Integrand size = 26, antiderivative size = 71 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {(b d-a e) (a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 b^2}+\frac {e \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b^2} \] Output:
1/6*(-a*e+b*d)*(b*x+a)^5*((b*x+a)^2)^(1/2)/b^2+1/7*e*(b^2*x^2+2*a*b*x+a^2) ^(7/2)/b^2
Time = 1.04 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.70 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {x \sqrt {(a+b x)^2} \left (21 a^5 (2 d+e x)+35 a^4 b x (3 d+2 e x)+35 a^3 b^2 x^2 (4 d+3 e x)+21 a^2 b^3 x^3 (5 d+4 e x)+7 a b^4 x^4 (6 d+5 e x)+b^5 x^5 (7 d+6 e x)\right )}{42 (a+b x)} \] Input:
Integrate[(d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
Output:
(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))
Time = 0.33 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1100, 1079, 17}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^{5/2} (d+e x) \, dx\) |
\(\Big \downarrow \) 1100 |
\(\displaystyle \frac {(b d-a e) \int \left (a^2+2 b x a+b^2 x^2\right )^{5/2}dx}{b}+\frac {e \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b^2}\) |
\(\Big \downarrow \) 1079 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) \int \left (x b^2+a b\right )^5dx}{b^6 (a+b x)}+\frac {e \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b^2}\) |
\(\Big \downarrow \) 17 |
\(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} (a+b x)^5 (b d-a e)}{6 b^2}+\frac {e \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 b^2}\) |
Input:
Int[(d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
Output:
((b*d - a*e)*(a + b*x)^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*b^2) + (e*(a^2 + 2*a*b*x + b^2*x^2)^(7/2))/(7*b^2)
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x + c *x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])) Int[(b/2 + c *x)^(2*p), x], x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0]
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] + Simp[(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] && EqQ[b^2 - 4*a*c, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(137\) vs. \(2(54)=108\).
Time = 0.96 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.94
method | result | size |
gosper | \(\frac {x \left (6 b^{5} e \,x^{6}+35 x^{5} a \,b^{4} e +7 d \,x^{5} b^{5}+84 a^{2} b^{3} e \,x^{4}+42 a \,b^{4} d \,x^{4}+105 x^{3} a^{3} b^{2} e +105 a^{2} b^{3} d \,x^{3}+70 x^{2} a^{4} b e +140 a^{3} b^{2} d \,x^{2}+21 x \,a^{5} e +105 a^{4} b d x +42 a^{5} d \right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{42 \left (b x +a \right )^{5}}\) | \(138\) |
default | \(\frac {x \left (6 b^{5} e \,x^{6}+35 x^{5} a \,b^{4} e +7 d \,x^{5} b^{5}+84 a^{2} b^{3} e \,x^{4}+42 a \,b^{4} d \,x^{4}+105 x^{3} a^{3} b^{2} e +105 a^{2} b^{3} d \,x^{3}+70 x^{2} a^{4} b e +140 a^{3} b^{2} d \,x^{2}+21 x \,a^{5} e +105 a^{4} b d x +42 a^{5} d \right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{42 \left (b x +a \right )^{5}}\) | \(138\) |
orering | \(\frac {x \left (6 b^{5} e \,x^{6}+35 x^{5} a \,b^{4} e +7 d \,x^{5} b^{5}+84 a^{2} b^{3} e \,x^{4}+42 a \,b^{4} d \,x^{4}+105 x^{3} a^{3} b^{2} e +105 a^{2} b^{3} d \,x^{3}+70 x^{2} a^{4} b e +140 a^{3} b^{2} d \,x^{2}+21 x \,a^{5} e +105 a^{4} b d x +42 a^{5} d \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {5}{2}}}{42 \left (b x +a \right )^{5}}\) | \(147\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{5} e \,x^{7}}{7 b x +7 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 a \,b^{4} e +b^{5} d \right ) x^{6}}{6 b x +6 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 a^{2} b^{3} e +5 a \,b^{4} d \right ) x^{5}}{5 b x +5 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (10 a^{3} b^{2} e +10 a^{2} b^{3} d \right ) x^{4}}{4 b x +4 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (5 a^{4} b e +10 a^{3} b^{2} d \right ) x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (a^{5} e +5 a^{4} b d \right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, a^{5} d x}{b x +a}\) | \(233\) |
Input:
int((e*x+d)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
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
Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (54) = 108\).
