\(\int \frac {(a^2+2 a b x+b^2 x^2)^2}{(d+e x)^{5/2}} \, dx\) [1635]

Optimal result

Integrand size = 28, antiderivative size = 125 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{5/2}} \, dx=-\frac {2 (b d-a e)^4}{3 e^5 (d+e x)^{3/2}}+\frac {8 b (b d-a e)^3}{e^5 \sqrt {d+e x}}+\frac {12 b^2 (b d-a e)^2 \sqrt {d+e x}}{e^5}-\frac {8 b^3 (b d-a e) (d+e x)^{3/2}}{3 e^5}+\frac {2 b^4 (d+e x)^{5/2}}{5 e^5} \]

[Out]

-2/3*(-a*e+b*d)^4/e^5/(e*x+d)^(3/2)-8/3*b^3*(-a*e+b*d)*(e*x+d)^(3/2)/e^5+2/5*b^4*(e*x+d)^(5/2)/e^5+8*b*(-a*e+b
*d)^3/e^5/(e*x+d)^(1/2)+12*b^2*(-a*e+b*d)^2*(e*x+d)^(1/2)/e^5

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 125, 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} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{5/2}} \, dx=-\frac {8 b^3 (d+e x)^{3/2} (b d-a e)}{3 e^5}+\frac {12 b^2 \sqrt {d+e x} (b d-a e)^2}{e^5}+\frac {8 b (b d-a e)^3}{e^5 \sqrt {d+e x}}-\frac {2 (b d-a e)^4}{3 e^5 (d+e x)^{3/2}}+\frac {2 b^4 (d+e x)^{5/2}}{5 e^5} \]

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^(5/2),x]

[Out]

(-2*(b*d - a*e)^4)/(3*e^5*(d + e*x)^(3/2)) + (8*b*(b*d - a*e)^3)/(e^5*Sqrt[d + e*x]) + (12*b^2*(b*d - a*e)^2*S
qrt[d + e*x])/e^5 - (8*b^3*(b*d - a*e)*(d + e*x)^(3/2))/(3*e^5) + (2*b^4*(d + e*x)^(5/2))/(5*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)^{5/2}} \, dx \\ & = \int \left (\frac {(-b d+a e)^4}{e^4 (d+e x)^{5/2}}-\frac {4 b (b d-a e)^3}{e^4 (d+e x)^{3/2}}+\frac {6 b^2 (b d-a e)^2}{e^4 \sqrt {d+e x}}-\frac {4 b^3 (b d-a e) \sqrt {d+e x}}{e^4}+\frac {b^4 (d+e x)^{3/2}}{e^4}\right ) \, dx \\ & = -\frac {2 (b d-a e)^4}{3 e^5 (d+e x)^{3/2}}+\frac {8 b (b d-a e)^3}{e^5 \sqrt {d+e x}}+\frac {12 b^2 (b d-a e)^2 \sqrt {d+e x}}{e^5}-\frac {8 b^3 (b d-a e) (d+e x)^{3/2}}{3 e^5}+\frac {2 b^4 (d+e x)^{5/2}}{5 e^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.22 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 \left (-5 a^4 e^4-20 a^3 b e^3 (2 d+3 e x)+30 a^2 b^2 e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+20 a b^3 e \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )+b^4 \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )\right )}{15 e^5 (d+e x)^{3/2}} \]

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^(5/2),x]

[Out]

(2*(-5*a^4*e^4 - 20*a^3*b*e^3*(2*d + 3*e*x) + 30*a^2*b^2*e^2*(8*d^2 + 12*d*e*x + 3*e^2*x^2) + 20*a*b^3*e*(-16*
d^3 - 24*d^2*e*x - 6*d*e^2*x^2 + e^3*x^3) + b^4*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*
x^4)))/(15*e^5*(d + e*x)^(3/2))

Maple [A] (verified)

