3.21.98 \(\int \frac {(3+5 x)^3}{(1-2 x)^{3/2}} \, dx\) [2098]

Optimal. Leaf size=53 \[ \frac {1331}{8 \sqrt {1-2 x}}+\frac {1815}{8} \sqrt {1-2 x}-\frac {275}{8} (1-2 x)^{3/2}+\frac {25}{8} (1-2 x)^{5/2} \]

[Out]

-275/8*(1-2*x)^(3/2)+25/8*(1-2*x)^(5/2)+1331/8/(1-2*x)^(1/2)+1815/8*(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 = {45} \begin {gather*} \frac {25}{8} (1-2 x)^{5/2}-\frac {275}{8} (1-2 x)^{3/2}+\frac {1815}{8} \sqrt {1-2 x}+\frac {1331}{8 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^3/(1 - 2*x)^(3/2),x]

[Out]

1331/(8*Sqrt[1 - 2*x]) + (1815*Sqrt[1 - 2*x])/8 - (275*(1 - 2*x)^(3/2))/8 + (25*(1 - 2*x)^(5/2))/8

Rule 45

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)^{3/2}} \, dx &=\int \left (\frac {1331}{8 (1-2 x)^{3/2}}-\frac {1815}{8 \sqrt {1-2 x}}+\frac {825}{8} \sqrt {1-2 x}-\frac {125}{8} (1-2 x)^{3/2}\right ) \, dx\\ &=\frac {1331}{8 \sqrt {1-2 x}}+\frac {1815}{8} \sqrt {1-2 x}-\frac {275}{8} (1-2 x)^{3/2}+\frac {25}{8} (1-2 x)^{5/2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 25, normalized size = 0.47 \begin {gather*} \frac {362-335 x-100 x^2-25 x^3}{\sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(362 - 335*x - 100*x^2 - 25*x^3)/Sqrt[1 - 2*x]

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Maple [A]
time = 0.15, size = 38, normalized size = 0.72

method	result	size
gosper	\(-\frac {25 x^{3}+100 x^{2}+335 x -362}{\sqrt {1-2 x}}\)	\(25\)
risch	\(-\frac {25 x^{3}+100 x^{2}+335 x -362}{\sqrt {1-2 x}}\)	\(25\)
trager	\(\frac {\left (25 x^{3}+100 x^{2}+335 x -362\right ) \sqrt {1-2 x}}{-1+2 x}\)	\(31\)
derivativedivides	\(-\frac {275 \left (1-2 x \right )^{\frac {3}{2}}}{8}+\frac {25 \left (1-2 x \right )^{\frac {5}{2}}}{8}+\frac {1331}{8 \sqrt {1-2 x}}+\frac {1815 \sqrt {1-2 x}}{8}\)	\(38\)
default	\(-\frac {275 \left (1-2 x \right )^{\frac {3}{2}}}{8}+\frac {25 \left (1-2 x \right )^{\frac {5}{2}}}{8}+\frac {1331}{8 \sqrt {1-2 x}}+\frac {1815 \sqrt {1-2 x}}{8}\)	\(38\)
meijerg	\(-\frac {27 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-135 \sqrt {\pi }+\frac {135 \sqrt {\pi }\, \left (-8 x +8\right )}{8 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {225 \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-8 x^{2}-16 x +16\right )}{6 \sqrt {1-2 x}}\right )}{4 \sqrt {\pi }}+\frac {-50 \sqrt {\pi }+\frac {25 \sqrt {\pi }\, \left (-64 x^{3}-64 x^{2}-128 x +128\right )}{64 \sqrt {1-2 x}}}{\sqrt {\pi }}\)	\(122\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-275/8*(1-2*x)^(3/2)+25/8*(1-2*x)^(5/2)+1331/8/(1-2*x)^(1/2)+1815/8*(1-2*x)^(1/2)

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Maxima [A]
time = 0.28, size = 37, normalized size = 0.70 \begin {gather*} \frac {25}{8} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {275}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1815}{8} \, \sqrt {-2 \, x + 1} + \frac {1331}{8 \, \sqrt {-2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/8*(-2*x + 1)^(5/2) - 275/8*(-2*x + 1)^(3/2) + 1815/8*sqrt(-2*x + 1) + 1331/8/sqrt(-2*x + 1)

