3.2162 \(\int \frac {(3+5 x)^3}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=53 \[ \frac {125}{24} (1-2 x)^{3/2}-\frac {825}{8} \sqrt {1-2 x}-\frac {1815}{8 \sqrt {1-2 x}}+\frac {1331}{24 (1-2 x)^{3/2}} \]

[Out]

1331/24/(1-2*x)^(3/2)+125/24*(1-2*x)^(3/2)-1815/8/(1-2*x)^(1/2)-825/8*(1-2*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 = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {43} \[ \frac {125}{24} (1-2 x)^{3/2}-\frac {825}{8} \sqrt {1-2 x}-\frac {1815}{8 \sqrt {1-2 x}}+\frac {1331}{24 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^3/(1 - 2*x)^(5/2),x]

[Out]

1331/(24*(1 - 2*x)^(3/2)) - 1815/(8*Sqrt[1 - 2*x]) - (825*Sqrt[1 - 2*x])/8 + (125*(1 - 2*x)^(3/2))/24

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {(3+5 x)^3}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac {1331}{8 (1-2 x)^{5/2}}-\frac {1815}{8 (1-2 x)^{3/2}}+\frac {825}{8 \sqrt {1-2 x}}-\frac {125}{8} \sqrt {1-2 x}\right ) \, dx\\ &=\frac {1331}{24 (1-2 x)^{3/2}}-\frac {1815}{8 \sqrt {1-2 x}}-\frac {825}{8} \sqrt {1-2 x}+\frac {125}{24} (1-2 x)^{3/2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 28, normalized size = 0.53 \[ -\frac {125 x^3+1050 x^2-2505 x+808}{3 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(808 - 2505*x + 1050*x^2 + 125*x^3)/(1 - 2*x)^(3/2)

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fricas [A]  time = 0.78, size = 36, normalized size = 0.68 \[ -\frac {{\left (125 \, x^{3} + 1050 \, x^{2} - 2505 \, x + 808\right )} \sqrt {-2 \, x + 1}}{3 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(125*x^3 + 1050*x^2 - 2505*x + 808)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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giac [A]  time = 1.23, size = 40, normalized size = 0.75 \[ \frac {125}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {825}{8} \, \sqrt {-2 \, x + 1} - \frac {121 \, {\left (45 \, x - 17\right )}}{12 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/24*(-2*x + 1)^(3/2) - 825/8*sqrt(-2*x + 1) - 121/12*(45*x - 17)/((2*x - 1)*sqrt(-2*x + 1))

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maple [A]  time = 0.00, size = 25, normalized size = 0.47 \[ -\frac {125 x^{3}+1050 x^{2}-2505 x +808}{3 \left (-2 x +1\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(125*x^3+1050*x^2-2505*x+808)/(-2*x+1)^(3/2)

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maxima [A]  time = 0.61, size = 33, normalized size = 0.62 \[ \frac {125}{24} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {825}{8} \, \sqrt {-2 \, x + 1} + \frac {121 \, {\left (45 \, x - 17\right )}}{12 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/24*(-2*x + 1)^(3/2) - 825/8*sqrt(-2*x + 1) + 121/12*(45*x - 17)/(-2*x + 1)^(3/2)

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mupad [B]  time = 0.03, size = 38, normalized size = 0.72 \[ \frac {2475\,{\left (2\,x-1\right )}^2-10890\,x+125\,{\left (2\,x-1\right )}^3+4114}{\sqrt {1-2\,x}\,\left (48\,x-24\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + 3)^3/(1 - 2*x)^(5/2),x)

[Out]

(2475*(2*x - 1)^2 - 10890*x + 125*(2*x - 1)^3 + 4114)/((1 - 2*x)^(1/2)*(48*x - 24))

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sympy [B]  time = 0.68, size = 102, normalized size = 1.92 \[ \frac {125 x^{3}}{6 x \sqrt {1 - 2 x} - 3 \sqrt {1 - 2 x}} + \frac {1050 x^{2}}{6 x \sqrt {1 - 2 x} - 3 \sqrt {1 - 2 x}} - \frac {2505 x}{6 x \sqrt {1 - 2 x} - 3 \sqrt {1 - 2 x}} + \frac {808}{6 x \sqrt {1 - 2 x} - 3 \sqrt {1 - 2 x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125*x**3/(6*x*sqrt(1 - 2*x) - 3*sqrt(1 - 2*x)) + 1050*x**2/(6*x*sqrt(1 - 2*x) - 3*sqrt(1 - 2*x)) - 2505*x/(6*x
*sqrt(1 - 2*x) - 3*sqrt(1 - 2*x)) + 808/(6*x*sqrt(1 - 2*x) - 3*sqrt(1 - 2*x))

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