3.1139 \(\int \frac{(A+B x) (d+e x)^2}{b x+c x^2} \, dx\)

Optimal. Leaf size=77 \[ \frac{e x (A c e-b B e+2 B c d)}{c^2}+\frac{(b B-A c) (c d-b e)^2 \log (b+c x)}{b c^3}+\frac{A d^2 \log (x)}{b}+\frac{B e^2 x^2}{2 c} \]

[Out]

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

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Rubi [A]  time = 0.0863375, antiderivative size = 77, normalized size of antiderivative = 1., 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} \[ \frac{e x (A c e-b B e+2 B c d)}{c^2}+\frac{(b B-A c) (c d-b e)^2 \log (b+c x)}{b c^3}+\frac{A d^2 \log (x)}{b}+\frac{B e^2 x^2}{2 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0497905, size = 74, normalized size = 0.96 \[ \frac{b c e x (2 A c e+B (-2 b e+4 c d+c e x))+2 (b B-A c) (c d-b e)^2 \log (b+c x)+2 A c^3 d^2 \log (x)}{2 b c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 144, normalized size = 1.9 \begin{align*}{\frac{B{e}^{2}{x}^{2}}{2\,c}}+{\frac{{e}^{2}Ax}{c}}-{\frac{B{e}^{2}bx}{{c}^{2}}}+2\,{\frac{Bdex}{c}}+{\frac{A{d}^{2}\ln \left ( x \right ) }{b}}-{\frac{b\ln \left ( cx+b \right ) A{e}^{2}}{{c}^{2}}}+2\,{\frac{\ln \left ( cx+b \right ) Ade}{c}}-{\frac{\ln \left ( cx+b \right ) A{d}^{2}}{b}}+{\frac{{b}^{2}\ln \left ( cx+b \right ) B{e}^{2}}{{c}^{3}}}-2\,{\frac{b\ln \left ( cx+b \right ) Bde}{{c}^{2}}}+{\frac{\ln \left ( cx+b \right ) B{d}^{2}}{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.16851, size = 155, normalized size = 2.01 \begin{align*} \frac{A d^{2} \log \left (x\right )}{b} + \frac{B c e^{2} x^{2} + 2 \,{\left (2 \, B c d e -{\left (B b - A c\right )} e^{2}\right )} x}{2 \, c^{2}} + \frac{{\left ({\left (B b c^{2} - A c^{3}\right )} d^{2} - 2 \,{\left (B b^{2} c - A b c^{2}\right )} d e +{\left (B b^{3} - A b^{2} c\right )} e^{2}\right )} \log \left (c x + b\right )}{b c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.44331, size = 261, normalized size = 3.39 \begin{align*} \frac{B b c^{2} e^{2} x^{2} + 2 \, A c^{3} d^{2} \log \left (x\right ) + 2 \,{\left (2 \, B b c^{2} d e -{\left (B b^{2} c - A b c^{2}\right )} e^{2}\right )} x + 2 \,{\left ({\left (B b c^{2} - A c^{3}\right )} d^{2} - 2 \,{\left (B b^{2} c - A b c^{2}\right )} d e +{\left (B b^{3} - A b^{2} c\right )} e^{2}\right )} \log \left (c x + b\right )}{2 \, b c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 3.50883, size = 163, normalized size = 2.12 \begin{align*} \frac{A d^{2} \log{\left (x \right )}}{b} + \frac{B e^{2} x^{2}}{2 c} - \frac{x \left (- A c e^{2} + B b e^{2} - 2 B c d e\right )}{c^{2}} + \frac{\left (- A c + B b\right ) \left (b e - c d\right )^{2} \log{\left (x + \frac{- A b c^{2} d^{2} + \frac{b \left (- A c + B b\right ) \left (b e - c d\right )^{2}}{c}}{- A b^{2} c e^{2} + 2 A b c^{2} d e - 2 A c^{3} d^{2} + B b^{3} e^{2} - 2 B b^{2} c d e + B b c^{2} d^{2}} \right )}}{b c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18164, size = 158, normalized size = 2.05 \begin{align*} \frac{A d^{2} \log \left ({\left | x \right |}\right )}{b} + \frac{B c x^{2} e^{2} + 4 \, B c d x e - 2 \, B b x e^{2} + 2 \, A c x e^{2}}{2 \, c^{2}} + \frac{{\left (B b c^{2} d^{2} - A c^{3} d^{2} - 2 \, B b^{2} c d e + 2 \, A b c^{2} d e + B b^{3} e^{2} - A b^{2} c e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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