3.32.90 \(\int \frac {(1-2 x) (2+3 x)^m}{3+5 x} \, dx\) [3190]

Optimal. Leaf size=52 \[ -\frac {2 (2+3 x)^{1+m}}{15 (1+m)}-\frac {11 (2+3 x)^{1+m} \, _2F_1(1,1+m;2+m;5 (2+3 x))}{5 (1+m)} \]

[Out]

-2/15*(2+3*x)^(1+m)/(1+m)-11/5*(2+3*x)^(1+m)*hypergeom([1, 1+m],[2+m],10+15*x)/(1+m)

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Rubi [A]
time = 0.01, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {81, 70} \begin {gather*} -\frac {11 (3 x+2)^{m+1} \, _2F_1(1,m+1;m+2;5 (3 x+2))}{5 (m+1)}-\frac {2 (3 x+2)^{m+1}}{15 (m+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(2 + 3*x)^(1 + m))/(15*(1 + m)) - (11*(2 + 3*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, 5*(2 + 3*x)])/(
5*(1 + m))

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/(b^(
n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rule 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rubi steps

\begin {align*} \int \frac {(1-2 x) (2+3 x)^m}{3+5 x} \, dx &=-\frac {2 (2+3 x)^{1+m}}{15 (1+m)}+\frac {11}{5} \int \frac {(2+3 x)^m}{3+5 x} \, dx\\ &=-\frac {2 (2+3 x)^{1+m}}{15 (1+m)}-\frac {11 (2+3 x)^{1+m} \, _2F_1(1,1+m;2+m;5 (2+3 x))}{5 (1+m)}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 37, normalized size = 0.71 \begin {gather*} -\frac {(2+3 x)^{1+m} (2+33 \, _2F_1(1,1+m;2+m;5 (2+3 x)))}{15 (1+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/15*((2 + 3*x)^(1 + m)*(2 + 33*Hypergeometric2F1[1, 1 + m, 2 + m, 5*(2 + 3*x)]))/(1 + m)

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (1-2 x \right ) \left (2+3 x \right )^{m}}{3+5 x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(3*x + 2)^m*(2*x - 1)/(5*x + 3), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {\left (3 x + 2\right )^{m}}{5 x + 3}\right )\, dx - \int \frac {2 x \left (3 x + 2\right )^{m}}{5 x + 3}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-(3*x + 2)**m/(5*x + 3), x) - Integral(2*x*(3*x + 2)**m/(5*x + 3), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} -\int \frac {\left (2\,x-1\right )\,{\left (3\,x+2\right )}^m}{5\,x+3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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