\(\int (d+e x) (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\) [1573]

Optimal result

Integrand size = 26, antiderivative size = 69 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {(b d-a e) (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{6 b^2}+\frac {e \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b^2} \]

[Out]

1/6*(-a*e+b*d)*(b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/b^2+1/7*e*(b^2*x^2+2*a*b*x+a^2)^(7/2)/b^2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {654, 623} \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} (b d-a e)}{6 b^2}+\frac {e \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b^2} \]

[In]

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

[Out]

((b*d - a*e)*(a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(6*b^2) + (e*(a^2 + 2*a*b*x + b^2*x^2)^(7/2))/(7*b^2)

Rule 623

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

Rule 654

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*((a + b*x + c*x^2)^(p +
1)/(2*c*(p + 1))), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {e \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b^2}+\frac {\left (2 b^2 d-2 a b e\right ) \int \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx}{2 b^2} \\ & = \frac {(b d-a e) (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{6 b^2}+\frac {e \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.03 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.75 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (21 a^5 (2 d+e x)+35 a^4 b x (3 d+2 e x)+35 a^3 b^2 x^2 (4 d+3 e x)+21 a^2 b^3 x^3 (5 d+4 e x)+7 a b^4 x^4 (6 d+5 e x)+b^5 x^5 (7 d+6 e x)\right )}{42 (a+b x)} \]

[In]

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

[Out]

(x*Sqrt[(a + b*x)^2]*(21*a^5*(2*d + e*x) + 35*a^4*b*x*(3*d + 2*e*x) + 35*a^3*b^2*x^2*(4*d + 3*e*x) + 21*a^2*b^
3*x^3*(5*d + 4*e*x) + 7*a*b^4*x^4*(6*d + 5*e*x) + b^5*x^5*(7*d + 6*e*x)))/(42*(a + b*x))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(137\) vs. \(2(61)=122\).

Time = 2.49 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.00

method	result	size
gosper	\(\frac {x \left (6 b^{5} e \,x^{6}+35 x^{5} e a \,b^{4}+7 x^{5} b^{5} d +84 a^{2} b^{3} e \,x^{4}+42 a \,b^{4} d \,x^{4}+105 a^{3} b^{2} e \,x^{3}+105 a^{2} b^{3} d \,x^{3}+70 x^{2} a^{4} e b +140 x^{2} a^{3} d \,b^{2}+21 x e \,a^{5}+105 x \,a^{4} b d +42 d \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{42 \left (b x +a \right )^{5}}\)	\(138\)
default	\(\frac {x \left (6 b^{5} e \,x^{6}+35 x^{5} e a \,b^{4}+7 x^{5} b^{5} d +84 a^{2} b^{3} e \,x^{4}+42 a \,b^{4} d \,x^{4}+105 a^{3} b^{2} e \,x^{3}+105 a^{2} b^{3} d \,x^{3}+70 x^{2} a^{4} e b +140 x^{2} a^{3} d \,b^{2}+21 x e \,a^{5}+105 x \,a^{4} b d +42 d \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{42 \left (b x +a \right )^{5}}\)	\(138\)
risch	\(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{5} e \,x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 e a \,b^{4}+b^{5} d \right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 e \,a^{2} b^{3}+5 d a \,b^{4}\right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 a^{3} b^{2} e +10 a^{2} b^{3} d \right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 a^{4} e b +10 a^{3} d \,b^{2}\right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (e \,a^{5}+5 a^{4} b d \right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, d \,a^{5} x}{b x +a}\)	\(233\)

[In]

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

[Out]

1/42*x*(6*b^5*e*x^6+35*a*b^4*e*x^5+7*b^5*d*x^5+84*a^2*b^3*e*x^4+42*a*b^4*d*x^4+105*a^3*b^2*e*x^3+105*a^2*b^3*d
*x^3+70*a^4*b*e*x^2+140*a^3*b^2*d*x^2+21*a^5*e*x+105*a^4*b*d*x+42*a^5*d)*((b*x+a)^2)^(5/2)/(b*x+a)^5

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.67 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{7} \, b^{5} e x^{7} + a^{5} d x + \frac {1}{6} \, {\left (b^{5} d + 5 \, a b^{4} e\right )} x^{6} + {\left (a b^{4} d + 2 \, a^{2} b^{3} e\right )} x^{5} + \frac {5}{2} \, {\left (a^{2} b^{3} d + a^{3} b^{2} e\right )} x^{4} + \frac {5}{3} \, {\left (2 \, a^{3} b^{2} d + a^{4} b e\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b d + a^{5} e\right )} x^{2} \]

[In]

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

[Out]

1/7*b^5*e*x^7 + a^5*d*x + 1/6*(b^5*d + 5*a*b^4*e)*x^6 + (a*b^4*d + 2*a^2*b^3*e)*x^5 + 5/2*(a^2*b^3*d + a^3*b^2
*e)*x^4 + 5/3*(2*a^3*b^2*d + a^4*b*e)*x^3 + 1/2*(5*a^4*b*d + a^5*e)*x^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2147 vs. \(2 (65) = 130\).

