3.12.47 \(\int (1-2 x) (2+3 x)^4 (3+5 x) \, dx\) [1147]

Optimal. Leaf size=34 \[ -\frac {7}{135} (2+3 x)^5+\frac {37}{162} (2+3 x)^6-\frac {10}{189} (2+3 x)^7 \]

[Out]

-7/135*(2+3*x)^5+37/162*(2+3*x)^6-10/189*(2+3*x)^7

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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}{189} (3 x+2)^7+\frac {37}{162} (3 x+2)^6-\frac {7}{135} (3 x+2)^5 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)*(2 + 3*x)^4*(3 + 5*x),x]

[Out]

(-7*(2 + 3*x)^5)/135 + (37*(2 + 3*x)^6)/162 - (10*(2 + 3*x)^7)/189

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)^4 (3+5 x) \, dx &=\int \left (-\frac {7}{9} (2+3 x)^4+\frac {37}{9} (2+3 x)^5-\frac {10}{9} (2+3 x)^6\right ) \, dx\\ &=-\frac {7}{135} (2+3 x)^5+\frac {37}{162} (2+3 x)^6-\frac {10}{189} (2+3 x)^7\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 42, normalized size = 1.24 \begin {gather*} 48 x+136 x^2+\frac {392 x^3}{3}-132 x^4-\frac {2133 x^5}{5}-\frac {747 x^6}{2}-\frac {810 x^7}{7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)*(2 + 3*x)^4*(3 + 5*x),x]

[Out]

48*x + 136*x^2 + (392*x^3)/3 - 132*x^4 - (2133*x^5)/5 - (747*x^6)/2 - (810*x^7)/7

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Maple [A]
time = 0.10, size = 35, normalized size = 1.03

method	result	size
gosper	\(-\frac {x \left (24300 x^{6}+78435 x^{5}+89586 x^{4}+27720 x^{3}-27440 x^{2}-28560 x -10080\right )}{210}\)	\(34\)
default	\(-\frac {810}{7} x^{7}-\frac {747}{2} x^{6}-\frac {2133}{5} x^{5}-132 x^{4}+\frac {392}{3} x^{3}+136 x^{2}+48 x\)	\(35\)
norman	\(-\frac {810}{7} x^{7}-\frac {747}{2} x^{6}-\frac {2133}{5} x^{5}-132 x^{4}+\frac {392}{3} x^{3}+136 x^{2}+48 x\)	\(35\)
risch	\(-\frac {810}{7} x^{7}-\frac {747}{2} x^{6}-\frac {2133}{5} x^{5}-132 x^{4}+\frac {392}{3} x^{3}+136 x^{2}+48 x\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)*(2+3*x)^4*(3+5*x),x,method=_RETURNVERBOSE)

[Out]

-810/7*x^7-747/2*x^6-2133/5*x^5-132*x^4+392/3*x^3+136*x^2+48*x

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Maxima [A]
time = 0.40, size = 34, normalized size = 1.00 \begin {gather*} -\frac {810}{7} \, x^{7} - \frac {747}{2} \, x^{6} - \frac {2133}{5} \, x^{5} - 132 \, x^{4} + \frac {392}{3} \, x^{3} + 136 \, x^{2} + 48 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(2+3*x)^4*(3+5*x),x, algorithm="maxima")

[Out]

-810/7*x^7 - 747/2*x^6 - 2133/5*x^5 - 132*x^4 + 392/3*x^3 + 136*x^2 + 48*x

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Fricas [A]
time = 0.45, size = 34, normalized size = 1.00 \begin {gather*} -\frac {810}{7} \, x^{7} - \frac {747}{2} \, x^{6} - \frac {2133}{5} \, x^{5} - 132 \, x^{4} + \frac {392}{3} \, x^{3} + 136 \, x^{2} + 48 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(2+3*x)^4*(3+5*x),x, algorithm="fricas")

[Out]

-810/7*x^7 - 747/2*x^6 - 2133/5*x^5 - 132*x^4 + 392/3*x^3 + 136*x^2 + 48*x

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Sympy [A]
time = 0.01, size = 39, normalized size = 1.15 \begin {gather*} - \frac {810 x^{7}}{7} - \frac {747 x^{6}}{2} - \frac {2133 x^{5}}{5} - 132 x^{4} + \frac {392 x^{3}}{3} + 136 x^{2} + 48 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(2+3*x)**4*(3+5*x),x)

[Out]

-810*x**7/7 - 747*x**6/2 - 2133*x**5/5 - 132*x**4 + 392*x**3/3 + 136*x**2 + 48*x

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Giac [A]
time = 1.75, size = 34, normalized size = 1.00 \begin {gather*} -\frac {810}{7} \, x^{7} - \frac {747}{2} \, x^{6} - \frac {2133}{5} \, x^{5} - 132 \, x^{4} + \frac {392}{3} \, x^{3} + 136 \, x^{2} + 48 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(2+3*x)^4*(3+5*x),x, algorithm="giac")

[Out]

-810/7*x^7 - 747/2*x^6 - 2133/5*x^5 - 132*x^4 + 392/3*x^3 + 136*x^2 + 48*x

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Mupad [B]
time = 0.02, size = 34, normalized size = 1.00 \begin {gather*} -\frac {810\,x^7}{7}-\frac {747\,x^6}{2}-\frac {2133\,x^5}{5}-132\,x^4+\frac {392\,x^3}{3}+136\,x^2+48\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - 1)*(3*x + 2)^4*(5*x + 3),x)

[Out]

48*x + 136*x^2 + (392*x^3)/3 - 132*x^4 - (2133*x^5)/5 - (747*x^6)/2 - (810*x^7)/7

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