3.13.39 \(\int (1-2 x)^2 (2+3 x)^5 (3+5 x) \, dx\) [1239]

Optimal. Leaf size=45 \[ -\frac {49}{486} (2+3 x)^6+\frac {13}{27} (2+3 x)^7-\frac {2}{9} (2+3 x)^8+\frac {20}{729} (2+3 x)^9 \]

[Out]

-49/486*(2+3*x)^6+13/27*(2+3*x)^7-2/9*(2+3*x)^8+20/729*(2+3*x)^9

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Rubi [A]
time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \begin {gather*} \frac {20}{729} (3 x+2)^9-\frac {2}{9} (3 x+2)^8+\frac {13}{27} (3 x+2)^7-\frac {49}{486} (3 x+2)^6 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-49*(2 + 3*x)^6)/486 + (13*(2 + 3*x)^7)/27 - (2*(2 + 3*x)^8)/9 + (20*(2 + 3*x)^9)/729

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)^2 (2+3 x)^5 (3+5 x) \, dx &=\int \left (-\frac {49}{27} (2+3 x)^5+\frac {91}{9} (2+3 x)^6-\frac {16}{3} (2+3 x)^7+\frac {20}{27} (2+3 x)^8\right ) \, dx\\ &=-\frac {49}{486} (2+3 x)^6+\frac {13}{27} (2+3 x)^7-\frac {2}{9} (2+3 x)^8+\frac {20}{729} (2+3 x)^9\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 48, normalized size = 1.07 \begin {gather*} 96 x+248 x^2+\frac {224 x^3}{3}-770 x^4-1218 x^5+\frac {273 x^6}{2}+1917 x^7+1782 x^8+540 x^9 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

96*x + 248*x^2 + (224*x^3)/3 - 770*x^4 - 1218*x^5 + (273*x^6)/2 + 1917*x^7 + 1782*x^8 + 540*x^9

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Maple [A]
time = 0.10, size = 45, normalized size = 1.00

method	result	size
gosper	\(\frac {x \left (3240 x^{8}+10692 x^{7}+11502 x^{6}+819 x^{5}-7308 x^{4}-4620 x^{3}+448 x^{2}+1488 x +576\right )}{6}\)	\(44\)
default	\(540 x^{9}+1782 x^{8}+1917 x^{7}+\frac {273}{2} x^{6}-1218 x^{5}-770 x^{4}+\frac {224}{3} x^{3}+248 x^{2}+96 x\)	\(45\)
norman	\(540 x^{9}+1782 x^{8}+1917 x^{7}+\frac {273}{2} x^{6}-1218 x^{5}-770 x^{4}+\frac {224}{3} x^{3}+248 x^{2}+96 x\)	\(45\)
risch	\(540 x^{9}+1782 x^{8}+1917 x^{7}+\frac {273}{2} x^{6}-1218 x^{5}-770 x^{4}+\frac {224}{3} x^{3}+248 x^{2}+96 x\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

540*x^9+1782*x^8+1917*x^7+273/2*x^6-1218*x^5-770*x^4+224/3*x^3+248*x^2+96*x

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Maxima [A]
time = 0.30, size = 44, normalized size = 0.98 \begin {gather*} 540 \, x^{9} + 1782 \, x^{8} + 1917 \, x^{7} + \frac {273}{2} \, x^{6} - 1218 \, x^{5} - 770 \, x^{4} + \frac {224}{3} \, x^{3} + 248 \, x^{2} + 96 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

540*x^9 + 1782*x^8 + 1917*x^7 + 273/2*x^6 - 1218*x^5 - 770*x^4 + 224/3*x^3 + 248*x^2 + 96*x

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Fricas [A]
time = 0.61, size = 44, normalized size = 0.98 \begin {gather*} 540 \, x^{9} + 1782 \, x^{8} + 1917 \, x^{7} + \frac {273}{2} \, x^{6} - 1218 \, x^{5} - 770 \, x^{4} + \frac {224}{3} \, x^{3} + 248 \, x^{2} + 96 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

540*x^9 + 1782*x^8 + 1917*x^7 + 273/2*x^6 - 1218*x^5 - 770*x^4 + 224/3*x^3 + 248*x^2 + 96*x

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Sympy [A]
time = 0.06, size = 46, normalized size = 1.02 \begin {gather*} 540 x^{9} + 1782 x^{8} + 1917 x^{7} + \frac {273 x^{6}}{2} - 1218 x^{5} - 770 x^{4} + \frac {224 x^{3}}{3} + 248 x^{2} + 96 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

540*x**9 + 1782*x**8 + 1917*x**7 + 273*x**6/2 - 1218*x**5 - 770*x**4 + 224*x**3/3 + 248*x**2 + 96*x

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Giac [A]
time = 0.59, size = 44, normalized size = 0.98 \begin {gather*} 540 \, x^{9} + 1782 \, x^{8} + 1917 \, x^{7} + \frac {273}{2} \, x^{6} - 1218 \, x^{5} - 770 \, x^{4} + \frac {224}{3} \, x^{3} + 248 \, x^{2} + 96 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

540*x^9 + 1782*x^8 + 1917*x^7 + 273/2*x^6 - 1218*x^5 - 770*x^4 + 224/3*x^3 + 248*x^2 + 96*x

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Mupad [B]
time = 0.03, size = 44, normalized size = 0.98 \begin {gather*} 540\,x^9+1782\,x^8+1917\,x^7+\frac {273\,x^6}{2}-1218\,x^5-770\,x^4+\frac {224\,x^3}{3}+248\,x^2+96\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

96*x + 248*x^2 + (224*x^3)/3 - 770*x^4 - 1218*x^5 + (273*x^6)/2 + 1917*x^7 + 1782*x^8 + 540*x^9

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