3.1812 \(\int \frac{(a+b x)^4}{(a c+(b c+a d) x+b d x^2)^2} \, dx\)

Optimal. Leaf size=51 \[ -\frac{(b c-a d)^2}{d^3 (c+d x)}-\frac{2 b (b c-a d) \log (c+d x)}{d^3}+\frac{b^2 x}{d^2} \]

[Out]

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

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Rubi [A]  time = 0.0434578, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {626, 43} \[ -\frac{(b c-a d)^2}{d^3 (c+d x)}-\frac{2 b (b c-a d) \log (c+d x)}{d^3}+\frac{b^2 x}{d^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^4/(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]

[Out]

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

Rule 626

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0345981, size = 47, normalized size = 0.92 \[ \frac{-\frac{(b c-a d)^2}{c+d x}+2 b (a d-b c) \log (c+d x)+b^2 d x}{d^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^4/(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]

[Out]

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

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Maple [A]  time = 0.044, size = 86, normalized size = 1.7 \begin{align*}{\frac{{b}^{2}x}{{d}^{2}}}+2\,{\frac{b\ln \left ( dx+c \right ) a}{{d}^{2}}}-2\,{\frac{{b}^{2}\ln \left ( dx+c \right ) c}{{d}^{3}}}-{\frac{{a}^{2}}{d \left ( dx+c \right ) }}+2\,{\frac{abc}{{d}^{2} \left ( dx+c \right ) }}-{\frac{{b}^{2}{c}^{2}}{{d}^{3} \left ( dx+c \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^4/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x)

[Out]

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

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Maxima [A]  time = 1.12773, size = 90, normalized size = 1.76 \begin{align*} \frac{b^{2} x}{d^{2}} - \frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{d^{4} x + c d^{3}} - \frac{2 \,{\left (b^{2} c - a b d\right )} \log \left (d x + c\right )}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.47149, size = 184, normalized size = 3.61 \begin{align*} \frac{b^{2} d^{2} x^{2} + b^{2} c d x - b^{2} c^{2} + 2 \, a b c d - a^{2} d^{2} - 2 \,{\left (b^{2} c^{2} - a b c d +{\left (b^{2} c d - a b d^{2}\right )} x\right )} \log \left (d x + c\right )}{d^{4} x + c d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.744323, size = 60, normalized size = 1.18 \begin{align*} \frac{b^{2} x}{d^{2}} + \frac{2 b \left (a d - b c\right ) \log{\left (c + d x \right )}}{d^{3}} - \frac{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}}{c d^{3} + d^{4} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**4/(a*c+(a*d+b*c)*x+b*d*x**2)**2,x)

[Out]

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

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Giac [A]  time = 1.26112, size = 88, normalized size = 1.73 \begin{align*} \frac{b^{2} x}{d^{2}} - \frac{2 \,{\left (b^{2} c - a b d\right )} \log \left ({\left | d x + c \right |}\right )}{d^{3}} - \frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{{\left (d x + c\right )} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(a*c+(a*d+b*c)*x+b*d*x^2)^2,x, algorithm="giac")

[Out]

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