3.132 \(\int \frac {(1+2 x) (1+3 x+4 x^2)}{(2+3 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=41 \[ \frac {2-51 x}{54 \left (3 x^2+2\right )^{3/2}}-\frac {16-13 x}{18 \sqrt {3 x^2+2}} \]

[Out]

1/54*(2-51*x)/(3*x^2+2)^(3/2)+1/18*(-16+13*x)/(3*x^2+2)^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1814, 637} \[ \frac {2-51 x}{54 \left (3 x^2+2\right )^{3/2}}-\frac {16-13 x}{18 \sqrt {3 x^2+2}} \]

Antiderivative was successfully verified.

[In]

Int[((1 + 2*x)*(1 + 3*x + 4*x^2))/(2 + 3*x^2)^(5/2),x]

[Out]

(2 - 51*x)/(54*(2 + 3*x^2)^(3/2)) - (16 - 13*x)/(18*Sqrt[2 + 3*x^2])

Rule 637

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(-(a*e) + c*d*x)/(a*c*Sqrt[a + c*x^2]),
x] /; FreeQ[{a, c, d, e}, x]

Rule 1814

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a
*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {(1+2 x) \left (1+3 x+4 x^2\right )}{\left (2+3 x^2\right )^{5/2}} \, dx &=\frac {2-51 x}{54 \left (2+3 x^2\right )^{3/2}}-\frac {1}{6} \int \frac {-\frac {26}{3}-16 x}{\left (2+3 x^2\right )^{3/2}} \, dx\\ &=\frac {2-51 x}{54 \left (2+3 x^2\right )^{3/2}}-\frac {16-13 x}{18 \sqrt {2+3 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 30, normalized size = 0.73 \[ \frac {117 x^3-144 x^2+27 x-94}{54 \left (3 x^2+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 + 2*x)*(1 + 3*x + 4*x^2))/(2 + 3*x^2)^(5/2),x]

[Out]

(-94 + 27*x - 144*x^2 + 117*x^3)/(54*(2 + 3*x^2)^(3/2))

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fricas [A]  time = 0.77, size = 40, normalized size = 0.98 \[ \frac {{\left (117 \, x^{3} - 144 \, x^{2} + 27 \, x - 94\right )} \sqrt {3 \, x^{2} + 2}}{54 \, {\left (9 \, x^{4} + 12 \, x^{2} + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)*(4*x^2+3*x+1)/(3*x^2+2)^(5/2),x, algorithm="fricas")

[Out]

1/54*(117*x^3 - 144*x^2 + 27*x - 94)*sqrt(3*x^2 + 2)/(9*x^4 + 12*x^2 + 4)

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giac [A]  time = 0.34, size = 25, normalized size = 0.61 \[ \frac {9 \, {\left ({\left (13 \, x - 16\right )} x + 3\right )} x - 94}{54 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)*(4*x^2+3*x+1)/(3*x^2+2)^(5/2),x, algorithm="giac")

[Out]

1/54*(9*((13*x - 16)*x + 3)*x - 94)/(3*x^2 + 2)^(3/2)

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maple [A]  time = 0.00, size = 27, normalized size = 0.66 \[ \frac {117 x^{3}-144 x^{2}+27 x -94}{54 \left (3 x^{2}+2\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+2*x)*(4*x^2+3*x+1)/(3*x^2+2)^(5/2),x)

[Out]

1/54*(117*x^3-144*x^2+27*x-94)/(3*x^2+2)^(3/2)

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maxima [A]  time = 0.42, size = 50, normalized size = 1.22 \[ \frac {13 \, x}{18 \, \sqrt {3 \, x^{2} + 2}} - \frac {8 \, x^{2}}{3 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {17 \, x}{18 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {47}{27 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)*(4*x^2+3*x+1)/(3*x^2+2)^(5/2),x, algorithm="maxima")

[Out]

13/18*x/sqrt(3*x^2 + 2) - 8/3*x^2/(3*x^2 + 2)^(3/2) - 17/18*x/(3*x^2 + 2)^(3/2) - 47/27/(3*x^2 + 2)^(3/2)

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mupad [B]  time = 4.11, size = 185, normalized size = 4.51 \[ \frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {-\frac {17}{16}+\frac {\sqrt {6}\,1{}\mathrm {i}}{48}}{x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}+\frac {\sqrt {6}\,\left (-\frac {17}{24}+\frac {\sqrt {6}\,1{}\mathrm {i}}{72}\right )\,1{}\mathrm {i}}{2\,{\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {\frac {17}{16}+\frac {\sqrt {6}\,1{}\mathrm {i}}{48}}{x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}-\frac {\sqrt {6}\,\left (\frac {17}{24}+\frac {\sqrt {6}\,1{}\mathrm {i}}{72}\right )\,1{}\mathrm {i}}{2\,{\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-192+\sqrt {6}\,69{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{2592\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (192+\sqrt {6}\,69{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{2592\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x + 1)*(3*x + 4*x^2 + 1))/(3*x^2 + 2)^(5/2),x)

[Out]

(3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*1i)/48 - 17/16)/(x + (6^(1/2)*1i)/3) + (6^(1/2)*((6^(1/2)*1i)/72 - 17/24
)*1i)/(2*(x + (6^(1/2)*1i)/3)^2)))/27 - (3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*1i)/48 + 17/16)/(x - (6^(1/2)*1i
)/3) - (6^(1/2)*((6^(1/2)*1i)/72 + 17/24)*1i)/(2*(x - (6^(1/2)*1i)/3)^2)))/27 - (3^(1/2)*6^(1/2)*(6^(1/2)*69i
- 192)*(x^2 + 2/3)^(1/2)*1i)/(2592*(x - (6^(1/2)*1i)/3)) - (3^(1/2)*6^(1/2)*(6^(1/2)*69i + 192)*(x^2 + 2/3)^(1
/2)*1i)/(2592*(x + (6^(1/2)*1i)/3))

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sympy [B]  time = 77.50, size = 180, normalized size = 4.39 \[ \frac {10 x^{3}}{18 x^{2} \sqrt {3 x^{2} + 2} + 12 \sqrt {3 x^{2} + 2}} + \frac {x^{3}}{6 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}} - \frac {72 x^{2}}{81 x^{2} \sqrt {3 x^{2} + 2} + 54 \sqrt {3 x^{2} + 2}} + \frac {x}{6 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}} - \frac {32}{81 x^{2} \sqrt {3 x^{2} + 2} + 54 \sqrt {3 x^{2} + 2}} - \frac {5}{27 x^{2} \sqrt {3 x^{2} + 2} + 18 \sqrt {3 x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)*(4*x**2+3*x+1)/(3*x**2+2)**(5/2),x)

[Out]

10*x**3/(18*x**2*sqrt(3*x**2 + 2) + 12*sqrt(3*x**2 + 2)) + x**3/(6*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2))
- 72*x**2/(81*x**2*sqrt(3*x**2 + 2) + 54*sqrt(3*x**2 + 2)) + x/(6*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2))
- 32/(81*x**2*sqrt(3*x**2 + 2) + 54*sqrt(3*x**2 + 2)) - 5/(27*x**2*sqrt(3*x**2 + 2) + 18*sqrt(3*x**2 + 2))

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