\(\int \frac {(a+b x)^4}{(a c+(b c+a d) x+b d x^2)^2} \, dx\) [1812]

Optimal result

Integrand size = 29, antiderivative size = 51 \[ \int \frac {(a+b x)^4}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\frac {b^2 x}{d^2}-\frac {(b c-a d)^2}{d^3 (c+d x)}-\frac {2 b (b c-a d) \log (c+d x)}{d^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {640, 45} \[ \int \frac {(a+b x)^4}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=-\frac {(b c-a d)^2}{d^3 (c+d x)}-\frac {2 b (b c-a d) \log (c+d x)}{d^3}+\frac {b^2 x}{d^2} \]

[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 45

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])

Rule 640

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c/e)*x)^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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^4}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\frac {b^2 d x-\frac {(b c-a d)^2}{c+d x}+2 b (-b c+a d) \log (c+d x)}{d^3} \]

[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

Maple [A] (verified)

Time = 2.64 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.24

method	result	size
default	\(\frac {b^{2} x}{d^{2}}+\frac {2 b \left (a d -b c \right ) \ln \left (d x +c \right )}{d^{3}}-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{d^{3} \left (d x +c \right )}\)	\(63\)
risch	\(\frac {b^{2} x}{d^{2}}+\frac {2 b \ln \left (d x +c \right ) a}{d^{2}}-\frac {2 b^{2} \ln \left (d x +c \right ) c}{d^{3}}-\frac {a^{2}}{d \left (d x +c \right )}+\frac {2 a b c}{d^{2} \left (d x +c \right )}-\frac {b^{2} c^{2}}{d^{3} \left (d x +c \right )}\)	\(86\)
parallelrisch	\(\frac {2 \ln \left (d x +c \right ) x a b \,d^{2}-2 \ln \left (d x +c \right ) x \,b^{2} c d +d^{2} x^{2} b^{2}+2 \ln \left (d x +c \right ) a b c d -2 \ln \left (d x +c \right ) b^{2} c^{2}-a^{2} d^{2}+2 a b c d -2 b^{2} c^{2}}{d^{3} \left (d x +c \right )}\)	\(99\)
norman	\(\frac {\frac {b^{3} x^{3}}{d}-\frac {a \left (a^{2} b \,d^{2}-a \,b^{2} c d +2 c^{2} b^{3}\right )}{d^{3} b}-\frac {\left (2 b^{2} d^{2} a^{2}-a \,b^{3} c d +2 b^{4} c^{2}\right ) x}{d^{3} b}}{\left (b x +a \right ) \left (d x +c \right )}+\frac {2 b \left (a d -b c \right ) \ln \left (d x +c \right )}{d^{3}}\)	\(119\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.80 \[ \int \frac {(a+b x)^4}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\frac {b^{2} d^{2} x^{2} + b^{2} c d x - b^{2} c^{2} + 2 \, a b c d - a^{2} d^{2} - 2 \, {\left (b^{2} c^{2} - a b c d + {\left (b^{2} c d - a b d^{2}\right )} x\right )} \log \left (d x + c\right )}{d^{4} x + c d^{3}} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.18 \[ \int \frac {(a+b x)^4}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\frac {b^{2} x}{d^{2}} + \frac {2 b \left (a d - b c\right ) \log {\left (c + d x \right )}}{d^{3}} + \frac {- a^{2} d^{2} + 2 a b c d - b^{2} c^{2}}{c d^{3} + d^{4} x} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.31 \[ \int \frac {(a+b x)^4}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\frac {b^{2} x}{d^{2}} - \frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{d^{4} x + c d^{3}} - \frac {2 \, {\left (b^{2} c - a b d\right )} \log \left (d x + c\right )}{d^{3}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.27 \[ \int \frac {(a+b x)^4}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\frac {b^{2} x}{d^{2}} - \frac {2 \, {\left (b^{2} c - a b d\right )} \log \left ({\left | d x + c \right |}\right )}{d^{3}} - \frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{{\left (d x + c\right )} d^{3}} \]

[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)

Mupad [B] (verification not implemented)

Time = 9.86 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.39 \[ \int \frac {(a+b x)^4}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx=\frac {b^2\,x}{d^2}-\frac {a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}{d\,\left (x\,d^3+c\,d^2\right )}-\frac {\ln \left (c+d\,x\right )\,\left (2\,b^2\,c-2\,a\,b\,d\right )}{d^3} \]

[In]

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

[Out]

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