Integrand size = 31, antiderivative size = 100 \[ \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=-\frac {2 (b d-a e)^3 (d+e x)^{5/2}}{5 e^4}+\frac {6 b (b d-a e)^2 (d+e x)^{7/2}}{7 e^4}-\frac {2 b^2 (b d-a e) (d+e x)^{9/2}}{3 e^4}+\frac {2 b^3 (d+e x)^{11/2}}{11 e^4} \]
-2/5*(-a*e+b*d)^3*(e*x+d)^(5/2)/e^4+6/7*b*(-a*e+b*d)^2*(e*x+d)^(7/2)/e^4-2 /3*b^2*(-a*e+b*d)*(e*x+d)^(9/2)/e^4+2/11*b^3*(e*x+d)^(11/2)/e^4
Time = 0.03 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.02 \[ \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {2 (d+e x)^{5/2} \left (231 a^3 e^3+99 a^2 b e^2 (-2 d+5 e x)+11 a b^2 e \left (8 d^2-20 d e x+35 e^2 x^2\right )+b^3 \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )\right )}{1155 e^4} \]
(2*(d + e*x)^(5/2)*(231*a^3*e^3 + 99*a^2*b*e^2*(-2*d + 5*e*x) + 11*a*b^2*e *(8*d^2 - 20*d*e*x + 35*e^2*x^2) + b^3*(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^ 2 + 105*e^3*x^3)))/(1155*e^4)
Time = 0.25 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1184, 27, 53, 2009}
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.
| \(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right ) (d+e x)^{3/2} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int b^2 (a+b x)^3 (d+e x)^{3/2}dx}{b^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int (a+b x)^3 (d+e x)^{3/2}dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {3 b^2 (d+e x)^{7/2} (b d-a e)}{e^3}+\frac {3 b (d+e x)^{5/2} (b d-a e)^2}{e^3}+\frac {(d+e x)^{3/2} (a e-b d)^3}{e^3}+\frac {b^3 (d+e x)^{9/2}}{e^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b^2 (d+e x)^{9/2} (b d-a e)}{3 e^4}+\frac {6 b (d+e x)^{7/2} (b d-a e)^2}{7 e^4}-\frac {2 (d+e x)^{5/2} (b d-a e)^3}{5 e^4}+\frac {2 b^3 (d+e x)^{11/2}}{11 e^4}\) |
(-2*(b*d - a*e)^3*(d + e*x)^(5/2))/(5*e^4) + (6*b*(b*d - a*e)^2*(d + e*x)^ (7/2))/(7*e^4) - (2*b^2*(b*d - a*e)*(d + e*x)^(9/2))/(3*e^4) + (2*b^3*(d + e*x)^(11/2))/(11*e^4)
3.21.44.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] && 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])
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 0.26 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.93
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (\left (\frac {5}{3} a \,b^{2} x^{2}+\frac {15}{7} b \,a^{2} x +\frac {5}{11} x^{3} b^{3}+a^{3}\right ) e^{3}-\frac {6 \left (\frac {35}{99} b^{2} x^{2}+\frac {10}{9} a b x +a^{2}\right ) b d \,e^{2}}{7}+\frac {8 \left (\frac {5 b x}{11}+a \right ) b^{2} d^{2} e}{21}-\frac {16 b^{3} d^{3}}{231}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5 e^{4}}\) | \(93\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (105 b^{3} x^{3} e^{3}+385 x^{2} a \,b^{2} e^{3}-70 x^{2} b^{3} d \,e^{2}+495 x \,a^{2} b \,e^{3}-220 x a \,b^{2} d \,e^{2}+40 x \,b^{3} d^{2} e +231 a^{3} e^{3}-198 a^{2} b d \,e^{2}+88 a \,b^{2} d^{2} e -16 b^{3} d^{3}\right )}{1155 e^{4}}\) | \(116\) |
