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

Optimal. Leaf size=53 \[ \frac {25}{8} (1-2 x)^{3/2}-\frac {505}{8} \sqrt {1-2 x}-\frac {1133}{8 \sqrt {1-2 x}}+\frac {847}{24 (1-2 x)^{3/2}} \]

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Rubi [A]  time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {77} \begin {gather*} \frac {25}{8} (1-2 x)^{3/2}-\frac {505}{8} \sqrt {1-2 x}-\frac {1133}{8 \sqrt {1-2 x}}+\frac {847}{24 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

847/(24*(1 - 2*x)^(3/2)) - 1133/(8*Sqrt[1 - 2*x]) - (505*Sqrt[1 - 2*x])/8 + (25*(1 - 2*x)^(3/2))/8

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int \frac {(2+3 x) (3+5 x)^2}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac {847}{8 (1-2 x)^{5/2}}-\frac {1133}{8 (1-2 x)^{3/2}}+\frac {505}{8 \sqrt {1-2 x}}-\frac {75}{8} \sqrt {1-2 x}\right ) \, dx\\ &=\frac {847}{24 (1-2 x)^{3/2}}-\frac {1133}{8 \sqrt {1-2 x}}-\frac {505}{8} \sqrt {1-2 x}+\frac {25}{8} (1-2 x)^{3/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 28, normalized size = 0.53 \begin {gather*} -\frac {75 x^3+645 x^2-1551 x+499}{3 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(499 - 1551*x + 645*x^2 + 75*x^3)/(1 - 2*x)^(3/2)

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IntegrateAlgebraic [A]  time = 0.02, size = 40, normalized size = 0.75 \begin {gather*} \frac {75 (1-2 x)^3-1515 (1-2 x)^2-3399 (1-2 x)+847}{24 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((2 + 3*x)*(3 + 5*x)^2)/(1 - 2*x)^(5/2),x]

[Out]

(847 - 3399*(1 - 2*x) - 1515*(1 - 2*x)^2 + 75*(1 - 2*x)^3)/(24*(1 - 2*x)^(3/2))

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fricas [A]  time = 1.13, size = 36, normalized size = 0.68 \begin {gather*} -\frac {{\left (75 \, x^{3} + 645 \, x^{2} - 1551 \, x + 499\right )} \sqrt {-2 \, x + 1}}{3 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(75*x^3 + 645*x^2 - 1551*x + 499)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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giac [A]  time = 1.20, size = 40, normalized size = 0.75 \begin {gather*} \frac {25}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {505}{8} \, \sqrt {-2 \, x + 1} - \frac {11 \, {\left (309 \, x - 116\right )}}{12 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/8*(-2*x + 1)^(3/2) - 505/8*sqrt(-2*x + 1) - 11/12*(309*x - 116)/((2*x - 1)*sqrt(-2*x + 1))

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maple [A]  time = 0.00, size = 25, normalized size = 0.47 \begin {gather*} -\frac {75 x^{3}+645 x^{2}-1551 x +499}{3 \left (-2 x +1\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(75*x^3+645*x^2-1551*x+499)/(-2*x+1)^(3/2)

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maxima [A]  time = 0.55, size = 33, normalized size = 0.62 \begin {gather*} \frac {25}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {505}{8} \, \sqrt {-2 \, x + 1} + \frac {11 \, {\left (309 \, x - 116\right )}}{12 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/8*(-2*x + 1)^(3/2) - 505/8*sqrt(-2*x + 1) + 11/12*(309*x - 116)/(-2*x + 1)^(3/2)

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mupad [B]  time = 1.19, size = 38, normalized size = 0.72 \begin {gather*} \frac {1515\,{\left (2\,x-1\right )}^2-6798\,x+75\,{\left (2\,x-1\right )}^3+2552}{\sqrt {1-2\,x}\,\left (48\,x-24\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1515*(2*x - 1)^2 - 6798*x + 75*(2*x - 1)^3 + 2552)/((1 - 2*x)^(1/2)*(48*x - 24))

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sympy [B]  time = 0.69, size = 102, normalized size = 1.92 \begin {gather*} \frac {75 x^{3}}{6 x \sqrt {1 - 2 x} - 3 \sqrt {1 - 2 x}} + \frac {645 x^{2}}{6 x \sqrt {1 - 2 x} - 3 \sqrt {1 - 2 x}} - \frac {1551 x}{6 x \sqrt {1 - 2 x} - 3 \sqrt {1 - 2 x}} + \frac {499}{6 x \sqrt {1 - 2 x} - 3 \sqrt {1 - 2 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

75*x**3/(6*x*sqrt(1 - 2*x) - 3*sqrt(1 - 2*x)) + 645*x**2/(6*x*sqrt(1 - 2*x) - 3*sqrt(1 - 2*x)) - 1551*x/(6*x*s
qrt(1 - 2*x) - 3*sqrt(1 - 2*x)) + 499/(6*x*sqrt(1 - 2*x) - 3*sqrt(1 - 2*x))

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