\(\int \frac {(a^2+2 a b x+b^2 x^2)^2}{(d+e x)^{3/2}} \, dx\) [1634]

Optimal result

Integrand size = 28, antiderivative size = 123 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{3/2}} \, dx=-\frac {2 (b d-a e)^4}{e^5 \sqrt {d+e x}}-\frac {8 b (b d-a e)^3 \sqrt {d+e x}}{e^5}+\frac {4 b^2 (b d-a e)^2 (d+e x)^{3/2}}{e^5}-\frac {8 b^3 (b d-a e) (d+e x)^{5/2}}{5 e^5}+\frac {2 b^4 (d+e x)^{7/2}}{7 e^5} \]

[Out]

4*b^2*(-a*e+b*d)^2*(e*x+d)^(3/2)/e^5-8/5*b^3*(-a*e+b*d)*(e*x+d)^(5/2)/e^5+2/7*b^4*(e*x+d)^(7/2)/e^5-2*(-a*e+b*
d)^4/e^5/(e*x+d)^(1/2)-8*b*(-a*e+b*d)^3*(e*x+d)^(1/2)/e^5

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 123, 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 = {27, 45} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{3/2}} \, dx=-\frac {8 b^3 (d+e x)^{5/2} (b d-a e)}{5 e^5}+\frac {4 b^2 (d+e x)^{3/2} (b d-a e)^2}{e^5}-\frac {8 b \sqrt {d+e x} (b d-a e)^3}{e^5}-\frac {2 (b d-a e)^4}{e^5 \sqrt {d+e x}}+\frac {2 b^4 (d+e x)^{7/2}}{7 e^5} \]

[In]

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

[Out]

(-2*(b*d - a*e)^4)/(e^5*Sqrt[d + e*x]) - (8*b*(b*d - a*e)^3*Sqrt[d + e*x])/e^5 + (4*b^2*(b*d - a*e)^2*(d + e*x
)^(3/2))/e^5 - (8*b^3*(b*d - a*e)*(d + e*x)^(5/2))/(5*e^5) + (2*b^4*(d + e*x)^(7/2))/(7*e^5)

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)^4}{(d+e x)^{3/2}} \, dx \\ & = \int \left (\frac {(-b d+a e)^4}{e^4 (d+e x)^{3/2}}-\frac {4 b (b d-a e)^3}{e^4 \sqrt {d+e x}}+\frac {6 b^2 (b d-a e)^2 \sqrt {d+e x}}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^{3/2}}{e^4}+\frac {b^4 (d+e x)^{5/2}}{e^4}\right ) \, dx \\ & = -\frac {2 (b d-a e)^4}{e^5 \sqrt {d+e x}}-\frac {8 b (b d-a e)^3 \sqrt {d+e x}}{e^5}+\frac {4 b^2 (b d-a e)^2 (d+e x)^{3/2}}{e^5}-\frac {8 b^3 (b d-a e) (d+e x)^{5/2}}{5 e^5}+\frac {2 b^4 (d+e x)^{7/2}}{7 e^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.23 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2 \left (-35 a^4 e^4+140 a^3 b e^3 (2 d+e x)+70 a^2 b^2 e^2 \left (-8 d^2-4 d e x+e^2 x^2\right )+28 a b^3 e \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+b^4 \left (-128 d^4-64 d^3 e x+16 d^2 e^2 x^2-8 d e^3 x^3+5 e^4 x^4\right )\right )}{35 e^5 \sqrt {d+e x}} \]

[In]

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

[Out]

(2*(-35*a^4*e^4 + 140*a^3*b*e^3*(2*d + e*x) + 70*a^2*b^2*e^2*(-8*d^2 - 4*d*e*x + e^2*x^2) + 28*a*b^3*e*(16*d^3
+ 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) + b^4*(-128*d^4 - 64*d^3*e*x + 16*d^2*e^2*x^2 - 8*d*e^3*x^3 + 5*e^4*x^4)
))/(35*e^5*Sqrt[d + e*x])

Maple [A] (verified)

