3.20.45 \(\int (1-2 x)^{5/2} (2+3 x) (3+5 x)^2 \, dx\) [1945]

Optimal. Leaf size=53 \[ -\frac {121}{8} (1-2 x)^{7/2}+\frac {1133}{72} (1-2 x)^{9/2}-\frac {505}{88} (1-2 x)^{11/2}+\frac {75}{104} (1-2 x)^{13/2} \]

[Out]

-121/8*(1-2*x)^(7/2)+1133/72*(1-2*x)^(9/2)-505/88*(1-2*x)^(11/2)+75/104*(1-2*x)^(13/2)

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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 = {78} \begin {gather*} \frac {75}{104} (1-2 x)^{13/2}-\frac {505}{88} (1-2 x)^{11/2}+\frac {1133}{72} (1-2 x)^{9/2}-\frac {121}{8} (1-2 x)^{7/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-121*(1 - 2*x)^(7/2))/8 + (1133*(1 - 2*x)^(9/2))/72 - (505*(1 - 2*x)^(11/2))/88 + (75*(1 - 2*x)^(13/2))/104

Rule 78

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 (1-2 x)^{5/2} (2+3 x) (3+5 x)^2 \, dx &=\int \left (\frac {847}{8} (1-2 x)^{5/2}-\frac {1133}{8} (1-2 x)^{7/2}+\frac {505}{8} (1-2 x)^{9/2}-\frac {75}{8} (1-2 x)^{11/2}\right ) \, dx\\ &=-\frac {121}{8} (1-2 x)^{7/2}+\frac {1133}{72} (1-2 x)^{9/2}-\frac {505}{88} (1-2 x)^{11/2}+\frac {75}{104} (1-2 x)^{13/2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 28, normalized size = 0.53 \begin {gather*} -\frac {(1-2 x)^{7/2} \left (5671+16531 x+18405 x^2+7425 x^3\right )}{1287} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/1287*((1 - 2*x)^(7/2)*(5671 + 16531*x + 18405*x^2 + 7425*x^3))

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Maple [A]
time = 0.10, size = 38, normalized size = 0.72

method	result	size
gosper	\(-\frac {\left (7425 x^{3}+18405 x^{2}+16531 x +5671\right ) \left (1-2 x \right )^{\frac {7}{2}}}{1287}\)	\(25\)
derivativedivides	\(-\frac {121 \left (1-2 x \right )^{\frac {7}{2}}}{8}+\frac {1133 \left (1-2 x \right )^{\frac {9}{2}}}{72}-\frac {505 \left (1-2 x \right )^{\frac {11}{2}}}{88}+\frac {75 \left (1-2 x \right )^{\frac {13}{2}}}{104}\)	\(38\)
default	\(-\frac {121 \left (1-2 x \right )^{\frac {7}{2}}}{8}+\frac {1133 \left (1-2 x \right )^{\frac {9}{2}}}{72}-\frac {505 \left (1-2 x \right )^{\frac {11}{2}}}{88}+\frac {75 \left (1-2 x \right )^{\frac {13}{2}}}{104}\)	\(38\)
trager	\(\left (\frac {600}{13} x^{6}+\frac {6460}{143} x^{5}-\frac {44062}{1287} x^{4}-\frac {49999}{1287} x^{3}+\frac {4243}{429} x^{2}+\frac {17495}{1287} x -\frac {5671}{1287}\right ) \sqrt {1-2 x}\)	\(39\)
risch	\(-\frac {\left (59400 x^{6}+58140 x^{5}-44062 x^{4}-49999 x^{3}+12729 x^{2}+17495 x -5671\right ) \left (-1+2 x \right )}{1287 \sqrt {1-2 x}}\)	\(45\)
meijerg	\(\frac {\frac {18 \sqrt {\pi }}{7}-\frac {9 \sqrt {\pi }\, \left (-16 x^{3}+24 x^{2}-12 x +2\right ) \sqrt {1-2 x}}{7}}{\sqrt {\pi }}-\frac {1305 \left (-\frac {32 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (-448 x^{4}+608 x^{3}-240 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{945}\right )}{32 \sqrt {\pi }}+\frac {\frac {40 \sqrt {\pi }}{99}-\frac {5 \sqrt {\pi }\, \left (-4032 x^{5}+5152 x^{4}-1808 x^{3}+24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{198}}{\sqrt {\pi }}-\frac {1125 \left (-\frac {256 \sqrt {\pi }}{45045}+\frac {2 \sqrt {\pi }\, \left (-118272 x^{6}+145152 x^{5}-47488 x^{4}+320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{45045}\right )}{128 \sqrt {\pi }}\)	\(184\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-121/8*(1-2*x)^(7/2)+1133/72*(1-2*x)^(9/2)-505/88*(1-2*x)^(11/2)+75/104*(1-2*x)^(13/2)

