\(\int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^2} \, dx\) [1457]

Optimal result

Integrand size = 24, antiderivative size = 51 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^2} \, dx=\frac {b^2 x}{e^2}-\frac {(b d-a e)^2}{e^3 (d+e x)}-\frac {2 b (b d-a e) \log (d+e x)}{e^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {27, 45} \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^2} \, dx=-\frac {(b d-a e)^2}{e^3 (d+e x)}-\frac {2 b (b d-a e) \log (d+e x)}{e^3}+\frac {b^2 x}{e^2} \]

[In]

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

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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])

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.92 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^2} \, dx=\frac {b^2 e x-\frac {(b d-a e)^2}{d+e x}+2 b (-b d+a e) \log (d+e x)}{e^3} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.24 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.24

method	result	size
default	\(\frac {b^{2} x}{e^{2}}-\frac {a^{2} e^{2}-2 a b d e +b^{2} d^{2}}{e^{3} \left (e x +d \right )}+\frac {2 b \left (a e -b d \right ) \ln \left (e x +d \right )}{e^{3}}\)	\(63\)
norman	\(\frac {\frac {b^{2} x^{2}}{e}-\frac {a^{2} e^{2}-2 a b d e +2 b^{2} d^{2}}{e^{3}}}{e x +d}+\frac {2 b \left (a e -b d \right ) \ln \left (e x +d \right )}{e^{3}}\)	\(68\)
risch	\(\frac {b^{2} x}{e^{2}}-\frac {a^{2}}{e \left (e x +d \right )}+\frac {2 a b d}{e^{2} \left (e x +d \right )}-\frac {b^{2} d^{2}}{e^{3} \left (e x +d \right )}+\frac {2 b \ln \left (e x +d \right ) a}{e^{2}}-\frac {2 b^{2} \ln \left (e x +d \right ) d}{e^{3}}\)	\(86\)
parallelrisch	\(\frac {2 \ln \left (e x +d \right ) x a b \,e^{2}-2 \ln \left (e x +d \right ) x \,b^{2} d e +x^{2} b^{2} e^{2}+2 \ln \left (e x +d \right ) a b d e -2 \ln \left (e x +d \right ) b^{2} d^{2}-a^{2} e^{2}+2 a b d e -2 b^{2} d^{2}}{e^{3} \left (e x +d \right )}\)	\(99\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.80 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^2} \, dx=\frac {b^{2} e^{2} x^{2} + b^{2} d e x - b^{2} d^{2} + 2 \, a b d e - a^{2} e^{2} - 2 \, {\left (b^{2} d^{2} - a b d e + {\left (b^{2} d e - a b e^{2}\right )} x\right )} \log \left (e x + d\right )}{e^{4} x + d e^{3}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.18 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^2} \, dx=\frac {b^{2} x}{e^{2}} + \frac {2 b \left (a e - b d\right ) \log {\left (d + e x \right )}}{e^{3}} + \frac {- a^{2} e^{2} + 2 a b d e - b^{2} d^{2}}{d e^{3} + e^{4} x} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (51) = 102\).

Time = 0.26 (sec) , antiderivative size = 114, normalized size of antiderivative = 2.24 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^2} \, dx=b^{2} {\left (\frac {2 \, d \log \left (\frac {{\left | e x + d \right |}}{{\left (e x + d\right )}^{2} {\left | e \right |}}\right )}{e^{3}} + \frac {e x + d}{e^{3}} - \frac {d^{2}}{{\left (e x + d\right )} e^{3}}\right )} - \frac {2 \, a b {\left (\frac {\log \left (\frac {{\left | e x + d \right |}}{{\left (e x + d\right )}^{2} {\left | e \right |}}\right )}{e} - \frac {d}{{\left (e x + d\right )} e}\right )}}{e} - \frac {a^{2}}{{\left (e x + d\right )} e} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.39 \[ \int \frac {a^2+2 a b x+b^2 x^2}{(d+e x)^2} \, dx=\frac {b^2\,x}{e^2}-\frac {a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2}{e\,\left (x\,e^3+d\,e^2\right )}-\frac {\ln \left (d+e\,x\right )\,\left (2\,b^2\,d-2\,a\,b\,e\right )}{e^3} \]

[In]

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

[Out]

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