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

Optimal. Leaf size=104 \[ \frac {x \left (-c e (3 b d-2 a e)+b^2 e^2+2 c^2 d^2\right )}{e^3}-\frac {(2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right )}{e^4}-\frac {c x^2 (2 c d-3 b e)}{2 e^2}+\frac {2 c^2 x^3}{3 e} \]

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Rubi [A]  time = 0.11, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {771} \begin {gather*} \frac {x \left (-c e (3 b d-2 a e)+b^2 e^2+2 c^2 d^2\right )}{e^3}-\frac {(2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right )}{e^4}-\frac {c x^2 (2 c d-3 b e)}{2 e^2}+\frac {2 c^2 x^3}{3 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 95, normalized size = 0.91 \begin {gather*} \frac {e x \left (3 c e (4 a e-6 b d+3 b e x)+6 b^2 e^2+2 c^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )-6 (2 c d-b e) \log (d+e x) \left (e (a e-b d)+c d^2\right )}{6 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*x*(6*b^2*e^2 + 3*c*e*(-6*b*d + 4*a*e + 3*b*e*x) + 2*c^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) - 6*(2*c*d - b*e)*(c
*d^2 + e*(-(b*d) + a*e))*Log[d + e*x])/(6*e^4)

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 117, normalized size = 1.12 \begin {gather*} \frac {4 \, c^{2} e^{3} x^{3} - 3 \, {\left (2 \, c^{2} d e^{2} - 3 \, b c e^{3}\right )} x^{2} + 6 \, {\left (2 \, c^{2} d^{2} e - 3 \, b c d e^{2} + {\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x - 6 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} \log \left (e x + d\right )}{6 \, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(4*c^2*e^3*x^3 - 3*(2*c^2*d*e^2 - 3*b*c*e^3)*x^2 + 6*(2*c^2*d^2*e - 3*b*c*d*e^2 + (b^2 + 2*a*c)*e^3)*x - 6
*(2*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*log(e*x + d))/e^4

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giac [A]  time = 0.15, size = 118, normalized size = 1.13 \begin {gather*} -{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2} + 2 \, a c d e^{2} - a b e^{3}\right )} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{6} \, {\left (4 \, c^{2} x^{3} e^{2} - 6 \, c^{2} d x^{2} e + 12 \, c^{2} d^{2} x + 9 \, b c x^{2} e^{2} - 18 \, b c d x e + 6 \, b^{2} x e^{2} + 12 \, a c x e^{2}\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*c^2*d^3 - 3*b*c*d^2*e + b^2*d*e^2 + 2*a*c*d*e^2 - a*b*e^3)*e^(-4)*log(abs(x*e + d)) + 1/6*(4*c^2*x^3*e^2 -
6*c^2*d*x^2*e + 12*c^2*d^2*x + 9*b*c*x^2*e^2 - 18*b*c*d*x*e + 6*b^2*x*e^2 + 12*a*c*x*e^2)*e^(-3)

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maple [A]  time = 0.05, size = 146, normalized size = 1.40 \begin {gather*} \frac {2 c^{2} x^{3}}{3 e}+\frac {3 b c \,x^{2}}{2 e}-\frac {c^{2} d \,x^{2}}{e^{2}}+\frac {a b \ln \left (e x +d \right )}{e}-\frac {2 a c d \ln \left (e x +d \right )}{e^{2}}+\frac {2 a c x}{e}-\frac {b^{2} d \ln \left (e x +d \right )}{e^{2}}+\frac {b^{2} x}{e}+\frac {3 b c \,d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {3 b c d x}{e^{2}}-\frac {2 c^{2} d^{3} \ln \left (e x +d \right )}{e^{4}}+\frac {2 c^{2} d^{2} x}{e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*c^2*x^3/e+3/2/e*x^2*b*c-1/e^2*x^2*c^2*d+2/e*x*a*c+1/e*x*b^2-3/e^2*x*b*c*d+2/e^3*x*c^2*d^2+1/e*ln(e*x+d)*a*
b-2/e^2*ln(e*x+d)*a*c*d-1/e^2*ln(e*x+d)*b^2*d+3/e^3*ln(e*x+d)*b*c*d^2-2/e^4*ln(e*x+d)*c^2*d^3

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maxima [A]  time = 0.53, size = 116, normalized size = 1.12 \begin {gather*} \frac {4 \, c^{2} e^{2} x^{3} - 3 \, {\left (2 \, c^{2} d e - 3 \, b c e^{2}\right )} x^{2} + 6 \, {\left (2 \, c^{2} d^{2} - 3 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} x}{6 \, e^{3}} - \frac {{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} \log \left (e x + d\right )}{e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(4*c^2*e^2*x^3 - 3*(2*c^2*d*e - 3*b*c*e^2)*x^2 + 6*(2*c^2*d^2 - 3*b*c*d*e + (b^2 + 2*a*c)*e^2)*x)/e^3 - (2
*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*log(e*x + d)/e^4

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mupad [B]  time = 1.78, size = 121, normalized size = 1.16 \begin {gather*} x\,\left (\frac {b^2+2\,a\,c}{e}+\frac {d\,\left (\frac {2\,c^2\,d}{e^2}-\frac {3\,b\,c}{e}\right )}{e}\right )-x^2\,\left (\frac {c^2\,d}{e^2}-\frac {3\,b\,c}{2\,e}\right )-\frac {\ln \left (d+e\,x\right )\,\left (b^2\,d\,e^2-3\,b\,c\,d^2\,e-a\,b\,e^3+2\,c^2\,d^3+2\,a\,c\,d\,e^2\right )}{e^4}+\frac {2\,c^2\,x^3}{3\,e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.35, size = 100, normalized size = 0.96 \begin {gather*} \frac {2 c^{2} x^{3}}{3 e} + x^{2} \left (\frac {3 b c}{2 e} - \frac {c^{2} d}{e^{2}}\right ) + x \left (\frac {2 a c}{e} + \frac {b^{2}}{e} - \frac {3 b c d}{e^{2}} + \frac {2 c^{2} d^{2}}{e^{3}}\right ) + \frac {\left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right ) \log {\left (d + e x \right )}}{e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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