Time = 2.46 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.18

method	result	size
pseudoelliptic	\(\frac {\frac {2 \left (5 b^{4} x^{4}+28 a \,b^{3} x^{3}+70 a^{2} b^{2} x^{2}+140 a^{3} b x -35 a^{4}\right ) e^{4}}{35}+16 b \left (-\frac {1}{35} b^{3} x^{3}-\frac {1}{5} a \,b^{2} x^{2}-a^{2} b x +a^{3}\right ) d \,e^{3}-32 b^{2} \left (-\frac {1}{35} b^{2} x^{2}-\frac {2}{5} a b x +a^{2}\right ) d^{2} e^{2}+\frac {128 \left (-\frac {b x}{7}+a \right ) b^{3} d^{3} e}{5}-\frac {256 b^{4} d^{4}}{35}}{\sqrt {e x +d}\, e^{5}}\)	\(145\)
risch	\(\frac {2 b \left (5 e^{3} x^{3} b^{3}+28 x^{2} a \,b^{2} e^{3}-13 x^{2} b^{3} d \,e^{2}+70 a^{2} b \,e^{3} x -84 x a \,b^{2} d \,e^{2}+29 b^{3} d^{2} e x +140 a^{3} e^{3}-350 a^{2} b d \,e^{2}+308 a \,b^{2} d^{2} e -93 b^{3} d^{3}\right ) \sqrt {e x +d}}{35 e^{5}}-\frac {2 \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{e^{5} \sqrt {e x +d}}\)	\(179\)
gosper	\(-\frac {2 \left (-5 b^{4} x^{4} e^{4}-28 x^{3} a \,b^{3} e^{4}+8 x^{3} b^{4} d \,e^{3}-70 x^{2} a^{2} b^{2} e^{4}+56 x^{2} a \,b^{3} d \,e^{3}-16 x^{2} b^{4} d^{2} e^{2}-140 x \,a^{3} b \,e^{4}+280 x \,a^{2} b^{2} d \,e^{3}-224 x a \,b^{3} d^{2} e^{2}+64 x \,b^{4} d^{3} e +35 e^{4} a^{4}-280 b \,e^{3} d \,a^{3}+560 b^{2} e^{2} d^{2} a^{2}-448 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right )}{35 \sqrt {e x +d}\, e^{5}}\)	\(186\)
trager	\(-\frac {2 \left (-5 b^{4} x^{4} e^{4}-28 x^{3} a \,b^{3} e^{4}+8 x^{3} b^{4} d \,e^{3}-70 x^{2} a^{2} b^{2} e^{4}+56 x^{2} a \,b^{3} d \,e^{3}-16 x^{2} b^{4} d^{2} e^{2}-140 x \,a^{3} b \,e^{4}+280 x \,a^{2} b^{2} d \,e^{3}-224 x a \,b^{3} d^{2} e^{2}+64 x \,b^{4} d^{3} e +35 e^{4} a^{4}-280 b \,e^{3} d \,a^{3}+560 b^{2} e^{2} d^{2} a^{2}-448 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right )}{35 \sqrt {e x +d}\, e^{5}}\)	\(186\)
derivativedivides	\(\frac {\frac {2 b^{4} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {8 a \,b^{3} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {8 b^{4} d \left (e x +d \right )^{\frac {5}{2}}}{5}+4 a^{2} b^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}-8 a \,b^{3} d e \left (e x +d \right )^{\frac {3}{2}}+4 b^{4} d^{2} \left (e x +d \right )^{\frac {3}{2}}+8 a^{3} b \,e^{3} \sqrt {e x +d}-24 a^{2} b^{2} d \,e^{2} \sqrt {e x +d}+24 a \,b^{3} d^{2} e \sqrt {e x +d}-8 b^{4} d^{3} \sqrt {e x +d}-\frac {2 \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{\sqrt {e x +d}}}{e^{5}}\)	\(219\)
default	\(\frac {\frac {2 b^{4} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {8 a \,b^{3} e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {8 b^{4} d \left (e x +d \right )^{\frac {5}{2}}}{5}+4 a^{2} b^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}-8 a \,b^{3} d e \left (e x +d \right )^{\frac {3}{2}}+4 b^{4} d^{2} \left (e x +d \right )^{\frac {3}{2}}+8 a^{3} b \,e^{3} \sqrt {e x +d}-24 a^{2} b^{2} d \,e^{2} \sqrt {e x +d}+24 a \,b^{3} d^{2} e \sqrt {e x +d}-8 b^{4} d^{3} \sqrt {e x +d}-\frac {2 \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{\sqrt {e x +d}}}{e^{5}}\)	\(219\)