Time = 2.76 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.02

method	result	size
risch	\(\frac {2 b^{2} \left (3 x^{2} b^{2} e^{2}+20 x a b \,e^{2}-14 b^{2} d e x +90 a^{2} e^{2}-160 a b d e +73 b^{2} d^{2}\right ) \sqrt {e x +d}}{15 e^{5}}-\frac {2 \left (12 b e x +a e +11 b d \right ) \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{3 e^{5} \left (e x +d \right )^{\frac {3}{2}}}\)	\(128\)
pseudoelliptic	\(-\frac {2 \left (\left (-\frac {3}{5} b^{4} x^{4}-4 a \,b^{3} x^{3}-18 a^{2} b^{2} x^{2}+12 a^{3} b x +a^{4}\right ) e^{4}+8 b \left (\frac {1}{5} b^{3} x^{3}+3 a \,b^{2} x^{2}-9 a^{2} b x +a^{3}\right ) d \,e^{3}-48 b^{2} \left (\frac {1}{5} b^{2} x^{2}-2 a b x +a^{2}\right ) d^{2} e^{2}+64 b^{3} \left (-\frac {3 b x}{5}+a \right ) d^{3} e -\frac {128 b^{4} d^{4}}{5}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{5}}\)	\(143\)
gosper	\(-\frac {2 \left (-3 b^{4} x^{4} e^{4}-20 x^{3} a \,b^{3} e^{4}+8 x^{3} b^{4} d \,e^{3}-90 x^{2} a^{2} b^{2} e^{4}+120 x^{2} a \,b^{3} d \,e^{3}-48 x^{2} b^{4} d^{2} e^{2}+60 x \,a^{3} b \,e^{4}-360 x \,a^{2} b^{2} d \,e^{3}+480 x a \,b^{3} d^{2} e^{2}-192 x \,b^{4} d^{3} e +5 e^{4} a^{4}+40 b \,e^{3} d \,a^{3}-240 b^{2} e^{2} d^{2} a^{2}+320 a \,b^{3} d^{3} e -128 b^{4} d^{4}\right )}{15 \left (e x +d \right )^{\frac {3}{2}} e^{5}}\)	\(186\)
trager	\(-\frac {2 \left (-3 b^{4} x^{4} e^{4}-20 x^{3} a \,b^{3} e^{4}+8 x^{3} b^{4} d \,e^{3}-90 x^{2} a^{2} b^{2} e^{4}+120 x^{2} a \,b^{3} d \,e^{3}-48 x^{2} b^{4} d^{2} e^{2}+60 x \,a^{3} b \,e^{4}-360 x \,a^{2} b^{2} d \,e^{3}+480 x a \,b^{3} d^{2} e^{2}-192 x \,b^{4} d^{3} e +5 e^{4} a^{4}+40 b \,e^{3} d \,a^{3}-240 b^{2} e^{2} d^{2} a^{2}+320 a \,b^{3} d^{3} e -128 b^{4} d^{4}\right )}{15 \left (e x +d \right )^{\frac {3}{2}} e^{5}}\)	\(186\)
derivativedivides	\(\frac {\frac {2 b^{4} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {8 a \,b^{3} e \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {8 b^{4} d \left (e x +d \right )^{\frac {3}{2}}}{3}+12 a^{2} b^{2} e^{2} \sqrt {e x +d}-24 a \,b^{3} d e \sqrt {e x +d}+12 b^{4} d^{2} \sqrt {e x +d}-\frac {8 b \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{\sqrt {e x +d}}-\frac {2 \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{5}}\)	\(198\)
default	\(\frac {\frac {2 b^{4} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {8 a \,b^{3} e \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {8 b^{4} d \left (e x +d \right )^{\frac {3}{2}}}{3}+12 a^{2} b^{2} e^{2} \sqrt {e x +d}-24 a \,b^{3} d e \sqrt {e x +d}+12 b^{4} d^{2} \sqrt {e x +d}-\frac {8 b \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{\sqrt {e x +d}}-\frac {2 \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{5}}\)	\(198\)

[In]

int((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/15*b^2*(3*b^2*e^2*x^2+20*a*b*e^2*x-14*b^2*d*e*x+90*a^2*e^2-160*a*b*d*e+73*b^2*d^2)*(e*x+d)^(1/2)/e^5-2/3*(12
*b*e*x+a*e+11*b*d)*(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)/e^5/(e*x+d)^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.62 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (3 \, b^{4} e^{4} x^{4} + 128 \, b^{4} d^{4} - 320 \, a b^{3} d^{3} e + 240 \, a^{2} b^{2} d^{2} e^{2} - 40 \, a^{3} b d e^{3} - 5 \, a^{4} e^{4} - 4 \, {\left (2 \, b^{4} d e^{3} - 5 \, a b^{3} e^{4}\right )} x^{3} + 6 \, {\left (8 \, b^{4} d^{2} e^{2} - 20 \, a b^{3} d e^{3} + 15 \, a^{2} b^{2} e^{4}\right )} x^{2} + 12 \, {\left (16 \, b^{4} d^{3} e - 40 \, a b^{3} d^{2} e^{2} + 30 \, a^{2} b^{2} d e^{3} - 5 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

2/15*(3*b^4*e^4*x^4 + 128*b^4*d^4 - 320*a*b^3*d^3*e + 240*a^2*b^2*d^2*e^2 - 40*a^3*b*d*e^3 - 5*a^4*e^4 - 4*(2*
b^4*d*e^3 - 5*a*b^3*e^4)*x^3 + 6*(8*b^4*d^2*e^2 - 20*a*b^3*d*e^3 + 15*a^2*b^2*e^4)*x^2 + 12*(16*b^4*d^3*e - 40
*a*b^3*d^2*e^2 + 30*a^2*b^2*d*e^3 - 5*a^3*b*e^4)*x)*sqrt(e*x + d)/(e^7*x^2 + 2*d*e^6*x + d^2*e^5)

Sympy [A] (verification not implemented)

