\(\int \frac {d+e x}{(a+b x+c x^2)^{3/2}} \, dx\) [2385]

Optimal result

Integrand size = 20, antiderivative size = 45 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \]

[Out]

-2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {650} \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \]

[In]

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

[Out]

(-2*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])

Rule 650

Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[-2*((b*d - 2*a*e + (2*c*
d - b*e)*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] &&
NeQ[b^2 - 4*a*c, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.96 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {-2 b d+4 a e-4 c d x+2 b e x}{\left (b^2-4 a c\right ) \sqrt {a+x (b+c x)}} \]

[In]

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

[Out]

(-2*b*d + 4*a*e - 4*c*d*x + 2*b*e*x)/((b^2 - 4*a*c)*Sqrt[a + x*(b + c*x)])

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00

method	result	size
gosper	\(-\frac {2 \left (b e x -2 c d x +2 a e -b d \right )}{\sqrt {c \,x^{2}+b x +a}\, \left (4 a c -b^{2}\right )}\)	\(45\)
trager	\(-\frac {2 \left (b e x -2 c d x +2 a e -b d \right )}{\sqrt {c \,x^{2}+b x +a}\, \left (4 a c -b^{2}\right )}\)	\(45\)
default	\(\frac {2 d \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+e \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )\)	\(91\)

[In]

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

[Out]

-2/(c*x^2+b*x+a)^(1/2)*(b*e*x-2*c*d*x+2*a*e-b*d)/(4*a*c-b^2)

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.64 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \, \sqrt {c x^{2} + b x + a} {\left (b d - 2 \, a e + {\left (2 \, c d - b e\right )} x\right )}}{a b^{2} - 4 \, a^{2} c + {\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} + {\left (b^{3} - 4 \, a b c\right )} x} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral((d + e*x)/(a + b*x + c*x**2)**(3/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((e*x+d)/(c*x^2+b*x+a)^(3/2),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(4*a*c-b^2>0)', see `assume?` f
or more deta

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.22 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left (\frac {{\left (2 \, c d - b e\right )} x}{b^{2} - 4 \, a c} + \frac {b d - 2 \, a e}{b^{2} - 4 \, a c}\right )}}{\sqrt {c x^{2} + b x + a}} \]

[In]

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

[Out]

-2*((2*c*d - b*e)*x/(b^2 - 4*a*c) + (b*d - 2*a*e)/(b^2 - 4*a*c))/sqrt(c*x^2 + b*x + a)

Mupad [B] (verification not implemented)

Time = 10.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {4\,a\,e-2\,b\,d+2\,b\,e\,x-4\,c\,d\,x}{\left (4\,a\,c-b^2\right )\,\sqrt {c\,x^2+b\,x+a}} \]

[In]

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

[Out]

-(4*a*e - 2*b*d + 2*b*e*x - 4*c*d*x)/((4*a*c - b^2)*(a + b*x + c*x^2)^(1/2))