[In]

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

[Out]

2/35*((5*b^4*x^4+28*a*b^3*x^3+70*a^2*b^2*x^2+140*a^3*b*x-35*a^4)*e^4+280*b*(-1/35*b^3*x^3-1/5*a*b^2*x^2-a^2*b*
x+a^3)*d*e^3-560*b^2*(-1/35*b^2*x^2-2/5*a*b*x+a^2)*d^2*e^2+448*(-1/7*b*x+a)*b^3*d^3*e-128*b^4*d^4)/(e*x+d)^(1/
2)/e^5

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.56 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (5 \, b^{4} e^{4} x^{4} - 128 \, b^{4} d^{4} + 448 \, a b^{3} d^{3} e - 560 \, a^{2} b^{2} d^{2} e^{2} + 280 \, a^{3} b d e^{3} - 35 \, a^{4} e^{4} - 4 \, {\left (2 \, b^{4} d e^{3} - 7 \, a b^{3} e^{4}\right )} x^{3} + 2 \, {\left (8 \, b^{4} d^{2} e^{2} - 28 \, a b^{3} d e^{3} + 35 \, a^{2} b^{2} e^{4}\right )} x^{2} - 4 \, {\left (16 \, b^{4} d^{3} e - 56 \, a b^{3} d^{2} e^{2} + 70 \, a^{2} b^{2} d e^{3} - 35 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{35 \, {\left (e^{6} x + d e^{5}\right )}} \]

[In]

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

[Out]

2/35*(5*b^4*e^4*x^4 - 128*b^4*d^4 + 448*a*b^3*d^3*e - 560*a^2*b^2*d^2*e^2 + 280*a^3*b*d*e^3 - 35*a^4*e^4 - 4*(
2*b^4*d*e^3 - 7*a*b^3*e^4)*x^3 + 2*(8*b^4*d^2*e^2 - 28*a*b^3*d*e^3 + 35*a^2*b^2*e^4)*x^2 - 4*(16*b^4*d^3*e - 5
6*a*b^3*d^2*e^2 + 70*a^2*b^2*d*e^3 - 35*a^3*b*e^4)*x)*sqrt(e*x + d)/(e^6*x + d*e^5)

Sympy [A] (verification not implemented)

Time = 3.75 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.82 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {b^{4} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{4}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (4 a b^{3} e - 4 b^{4} d\right )}{5 e^{4}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (6 a^{2} b^{2} e^{2} - 12 a b^{3} d e + 6 b^{4} d^{2}\right )}{3 e^{4}} + \frac {\sqrt {d + e x} \left (4 a^{3} b e^{3} - 12 a^{2} b^{2} d e^{2} + 12 a b^{3} d^{2} e - 4 b^{4} d^{3}\right )}{e^{4}} - \frac {\left (a e - b d\right )^{4}}{e^{4} \sqrt {d + e x}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {a^{4} x + 2 a^{3} b x^{2} + 2 a^{2} b^{2} x^{3} + a b^{3} x^{4} + \frac {b^{4} x^{5}}{5}}{d^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((2*(b**4*(d + e*x)**(7/2)/(7*e**4) + (d + e*x)**(5/2)*(4*a*b**3*e - 4*b**4*d)/(5*e**4) + (d + e*x)**
(3/2)*(6*a**2*b**2*e**2 - 12*a*b**3*d*e + 6*b**4*d**2)/(3*e**4) + sqrt(d + e*x)*(4*a**3*b*e**3 - 12*a**2*b**2*
d*e**2 + 12*a*b**3*d**2*e - 4*b**4*d**3)/e**4 - (a*e - b*d)**4/(e**4*sqrt(d + e*x)))/e, Ne(e, 0)), ((a**4*x +
2*a**3*b*x**2 + 2*a**2*b**2*x**3 + a*b**3*x**4 + b**4*x**5/5)/d**(3/2), True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.54 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {5 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{4} - 28 \, {\left (b^{4} d - a b^{3} e\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 70 \, {\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 140 \, {\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )} \sqrt {e x + d}}{e^{4}} - \frac {35 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}}{\sqrt {e x + d} e^{4}}\right )}}{35 \, e} \]

