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3.93 \(\int \frac{\left (x+x^2\right )^{3/2}}{1+x^2} \, dx\)

Optimal. Leaf size=130 \[ \frac{1}{4} \sqrt{x^2+x} (2 x+5)+\sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{-x+\sqrt{2}+1}{\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{x^2+x}}\right )-\sqrt{\sqrt{2}-1} \tanh ^{-1}\left (\frac{-x-\sqrt{2}+1}{\sqrt{2 \left (\sqrt{2}-1\right )} \sqrt{x^2+x}}\right )-\frac{5}{4} \tanh ^{-1}\left (\frac{x}{\sqrt{x^2+x}}\right ) \]

[Out]

((5 + 2*x)*Sqrt[x + x^2])/4 + Sqrt[1 + Sqrt[2]]*ArcTan[(1 + Sqrt[2] - x)/(Sqrt[2
*(1 + Sqrt[2])]*Sqrt[x + x^2])] - Sqrt[-1 + Sqrt[2]]*ArcTanh[(1 - Sqrt[2] - x)/(
Sqrt[2*(-1 + Sqrt[2])]*Sqrt[x + x^2])] - (5*ArcTanh[x/Sqrt[x + x^2]])/4

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Rubi [A]  time = 0.38194, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529 \[ \frac{1}{4} \sqrt{x^2+x} (2 x+5)+\sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{-x+\sqrt{2}+1}{\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{x^2+x}}\right )-\sqrt{\sqrt{2}-1} \tanh ^{-1}\left (\frac{-x-\sqrt{2}+1}{\sqrt{2 \left (\sqrt{2}-1\right )} \sqrt{x^2+x}}\right )-\frac{5}{4} \tanh ^{-1}\left (\frac{x}{\sqrt{x^2+x}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(x + x^2)^(3/2)/(1 + x^2),x]

[Out]

((5 + 2*x)*Sqrt[x + x^2])/4 + Sqrt[1 + Sqrt[2]]*ArcTan[(1 + Sqrt[2] - x)/(Sqrt[2
*(1 + Sqrt[2])]*Sqrt[x + x^2])] - Sqrt[-1 + Sqrt[2]]*ArcTanh[(1 - Sqrt[2] - x)/(
Sqrt[2*(-1 + Sqrt[2])]*Sqrt[x + x^2])] - (5*ArcTanh[x/Sqrt[x + x^2]])/4

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Rubi in Sympy [A]  time = 39.4762, size = 141, normalized size = 1.08 \[ \frac{\left (x + \frac{5}{2}\right ) \sqrt{x^{2} + x}}{2} - \frac{\left (2 + 2 \sqrt{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (x - \sqrt{2} - 1\right )}{2 \sqrt{1 + \sqrt{2}} \sqrt{x^{2} + x}} \right )}}{2 \sqrt{1 + \sqrt{2}}} - \frac{5 \operatorname{atanh}{\left (\frac{x}{\sqrt{x^{2} + x}} \right )}}{4} - \frac{\left (- 2 \sqrt{2} + 2\right ) \operatorname{atanh}{\left (\frac{\sqrt{2} \left (x - 1 + \sqrt{2}\right )}{2 \sqrt{-1 + \sqrt{2}} \sqrt{x^{2} + x}} \right )}}{2 \sqrt{-1 + \sqrt{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2+x)**(3/2)/(x**2+1),x)

[Out]

(x + 5/2)*sqrt(x**2 + x)/2 - (2 + 2*sqrt(2))*atan(sqrt(2)*(x - sqrt(2) - 1)/(2*s
qrt(1 + sqrt(2))*sqrt(x**2 + x)))/(2*sqrt(1 + sqrt(2))) - 5*atanh(x/sqrt(x**2 +
x))/4 - (-2*sqrt(2) + 2)*atanh(sqrt(2)*(x - 1 + sqrt(2))/(2*sqrt(-1 + sqrt(2))*s
qrt(x**2 + x)))/(2*sqrt(-1 + sqrt(2)))

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Mathematica [C]  time = 0.199655, size = 124, normalized size = 0.95 \[ \frac{\sqrt{x} \sqrt{x+1} \left (2 \sqrt{x+1} x^{3/2}+5 \sqrt{x+1} \sqrt{x}-4 \sqrt{2-2 i} \tan ^{-1}\left ((1-i)^{3/2} \sqrt{\frac{x}{2 x+2}}\right )-4 \sqrt{2+2 i} \tan ^{-1}\left ((1+i)^{3/2} \sqrt{\frac{x}{2 x+2}}\right )-5 \sinh ^{-1}\left (\sqrt{x}\right )\right )}{4 \sqrt{x (x+1)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x + x^2)^(3/2)/(1 + x^2),x]

[Out]

