3.478 \(\int \frac {-7+4 x^2}{3+2 x} \, dx\)

Optimal. Leaf size=13 \[ x^2-3 x+\log (2 x+3) \]

[Out]

-3*x+x^2+ln(3+2*x)

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Rubi [A]  time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {697} \[ x^2-3 x+\log (2 x+3) \]

Antiderivative was successfully verified.

[In]

Int[(-7 + 4*x^2)/(3 + 2*x),x]

[Out]

-3*x + x^2 + Log[3 + 2*x]

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {-7+4 x^2}{3+2 x} \, dx &=\int \left (-3+2 x+\frac {2}{3+2 x}\right ) \, dx\\ &=-3 x+x^2+\log (3+2 x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.23 \[ x^2-3 x+\log (2 x+3)-\frac {27}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[(-7 + 4*x^2)/(3 + 2*x),x]

[Out]

-27/4 - 3*x + x^2 + Log[3 + 2*x]

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fricas [A]  time = 0.84, size = 13, normalized size = 1.00 \[ x^{2} - 3 \, x + \log \left (2 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2-7)/(3+2*x),x, algorithm="fricas")

[Out]

x^2 - 3*x + log(2*x + 3)

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giac [A]  time = 0.36, size = 14, normalized size = 1.08 \[ x^{2} - 3 \, x + \log \left ({\left | 2 \, x + 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2-7)/(3+2*x),x, algorithm="giac")

[Out]

x^2 - 3*x + log(abs(2*x + 3))

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maple [A]  time = 0.00, size = 14, normalized size = 1.08 \[ x^{2}-3 x +\ln \left (2 x +3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2-7)/(2*x+3),x)

[Out]

-3*x+x^2+ln(2*x+3)

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maxima [A]  time = 0.64, size = 13, normalized size = 1.00 \[ x^{2} - 3 \, x + \log \left (2 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2-7)/(3+2*x),x, algorithm="maxima")

[Out]

x^2 - 3*x + log(2*x + 3)

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mupad [B]  time = 2.21, size = 11, normalized size = 0.85 \[ \ln \left (x+\frac {3}{2}\right )-3\,x+x^2 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2 - 7)/(2*x + 3),x)

[Out]

log(x + 3/2) - 3*x + x^2

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sympy [A]  time = 0.08, size = 12, normalized size = 0.92 \[ x^{2} - 3 x + \log {\left (2 x + 3 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**2-7)/(3+2*x),x)

[Out]

x**2 - 3*x + log(2*x + 3)

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