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

Optimal. Leaf size=66 \[ -\frac {75}{16} (1-2 x)^{5/2}+\frac {1675}{24} (1-2 x)^{3/2}-\frac {2805}{4} \sqrt {1-2 x}-\frac {8349}{8 \sqrt {1-2 x}}+\frac {9317}{48 (1-2 x)^{3/2}} \]

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Rubi [A]  time = 0.01, antiderivative size = 66, 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 {75}{16} (1-2 x)^{5/2}+\frac {1675}{24} (1-2 x)^{3/2}-\frac {2805}{4} \sqrt {1-2 x}-\frac {8349}{8 \sqrt {1-2 x}}+\frac {9317}{48 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

9317/(48*(1 - 2*x)^(3/2)) - 8349/(8*Sqrt[1 - 2*x]) - (2805*Sqrt[1 - 2*x])/4 + (1675*(1 - 2*x)^(3/2))/24 - (75*
(1 - 2*x)^(5/2))/16

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)^3}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac {9317}{16 (1-2 x)^{5/2}}-\frac {8349}{8 (1-2 x)^{3/2}}+\frac {2805}{4 \sqrt {1-2 x}}-\frac {1675}{8} \sqrt {1-2 x}+\frac {375}{16} (1-2 x)^{3/2}\right ) \, dx\\ &=\frac {9317}{48 (1-2 x)^{3/2}}-\frac {8349}{8 \sqrt {1-2 x}}-\frac {2805}{4} \sqrt {1-2 x}+\frac {1675}{24} (1-2 x)^{3/2}-\frac {75}{16} (1-2 x)^{5/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 33, normalized size = 0.50 \begin {gather*} -\frac {225 x^4+1225 x^3+6240 x^2-13533 x+4457}{3 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(4457 - 13533*x + 6240*x^2 + 1225*x^3 + 225*x^4)/(1 - 2*x)^(3/2)

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IntegrateAlgebraic [A]  time = 0.02, size = 49, normalized size = 0.74 \begin {gather*} \frac {-225 (1-2 x)^4+3350 (1-2 x)^3-33660 (1-2 x)^2-50094 (1-2 x)+9317}{48 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(9317 - 50094*(1 - 2*x) - 33660*(1 - 2*x)^2 + 3350*(1 - 2*x)^3 - 225*(1 - 2*x)^4)/(48*(1 - 2*x)^(3/2))

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fricas [A]  time = 0.82, size = 41, normalized size = 0.62 \begin {gather*} -\frac {{\left (225 \, x^{4} + 1225 \, x^{3} + 6240 \, x^{2} - 13533 \, x + 4457\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)^3/(1-2*x)^(5/2),x, algorithm="fricas")

[Out]

-1/3*(225*x^4 + 1225*x^3 + 6240*x^2 - 13533*x + 4457)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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giac [A]  time = 1.19, size = 56, normalized size = 0.85 \begin {gather*} -\frac {75}{16} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {1675}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {2805}{4} \, \sqrt {-2 \, x + 1} - \frac {121 \, {\left (828 \, x - 337\right )}}{48 \, {\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)^3/(1-2*x)^(5/2),x, algorithm="giac")

[Out]

-75/16*(2*x - 1)^2*sqrt(-2*x + 1) + 1675/24*(-2*x + 1)^(3/2) - 2805/4*sqrt(-2*x + 1) - 121/48*(828*x - 337)/((
2*x - 1)*sqrt(-2*x + 1))

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maple [A]  time = 0.00, size = 30, normalized size = 0.45 \begin {gather*} -\frac {225 x^{4}+1225 x^{3}+6240 x^{2}-13533 x +4457}{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)^3/(-2*x+1)^(5/2),x)

[Out]

-1/3*(225*x^4+1225*x^3+6240*x^2-13533*x+4457)/(-2*x+1)^(3/2)

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maxima [A]  time = 0.51, size = 42, normalized size = 0.64 \begin {gather*} -\frac {75}{16} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {1675}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {2805}{4} \, \sqrt {-2 \, x + 1} + \frac {121 \, {\left (828 \, x - 337\right )}}{48 \, {\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)^3/(1-2*x)^(5/2),x, algorithm="maxima")

[Out]

-75/16*(-2*x + 1)^(5/2) + 1675/24*(-2*x + 1)^(3/2) - 2805/4*sqrt(-2*x + 1) + 121/48*(828*x - 337)/(-2*x + 1)^(
3/2)

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mupad [B]  time = 0.04, size = 41, normalized size = 0.62 \begin {gather*} \frac {\frac {8349\,x}{4}-\frac {40777}{48}}{{\left (1-2\,x\right )}^{3/2}}-\frac {2805\,\sqrt {1-2\,x}}{4}+\frac {1675\,{\left (1-2\,x\right )}^{3/2}}{24}-\frac {75\,{\left (1-2\,x\right )}^{5/2}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((8349*x)/4 - 40777/48)/(1 - 2*x)^(3/2) - (2805*(1 - 2*x)^(1/2))/4 + (1675*(1 - 2*x)^(3/2))/24 - (75*(1 - 2*x)
^(5/2))/16

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sympy [A]  time = 23.39, size = 58, normalized size = 0.88 \begin {gather*} - \frac {75 \left (1 - 2 x\right )^{\frac {5}{2}}}{16} + \frac {1675 \left (1 - 2 x\right )^{\frac {3}{2}}}{24} - \frac {2805 \sqrt {1 - 2 x}}{4} - \frac {8349}{8 \sqrt {1 - 2 x}} + \frac {9317}{48 \left (1 - 2 x\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-75*(1 - 2*x)**(5/2)/16 + 1675*(1 - 2*x)**(3/2)/24 - 2805*sqrt(1 - 2*x)/4 - 8349/(8*sqrt(1 - 2*x)) + 9317/(48*
(1 - 2*x)**(3/2))

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