Time = 0.08 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.62 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{7} \, b^{5} e x^{7} + a^{5} d x + \frac {1}{6} \, {\left (b^{5} d + 5 \, a b^{4} e\right )} x^{6} + {\left (a b^{4} d + 2 \, a^{2} b^{3} e\right )} x^{5} + \frac {5}{2} \, {\left (a^{2} b^{3} d + a^{3} b^{2} e\right )} x^{4} + \frac {5}{3} \, {\left (2 \, a^{3} b^{2} d + a^{4} b e\right )} x^{3} + \frac {1}{2} \, {\left (5 \, a^{4} b d + a^{5} e\right )} x^{2} \] Input:
integrate((e*x+d)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")
Output:
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
Leaf count of result is larger than twice the leaf count of optimal. 2147 vs. \(2 (56) = 112\).
Time = 0.78 (sec) , antiderivative size = 2147, normalized size of antiderivative = 30.24 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
Output:
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*(1 5*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 - 1 1*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*(1 5*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 - 1 1*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...
Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (54) = 108\).
Time = 0.04 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.76 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{6} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} d x - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a e x}{6 \, b} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a d}{6 \, b} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} a^{2} e}{6 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} e}{7 \, b^{2}} \] Input:
integrate((e*x+d)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")
Output:
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
Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (54) = 108\).
Time = 0.32 (sec) , antiderivative size = 218, normalized size of antiderivative = 3.07 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {1}{7} \, b^{5} e x^{7} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{6} \, b^{5} d x^{6} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{6} \, a b^{4} e x^{6} \mathrm {sgn}\left (b x + a\right ) + a b^{4} d x^{5} \mathrm {sgn}\left (b x + a\right ) + 2 \, a^{2} b^{3} e x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{2} b^{3} d x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{3} b^{2} e x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, a^{3} b^{2} d x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{3} \, a^{4} b e x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {5}{2} \, a^{4} b d x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, a^{5} e x^{2} \mathrm {sgn}\left (b x + a\right ) + a^{5} d x \mathrm {sgn}\left (b x + a\right ) + \frac {{\left (7 \, a^{6} b d - a^{7} e\right )} \mathrm {sgn}\left (b x + a\right )}{42 \, b^{2}} \] Input:
integrate((e*x+d)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")
Output:
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*sg n(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
Timed out. \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int \left (d+e\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \] Input:
int((d + e*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)
Output:
int((d + e*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)
Time = 0.23 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.70 \[ \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {x \left (6 b^{5} e \,x^{6}+35 a \,b^{4} e \,x^{5}+7 b^{5} d \,x^{5}+84 a^{2} b^{3} e \,x^{4}+42 a \,b^{4} d \,x^{4}+105 a^{3} b^{2} e \,x^{3}+105 a^{2} b^{3} d \,x^{3}+70 a^{4} b e \,x^{2}+140 a^{3} b^{2} d \,x^{2}+21 a^{5} e x +105 a^{4} b d x +42 a^{5} d \right )}{42} \] Input:
int((e*x+d)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)
Output:
(x*(42*a**5*d + 21*a**5*e*x + 105*a**4*b*d*x + 70*a**4*b*e*x**2 + 140*a**3 *b**2*d*x**2 + 105*a**3*b**2*e*x**3 + 105*a**2*b**3*d*x**3 + 84*a**2*b**3* e*x**4 + 42*a*b**4*d*x**4 + 35*a*b**4*e*x**5 + 7*b**5*d*x**5 + 6*b**5*e*x* *6))/42