[In]

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

[Out]

2/35*((5*(e*x + d)^(7/2)*b^4 - 28*(b^4*d - a*b^3*e)*(e*x + d)^(5/2) + 70*(b^4*d^2 - 2*a*b^3*d*e + a^2*b^2*e^2)
*(e*x + d)^(3/2) - 140*(b^4*d^3 - 3*a*b^3*d^2*e + 3*a^2*b^2*d*e^2 - a^3*b*e^3)*sqrt(e*x + d))/e^4 - 35*(b^4*d^
4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)/(sqrt(e*x + d)*e^4))/e

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (109) = 218\).

Time = 0.28 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.95 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{3/2}} \, dx=-\frac {2 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}}{\sqrt {e x + d} e^{5}} + \frac {2 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{4} e^{30} - 28 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{4} d e^{30} + 70 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{4} d^{2} e^{30} - 140 \, \sqrt {e x + d} b^{4} d^{3} e^{30} + 28 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{3} e^{31} - 140 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{3} d e^{31} + 420 \, \sqrt {e x + d} a b^{3} d^{2} e^{31} + 70 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{2} e^{32} - 420 \, \sqrt {e x + d} a^{2} b^{2} d e^{32} + 140 \, \sqrt {e x + d} a^{3} b e^{33}\right )}}{35 \, e^{35}} \]

[In]

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

[Out]

-2*(b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)/(sqrt(e*x + d)*e^5) + 2/35*(5*(e*x
+ d)^(7/2)*b^4*e^30 - 28*(e*x + d)^(5/2)*b^4*d*e^30 + 70*(e*x + d)^(3/2)*b^4*d^2*e^30 - 140*sqrt(e*x + d)*b^4*
d^3*e^30 + 28*(e*x + d)^(5/2)*a*b^3*e^31 - 140*(e*x + d)^(3/2)*a*b^3*d*e^31 + 420*sqrt(e*x + d)*a*b^3*d^2*e^31
+ 70*(e*x + d)^(3/2)*a^2*b^2*e^32 - 420*sqrt(e*x + d)*a^2*b^2*d*e^32 + 140*sqrt(e*x + d)*a^3*b*e^33)/e^35

Mupad [B] (verification not implemented)

Time = 9.59 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.24 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2\,b^4\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5}-\frac {\left (8\,b^4\,d-8\,a\,b^3\,e\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5}-\frac {2\,a^4\,e^4-8\,a^3\,b\,d\,e^3+12\,a^2\,b^2\,d^2\,e^2-8\,a\,b^3\,d^3\,e+2\,b^4\,d^4}{e^5\,\sqrt {d+e\,x}}+\frac {4\,b^2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{3/2}}{e^5}+\frac {8\,b\,{\left (a\,e-b\,d\right )}^3\,\sqrt {d+e\,x}}{e^5} \]

[In]

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

[Out]

(2*b^4*(d + e*x)^(7/2))/(7*e^5) - ((8*b^4*d - 8*a*b^3*e)*(d + e*x)^(5/2))/(5*e^5) - (2*a^4*e^4 + 2*b^4*d^4 + 1
2*a^2*b^2*d^2*e^2 - 8*a*b^3*d^3*e - 8*a^3*b*d*e^3)/(e^5*(d + e*x)^(1/2)) + (4*b^2*(a*e - b*d)^2*(d + e*x)^(3/2
))/e^5 + (8*b*(a*e - b*d)^3*(d + e*x)^(1/2))/e^5