1)

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Fricas [A]  time = 0.271504, size = 34, normalized size = 1.1 \[ 2 \, \sqrt{x + 1} - 4 \,{\left (x + 1\right )}^{\frac{1}{4}} + 4 \, \log \left ({\left (x + 1\right )}^{\frac{1}{4}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x + 1) + (x + 1)^(1/4)),x, algorithm="fricas")

[Out]

2*sqrt(x + 1) - 4*(x + 1)^(1/4) + 4*log((x + 1)^(1/4) + 1)

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Sympy [A]  time = 0.588333, size = 27, normalized size = 0.87 \[ - 4 \sqrt [4]{x + 1} + 2 \sqrt{x + 1} + 4 \log{\left (\sqrt [4]{x + 1} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((1+x)**(1/4)+(1+x)**(1/2)),x)

[Out]

-4*(x + 1)**(1/4) + 2*sqrt(x + 1) + 4*log((x + 1)**(1/4) + 1)

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GIAC/XCAS [A]  time = 0.262126, size = 34, normalized size = 1.1 \[ 2 \, \sqrt{x + 1} - 4 \,{\left (x + 1\right )}^{\frac{1}{4}} + 4 \,{\rm ln}\left ({\left (x + 1\right )}^{\frac{1}{4}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x + 1) + (x + 1)^(1/4)),x, algorithm="giac")

[Out]

2*sqrt(x + 1) - 4*(x + 1)^(1/4) + 4*ln((x + 1)^(1/4) + 1)