Integrand size = 24, antiderivative size = 126 \[ \int (A+B x) (d+e x)^{5/2} \left (b x+c x^2\right ) \, dx=-\frac {2 d (B d-A e) (c d-b e) (d+e x)^{7/2}}{7 e^4}+\frac {2 (B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^{9/2}}{9 e^4}-\frac {2 (3 B c d-b B e-A c e) (d+e x)^{11/2}}{11 e^4}+\frac {2 B c (d+e x)^{13/2}}{13 e^4} \]
-2/7*d*(-A*e+B*d)*(-b*e+c*d)*(e*x+d)^(7/2)/e^4+2/9*(B*d*(-2*b*e+3*c*d)-A*e *(-b*e+2*c*d))*(e*x+d)^(9/2)/e^4-2/11*(-A*c*e-B*b*e+3*B*c*d)*(e*x+d)^(11/2 )/e^4+2/13*B*c*(e*x+d)^(13/2)/e^4
Time = 0.11 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.90 \[ \int (A+B x) (d+e x)^{5/2} \left (b x+c x^2\right ) \, dx=\frac {2 (d+e x)^{7/2} \left (13 A e \left (11 b e (-2 d+7 e x)+c \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )+B \left (13 b e \left (8 d^2-28 d e x+63 e^2 x^2\right )+c \left (-48 d^3+168 d^2 e x-378 d e^2 x^2+693 e^3 x^3\right )\right )\right )}{9009 e^4} \]
(2*(d + e*x)^(7/2)*(13*A*e*(11*b*e*(-2*d + 7*e*x) + c*(8*d^2 - 28*d*e*x + 63*e^2*x^2)) + B*(13*b*e*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + c*(-48*d^3 + 16 8*d^2*e*x - 378*d*e^2*x^2 + 693*e^3*x^3))))/(9009*e^4)
Time = 0.27 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1195, 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.
\(\displaystyle \int (A+B x) \left (b x+c x^2\right ) (d+e x)^{5/2} \, dx\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (\frac {(d+e x)^{9/2} (A c e+b B e-3 B c d)}{e^3}+\frac {(d+e x)^{7/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{e^3}-\frac {d (d+e x)^{5/2} (B d-A e) (c d-b e)}{e^3}+\frac {B c (d+e x)^{11/2}}{e^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 (d+e x)^{11/2} (-A c e-b B e+3 B c d)}{11 e^4}+\frac {2 (d+e x)^{9/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{9 e^4}-\frac {2 d (d+e x)^{7/2} (B d-A e) (c d-b e)}{7 e^4}+\frac {2 B c (d+e x)^{13/2}}{13 e^4}\) |
(-2*d*(B*d - A*e)*(c*d - b*e)*(d + e*x)^(7/2))/(7*e^4) + (2*(B*d*(3*c*d - 2*b*e) - A*e*(2*c*d - b*e))*(d + e*x)^(9/2))/(9*e^4) - (2*(3*B*c*d - b*B*e - A*c*e)*(d + e*x)^(11/2))/(11*e^4) + (2*B*c*(d + e*x)^(13/2))/(13*e^4)
3.13.14.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Time = 0.38 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.73
method | result | size |
pseudoelliptic | \(-\frac {4 \left (-\frac {7 \left (\frac {9 B c \,x^{2}}{13}+\frac {9 \left (A c +B b \right ) x}{11}+A b \right ) x \,e^{3}}{2}+d \left (\frac {189 B c \,x^{2}}{143}+\frac {14 \left (A c +B b \right ) x}{11}+A b \right ) e^{2}-\frac {4 \left (\frac {21}{13} B c x +A c +B b \right ) d^{2} e}{11}+\frac {24 B c \,d^{3}}{143}\right ) \left (e x +d \right )^{\frac {7}{2}}}{63 e^{4}}\) | \(92\) |
default | \(\frac {\frac {2 B c \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (A e -2 B d \right ) c +B \left (b e -c d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (-A e +B d \right ) d c +\left (A e -2 B d \right ) \left (b e -c d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-A e +B d \right ) d \left (b e -c d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{4}}\) | \(112\) |
