3.12.81 \(\int (1-2 x) (2+3 x)^2 (3+5 x)^3 \, dx\) [1181]

Optimal. Leaf size=40 \[ 108 x+324 x^2+345 x^3-\frac {1111 x^4}{4}-1061 x^5-\frac {1975 x^6}{2}-\frac {2250 x^7}{7} \]

[Out]

108*x+324*x^2+345*x^3-1111/4*x^4-1061*x^5-1975/2*x^6-2250/7*x^7

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Rubi [A]
time = 0.01, antiderivative size = 40, 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 {2250 x^7}{7}-\frac {1975 x^6}{2}-1061 x^5-\frac {1111 x^4}{4}+345 x^3+324 x^2+108 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

108*x + 324*x^2 + 345*x^3 - (1111*x^4)/4 - 1061*x^5 - (1975*x^6)/2 - (2250*x^7)/7

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)^2 (3+5 x)^3 \, dx &=\int \left (108+648 x+1035 x^2-1111 x^3-5305 x^4-5925 x^5-2250 x^6\right ) \, dx\\ &=108 x+324 x^2+345 x^3-\frac {1111 x^4}{4}-1061 x^5-\frac {1975 x^6}{2}-\frac {2250 x^7}{7}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 40, normalized size = 1.00 \begin {gather*} 108 x+324 x^2+345 x^3-\frac {1111 x^4}{4}-1061 x^5-\frac {1975 x^6}{2}-\frac {2250 x^7}{7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

108*x + 324*x^2 + 345*x^3 - (1111*x^4)/4 - 1061*x^5 - (1975*x^6)/2 - (2250*x^7)/7

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Maple [A]
time = 0.07, size = 35, normalized size = 0.88

method	result	size
gosper	\(-\frac {x \left (9000 x^{6}+27650 x^{5}+29708 x^{4}+7777 x^{3}-9660 x^{2}-9072 x -3024\right )}{28}\)	\(34\)
default	\(108 x +324 x^{2}+345 x^{3}-\frac {1111}{4} x^{4}-1061 x^{5}-\frac {1975}{2} x^{6}-\frac {2250}{7} x^{7}\)	\(35\)
norman	\(108 x +324 x^{2}+345 x^{3}-\frac {1111}{4} x^{4}-1061 x^{5}-\frac {1975}{2} x^{6}-\frac {2250}{7} x^{7}\)	\(35\)
risch	\(108 x +324 x^{2}+345 x^{3}-\frac {1111}{4} x^{4}-1061 x^{5}-\frac {1975}{2} x^{6}-\frac {2250}{7} x^{7}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

108*x+324*x^2+345*x^3-1111/4*x^4-1061*x^5-1975/2*x^6-2250/7*x^7

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Maxima [A]
time = 0.34, size = 34, normalized size = 0.85 \begin {gather*} -\frac {2250}{7} \, x^{7} - \frac {1975}{2} \, x^{6} - 1061 \, x^{5} - \frac {1111}{4} \, x^{4} + 345 \, x^{3} + 324 \, x^{2} + 108 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2250/7*x^7 - 1975/2*x^6 - 1061*x^5 - 1111/4*x^4 + 345*x^3 + 324*x^2 + 108*x

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Fricas [A]
time = 0.53, size = 34, normalized size = 0.85 \begin {gather*} -\frac {2250}{7} \, x^{7} - \frac {1975}{2} \, x^{6} - 1061 \, x^{5} - \frac {1111}{4} \, x^{4} + 345 \, x^{3} + 324 \, x^{2} + 108 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2250/7*x^7 - 1975/2*x^6 - 1061*x^5 - 1111/4*x^4 + 345*x^3 + 324*x^2 + 108*x

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Sympy [A]
time = 0.01, size = 37, normalized size = 0.92 \begin {gather*} - \frac {2250 x^{7}}{7} - \frac {1975 x^{6}}{2} - 1061 x^{5} - \frac {1111 x^{4}}{4} + 345 x^{3} + 324 x^{2} + 108 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2250*x**7/7 - 1975*x**6/2 - 1061*x**5 - 1111*x**4/4 + 345*x**3 + 324*x**2 + 108*x

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Giac [A]
time = 0.72, size = 34, normalized size = 0.85 \begin {gather*} -\frac {2250}{7} \, x^{7} - \frac {1975}{2} \, x^{6} - 1061 \, x^{5} - \frac {1111}{4} \, x^{4} + 345 \, x^{3} + 324 \, x^{2} + 108 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2250/7*x^7 - 1975/2*x^6 - 1061*x^5 - 1111/4*x^4 + 345*x^3 + 324*x^2 + 108*x

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Mupad [B]
time = 0.02, size = 34, normalized size = 0.85 \begin {gather*} -\frac {2250\,x^7}{7}-\frac {1975\,x^6}{2}-1061\,x^5-\frac {1111\,x^4}{4}+345\,x^3+324\,x^2+108\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

108*x + 324*x^2 + 345*x^3 - (1111*x^4)/4 - 1061*x^5 - (1975*x^6)/2 - (2250*x^7)/7

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