3.1108 \(\int \frac{(A+B x) (b x+c x^2)}{d+e x} \, dx\)

Optimal. Leaf size=87 \[ -\frac{x^2 (-A c e-b B e+B c d)}{2 e^2}+\frac{x (B d-A e) (c d-b e)}{e^3}-\frac{d (B d-A e) (c d-b e) \log (d+e x)}{e^4}+\frac{B c x^3}{3 e} \]

[Out]

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

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Rubi [A]  time = 0.0978763, antiderivative size = 87, normalized size of antiderivative = 1., 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 = {771} \[ -\frac{x^2 (-A c e-b B e+B c d)}{2 e^2}+\frac{x (B d-A e) (c d-b e)}{e^3}-\frac{d (B d-A e) (c d-b e) \log (d+e x)}{e^4}+\frac{B c x^3}{3 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0475735, size = 88, normalized size = 1.01 \[ \frac{e x \left (3 A e (2 b e-2 c d+c e x)+3 b B e (e x-2 d)+B c \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )-6 d (B d-A e) (c d-b e) \log (d+e x)}{6 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 138, normalized size = 1.6 \begin{align*}{\frac{Bc{x}^{3}}{3\,e}}+{\frac{Ac{x}^{2}}{2\,e}}+{\frac{Bb{x}^{2}}{2\,e}}-{\frac{Bc{x}^{2}d}{2\,{e}^{2}}}+{\frac{Abx}{e}}-{\frac{Acdx}{{e}^{2}}}-{\frac{bBdx}{{e}^{2}}}+{\frac{Bc{d}^{2}x}{{e}^{3}}}-{\frac{d\ln \left ( ex+d \right ) Ab}{{e}^{2}}}+{\frac{{d}^{2}\ln \left ( ex+d \right ) Ac}{{e}^{3}}}+{\frac{{d}^{2}\ln \left ( ex+d \right ) Bb}{{e}^{3}}}-{\frac{{d}^{3}\ln \left ( ex+d \right ) Bc}{{e}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01976, size = 142, normalized size = 1.63 \begin{align*} \frac{2 \, B c e^{2} x^{3} - 3 \,{\left (B c d e -{\left (B b + A c\right )} e^{2}\right )} x^{2} + 6 \,{\left (B c d^{2} + A b e^{2} -{\left (B b + A c\right )} d e\right )} x}{6 \, e^{3}} - \frac{{\left (B c d^{3} + A b d e^{2} -{\left (B b + A c\right )} d^{2} e\right )} \log \left (e x + d\right )}{e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.47073, size = 230, normalized size = 2.64 \begin{align*} \frac{2 \, B c e^{3} x^{3} - 3 \,{\left (B c d e^{2} -{\left (B b + A c\right )} e^{3}\right )} x^{2} + 6 \,{\left (B c d^{2} e + A b e^{3} -{\left (B b + A c\right )} d e^{2}\right )} x - 6 \,{\left (B c d^{3} + A b d e^{2} -{\left (B b + A c\right )} d^{2} e\right )} \log \left (e x + d\right )}{6 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.675317, size = 90, normalized size = 1.03 \begin{align*} \frac{B c x^{3}}{3 e} + \frac{d \left (- A e + B d\right ) \left (b e - c d\right ) \log{\left (d + e x \right )}}{e^{4}} + \frac{x^{2} \left (A c e + B b e - B c d\right )}{2 e^{2}} - \frac{x \left (- A b e^{2} + A c d e + B b d e - B c d^{2}\right )}{e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.27036, size = 158, normalized size = 1.82 \begin{align*} -{\left (B c d^{3} - B b d^{2} e - A c d^{2} e + A b d e^{2}\right )} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (2 \, B c x^{3} e^{2} - 3 \, B c d x^{2} e + 6 \, B c d^{2} x + 3 \, B b x^{2} e^{2} + 3 \, A c x^{2} e^{2} - 6 \, B b d x e - 6 \, A c d x e + 6 \, A b x e^{2}\right )} e^{\left (-3\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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