\(\int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx\) [1963]

Optimal result

Integrand size = 33, antiderivative size = 41 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx=\frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 (b d-a e) (d+e x)^3} \]

[Out]

1/3*(b^2*x^2+2*a*b*x+a^2)^(3/2)/(-a*e+b*d)/(e*x+d)^3

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {781} \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx=\frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 (d+e x)^3 (b d-a e)} \]

[In]

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

[Out]

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

Rule 781

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_) + (g_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp
[(-f)*g*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(b*(p + 1)*(e*f - d*g))), x] /; FreeQ[{a, b, c, d, e, f,
g, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && EqQ[m + 2*p + 3, 0] && EqQ[2*c*f - b*g, 0]

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

Mathematica [A] (verified)

Time = 1.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.73 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx=-\frac {\sqrt {(a+b x)^2} \left (a^2 e^2+a b e (d+3 e x)+b^2 \left (d^2+3 d e x+3 e^2 x^2\right )\right )}{3 e^3 (a+b x) (d+e x)^3} \]

[In]

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

[Out]

-1/3*(Sqrt[(a + b*x)^2]*(a^2*e^2 + a*b*e*(d + 3*e*x) + b^2*(d^2 + 3*d*e*x + 3*e^2*x^2)))/(e^3*(a + b*x)*(d + e
*x)^3)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 2.

Time = 0.56 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.61

method	result	size
default	\(-\frac {\operatorname {csgn}\left (b x +a \right ) \left (3 b^{2} e^{2} x^{2}+3 a b \,e^{2} x +3 b^{2} d e x +e^{2} a^{2}+a b d e +b^{2} d^{2}\right )}{3 e^{3} \left (e x +d \right )^{3}}\)	\(66\)
gosper	\(-\frac {\left (3 b^{2} e^{2} x^{2}+3 a b \,e^{2} x +3 b^{2} d e x +e^{2} a^{2}+a b d e +b^{2} d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{3 \left (e x +d \right )^{3} e^{3} \left (b x +a \right )}\)	\(76\)
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b^{2} x^{2}}{e}-\frac {b \left (a e +b d \right ) x}{e^{2}}-\frac {e^{2} a^{2}+a b d e +b^{2} d^{2}}{3 e^{3}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{3}}\)	\(76\)

[In]

int((b*x+a)*((b*x+a)^2)^(1/2)/(e*x+d)^4,x,method=_RETURNVERBOSE)

[Out]

-1/3*csgn(b*x+a)*(3*b^2*e^2*x^2+3*a*b*e^2*x+3*b^2*d*e*x+a^2*e^2+a*b*d*e+b^2*d^2)/e^3/(e*x+d)^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (37) = 74\).

Time = 0.32 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.05 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx=-\frac {3 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + a b d e + a^{2} e^{2} + 3 \, {\left (b^{2} d e + a b e^{2}\right )} x}{3 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \]

[In]

integrate((b*x+a)*((b*x+a)^2)^(1/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

-1/3*(3*b^2*e^2*x^2 + b^2*d^2 + a*b*d*e + a^2*e^2 + 3*(b^2*d*e + a*b*e^2)*x)/(e^6*x^3 + 3*d*e^5*x^2 + 3*d^2*e^
4*x + d^3*e^3)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx=\text {Timed out} \]

[In]

integrate((b*x+a)*((b*x+a)**2)**(1/2)/(e*x+d)**4,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((b*x+a)*((b*x+a)^2)^(1/2)/(e*x+d)^4,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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (37) = 74\).

Time = 0.26 (sec) , antiderivative size = 122, normalized size of antiderivative = 2.98 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx=\frac {b^{3} \mathrm {sgn}\left (b x + a\right )}{3 \, {\left (b d e^{3} - a e^{4}\right )}} - \frac {3 \, b^{2} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, b^{2} d e x \mathrm {sgn}\left (b x + a\right ) + 3 \, a b e^{2} x \mathrm {sgn}\left (b x + a\right ) + b^{2} d^{2} \mathrm {sgn}\left (b x + a\right ) + a b d e \mathrm {sgn}\left (b x + a\right ) + a^{2} e^{2} \mathrm {sgn}\left (b x + a\right )}{3 \, {\left (e x + d\right )}^{3} e^{3}} \]

[In]

integrate((b*x+a)*((b*x+a)^2)^(1/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

1/3*b^3*sgn(b*x + a)/(b*d*e^3 - a*e^4) - 1/3*(3*b^2*e^2*x^2*sgn(b*x + a) + 3*b^2*d*e*x*sgn(b*x + a) + 3*a*b*e^
2*x*sgn(b*x + a) + b^2*d^2*sgn(b*x + a) + a*b*d*e*sgn(b*x + a) + a^2*e^2*sgn(b*x + a))/((e*x + d)^3*e^3)

Mupad [B] (verification not implemented)

Time = 10.91 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.83 \[ \int \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^4} \, dx=-\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (a^2\,e^2+a\,b\,d\,e+3\,a\,b\,e^2\,x+b^2\,d^2+3\,b^2\,d\,e\,x+3\,b^2\,e^2\,x^2\right )}{3\,e^3\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^3} \]

[In]

int((((a + b*x)^2)^(1/2)*(a + b*x))/(d + e*x)^4,x)

[Out]

-(((a + b*x)^2)^(1/2)*(a^2*e^2 + b^2*d^2 + 3*b^2*e^2*x^2 + 3*a*b*e^2*x + 3*b^2*d*e*x + a*b*d*e))/(3*e^3*(a + b
*x)*(d + e*x)^3)