3.12.45 \(\int (1-2 x) (2+3 x)^6 (3+5 x) \, dx\) [1145]

Optimal. Leaf size=34 \[ -\frac {1}{27} (2+3 x)^7+\frac {37}{216} (2+3 x)^8-\frac {10}{243} (2+3 x)^9 \]

[Out]

-1/27*(2+3*x)^7+37/216*(2+3*x)^8-10/243*(2+3*x)^9

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Rubi [A]
time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {78} \begin {gather*} -\frac {10}{243} (3 x+2)^9+\frac {37}{216} (3 x+2)^8-\frac {1}{27} (3 x+2)^7 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)*(2 + 3*x)^6*(3 + 5*x),x]

[Out]

-1/27*(2 + 3*x)^7 + (37*(2 + 3*x)^8)/216 - (10*(2 + 3*x)^9)/243

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+3 x)^6 (3+5 x) \, dx &=\int \left (-\frac {7}{9} (2+3 x)^6+\frac {37}{9} (2+3 x)^7-\frac {10}{9} (2+3 x)^8\right ) \, dx\\ &=-\frac {1}{27} (2+3 x)^7+\frac {37}{216} (2+3 x)^8-\frac {10}{243} (2+3 x)^9\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 48, normalized size = 1.41 \begin {gather*} 192 x+832 x^2+\frac {5264 x^3}{3}+1260 x^4-2268 x^5-6552 x^6-7047 x^7-\frac {29889 x^8}{8}-810 x^9 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)*(2 + 3*x)^6*(3 + 5*x),x]

[Out]

192*x + 832*x^2 + (5264*x^3)/3 + 1260*x^4 - 2268*x^5 - 6552*x^6 - 7047*x^7 - (29889*x^8)/8 - 810*x^9

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Maple [A]
time = 0.08, size = 45, normalized size = 1.32

method	result	size
gosper	\(-\frac {x \left (19440 x^{8}+89667 x^{7}+169128 x^{6}+157248 x^{5}+54432 x^{4}-30240 x^{3}-42112 x^{2}-19968 x -4608\right )}{24}\)	\(44\)
default	\(-810 x^{9}-\frac {29889}{8} x^{8}-7047 x^{7}-6552 x^{6}-2268 x^{5}+1260 x^{4}+\frac {5264}{3} x^{3}+832 x^{2}+192 x\)	\(45\)
norman	\(-810 x^{9}-\frac {29889}{8} x^{8}-7047 x^{7}-6552 x^{6}-2268 x^{5}+1260 x^{4}+\frac {5264}{3} x^{3}+832 x^{2}+192 x\)	\(45\)
risch	\(-810 x^{9}-\frac {29889}{8} x^{8}-7047 x^{7}-6552 x^{6}-2268 x^{5}+1260 x^{4}+\frac {5264}{3} x^{3}+832 x^{2}+192 x\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)*(2+3*x)^6*(3+5*x),x,method=_RETURNVERBOSE)

[Out]

-810*x^9-29889/8*x^8-7047*x^7-6552*x^6-2268*x^5+1260*x^4+5264/3*x^3+832*x^2+192*x

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Maxima [A]
time = 0.29, size = 44, normalized size = 1.29 \begin {gather*} -810 \, x^{9} - \frac {29889}{8} \, x^{8} - 7047 \, x^{7} - 6552 \, x^{6} - 2268 \, x^{5} + 1260 \, x^{4} + \frac {5264}{3} \, x^{3} + 832 \, x^{2} + 192 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(2+3*x)^6*(3+5*x),x, algorithm="maxima")

[Out]

-810*x^9 - 29889/8*x^8 - 7047*x^7 - 6552*x^6 - 2268*x^5 + 1260*x^4 + 5264/3*x^3 + 832*x^2 + 192*x

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Fricas [A]
time = 0.79, size = 44, normalized size = 1.29 \begin {gather*} -810 \, x^{9} - \frac {29889}{8} \, x^{8} - 7047 \, x^{7} - 6552 \, x^{6} - 2268 \, x^{5} + 1260 \, x^{4} + \frac {5264}{3} \, x^{3} + 832 \, x^{2} + 192 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(2+3*x)^6*(3+5*x),x, algorithm="fricas")

[Out]

-810*x^9 - 29889/8*x^8 - 7047*x^7 - 6552*x^6 - 2268*x^5 + 1260*x^4 + 5264/3*x^3 + 832*x^2 + 192*x

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Sympy [A]
time = 0.02, size = 46, normalized size = 1.35 \begin {gather*} - 810 x^{9} - \frac {29889 x^{8}}{8} - 7047 x^{7} - 6552 x^{6} - 2268 x^{5} + 1260 x^{4} + \frac {5264 x^{3}}{3} + 832 x^{2} + 192 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(2+3*x)**6*(3+5*x),x)

[Out]

-810*x**9 - 29889*x**8/8 - 7047*x**7 - 6552*x**6 - 2268*x**5 + 1260*x**4 + 5264*x**3/3 + 832*x**2 + 192*x

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Giac [A]
time = 2.65, size = 44, normalized size = 1.29 \begin {gather*} -810 \, x^{9} - \frac {29889}{8} \, x^{8} - 7047 \, x^{7} - 6552 \, x^{6} - 2268 \, x^{5} + 1260 \, x^{4} + \frac {5264}{3} \, x^{3} + 832 \, x^{2} + 192 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(2+3*x)^6*(3+5*x),x, algorithm="giac")

[Out]

-810*x^9 - 29889/8*x^8 - 7047*x^7 - 6552*x^6 - 2268*x^5 + 1260*x^4 + 5264/3*x^3 + 832*x^2 + 192*x

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Mupad [B]
time = 0.03, size = 44, normalized size = 1.29 \begin {gather*} -810\,x^9-\frac {29889\,x^8}{8}-7047\,x^7-6552\,x^6-2268\,x^5+1260\,x^4+\frac {5264\,x^3}{3}+832\,x^2+192\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - 1)*(3*x + 2)^6*(5*x + 3),x)

[Out]

192*x + 832*x^2 + (5264*x^3)/3 + 1260*x^4 - 2268*x^5 - 6552*x^6 - 7047*x^7 - (29889*x^8)/8 - 810*x^9

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