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Maxima [A]
time = 0.28, size = 37, normalized size = 0.70 \begin {gather*} \frac {75}{104} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {505}{88} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {1133}{72} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {121}{8} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

75/104*(-2*x + 1)^(13/2) - 505/88*(-2*x + 1)^(11/2) + 1133/72*(-2*x + 1)^(9/2) - 121/8*(-2*x + 1)^(7/2)

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Fricas [A]
time = 0.81, size = 39, normalized size = 0.74 \begin {gather*} \frac {1}{1287} \, {\left (59400 \, x^{6} + 58140 \, x^{5} - 44062 \, x^{4} - 49999 \, x^{3} + 12729 \, x^{2} + 17495 \, x - 5671\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1287*(59400*x^6 + 58140*x^5 - 44062*x^4 - 49999*x^3 + 12729*x^2 + 17495*x - 5671)*sqrt(-2*x + 1)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (46) = 92\).
time = 0.29, size = 100, normalized size = 1.89 \begin {gather*} \frac {600 x^{6} \sqrt {1 - 2 x}}{13} + \frac {6460 x^{5} \sqrt {1 - 2 x}}{143} - \frac {44062 x^{4} \sqrt {1 - 2 x}}{1287} - \frac {49999 x^{3} \sqrt {1 - 2 x}}{1287} + \frac {4243 x^{2} \sqrt {1 - 2 x}}{429} + \frac {17495 x \sqrt {1 - 2 x}}{1287} - \frac {5671 \sqrt {1 - 2 x}}{1287} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

600*x**6*sqrt(1 - 2*x)/13 + 6460*x**5*sqrt(1 - 2*x)/143 - 44062*x**4*sqrt(1 - 2*x)/1287 - 49999*x**3*sqrt(1 -
2*x)/1287 + 4243*x**2*sqrt(1 - 2*x)/429 + 17495*x*sqrt(1 - 2*x)/1287 - 5671*sqrt(1 - 2*x)/1287

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Giac [A]
time = 0.53, size = 65, normalized size = 1.23 \begin {gather*} \frac {75}{104} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {505}{88} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {1133}{72} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {121}{8} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

75/104*(2*x - 1)^6*sqrt(-2*x + 1) + 505/88*(2*x - 1)^5*sqrt(-2*x + 1) + 1133/72*(2*x - 1)^4*sqrt(-2*x + 1) + 1
21/8*(2*x - 1)^3*sqrt(-2*x + 1)

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Mupad [B]
time = 0.04, size = 37, normalized size = 0.70 \begin {gather*} \frac {1133\,{\left (1-2\,x\right )}^{9/2}}{72}-\frac {121\,{\left (1-2\,x\right )}^{7/2}}{8}-\frac {505\,{\left (1-2\,x\right )}^{11/2}}{88}+\frac {75\,{\left (1-2\,x\right )}^{13/2}}{104} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1133*(1 - 2*x)^(9/2))/72 - (121*(1 - 2*x)^(7/2))/8 - (505*(1 - 2*x)^(11/2))/88 + (75*(1 - 2*x)^(13/2))/104

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