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Fricas [A]
time = 0.85, size = 30, normalized size = 0.57 \begin {gather*} \frac {{\left (25 \, x^{3} + 100 \, x^{2} + 335 \, x - 362\right )} \sqrt {-2 \, x + 1}}{2 \, x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(25*x^3 + 100*x^2 + 335*x - 362)*sqrt(-2*x + 1)/(2*x - 1)

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Sympy [C] Result contains complex when optimal does not.
time = 1.06, size = 434, normalized size = 8.19 \begin {gather*} \begin {cases} \frac {125 \sqrt {55} i \left (x + \frac {3}{5}\right )^{3} \sqrt {10 x - 5}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} + \frac {275 \sqrt {55} i \left (x + \frac {3}{5}\right )^{2} \sqrt {10 x - 5}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} + \frac {1210 \sqrt {55} i \left (x + \frac {3}{5}\right ) \sqrt {10 x - 5}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} - \frac {26620 \sqrt {5} \left (x + \frac {3}{5}\right )}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} - \frac {2662 \sqrt {55} i \sqrt {10 x - 5}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} + \frac {29282 \sqrt {5}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\\frac {125 \sqrt {55} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{3}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} + \frac {275 \sqrt {55} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{2}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} + \frac {1210 \sqrt {55} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} - \frac {2662 \sqrt {55} \sqrt {5 - 10 x}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} - \frac {26620 \sqrt {5} \left (x + \frac {3}{5}\right )}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} + \frac {29282 \sqrt {5}}{50 \sqrt {11} \left (x + \frac {3}{5}\right ) - 55 \sqrt {11}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((125*sqrt(55)*I*(x + 3/5)**3*sqrt(10*x - 5)/(50*sqrt(11)*(x + 3/5) - 55*sqrt(11)) + 275*sqrt(55)*I*(
x + 3/5)**2*sqrt(10*x - 5)/(50*sqrt(11)*(x + 3/5) - 55*sqrt(11)) + 1210*sqrt(55)*I*(x + 3/5)*sqrt(10*x - 5)/(5
0*sqrt(11)*(x + 3/5) - 55*sqrt(11)) - 26620*sqrt(5)*(x + 3/5)/(50*sqrt(11)*(x + 3/5) - 55*sqrt(11)) - 2662*sqr
t(55)*I*sqrt(10*x - 5)/(50*sqrt(11)*(x + 3/5) - 55*sqrt(11)) + 29282*sqrt(5)/(50*sqrt(11)*(x + 3/5) - 55*sqrt(
11)), Abs(x + 3/5) > 11/10), (125*sqrt(55)*sqrt(5 - 10*x)*(x + 3/5)**3/(50*sqrt(11)*(x + 3/5) - 55*sqrt(11)) +
275*sqrt(55)*sqrt(5 - 10*x)*(x + 3/5)**2/(50*sqrt(11)*(x + 3/5) - 55*sqrt(11)) + 1210*sqrt(55)*sqrt(5 - 10*x)
*(x + 3/5)/(50*sqrt(11)*(x + 3/5) - 55*sqrt(11)) - 2662*sqrt(55)*sqrt(5 - 10*x)/(50*sqrt(11)*(x + 3/5) - 55*sq
rt(11)) - 26620*sqrt(5)*(x + 3/5)/(50*sqrt(11)*(x + 3/5) - 55*sqrt(11)) + 29282*sqrt(5)/(50*sqrt(11)*(x + 3/5)
- 55*sqrt(11)), True))

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Giac [A]
time = 1.06, size = 44, normalized size = 0.83 \begin {gather*} \frac {25}{8} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {275}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1815}{8} \, \sqrt {-2 \, x + 1} + \frac {1331}{8 \, \sqrt {-2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/8*(2*x - 1)^2*sqrt(-2*x + 1) - 275/8*(-2*x + 1)^(3/2) + 1815/8*sqrt(-2*x + 1) + 1331/8/sqrt(-2*x + 1)

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Mupad [B]
time = 0.03, size = 37, normalized size = 0.70 \begin {gather*} \frac {1331}{8\,\sqrt {1-2\,x}}+\frac {1815\,\sqrt {1-2\,x}}{8}-\frac {275\,{\left (1-2\,x\right )}^{3/2}}{8}+\frac {25\,{\left (1-2\,x\right )}^{5/2}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + 3)^3/(1 - 2*x)^(3/2),x)

[Out]

1331/(8*(1 - 2*x)^(1/2)) + (1815*(1 - 2*x)^(1/2))/8 - (275*(1 - 2*x)^(3/2))/8 + (25*(1 - 2*x)^(5/2))/8

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