| derivativedivides | \(\frac {\frac {2 b^{3} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a e -b d \right ) b^{2}+b \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a e -b d \right ) \left (2 a b e -2 b^{2} d \right )+b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{4}}\) | \(147\) |
| default | \(\frac {\frac {2 b^{3} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a e -b d \right ) b^{2}+b \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a e -b d \right ) \left (2 a b e -2 b^{2} d \right )+b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{4}}\) | \(147\) |
| trager | \(\frac {2 \left (105 b^{3} e^{5} x^{5}+385 b^{2} a \,e^{5} x^{4}+140 b^{3} d \,e^{4} x^{4}+495 a^{2} b \,e^{5} x^{3}+550 a \,b^{2} d \,e^{4} x^{3}+5 b^{3} d^{2} e^{3} x^{3}+231 a^{3} e^{5} x^{2}+792 b \,a^{2} d \,e^{4} x^{2}+33 b^{2} a \,d^{2} e^{3} x^{2}-6 b^{3} d^{3} e^{2} x^{2}+462 a^{3} d \,e^{4} x +99 a^{2} b \,d^{2} e^{3} x -44 a \,b^{2} d^{3} e^{2} x +8 b^{3} d^{4} e x +231 a^{3} d^{2} e^{3}-198 b \,a^{2} d^{3} e^{2}+88 b^{2} a \,d^{4} e -16 b^{3} d^{5}\right ) \sqrt {e x +d}}{1155 e^{4}}\) | \(228\) |
| risch | \(\frac {2 \left (105 b^{3} e^{5} x^{5}+385 b^{2} a \,e^{5} x^{4}+140 b^{3} d \,e^{4} x^{4}+495 a^{2} b \,e^{5} x^{3}+550 a \,b^{2} d \,e^{4} x^{3}+5 b^{3} d^{2} e^{3} x^{3}+231 a^{3} e^{5} x^{2}+792 b \,a^{2} d \,e^{4} x^{2}+33 b^{2} a \,d^{2} e^{3} x^{2}-6 b^{3} d^{3} e^{2} x^{2}+462 a^{3} d \,e^{4} x +99 a^{2} b \,d^{2} e^{3} x -44 a \,b^{2} d^{3} e^{2} x +8 b^{3} d^{4} e x +231 a^{3} d^{2} e^{3}-198 b \,a^{2} d^{3} e^{2}+88 b^{2} a \,d^{4} e -16 b^{3} d^{5}\right ) \sqrt {e x +d}}{1155 e^{4}}\) | \(228\) |
2/5*((5/3*a*b^2*x^2+15/7*b*a^2*x+5/11*x^3*b^3+a^3)*e^3-6/7*(35/99*b^2*x^2+ 10/9*a*b*x+a^2)*b*d*e^2+8/21*(5/11*b*x+a)*b^2*d^2*e-16/231*b^3*d^3)*(e*x+d )^(5/2)/e^4
Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (84) = 168\).
Time = 0.37 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.16 \[ \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {2 \, {\left (105 \, b^{3} e^{5} x^{5} - 16 \, b^{3} d^{5} + 88 \, a b^{2} d^{4} e - 198 \, a^{2} b d^{3} e^{2} + 231 \, a^{3} d^{2} e^{3} + 35 \, {\left (4 \, b^{3} d e^{4} + 11 \, a b^{2} e^{5}\right )} x^{4} + 5 \, {\left (b^{3} d^{2} e^{3} + 110 \, a b^{2} d e^{4} + 99 \, a^{2} b e^{5}\right )} x^{3} - 3 \, {\left (2 \, b^{3} d^{3} e^{2} - 11 \, a b^{2} d^{2} e^{3} - 264 \, a^{2} b d e^{4} - 77 \, a^{3} e^{5}\right )} x^{2} + {\left (8 \, b^{3} d^{4} e - 44 \, a b^{2} d^{3} e^{2} + 99 \, a^{2} b d^{2} e^{3} + 462 \, a^{3} d e^{4}\right )} x\right )} \sqrt {e x + d}}{1155 \, e^{4}} \]
2/1155*(105*b^3*e^5*x^5 - 16*b^3*d^5 + 88*a*b^2*d^4*e - 198*a^2*b*d^3*e^2 + 231*a^3*d^2*e^3 + 35*(4*b^3*d*e^4 + 11*a*b^2*e^5)*x^4 + 5*(b^3*d^2*e^3 + 110*a*b^2*d*e^4 + 99*a^2*b*e^5)*x^3 - 3*(2*b^3*d^3*e^2 - 11*a*b^2*d^2*e^3 - 264*a^2*b*d*e^4 - 77*a^3*e^5)*x^2 + (8*b^3*d^4*e - 44*a*b^2*d^3*e^2 + 9 9*a^2*b*d^2*e^3 + 462*a^3*d*e^4)*x)*sqrt(e*x + d)/e^4
Leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (92) = 184\).