Time = 0.82 (sec) , antiderivative size = 2147, normalized size of antiderivative = 31.12 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((sqrt(a**2 + 2*a*b*x + b**2*x**2)*(b**4*e*x**6/7 + x**5*(29*a*b**5*e/7 + b**6*d)/(6*b**2) + x**4*(99
*a**2*b**4*e/7 + 6*a*b**5*d - 11*a*(29*a*b**5*e/7 + b**6*d)/(6*b))/(5*b**2) + x**3*(20*a**3*b**3*e + 15*a**2*b
**4*d - 5*a**2*(29*a*b**5*e/7 + b**6*d)/(6*b**2) - 9*a*(99*a**2*b**4*e/7 + 6*a*b**5*d - 11*a*(29*a*b**5*e/7 +
b**6*d)/(6*b))/(5*b))/(4*b**2) + x**2*(15*a**4*b**2*e + 20*a**3*b**3*d - 4*a**2*(99*a**2*b**4*e/7 + 6*a*b**5*d
- 11*a*(29*a*b**5*e/7 + b**6*d)/(6*b))/(5*b**2) - 7*a*(20*a**3*b**3*e + 15*a**2*b**4*d - 5*a**2*(29*a*b**5*e/
7 + b**6*d)/(6*b**2) - 9*a*(99*a**2*b**4*e/7 + 6*a*b**5*d - 11*a*(29*a*b**5*e/7 + b**6*d)/(6*b))/(5*b))/(4*b))
/(3*b**2) + x*(6*a**5*b*e + 15*a**4*b**2*d - 3*a**2*(20*a**3*b**3*e + 15*a**2*b**4*d - 5*a**2*(29*a*b**5*e/7 +
b**6*d)/(6*b**2) - 9*a*(99*a**2*b**4*e/7 + 6*a*b**5*d - 11*a*(29*a*b**5*e/7 + b**6*d)/(6*b))/(5*b))/(4*b**2)
- 5*a*(15*a**4*b**2*e + 20*a**3*b**3*d - 4*a**2*(99*a**2*b**4*e/7 + 6*a*b**5*d - 11*a*(29*a*b**5*e/7 + b**6*d)
/(6*b))/(5*b**2) - 7*a*(20*a**3*b**3*e + 15*a**2*b**4*d - 5*a**2*(29*a*b**5*e/7 + b**6*d)/(6*b**2) - 9*a*(99*a
**2*b**4*e/7 + 6*a*b**5*d - 11*a*(29*a*b**5*e/7 + b**6*d)/(6*b))/(5*b))/(4*b))/(3*b))/(2*b**2) + (a**6*e + 6*a
**5*b*d - 2*a**2*(15*a**4*b**2*e + 20*a**3*b**3*d - 4*a**2*(99*a**2*b**4*e/7 + 6*a*b**5*d - 11*a*(29*a*b**5*e/
7 + b**6*d)/(6*b))/(5*b**2) - 7*a*(20*a**3*b**3*e + 15*a**2*b**4*d - 5*a**2*(29*a*b**5*e/7 + b**6*d)/(6*b**2)
- 9*a*(99*a**2*b**4*e/7 + 6*a*b**5*d - 11*a*(29*a*b**5*e/7 + b**6*d)/(6*b))/(5*b))/(4*b))/(3*b**2) - 3*a*(6*a*
*5*b*e + 15*a**4*b**2*d - 3*a**2*(20*a**3*b**3*e + 15*a**2*b**4*d - 5*a**2*(29*a*b**5*e/7 + b**6*d)/(6*b**2) -
9*a*(99*a**2*b**4*e/7 + 6*a*b**5*d - 11*a*(29*a*b**5*e/7 + b**6*d)/(6*b))/(5*b))/(4*b**2) - 5*a*(15*a**4*b**2
*e + 20*a**3*b**3*d - 4*a**2*(99*a**2*b**4*e/7 + 6*a*b**5*d - 11*a*(29*a*b**5*e/7 + b**6*d)/(6*b))/(5*b**2) -
7*a*(20*a**3*b**3*e + 15*a**2*b**4*d - 5*a**2*(29*a*b**5*e/7 + b**6*d)/(6*b**2) - 9*a*(99*a**2*b**4*e/7 + 6*a*
b**5*d - 11*a*(29*a*b**5*e/7 + b**6*d)/(6*b))/(5*b))/(4*b))/(3*b))/(2*b))/b**2) + (a/b + x)*(a**6*d - a**2*(6*
a**5*b*e + 15*a**4*b**2*d - 3*a**2*(20*a**3*b**3*e + 15*a**2*b**4*d - 5*a**2*(29*a*b**5*e/7 + b**6*d)/(6*b**2)
- 9*a*(99*a**2*b**4*e/7 + 6*a*b**5*d - 11*a*(29*a*b**5*e/7 + b**6*d)/(6*b))/(5*b))/(4*b**2) - 5*a*(15*a**4*b*
*2*e + 20*a**3*b**3*d - 4*a**2*(99*a**2*b**4*e/7 + 6*a*b**5*d - 11*a*(29*a*b**5*e/7 + b**6*d)/(6*b))/(5*b**2)
- 7*a*(20*a**3*b**3*e + 15*a**2*b**4*d - 5*a**2*(29*a*b**5*e/7 + b**6*d)/(6*b**2) - 9*a*(99*a**2*b**4*e/7 + 6*
a*b**5*d - 11*a*(29*a*b**5*e/7 + b**6*d)/(6*b))/(5*b))/(4*b))/(3*b))/(2*b**2) - a*(a**6*e + 6*a**5*b*d - 2*a**
2*(15*a**4*b**2*e + 20*a**3*b**3*d - 4*a**2*(99*a**2*b**4*e/7 + 6*a*b**5*d - 11*a*(29*a*b**5*e/7 + b**6*d)/(6*
b))/(5*b**2) - 7*a*(20*a**3*b**3*e + 15*a**2*b**4*d - 5*a**2*(29*a*b**5*e/7 + b**6*d)/(6*b**2) - 9*a*(99*a**2*
b**4*e/7 + 6*a*b**5*d - 11*a*(29*a*b**5*e/7 + b**6*d)/(6*b))/(5*b))/(4*b))/(3*b**2) - 3*a*(6*a**5*b*e + 15*a**
4*b**2*d - 3*a**2*(20*a**3*b**3*e + 15*a**2*b**4*d - 5*a**2*(29*a*b**5*e/7 + b**6*d)/(6*b**2) - 9*a*(99*a**2*b
**4*e/7 + 6*a*b**5*d - 11*a*(29*a*b**5*e/7 + b**6*d)/(6*b))/(5*b))/(4*b**2) - 5*a*(15*a**4*b**2*e + 20*a**3*b*
*3*d - 4*a**2*(99*a**2*b**4*e/7 + 6*a*b**5*d - 11*a*(29*a*b**5*e/7 + b**6*d)/(6*b))/(5*b**2) - 7*a*(20*a**3*b*
*3*e + 15*a**2*b**4*d - 5*a**2*(29*a*b**5*e/7 + b**6*d)/(6*b**2) - 9*a*(99*a**2*b**4*e/7 + 6*a*b**5*d - 11*a*(
29*a*b**5*e/7 + b**6*d)/(6*b))/(5*b))/(4*b))/(3*b))/(2*b))/b)*log(a/b + x)/sqrt(b**2*(a/b + x)**2), Ne(b**2, 0
)), (((a**2 + 2*a*b*x)**(7/2)*(-a*e + 2*b*d)/(14*b) + e*(a**2 + 2*a*b*x)**(9/2)/(18*a*b))/(a*b), Ne(a*b, 0)),
((d*x + e*x**2/2)*(a**2)**(5/2), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (61) = 122\).

