3.2.32 \(\int \frac {(1+2 x) (1+3 x+4 x^2)}{(2+3 x^2)^{5/2}} \, dx\) [132]

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

[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 = 0.03, 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 = {1828, 651} \begin {gather*} \frac {2-51 x}{54 \left (3 x^2+2\right )^{3/2}}-\frac {16-13 x}{18 \sqrt {3 x^2+2}} \end {gather*}

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 651

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 1828

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 = 0.23, size = 30, normalized size = 0.73 \begin {gather*} \frac {-94+27 x-144 x^2+117 x^3}{54 \left (2+3 x^2\right )^{3/2}} \end {gather*}

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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Maple [A]
time = 0.07, size = 51, normalized size = 1.24

method	result	size
gosper	\(\frac {117 x^{3}-144 x^{2}+27 x -94}{54 \left (3 x^{2}+2\right )^{\frac {3}{2}}}\)	\(27\)
trager	\(\frac {117 x^{3}-144 x^{2}+27 x -94}{54 \left (3 x^{2}+2\right )^{\frac {3}{2}}}\)	\(27\)
risch	\(\frac {117 x^{3}-144 x^{2}+27 x -94}{54 \left (3 x^{2}+2\right )^{\frac {3}{2}}}\)	\(27\)
default	\(-\frac {8 x^{2}}{3 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {47}{27 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {17 x}{18 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {13 x}{18 \sqrt {3 x^{2}+2}}\)	\(51\)
meijerg	\(\frac {\sqrt {2}\, x \left (3 x^{2}+3\right )}{24 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}+\frac {5 \sqrt {2}\, x^{3}}{12 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}+\frac {5 \sqrt {2}\, \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}\right )}{18 \sqrt {\pi }}+\frac {8 \sqrt {2}\, \left (\sqrt {\pi }-\frac {\sqrt {\pi }\, \left (18 x^{2}+8\right )}{8 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}\right )}{27 \sqrt {\pi }}\)	\(102\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/3*x^2/(3*x^2+2)^(3/2)-47/27/(3*x^2+2)^(3/2)-17/18/(3*x^2+2)^(3/2)*x+13/18*x/(3*x^2+2)^(1/2)

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Maxima [A]
time = 0.30, size = 50, normalized size = 1.22 \begin {gather*} \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}}} \end {gather*}

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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Fricas [A]
time = 0.37, size = 40, normalized size = 0.98 \begin {gather*} \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 )}} \end {gather*}

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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (36) = 72\).
time = 19.00, size = 180, normalized size = 4.39 \begin {gather*} \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}} \end {gather*}

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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Giac [A]
time = 4.83, size = 25, normalized size = 0.61 \begin {gather*} \frac {9 \, {\left ({\left (13 \, x - 16\right )} x + 3\right )} x - 94}{54 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \end {gather*}

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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Mupad [B]
time = 4.11, size = 185, normalized size = 4.51 \begin {gather*} \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 )} \end {gather*}

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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