3.2530 \(\int \frac {(5-x) (2+5 x+3 x^2)}{\sqrt {3+2 x}} \, dx\)

Optimal. Leaf size=53 \[ -\frac {3}{56} (2 x+3)^{7/2}+\frac {47}{40} (2 x+3)^{5/2}-\frac {109}{24} (2 x+3)^{3/2}+\frac {65}{8} \sqrt {2 x+3} \]

[Out]

-109/24*(3+2*x)^(3/2)+47/40*(3+2*x)^(5/2)-3/56*(3+2*x)^(7/2)+65/8*(3+2*x)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {771} \[ -\frac {3}{56} (2 x+3)^{7/2}+\frac {47}{40} (2 x+3)^{5/2}-\frac {109}{24} (2 x+3)^{3/2}+\frac {65}{8} \sqrt {2 x+3} \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(2 + 5*x + 3*x^2))/Sqrt[3 + 2*x],x]

[Out]

(65*Sqrt[3 + 2*x])/8 - (109*(3 + 2*x)^(3/2))/24 + (47*(3 + 2*x)^(5/2))/40 - (3*(3 + 2*x)^(7/2))/56

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )}{\sqrt {3+2 x}} \, dx &=\int \left (\frac {65}{8 \sqrt {3+2 x}}-\frac {109}{8} \sqrt {3+2 x}+\frac {47}{8} (3+2 x)^{3/2}-\frac {3}{8} (3+2 x)^{5/2}\right ) \, dx\\ &=\frac {65}{8} \sqrt {3+2 x}-\frac {109}{24} (3+2 x)^{3/2}+\frac {47}{40} (3+2 x)^{5/2}-\frac {3}{56} (3+2 x)^{7/2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 28, normalized size = 0.53 \[ -\frac {1}{105} \sqrt {2 x+3} \left (45 x^3-291 x^2-223 x-381\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(2 + 5*x + 3*x^2))/Sqrt[3 + 2*x],x]

[Out]

-1/105*(Sqrt[3 + 2*x]*(-381 - 223*x - 291*x^2 + 45*x^3))

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fricas [A]  time = 0.59, size = 24, normalized size = 0.45 \[ -\frac {1}{105} \, {\left (45 \, x^{3} - 291 \, x^{2} - 223 \, x - 381\right )} \sqrt {2 \, x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(45*x^3 - 291*x^2 - 223*x - 381)*sqrt(2*x + 3)

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giac [A]  time = 0.16, size = 37, normalized size = 0.70 \[ -\frac {3}{56} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + \frac {47}{40} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} - \frac {109}{24} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {65}{8} \, \sqrt {2 \, x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/56*(2*x + 3)^(7/2) + 47/40*(2*x + 3)^(5/2) - 109/24*(2*x + 3)^(3/2) + 65/8*sqrt(2*x + 3)

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maple [A]  time = 0.00, size = 25, normalized size = 0.47 \[ -\frac {\left (45 x^{3}-291 x^{2}-223 x -381\right ) \sqrt {2 x +3}}{105} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/105*(45*x^3-291*x^2-223*x-381)*(2*x+3)^(1/2)

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maxima [A]  time = 0.49, size = 37, normalized size = 0.70 \[ -\frac {3}{56} \, {\left (2 \, x + 3\right )}^{\frac {7}{2}} + \frac {47}{40} \, {\left (2 \, x + 3\right )}^{\frac {5}{2}} - \frac {109}{24} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} + \frac {65}{8} \, \sqrt {2 \, x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/56*(2*x + 3)^(7/2) + 47/40*(2*x + 3)^(5/2) - 109/24*(2*x + 3)^(3/2) + 65/8*sqrt(2*x + 3)

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mupad [B]  time = 0.03, size = 37, normalized size = 0.70 \[ \frac {65\,\sqrt {2\,x+3}}{8}-\frac {109\,{\left (2\,x+3\right )}^{3/2}}{24}+\frac {47\,{\left (2\,x+3\right )}^{5/2}}{40}-\frac {3\,{\left (2\,x+3\right )}^{7/2}}{56} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(65*(2*x + 3)^(1/2))/8 - (109*(2*x + 3)^(3/2))/24 + (47*(2*x + 3)^(5/2))/40 - (3*(2*x + 3)^(7/2))/56

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sympy [A]  time = 41.28, size = 46, normalized size = 0.87 \[ - \frac {3 \left (2 x + 3\right )^{\frac {7}{2}}}{56} + \frac {47 \left (2 x + 3\right )^{\frac {5}{2}}}{40} - \frac {109 \left (2 x + 3\right )^{\frac {3}{2}}}{24} + \frac {65 \sqrt {2 x + 3}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*(2*x + 3)**(7/2)/56 + 47*(2*x + 3)**(5/2)/40 - 109*(2*x + 3)**(3/2)/24 + 65*sqrt(2*x + 3)/8

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