3.10.43 \(\int \frac {x-2 x^3}{\sqrt {2+3 x}} \, dx\) [943]

Optimal. Leaf size=53 \[ -\frac {4}{81} \sqrt {2+3 x}-\frac {10}{81} (2+3 x)^{3/2}+\frac {8}{135} (2+3 x)^{5/2}-\frac {4}{567} (2+3 x)^{7/2} \]

[Out]

-10/81*(2+3*x)^(3/2)+8/135*(2+3*x)^(5/2)-4/567*(2+3*x)^(7/2)-4/81*(2+3*x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1607, 786} \begin {gather*} -\frac {4}{567} (3 x+2)^{7/2}+\frac {8}{135} (3 x+2)^{5/2}-\frac {10}{81} (3 x+2)^{3/2}-\frac {4}{81} \sqrt {3 x+2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x - 2*x^3)/Sqrt[2 + 3*x],x]

[Out]

(-4*Sqrt[2 + 3*x])/81 - (10*(2 + 3*x)^(3/2))/81 + (8*(2 + 3*x)^(5/2))/135 - (4*(2 + 3*x)^(7/2))/567

Rule 786

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

\begin {align*} \int \frac {x-2 x^3}{\sqrt {2+3 x}} \, dx &=\int \frac {x \left (1-2 x^2\right )}{\sqrt {2+3 x}} \, dx\\ &=\int \left (-\frac {2}{27 \sqrt {2+3 x}}-\frac {5}{9} \sqrt {2+3 x}+\frac {4}{9} (2+3 x)^{3/2}-\frac {2}{27} (2+3 x)^{5/2}\right ) \, dx\\ &=-\frac {4}{81} \sqrt {2+3 x}-\frac {10}{81} (2+3 x)^{3/2}+\frac {8}{135} (2+3 x)^{5/2}-\frac {4}{567} (2+3 x)^{7/2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 28, normalized size = 0.53 \begin {gather*} -\frac {2 \sqrt {2+3 x} \left (164-123 x-216 x^2+270 x^3\right )}{2835} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x - 2*x^3)/Sqrt[2 + 3*x],x]

[Out]

(-2*Sqrt[2 + 3*x]*(164 - 123*x - 216*x^2 + 270*x^3))/2835

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Maple [A]
time = 0.22, size = 38, normalized size = 0.72

method	result	size
trager	\(\left (-\frac {4}{21} x^{3}+\frac {16}{105} x^{2}+\frac {82}{945} x -\frac {328}{2835}\right ) \sqrt {2+3 x}\)	\(24\)
gosper	\(-\frac {2 \left (270 x^{3}-216 x^{2}-123 x +164\right ) \sqrt {2+3 x}}{2835}\)	\(25\)
risch	\(-\frac {2 \left (270 x^{3}-216 x^{2}-123 x +164\right ) \sqrt {2+3 x}}{2835}\)	\(25\)
derivativedivides	\(-\frac {10 \left (2+3 x \right )^{\frac {3}{2}}}{81}+\frac {8 \left (2+3 x \right )^{\frac {5}{2}}}{135}-\frac {4 \left (2+3 x \right )^{\frac {7}{2}}}{567}-\frac {4 \sqrt {2+3 x}}{81}\)	\(38\)
default	\(-\frac {10 \left (2+3 x \right )^{\frac {3}{2}}}{81}+\frac {8 \left (2+3 x \right )^{\frac {5}{2}}}{135}-\frac {4 \left (2+3 x \right )^{\frac {7}{2}}}{567}-\frac {4 \sqrt {2+3 x}}{81}\)	\(38\)
meijerg	\(-\frac {16 \sqrt {2}\, \left (\frac {32 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (-135 x^{3}+108 x^{2}-96 x +128\right ) \sqrt {1+\frac {3 x}{2}}}{140}\right )}{81 \sqrt {\pi }}+\frac {2 \sqrt {2}\, \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-6 x +8\right ) \sqrt {1+\frac {3 x}{2}}}{6}\right )}{9 \sqrt {\pi }}\)	\(74\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-10/81*(2+3*x)^(3/2)+8/135*(2+3*x)^(5/2)-4/567*(2+3*x)^(7/2)-4/81*(2+3*x)^(1/2)

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Maxima [A]
time = 0.37, size = 37, normalized size = 0.70 \begin {gather*} -\frac {4}{567} \, {\left (3 \, x + 2\right )}^{\frac {7}{2}} + \frac {8}{135} \, {\left (3 \, x + 2\right )}^{\frac {5}{2}} - \frac {10}{81} \, {\left (3 \, x + 2\right )}^{\frac {3}{2}} - \frac {4}{81} \, \sqrt {3 \, x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/567*(3*x + 2)^(7/2) + 8/135*(3*x + 2)^(5/2) - 10/81*(3*x + 2)^(3/2) - 4/81*sqrt(3*x + 2)

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Fricas [A]
time = 0.34, size = 24, normalized size = 0.45 \begin {gather*} -\frac {2}{2835} \, {\left (270 \, x^{3} - 216 \, x^{2} - 123 \, x + 164\right )} \sqrt {3 \, x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/2835*(270*x^3 - 216*x^2 - 123*x + 164)*sqrt(3*x + 2)

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Sympy [A]
time = 5.97, size = 46, normalized size = 0.87 \begin {gather*} - \frac {4 \left (3 x + 2\right )^{\frac {7}{2}}}{567} + \frac {8 \left (3 x + 2\right )^{\frac {5}{2}}}{135} - \frac {10 \left (3 x + 2\right )^{\frac {3}{2}}}{81} - \frac {4 \sqrt {3 x + 2}}{81} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**3+x)/(2+3*x)**(1/2),x)

[Out]

-4*(3*x + 2)**(7/2)/567 + 8*(3*x + 2)**(5/2)/135 - 10*(3*x + 2)**(3/2)/81 - 4*sqrt(3*x + 2)/81

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Giac [A]
time = 5.29, size = 37, normalized size = 0.70 \begin {gather*} -\frac {4}{567} \, {\left (3 \, x + 2\right )}^{\frac {7}{2}} + \frac {8}{135} \, {\left (3 \, x + 2\right )}^{\frac {5}{2}} - \frac {10}{81} \, {\left (3 \, x + 2\right )}^{\frac {3}{2}} - \frac {4}{81} \, \sqrt {3 \, x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/567*(3*x + 2)^(7/2) + 8/135*(3*x + 2)^(5/2) - 10/81*(3*x + 2)^(3/2) - 4/81*sqrt(3*x + 2)

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Mupad [B]
time = 0.04, size = 37, normalized size = 0.70 \begin {gather*} \frac {8\,{\left (3\,x+2\right )}^{5/2}}{135}-\frac {10\,{\left (3\,x+2\right )}^{3/2}}{81}-\frac {4\,\sqrt {3\,x+2}}{81}-\frac {4\,{\left (3\,x+2\right )}^{7/2}}{567} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - 2*x^3)/(3*x + 2)^(1/2),x)

[Out]

(8*(3*x + 2)^(5/2))/135 - (10*(3*x + 2)^(3/2))/81 - (4*(3*x + 2)^(1/2))/81 - (4*(3*x + 2)^(7/2))/567

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