3.19.73 \(\int (1-2 x)^{3/2} (2+3 x) (3+5 x)^2 \, dx\) [1873]

Optimal. Leaf size=53 \[ -\frac {847}{40} (1-2 x)^{5/2}+\frac {1133}{56} (1-2 x)^{7/2}-\frac {505}{72} (1-2 x)^{9/2}+\frac {75}{88} (1-2 x)^{11/2} \]

[Out]

-847/40*(1-2*x)^(5/2)+1133/56*(1-2*x)^(7/2)-505/72*(1-2*x)^(9/2)+75/88*(1-2*x)^(11/2)

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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}{88} (1-2 x)^{11/2}-\frac {505}{72} (1-2 x)^{9/2}+\frac {1133}{56} (1-2 x)^{7/2}-\frac {847}{40} (1-2 x)^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-847*(1 - 2*x)^(5/2))/40 + (1133*(1 - 2*x)^(7/2))/56 - (505*(1 - 2*x)^(9/2))/72 + (75*(1 - 2*x)^(11/2))/88

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)^{3/2} (2+3 x) (3+5 x)^2 \, dx &=\int \left (\frac {847}{8} (1-2 x)^{3/2}-\frac {1133}{8} (1-2 x)^{5/2}+\frac {505}{8} (1-2 x)^{7/2}-\frac {75}{8} (1-2 x)^{9/2}\right ) \, dx\\ &=-\frac {847}{40} (1-2 x)^{5/2}+\frac {1133}{56} (1-2 x)^{7/2}-\frac {505}{72} (1-2 x)^{9/2}+\frac {75}{88} (1-2 x)^{11/2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 28, normalized size = 0.53 \begin {gather*} -\frac {(1-2 x)^{5/2} \left (24617+60715 x+61775 x^2+23625 x^3\right )}{3465} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3465*((1 - 2*x)^(5/2)*(24617 + 60715*x + 61775*x^2 + 23625*x^3))

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Maple [A]
time = 0.09, size = 38, normalized size = 0.72

method	result	size
gosper	\(-\frac {\left (23625 x^{3}+61775 x^{2}+60715 x +24617\right ) \left (1-2 x \right )^{\frac {5}{2}}}{3465}\)	\(25\)
trager	\(\left (-\frac {300}{11} x^{5}-\frac {4360}{99} x^{4}-\frac {3877}{693} x^{3}+\frac {27539}{1155} x^{2}+\frac {37753}{3465} x -\frac {24617}{3465}\right ) \sqrt {1-2 x}\)	\(34\)
derivativedivides	\(-\frac {847 \left (1-2 x \right )^{\frac {5}{2}}}{40}+\frac {1133 \left (1-2 x \right )^{\frac {7}{2}}}{56}-\frac {505 \left (1-2 x \right )^{\frac {9}{2}}}{72}+\frac {75 \left (1-2 x \right )^{\frac {11}{2}}}{88}\)	\(38\)
default	\(-\frac {847 \left (1-2 x \right )^{\frac {5}{2}}}{40}+\frac {1133 \left (1-2 x \right )^{\frac {7}{2}}}{56}-\frac {505 \left (1-2 x \right )^{\frac {9}{2}}}{72}+\frac {75 \left (1-2 x \right )^{\frac {11}{2}}}{88}\)	\(38\)
risch	\(\frac {\left (94500 x^{5}+152600 x^{4}+19385 x^{3}-82617 x^{2}-37753 x +24617\right ) \left (-1+2 x \right )}{3465 \sqrt {1-2 x}}\)	\(40\)
meijerg	\(-\frac {27 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (8 x^{2}-8 x +2\right ) \sqrt {1-2 x}}{15}\right )}{4 \sqrt {\pi }}+\frac {\frac {87 \sqrt {\pi }}{35}-\frac {87 \sqrt {\pi }\, \left (160 x^{3}-128 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{280}}{\sqrt {\pi }}-\frac {105 \left (-\frac {64 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (1120 x^{4}-800 x^{3}+24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{945}\right )}{8 \sqrt {\pi }}+\frac {\frac {10 \sqrt {\pi }}{77}-\frac {5 \sqrt {\pi }\, \left (26880 x^{5}-17920 x^{4}+320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{4928}}{\sqrt {\pi }}\)	\(164\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-847/40*(1-2*x)^(5/2)+1133/56*(1-2*x)^(7/2)-505/72*(1-2*x)^(9/2)+75/88*(1-2*x)^(11/2)

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Maxima [A]
time = 0.28, size = 37, normalized size = 0.70 \begin {gather*} \frac {75}{88} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {505}{72} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {1133}{56} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {847}{40} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

75/88*(-2*x + 1)^(11/2) - 505/72*(-2*x + 1)^(9/2) + 1133/56*(-2*x + 1)^(7/2) - 847/40*(-2*x + 1)^(5/2)

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Fricas [A]
time = 1.24, size = 34, normalized size = 0.64 \begin {gather*} -\frac {1}{3465} \, {\left (94500 \, x^{5} + 152600 \, x^{4} + 19385 \, x^{3} - 82617 \, x^{2} - 37753 \, x + 24617\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3465*(94500*x^5 + 152600*x^4 + 19385*x^3 - 82617*x^2 - 37753*x + 24617)*sqrt(-2*x + 1)

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Sympy [A]
time = 5.84, size = 46, normalized size = 0.87 \begin {gather*} \frac {75 \left (1 - 2 x\right )^{\frac {11}{2}}}{88} - \frac {505 \left (1 - 2 x\right )^{\frac {9}{2}}}{72} + \frac {1133 \left (1 - 2 x\right )^{\frac {7}{2}}}{56} - \frac {847 \left (1 - 2 x\right )^{\frac {5}{2}}}{40} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

75*(1 - 2*x)**(11/2)/88 - 505*(1 - 2*x)**(9/2)/72 + 1133*(1 - 2*x)**(7/2)/56 - 847*(1 - 2*x)**(5/2)/40

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Giac [A]
time = 1.85, size = 65, normalized size = 1.23 \begin {gather*} -\frac {75}{88} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {505}{72} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {1133}{56} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {847}{40} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-75/88*(2*x - 1)^5*sqrt(-2*x + 1) - 505/72*(2*x - 1)^4*sqrt(-2*x + 1) - 1133/56*(2*x - 1)^3*sqrt(-2*x + 1) - 8
47/40*(2*x - 1)^2*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 )}^{7/2}}{56}-\frac {847\,{\left (1-2\,x\right )}^{5/2}}{40}-\frac {505\,{\left (1-2\,x\right )}^{9/2}}{72}+\frac {75\,{\left (1-2\,x\right )}^{11/2}}{88} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1133*(1 - 2*x)^(7/2))/56 - (847*(1 - 2*x)^(5/2))/40 - (505*(1 - 2*x)^(9/2))/72 + (75*(1 - 2*x)^(11/2))/88

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