Time = 3.78 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.52 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{5/2}} \, dx=\begin {cases} \frac {2 \left (\frac {b^{4} \left (d + e x\right )^{\frac {5}{2}}}{5 e^{4}} - \frac {4 b \left (a e - b d\right )^{3}}{e^{4} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (4 a b^{3} e - 4 b^{4} d\right )}{3 e^{4}} + \frac {\sqrt {d + e x} \left (6 a^{2} b^{2} e^{2} - 12 a b^{3} d e + 6 b^{4} d^{2}\right )}{e^{4}} - \frac {\left (a e - b d\right )^{4}}{3 e^{4} \left (d + e x\right )^{\frac {3}{2}}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {a^{4} x + 2 a^{3} b x^{2} + 2 a^{2} b^{2} x^{3} + a b^{3} x^{4} + \frac {b^{4} x^{5}}{5}}{d^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**(5/2),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.50 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {3 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{4} - 20 \, {\left (b^{4} d - a b^{3} e\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 90 \, {\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} \sqrt {e x + d}}{e^{4}} - \frac {5 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4} - 12 \, {\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{4}}\right )}}{15 \, e} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

2/15*((3*(e*x + d)^(5/2)*b^4 - 20*(b^4*d - a*b^3*e)*(e*x + d)^(3/2) + 90*(b^4*d^2 - 2*a*b^3*d*e + a^2*b^2*e^2)
*sqrt(e*x + d))/e^4 - 5*(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 - 12*(b^4*d^3 -
3*a*b^3*d^2*e + 3*a^2*b^2*d*e^2 - a^3*b*e^3)*(e*x + d))/((e*x + d)^(3/2)*e^4))/e

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 229 vs. \(2 (109) = 218\).

Time = 0.28 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.83 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (12 \, {\left (e x + d\right )} b^{4} d^{3} - b^{4} d^{4} - 36 \, {\left (e x + d\right )} a b^{3} d^{2} e + 4 \, a b^{3} d^{3} e + 36 \, {\left (e x + d\right )} a^{2} b^{2} d e^{2} - 6 \, a^{2} b^{2} d^{2} e^{2} - 12 \, {\left (e x + d\right )} a^{3} b e^{3} + 4 \, a^{3} b d e^{3} - a^{4} e^{4}\right )}}{3 \, {\left (e x + d\right )}^{\frac {3}{2}} e^{5}} + \frac {2 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{4} e^{20} - 20 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{4} d e^{20} + 90 \, \sqrt {e x + d} b^{4} d^{2} e^{20} + 20 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{3} e^{21} - 180 \, \sqrt {e x + d} a b^{3} d e^{21} + 90 \, \sqrt {e x + d} a^{2} b^{2} e^{22}\right )}}{15 \, e^{25}} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

2/3*(12*(e*x + d)*b^4*d^3 - b^4*d^4 - 36*(e*x + d)*a*b^3*d^2*e + 4*a*b^3*d^3*e + 36*(e*x + d)*a^2*b^2*d*e^2 -
6*a^2*b^2*d^2*e^2 - 12*(e*x + d)*a^3*b*e^3 + 4*a^3*b*d*e^3 - a^4*e^4)/((e*x + d)^(3/2)*e^5) + 2/15*(3*(e*x + d
)^(5/2)*b^4*e^20 - 20*(e*x + d)^(3/2)*b^4*d*e^20 + 90*sqrt(e*x + d)*b^4*d^2*e^20 + 20*(e*x + d)^(3/2)*a*b^3*e^
21 - 180*sqrt(e*x + d)*a*b^3*d*e^21 + 90*sqrt(e*x + d)*a^2*b^2*e^22)/e^25

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {\left (d+e\,x\right )\,\left (-8\,a^3\,b\,e^3+24\,a^2\,b^2\,d\,e^2-24\,a\,b^3\,d^2\,e+8\,b^4\,d^3\right )-\frac {2\,a^4\,e^4}{3}-\frac {2\,b^4\,d^4}{3}-4\,a^2\,b^2\,d^2\,e^2+\frac {8\,a\,b^3\,d^3\,e}{3}+\frac {8\,a^3\,b\,d\,e^3}{3}}{e^5\,{\left (d+e\,x\right )}^{3/2}}+\frac {2\,b^4\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5}-\frac {\left (8\,b^4\,d-8\,a\,b^3\,e\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,e^5}+\frac {12\,b^2\,{\left (a\,e-b\,d\right )}^2\,\sqrt {d+e\,x}}{e^5} \]

[In]

int((a^2 + b^2*x^2 + 2*a*b*x)^2/(d + e*x)^(5/2),x)

[Out]

((d + e*x)*(8*b^4*d^3 - 8*a^3*b*e^3 + 24*a^2*b^2*d*e^2 - 24*a*b^3*d^2*e) - (2*a^4*e^4)/3 - (2*b^4*d^4)/3 - 4*a
^2*b^2*d^2*e^2 + (8*a*b^3*d^3*e)/3 + (8*a^3*b*d*e^3)/3)/(e^5*(d + e*x)^(3/2)) + (2*b^4*(d + e*x)^(5/2))/(5*e^5
) - ((8*b^4*d - 8*a*b^3*e)*(d + e*x)^(3/2))/(3*e^5) + (12*b^2*(a*e - b*d)^2*(d + e*x)^(1/2))/e^5