\(\int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^2} \, dx\) [1549]

Optimal result

Integrand size = 28, antiderivative size = 85 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^2} \, dx=\frac {(b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^2 (a+b x) (d+e x)}+\frac {b \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^2 (a+b x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {660, 45} \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^2} \, dx=\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{e^2 (a+b x) (d+e x)}+\frac {b \sqrt {a^2+2 a b x+b^2 x^2} \log (d+e x)}{e^2 (a+b x)} \]

[In]

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

[Out]

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

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 660

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

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(245\) vs. \(2(85)=170\).

Time = 0.76 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.88 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^2} \, dx=\frac {-\frac {e \sqrt {(a+b x)^2}}{d+e x}-\frac {\sqrt {a^2} e (-b d x+a (d+2 e x))}{a d (d+e x)}+\frac {\left (a+\sqrt {a^2}\right ) b \log \left (\sqrt {a^2}-b x-\sqrt {(a+b x)^2}\right )}{a}+\frac {\left (-a+\sqrt {a^2}\right ) b \log \left (\sqrt {a^2}+b x-\sqrt {(a+b x)^2}\right )}{a}-\frac {\left (a+\sqrt {a^2}\right ) b \log \left (2 a e x+d \left (\sqrt {a^2}-b x-\sqrt {(a+b x)^2}\right )\right )}{a}+\frac {\left (a-\sqrt {a^2}\right ) b \log \left (-2 a e x+d \left (\sqrt {a^2}+b x-\sqrt {(a+b x)^2}\right )\right )}{a}}{2 e^2} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 2.47 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.65

method	result	size
default	\(\frac {\operatorname {csgn}\left (b x +a \right ) \left (\ln \left (-b e x -b d \right ) b e x +\ln \left (-b e x -b d \right ) b d -a e +b d \right )}{e^{2} \left (e x +d \right )}\)	\(55\)
risch	\(-\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (a e -b d \right )}{\left (b x +a \right ) e^{2} \left (e x +d \right )}+\frac {b \ln \left (e x +d \right ) \sqrt {\left (b x +a \right )^{2}}}{e^{2} \left (b x +a \right )}\)	\(65\)

[In]

int(((b*x+a)^2)^(1/2)/(e*x+d)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.44 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^2} \, dx=\frac {b d - a e + {\left (b e x + b d\right )} \log \left (e x + d\right )}{e^{3} x + d e^{2}} \]

[In]

integrate(((b*x+a)^2)^(1/2)/(e*x+d)^2,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^2} \, dx=\int \frac {\sqrt {\left (a + b x\right )^{2}}}{\left (d + e x\right )^{2}}\, dx \]

[In]

integrate(((b*x+a)**2)**(1/2)/(e*x+d)**2,x)

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(((b*x+a)^2)^(1/2)/(e*x+d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for
more detail

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^2} \, dx=\frac {b \log \left ({\left | e x + d \right |}\right ) \mathrm {sgn}\left (b x + a\right )}{e^{2}} + \frac {b d \mathrm {sgn}\left (b x + a\right ) - a e \mathrm {sgn}\left (b x + a\right )}{{\left (e x + d\right )} e^{2}} \]

[In]

integrate(((b*x+a)^2)^(1/2)/(e*x+d)^2,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^2} \, dx=\int \frac {\sqrt {{\left (a+b\,x\right )}^2}}{{\left (d+e\,x\right )}^2} \,d x \]

[In]

int(((a + b*x)^2)^(1/2)/(d + e*x)^2,x)

[Out]

int(((a + b*x)^2)^(1/2)/(d + e*x)^2, x)