3.12.59 \(\int \frac {(b d+2 c d x)^3}{a+b x+c x^2} \, dx\) [1159]

Optimal. Leaf size=36 \[ d^3 (b+2 c x)^2+\left (b^2-4 a c\right ) d^3 \log \left (a+b x+c x^2\right ) \]

[Out]

d^3*(2*c*x+b)^2+(-4*a*c+b^2)*d^3*ln(c*x^2+b*x+a)

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Rubi [A]
time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {706, 642} \begin {gather*} d^3 \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )+d^3 (b+2 c x)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*d + 2*c*d*x)^3/(a + b*x + c*x^2),x]

[Out]

d^3*(b + 2*c*x)^2 + (b^2 - 4*a*c)*d^3*Log[a + b*x + c*x^2]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 706

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2*d*(d + e*x)^(m - 1
)*((a + b*x + c*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] + Dist[d^2*(m - 1)*((b^2 - 4*a*c)/(b^2*(m + 2*p + 1))), In
t[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[
2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] &
& RationalQ[p]) || OddQ[m])

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 33, normalized size = 0.92 \begin {gather*} d^3 \left (4 c x (b+c x)+\left (b^2-4 a c\right ) \log (a+x (b+c x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*d + 2*c*d*x)^3/(a + b*x + c*x^2),x]

[Out]

d^3*(4*c*x*(b + c*x) + (b^2 - 4*a*c)*Log[a + x*(b + c*x)])

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Maple [A]
time = 0.67, size = 39, normalized size = 1.08

method	result	size
default	\(d^{3} \left (4 c^{2} x^{2}+4 b c x +\left (-4 a c +b^{2}\right ) \ln \left (c \,x^{2}+b x +a \right )\right )\)	\(39\)
norman	\(4 c^{2} d^{3} x^{2}+4 b c \,d^{3} x +\left (-4 d^{3} a c +b^{2} d^{3}\right ) \ln \left (c \,x^{2}+b x +a \right )\)	\(48\)
risch	\(4 c^{2} d^{3} x^{2}+4 b c \,d^{3} x -4 \ln \left (c \,x^{2}+b x +a \right ) a c \,d^{3}+\ln \left (c \,x^{2}+b x +a \right ) b^{2} d^{3}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)^3/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)

[Out]

d^3*(4*c^2*x^2+4*b*c*x+(-4*a*c+b^2)*ln(c*x^2+b*x+a))

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Maxima [A]
time = 0.28, size = 43, normalized size = 1.19 \begin {gather*} 4 \, c^{2} d^{3} x^{2} + 4 \, b c d^{3} x + {\left (b^{2} - 4 \, a c\right )} d^{3} \log \left (c x^{2} + b x + a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^3/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

4*c^2*d^3*x^2 + 4*b*c*d^3*x + (b^2 - 4*a*c)*d^3*log(c*x^2 + b*x + a)

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Fricas [A]
time = 1.65, size = 43, normalized size = 1.19 \begin {gather*} 4 \, c^{2} d^{3} x^{2} + 4 \, b c d^{3} x + {\left (b^{2} - 4 \, a c\right )} d^{3} \log \left (c x^{2} + b x + a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^3/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

4*c^2*d^3*x^2 + 4*b*c*d^3*x + (b^2 - 4*a*c)*d^3*log(c*x^2 + b*x + a)

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Sympy [A]
time = 0.21, size = 44, normalized size = 1.22 \begin {gather*} 4 b c d^{3} x + 4 c^{2} d^{3} x^{2} - d^{3} \cdot \left (4 a c - b^{2}\right ) \log {\left (a + b x + c x^{2} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)**3/(c*x**2+b*x+a),x)

[Out]

4*b*c*d**3*x + 4*c**2*d**3*x**2 - d**3*(4*a*c - b**2)*log(a + b*x + c*x**2)

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Giac [A]
time = 0.78, size = 50, normalized size = 1.39 \begin {gather*} 4 \, {\left (c d x^{2} + b d x\right )} c d^{2} + {\left (b^{2} d^{3} - 4 \, a c d^{3}\right )} \log \left ({\left | c d x^{2} + b d x + a d \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^3/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

4*(c*d*x^2 + b*d*x)*c*d^2 + (b^2*d^3 - 4*a*c*d^3)*log(abs(c*d*x^2 + b*d*x + a*d))

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Mupad [B]
time = 0.44, size = 47, normalized size = 1.31 \begin {gather*} \ln \left (c\,x^2+b\,x+a\right )\,\left (b^2\,d^3-4\,a\,c\,d^3\right )+4\,c^2\,d^3\,x^2+4\,b\,c\,d^3\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*d + 2*c*d*x)^3/(a + b*x + c*x^2),x)

[Out]

log(a + b*x + c*x^2)*(b^2*d^3 - 4*a*c*d^3) + 4*c^2*d^3*x^2 + 4*b*c*d^3*x

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