Time = 0.20 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.81 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{6} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} d x - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a e x}{6 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d}{6 \, b} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} e}{6 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} e}{7 \, b^{2}} \]

[In]

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

[Out]

1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*d*x - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a*e*x/b + 1/6*(b^2*x^2 + 2*a*b*x
+ a^2)^(5/2)*a*d/b - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*a^2*e/b^2 + 1/7*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*e/b^
2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (61) = 122\).

Time = 0.27 (sec) , antiderivative size = 218, normalized size of antiderivative = 3.16 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{7} \, b^{5} e x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, b^{5} d x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{6} \, a b^{4} e x^{6} \mathrm {sgn}\left (b x + a\right ) + a b^{4} d x^{5} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{2} b^{3} e x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{2} b^{3} d x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{3} b^{2} e x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a^{3} b^{2} d x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, a^{4} b e x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{4} b d x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, a^{5} e x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{5} d x \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (7 \, a^{6} b d - a^{7} e\right )} \mathrm {sgn}\left (b x + a\right )}{42 \, b^{2}} \]

[In]

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

[Out]

1/7*b^5*e*x^7*sgn(b*x + a) + 1/6*b^5*d*x^6*sgn(b*x + a) + 5/6*a*b^4*e*x^6*sgn(b*x + a) + a*b^4*d*x^5*sgn(b*x +
a) + 2*a^2*b^3*e*x^5*sgn(b*x + a) + 5/2*a^2*b^3*d*x^4*sgn(b*x + a) + 5/2*a^3*b^2*e*x^4*sgn(b*x + a) + 10/3*a^
3*b^2*d*x^3*sgn(b*x + a) + 5/3*a^4*b*e*x^3*sgn(b*x + a) + 5/2*a^4*b*d*x^2*sgn(b*x + a) + 1/2*a^5*e*x^2*sgn(b*x
+ a) + a^5*d*x*sgn(b*x + a) + 1/42*(7*a^6*b*d - a^7*e)*sgn(b*x + a)/b^2

Mupad [F(-1)]

Timed out. \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int \left (d+e\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \]

[In]

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

[Out]

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