(Sqrt[x]*Sqrt[1 + x]*(5*Sqrt[x]*Sqrt[1 + x] + 2*x^(3/2)*Sqrt[1 + x] - 5*ArcSinh[
Sqrt[x]] - 4*Sqrt[2 - 2*I]*ArcTan[(1 - I)^(3/2)*Sqrt[x/(2 + 2*x)]] - 4*Sqrt[2 +
2*I]*ArcTan[(1 + I)^(3/2)*Sqrt[x/(2 + 2*x)]]))/(4*Sqrt[x*(1 + x)])

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Maple [B]  time = 0.08, size = 789, normalized size = 6.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+x)^(3/2)/(x^2+1),x)

[Out]

1/2*x*(x^2+x)^(1/2)+5/4*(x^2+x)^(1/2)-5/8*ln(1/2+x+(x^2+x)^(1/2))+1/2*(4*(-2^(1/
2)-1+x)^2/(1-x-2^(1/2))^2-3*2^(1/2)*(-2^(1/2)-1+x)^2/(1-x-2^(1/2))^2+4+3*2^(1/2)
)^(1/2)*2^(1/2)*((-2+2*2^(1/2))^(1/2)*arctan(1/2*(-2^(1/2)-1+x)/(1-x-2^(1/2))*(-
4+3*2^(1/2))*(24*(-2^(1/2)-1+x)^2/(1-x-2^(1/2))^2+17*2^(1/2)*(-2^(1/2)-1+x)^2/(1
-x-2^(1/2))^2-2^(1/2))*(-2+2*2^(1/2))^(1/2)*((-4+3*2^(1/2))*(-(-2^(1/2)-1+x)^2/(
1-x-2^(1/2))^2+12*2^(1/2)+17))^(1/2)/((-2^(1/2)-1+x)^4/(1-x-2^(1/2))^4-34*(-2^(1
/2)-1+x)^2/(1-x-2^(1/2))^2+1))*(1+2^(1/2))^(1/2)*2^(1/2)-2*(-2+2*2^(1/2))^(1/2)*
arctan(1/2*(-2^(1/2)-1+x)/(1-x-2^(1/2))*(-4+3*2^(1/2))*(24*(-2^(1/2)-1+x)^2/(1-x
-2^(1/2))^2+17*2^(1/2)*(-2^(1/2)-1+x)^2/(1-x-2^(1/2))^2-2^(1/2))*(-2+2*2^(1/2))^
(1/2)*((-4+3*2^(1/2))*(-(-2^(1/2)-1+x)^2/(1-x-2^(1/2))^2+12*2^(1/2)+17))^(1/2)/(
(-2^(1/2)-1+x)^4/(1-x-2^(1/2))^4-34*(-2^(1/2)-1+x)^2/(1-x-2^(1/2))^2+1))*(1+2^(1
/2))^(1/2)-4*arctanh(1/2*(4*(-2^(1/2)-1+x)^2/(1-x-2^(1/2))^2-3*2^(1/2)*(-2^(1/2)
-1+x)^2/(1-x-2^(1/2))^2+4+3*2^(1/2))^(1/2)/(1+2^(1/2))^(1/2))*2^(1/2)+6*arctanh(
1/2*(4*(-2^(1/2)-1+x)^2/(1-x-2^(1/2))^2-3*2^(1/2)*(-2^(1/2)-1+x)^2/(1-x-2^(1/2))
^2+4+3*2^(1/2))^(1/2)/(1+2^(1/2))^(1/2)))/(-(3*2^(1/2)*(-2^(1/2)-1+x)^2/(1-x-2^(
1/2))^2-4*(-2^(1/2)-1+x)^2/(1-x-2^(1/2))^2-3*2^(1/2)-4)/(1+(-2^(1/2)-1+x)/(1-x-2
^(1/2)))^2)^(1/2)/(1+(-2^(1/2)-1+x)/(1-x-2^(1/2)))/(-4+3*2^(1/2))/(1+2^(1/2))^(1
/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x^{2} + x\right )}^{\frac{3}{2}}}{x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + x)^(3/2)/(x^2 + 1),x, algorithm="maxima")

[Out]

integrate((x^2 + x)^(3/2)/(x^2 + 1), x)

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Fricas [A]  time = 0.336094, size = 1806, normalized size = 13.89 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + x)^(3/2)/(x^2 + 1),x, algorithm="fricas")

[Out]

