3.10.25 \(\int \frac {a+b x+c x^2}{b d+2 c d x} \, dx\)

Optimal. Leaf size=48 \[ -\frac {\left (b^2-4 a c\right ) \log (b+2 c x)}{8 c^2 d}+\frac {b x}{4 c d}+\frac {x^2}{4 d} \]

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Rubi [A]  time = 0.04, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {683} \begin {gather*} -\frac {\left (b^2-4 a c\right ) \log (b+2 c x)}{8 c^2 d}+\frac {b x}{4 c d}+\frac {x^2}{4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)/(b*d + 2*c*d*x),x]

[Out]

(b*x)/(4*c*d) + x^2/(4*d) - ((b^2 - 4*a*c)*Log[b + 2*c*x])/(8*c^2*d)

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps

\begin {align*} \int \frac {a+b x+c x^2}{b d+2 c d x} \, dx &=\int \left (\frac {b}{4 c d}+\frac {x}{2 d}+\frac {-b^2+4 a c}{4 c d (b+2 c x)}\right ) \, dx\\ &=\frac {b x}{4 c d}+\frac {x^2}{4 d}-\frac {\left (b^2-4 a c\right ) \log (b+2 c x)}{8 c^2 d}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 37, normalized size = 0.77 \begin {gather*} \frac {2 c x (b+c x)-\left (b^2-4 a c\right ) \log (b+2 c x)}{8 c^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)/(b*d + 2*c*d*x),x]

[Out]

(2*c*x*(b + c*x) - (b^2 - 4*a*c)*Log[b + 2*c*x])/(8*c^2*d)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b x+c x^2}{b d+2 c d x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x + c*x^2)/(b*d + 2*c*d*x),x]

[Out]

IntegrateAlgebraic[(a + b*x + c*x^2)/(b*d + 2*c*d*x), x]

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fricas [A]  time = 0.39, size = 39, normalized size = 0.81 \begin {gather*} \frac {2 \, c^{2} x^{2} + 2 \, b c x - {\left (b^{2} - 4 \, a c\right )} \log \left (2 \, c x + b\right )}{8 \, c^{2} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)/(2*c*d*x+b*d),x, algorithm="fricas")

[Out]

1/8*(2*c^2*x^2 + 2*b*c*x - (b^2 - 4*a*c)*log(2*c*x + b))/(c^2*d)

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giac [A]  time = 0.16, size = 47, normalized size = 0.98 \begin {gather*} -\frac {{\left (b^{2} - 4 \, a c\right )} \log \left ({\left | 2 \, c x + b \right |}\right )}{8 \, c^{2} d} + \frac {c^{2} d x^{2} + b c d x}{4 \, c^{2} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)/(2*c*d*x+b*d),x, algorithm="giac")

[Out]

-1/8*(b^2 - 4*a*c)*log(abs(2*c*x + b))/(c^2*d) + 1/4*(c^2*d*x^2 + b*c*d*x)/(c^2*d^2)

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maple [A]  time = 0.05, size = 54, normalized size = 1.12 \begin {gather*} \frac {x^{2}}{4 d}+\frac {a \ln \left (2 c x +b \right )}{2 c d}-\frac {b^{2} \ln \left (2 c x +b \right )}{8 c^{2} d}+\frac {b x}{4 c d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)/(2*c*d*x+b*d),x)

[Out]

1/4*x^2/d+1/4*b*x/c/d+1/2/d/c*ln(2*c*x+b)*a-1/8/d/c^2*ln(2*c*x+b)*b^2

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maxima [A]  time = 1.30, size = 41, normalized size = 0.85 \begin {gather*} \frac {c x^{2} + b x}{4 \, c d} - \frac {{\left (b^{2} - 4 \, a c\right )} \log \left (2 \, c x + b\right )}{8 \, c^{2} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)/(2*c*d*x+b*d),x, algorithm="maxima")

[Out]

1/4*(c*x^2 + b*x)/(c*d) - 1/8*(b^2 - 4*a*c)*log(2*c*x + b)/(c^2*d)

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mupad [B]  time = 0.44, size = 44, normalized size = 0.92 \begin {gather*} \frac {x^2}{4\,d}+\frac {b\,x}{4\,c\,d}+\frac {\ln \left (b+2\,c\,x\right )\,\left (4\,a\,c-b^2\right )}{8\,c^2\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)/(b*d + 2*c*d*x),x)

[Out]

x^2/(4*d) + (b*x)/(4*c*d) + (log(b + 2*c*x)*(4*a*c - b^2))/(8*c^2*d)

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sympy [A]  time = 0.21, size = 37, normalized size = 0.77 \begin {gather*} \frac {b x}{4 c d} + \frac {x^{2}}{4 d} + \frac {\left (4 a c - b^{2}\right ) \log {\left (b + 2 c x \right )}}{8 c^{2} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)/(2*c*d*x+b*d),x)

[Out]

b*x/(4*c*d) + x**2/(4*d) + (4*a*c - b**2)*log(b + 2*c*x)/(8*c**2*d)

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