3.1162 \(\int (1-2 x) (2+3 x)^4 (3+5 x)^2 \, dx\)

Optimal. Leaf size=45 \[ -\frac{25}{324} (3 x+2)^8+\frac{65}{189} (3 x+2)^7-\frac{4}{27} (3 x+2)^6+\frac{7}{405} (3 x+2)^5 \]

[Out]

(7*(2 + 3*x)^5)/405 - (4*(2 + 3*x)^6)/27 + (65*(2 + 3*x)^7)/189 - (25*(2 + 3*x)^8)/324

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Rubi [A]  time = 0.0201711, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ -\frac{25}{324} (3 x+2)^8+\frac{65}{189} (3 x+2)^7-\frac{4}{27} (3 x+2)^6+\frac{7}{405} (3 x+2)^5 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*(2 + 3*x)^5)/405 - (4*(2 + 3*x)^6)/27 + (65*(2 + 3*x)^7)/189 - (25*(2 + 3*x)^8)/324

Rule 77

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)^2 \, dx &=\int \left (\frac{7}{27} (2+3 x)^4-\frac{8}{3} (2+3 x)^5+\frac{65}{9} (2+3 x)^6-\frac{50}{27} (2+3 x)^7\right ) \, dx\\ &=\frac{7}{405} (2+3 x)^5-\frac{4}{27} (2+3 x)^6+\frac{65}{189} (2+3 x)^7-\frac{25}{324} (2+3 x)^8\\ \end{align*}

Mathematica [A]  time = 0.0011451, size = 47, normalized size = 1.04 \[ -\frac{2025 x^8}{4}-\frac{13635 x^7}{7}-2898 x^6-\frac{9039 x^5}{5}+94 x^4+\frac{2536 x^3}{3}+528 x^2+144 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

144*x + 528*x^2 + (2536*x^3)/3 + 94*x^4 - (9039*x^5)/5 - 2898*x^6 - (13635*x^7)/7 - (2025*x^8)/4

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Maple [A]  time = 0.001, size = 40, normalized size = 0.9 \begin{align*} -{\frac{2025\,{x}^{8}}{4}}-{\frac{13635\,{x}^{7}}{7}}-2898\,{x}^{6}-{\frac{9039\,{x}^{5}}{5}}+94\,{x}^{4}+{\frac{2536\,{x}^{3}}{3}}+528\,{x}^{2}+144\,x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2025/4*x^8-13635/7*x^7-2898*x^6-9039/5*x^5+94*x^4+2536/3*x^3+528*x^2+144*x

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Maxima [A]  time = 1.14081, size = 53, normalized size = 1.18 \begin{align*} -\frac{2025}{4} \, x^{8} - \frac{13635}{7} \, x^{7} - 2898 \, x^{6} - \frac{9039}{5} \, x^{5} + 94 \, x^{4} + \frac{2536}{3} \, x^{3} + 528 \, x^{2} + 144 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2025/4*x^8 - 13635/7*x^7 - 2898*x^6 - 9039/5*x^5 + 94*x^4 + 2536/3*x^3 + 528*x^2 + 144*x

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Fricas [A]  time = 1.29925, size = 123, normalized size = 2.73 \begin{align*} -\frac{2025}{4} x^{8} - \frac{13635}{7} x^{7} - 2898 x^{6} - \frac{9039}{5} x^{5} + 94 x^{4} + \frac{2536}{3} x^{3} + 528 x^{2} + 144 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2025/4*x^8 - 13635/7*x^7 - 2898*x^6 - 9039/5*x^5 + 94*x^4 + 2536/3*x^3 + 528*x^2 + 144*x

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Sympy [A]  time = 0.06363, size = 44, normalized size = 0.98 \begin{align*} - \frac{2025 x^{8}}{4} - \frac{13635 x^{7}}{7} - 2898 x^{6} - \frac{9039 x^{5}}{5} + 94 x^{4} + \frac{2536 x^{3}}{3} + 528 x^{2} + 144 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2025*x**8/4 - 13635*x**7/7 - 2898*x**6 - 9039*x**5/5 + 94*x**4 + 2536*x**3/3 + 528*x**2 + 144*x

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Giac [A]  time = 1.77101, size = 53, normalized size = 1.18 \begin{align*} -\frac{2025}{4} \, x^{8} - \frac{13635}{7} \, x^{7} - 2898 \, x^{6} - \frac{9039}{5} \, x^{5} + 94 \, x^{4} + \frac{2536}{3} \, x^{3} + 528 \, x^{2} + 144 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2025/4*x^8 - 13635/7*x^7 - 2898*x^6 - 9039/5*x^5 + 94*x^4 + 2536/3*x^3 + 528*x^2 + 144*x