Time = 1.12 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.96 \[ \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\begin {cases} \frac {2 \left (\frac {b^{3} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{3}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (3 a b^{2} e - 3 b^{3} d\right )}{9 e^{3}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (3 a^{2} b e^{2} - 6 a b^{2} d e + 3 b^{3} d^{2}\right )}{7 e^{3}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (a^{3} e^{3} - 3 a^{2} b d e^{2} + 3 a b^{2} d^{2} e - b^{3} d^{3}\right )}{5 e^{3}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {3}{2}} \left (\begin {cases} a^{3} x & \text {for}\: b = 0 \\\frac {a^{3} b x + \frac {3 a^{2} b^{2} x^{2}}{2} + a b^{3} x^{3} + \frac {b^{4} x^{4}}{4}}{b} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*(b**3*(d + e*x)**(11/2)/(11*e**3) + (d + e*x)**(9/2)*(3*a*b** 2*e - 3*b**3*d)/(9*e**3) + (d + e*x)**(7/2)*(3*a**2*b*e**2 - 6*a*b**2*d*e + 3*b**3*d**2)/(7*e**3) + (d + e*x)**(5/2)*(a**3*e**3 - 3*a**2*b*d*e**2 + 3*a*b**2*d**2*e - b**3*d**3)/(5*e**3))/e, Ne(e, 0)), (d**(3/2)*Piecewise(( a**3*x, Eq(b, 0)), ((a**3*b*x + 3*a**2*b**2*x**2/2 + a*b**3*x**3 + b**4*x* *4/4)/b, True)), True))
Timed out. \[ \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 566 vs. \(2 (84) = 168\).
Time = 0.29 (sec) , antiderivative size = 566, normalized size of antiderivative = 5.66 \[ \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {2 \, {\left (3465 \, \sqrt {e x + d} a^{3} d^{2} + 2310 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a^{3} d + \frac {3465 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a^{2} b d^{2}}{e} + 231 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a^{3} + \frac {693 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a b^{2} d^{2}}{e^{2}} + \frac {1386 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a^{2} b d}{e} + \frac {99 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} b^{3} d^{2}}{e^{3}} + \frac {594 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a b^{2} d}{e^{2}} + \frac {297 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a^{2} b}{e} + \frac {22 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} b^{3} d}{e^{3}} + \frac {33 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} a b^{2}}{e^{2}} + \frac {5 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} b^{3}}{e^{3}}\right )}}{3465 \, e} \]
2/3465*(3465*sqrt(e*x + d)*a^3*d^2 + 2310*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^3*d + 3465*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^2*b*d^2/e + 231 *(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^3 + 6 93*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a*b^2 *d^2/e^2 + 1386*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^2*b*d/e + 99*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*b^3*d^2/e^3 + 594*(5*(e*x + d)^(7/ 2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3) *a*b^2*d/e^2 + 297*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d )^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a^2*b/e + 22*(35*(e*x + d)^(9/2) - 180 *(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 3 15*sqrt(e*x + d)*d^4)*b^3*d/e^3 + 33*(35*(e*x + d)^(9/2) - 180*(e*x + d)^( 7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*a*b^2/e^2 + 5*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990 *(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*b^3/e^3)/e
Time = 0.06 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.87 \[ \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {2\,b^3\,{\left (d+e\,x\right )}^{11/2}}{11\,e^4}-\frac {\left (6\,b^3\,d-6\,a\,b^2\,e\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^4}+\frac {2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{5/2}}{5\,e^4}+\frac {6\,b\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^4} \]