\(\int \frac {(d+e x)^2}{(c d^2+2 c d e x+c e^2 x^2)^{3/2}} \, dx\) [1073]

Optimal result

Integrand size = 32, antiderivative size = 42 \[ \int \frac {(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {(d+e x) \log (d+e x)}{c e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]

[Out]

(e*x+d)*ln(e*x+d)/c/e/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {656, 622, 31} \[ \int \frac {(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {(d+e x) \log (d+e x)}{c e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]

[In]

Int[(d + e*x)^2/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2),x]

[Out]

((d + e*x)*Log[d + e*x])/(c*e*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 622

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(b/2 + c*x)/Sqrt[a + b*x + c*x^2], Int[1/(b/2
+ c*x), x], x] /; FreeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 656

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

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{\sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx}{c} \\ & = \frac {\left (c d e+c e^2 x\right ) \int \frac {1}{c d e+c e^2 x} \, dx}{c \sqrt {c d^2+2 c d e x+c e^2 x^2}} \\ & = \frac {(d+e x) \log (d+e x)}{c e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.74 \[ \int \frac {(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {(d+e x) \log (d+e x)}{c e \sqrt {c (d+e x)^2}} \]

[In]

Integrate[(d + e*x)^2/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2),x]

[Out]

((d + e*x)*Log[d + e*x])/(c*e*Sqrt[c*(d + e*x)^2])

Maple [A] (verified)

Time = 2.59 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.71

method	result	size
risch	\(\frac {\left (e x +d \right ) \ln \left (e x +d \right )}{c \sqrt {c \left (e x +d \right )^{2}}\, e}\)	\(30\)
default	\(\frac {\left (e x +d \right )^{3} \ln \left (e x +d \right )}{\left (c \,x^{2} e^{2}+2 x c d e +c \,d^{2}\right )^{\frac {3}{2}} e}\)	\(40\)

[In]

int((e*x+d)^2/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/c/(c*(e*x+d)^2)^(1/2)*(e*x+d)*ln(e*x+d)/e

Fricas [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.10 \[ \int \frac {(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} \log \left (e x + d\right )}{c^{2} e^{2} x + c^{2} d e} \]

[In]

integrate((e*x+d)^2/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x, algorithm="fricas")

[Out]

sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*log(e*x + d)/(c^2*e^2*x + c^2*d*e)

Sympy [A] (verification not implemented)

Time = 1.83 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.64 \[ \int \frac {(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {\left (\frac {d}{e} + x\right ) \log {\left (\frac {d}{e} + x \right )}}{c \sqrt {c e^{2} \left (\frac {d}{e} + x\right )^{2}}} \]

[In]

integrate((e*x+d)**2/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(3/2),x)

[Out]

(d/e + x)*log(d/e + x)/(c*sqrt(c*e**2*(d/e + x)**2))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (40) = 80\).

Time = 0.20 (sec) , antiderivative size = 123, normalized size of antiderivative = 2.93 \[ \int \frac {(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {3 \, c^{2} d^{2} e^{4}}{2 \, \left (c e^{2}\right )^{\frac {7}{2}} {\left (x + \frac {d}{e}\right )}^{2}} + \frac {2 \, c d e^{3} x}{\left (c e^{2}\right )^{\frac {5}{2}} {\left (x + \frac {d}{e}\right )}^{2}} + \frac {e^{2} \log \left (x + \frac {d}{e}\right )}{\left (c e^{2}\right )^{\frac {3}{2}}} - \frac {2 \, d}{\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c e} + \frac {d^{2}}{2 \, \left (c e^{2}\right )^{\frac {3}{2}} {\left (x + \frac {d}{e}\right )}^{2}} \]

[In]

integrate((e*x+d)^2/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x, algorithm="maxima")

[Out]

3/2*c^2*d^2*e^4/((c*e^2)^(7/2)*(x + d/e)^2) + 2*c*d*e^3*x/((c*e^2)^(5/2)*(x + d/e)^2) + e^2*log(x + d/e)/(c*e^
2)^(3/2) - 2*d/(sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*c*e) + 1/2*d^2/((c*e^2)^(3/2)*(x + d/e)^2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.52 \[ \int \frac {(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\frac {\log \left ({\left | e x + d \right |}\right )}{c^{\frac {3}{2}} e \mathrm {sgn}\left (e x + d\right )} \]

[In]

integrate((e*x+d)^2/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(3/2),x, algorithm="giac")

[Out]

log(abs(e*x + d))/(c^(3/2)*e*sgn(e*x + d))

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^2}{{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{3/2}} \,d x \]

[In]

int((d + e*x)^2/(c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(3/2),x)

[Out]

int((d + e*x)^2/(c*d^2 + c*e^2*x^2 + 2*c*d*e*x)^(3/2), x)