1/32*(4*sqrt(2)*(32*x^3 + 112*x^2 - sqrt(2)*(32*x^3 + 112*x^2 + 66*x + 1) + 66*x
 + 1)*sqrt(x^2 + x)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) - sqrt(2)*(128*x^4 + 512
*x^3 + 472*x^2 - sqrt(2)*(128*x^4 + 512*x^3 + 472*x^2 + 88*x - 9) + 88*x - 9)*sq
rt((sqrt(2) - 2)/(2*sqrt(2) - 3)) + 64*(4*8^(1/4)*sqrt(x^2 + x)*(2*x + 1) - 8^(1
/4)*(8*x^2 + 8*x + 1))*arctan((2*sqrt(2)*(sqrt(2) - 1)*sqrt((sqrt(2) - 2)/(2*sqr
t(2) - 3)) + 8^(1/4)*sqrt(2))/(2*sqrt(2)*sqrt(x^2 + x)*(sqrt(2) - 1)*sqrt((sqrt(
2) - 2)/(2*sqrt(2) - 3)) - 2*sqrt(2)*(sqrt(2)*x - x)*sqrt((sqrt(2) - 2)/(2*sqrt(
2) - 3)) + 2*(sqrt(2) - 1)*sqrt((8^(1/4)*sqrt(2)*(sqrt(2)*(17*x + 7) - 24*x - 10
)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) - 40*x^2 + 14*sqrt(2)*(2*x^2 + x + 1) - (8
^(1/4)*sqrt(2)*(17*sqrt(2) - 24)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) + 28*sqrt(2
)*x - 40*x)*sqrt(x^2 + x) + 2*sqrt(2)*(7*sqrt(2) - 10) - 20*x - 20)/(7*sqrt(2) -
 10))*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) - 8^(1/4)*(sqrt(2) - 2))) - 64*(4*8^(1
/4)*sqrt(x^2 + x)*(2*x + 1) - 8^(1/4)*(8*x^2 + 8*x + 1))*arctan((2*sqrt(2)*(sqrt
(2) - 1)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) - 8^(1/4)*sqrt(2))/(2*sqrt(2)*sqrt(
x^2 + x)*(sqrt(2) - 1)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) - 2*sqrt(2)*(sqrt(2)*
x - x)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) + 2*(sqrt(2) - 1)*sqrt(-(8^(1/4)*sqrt
(2)*(sqrt(2)*(17*x + 7) - 24*x - 10)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) + 40*x^
2 - 14*sqrt(2)*(2*x^2 + x + 1) - (8^(1/4)*sqrt(2)*(17*sqrt(2) - 24)*sqrt((sqrt(2
) - 2)/(2*sqrt(2) - 3)) - 28*sqrt(2)*x + 40*x)*sqrt(x^2 + x) - 2*sqrt(2)*(7*sqrt
(2) - 10) + 20*x + 20)/(7*sqrt(2) - 10))*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) + 8
^(1/4)*(sqrt(2) - 2))) + 20*(4*sqrt(2)*sqrt(x^2 + x)*(sqrt(2)*(2*x + 1) - 2*x -
1)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) + sqrt(2)*(8*x^2 - sqrt(2)*(8*x^2 + 8*x +
 1) + 8*x + 1)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)))*log(-2*x + 2*sqrt(x^2 + x) -
 1) - 16*(4*8^(1/4)*sqrt(x^2 + x)*(sqrt(2)*(2*x + 1) - 2*x - 1) + 8^(1/4)*(8*x^2
 - sqrt(2)*(8*x^2 + 8*x + 1) + 8*x + 1))*log(-2*(8^(1/4)*sqrt(2)*(sqrt(2)*(17*x
+ 7) - 24*x - 10)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) + 40*x^2 - 14*sqrt(2)*(2*x
^2 + x + 1) - (8^(1/4)*sqrt(2)*(17*sqrt(2) - 24)*sqrt((sqrt(2) - 2)/(2*sqrt(2) -
 3)) - 28*sqrt(2)*x + 40*x)*sqrt(x^2 + x) - 2*sqrt(2)*(7*sqrt(2) - 10) + 20*x +
20)/(7*sqrt(2) - 10)) + 16*(4*8^(1/4)*sqrt(x^2 + x)*(sqrt(2)*(2*x + 1) - 2*x - 1
) + 8^(1/4)*(8*x^2 - sqrt(2)*(8*x^2 + 8*x + 1) + 8*x + 1))*log(2*(8^(1/4)*sqrt(2
)*(sqrt(2)*(17*x + 7) - 24*x - 10)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) - 40*x^2
+ 14*sqrt(2)*(2*x^2 + x + 1) - (8^(1/4)*sqrt(2)*(17*sqrt(2) - 24)*sqrt((sqrt(2)
- 2)/(2*sqrt(2) - 3)) + 28*sqrt(2)*x - 40*x)*sqrt(x^2 + x) + 2*sqrt(2)*(7*sqrt(2
) - 10) - 20*x - 20)/(7*sqrt(2) - 10)))/(4*sqrt(2)*sqrt(x^2 + x)*(sqrt(2)*(2*x +
 1) - 2*x - 1)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) + sqrt(2)*(8*x^2 - sqrt(2)*(8
*x^2 + 8*x + 1) + 8*x + 1)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (x + 1\right )\right )^{\frac{3}{2}}}{x^{2} + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2+x)**(3/2)/(x**2+1),x)

[Out]

Integral((x*(x + 1))**(3/2)/(x**2 + 1), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x^{2} + x\right )}^{\frac{3}{2}}}{x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + x)^(3/2)/(x^2 + 1),x, algorithm="giac")

[Out]

integrate((x^2 + x)^(3/2)/(x^2 + 1), x)

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