derivativedivides | \(\frac {\frac {2 B c \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (A e -2 B d \right ) c +B \left (b e -c d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (-\left (A e -B d \right ) d c +\left (A e -2 B d \right ) \left (b e -c d \right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}-\frac {2 \left (A e -B d \right ) d \left (b e -c d \right ) \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{4}}\) | \(113\) |
gosper | \(-\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (-693 B c \,x^{3} e^{3}-819 A c \,e^{3} x^{2}-819 B \,x^{2} b \,e^{3}+378 B \,x^{2} c d \,e^{2}-1001 A b \,e^{3} x +364 A c d \,e^{2} x +364 B x b d \,e^{2}-168 B c \,d^{2} e x +286 A b d \,e^{2}-104 A c \,d^{2} e -104 B b \,d^{2} e +48 B c \,d^{3}\right )}{9009 e^{4}}\) | \(121\) |
trager | \(-\frac {2 \left (-693 B \,e^{6} c \,x^{6}-819 A c \,e^{6} x^{5}-819 B b \,e^{6} x^{5}-1701 B c d \,e^{5} x^{5}-1001 A b \,e^{6} x^{4}-2093 A c d \,e^{5} x^{4}-2093 B b d \,e^{5} x^{4}-1113 B c \,d^{2} e^{4} x^{4}-2717 A b d \,e^{5} x^{3}-1469 A c \,d^{2} e^{4} x^{3}-1469 B b \,d^{2} e^{4} x^{3}-15 B c \,d^{3} e^{3} x^{3}-2145 A b \,d^{2} e^{4} x^{2}-39 A c \,d^{3} e^{3} x^{2}-39 B b \,d^{3} e^{3} x^{2}+18 B c \,d^{4} e^{2} x^{2}-143 A b \,d^{3} e^{3} x +52 A c \,d^{4} e^{2} x +52 B b \,d^{4} e^{2} x -24 B c \,d^{5} e x +286 A b \,d^{4} e^{2}-104 A c \,d^{5} e -104 B b \,d^{5} e +48 B c \,d^{6}\right ) \sqrt {e x +d}}{9009 e^{4}}\) | \(277\) |
risch | \(-\frac {2 \left (-693 B \,e^{6} c \,x^{6}-819 A c \,e^{6} x^{5}-819 B b \,e^{6} x^{5}-1701 B c d \,e^{5} x^{5}-1001 A b \,e^{6} x^{4}-2093 A c d \,e^{5} x^{4}-2093 B b d \,e^{5} x^{4}-1113 B c \,d^{2} e^{4} x^{4}-2717 A b d \,e^{5} x^{3}-1469 A c \,d^{2} e^{4} x^{3}-1469 B b \,d^{2} e^{4} x^{3}-15 B c \,d^{3} e^{3} x^{3}-2145 A b \,d^{2} e^{4} x^{2}-39 A c \,d^{3} e^{3} x^{2}-39 B b \,d^{3} e^{3} x^{2}+18 B c \,d^{4} e^{2} x^{2}-143 A b \,d^{3} e^{3} x +52 A c \,d^{4} e^{2} x +52 B b \,d^{4} e^{2} x -24 B c \,d^{5} e x +286 A b \,d^{4} e^{2}-104 A c \,d^{5} e -104 B b \,d^{5} e +48 B c \,d^{6}\right ) \sqrt {e x +d}}{9009 e^{4}}\) | \(277\) |
-4/63*(-7/2*(9/13*B*c*x^2+9/11*(A*c+B*b)*x+A*b)*x*e^3+d*(189/143*B*c*x^2+1 4/11*(A*c+B*b)*x+A*b)*e^2-4/11*(21/13*B*c*x+A*c+B*b)*d^2*e+24/143*B*c*d^3) *(e*x+d)^(7/2)/e^4
Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (110) = 220\).
Time = 0.28 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.83 \[ \int (A+B x) (d+e x)^{5/2} \left (b x+c x^2\right ) \, dx=\frac {2 \, {\left (693 \, B c e^{6} x^{6} - 48 \, B c d^{6} - 286 \, A b d^{4} e^{2} + 104 \, {\left (B b + A c\right )} d^{5} e + 63 \, {\left (27 \, B c d e^{5} + 13 \, {\left (B b + A c\right )} e^{6}\right )} x^{5} + 7 \, {\left (159 \, B c d^{2} e^{4} + 143 \, A b e^{6} + 299 \, {\left (B b + A c\right )} d e^{5}\right )} x^{4} + {\left (15 \, B c d^{3} e^{3} + 2717 \, A b d e^{5} + 1469 \, {\left (B b + A c\right )} d^{2} e^{4}\right )} x^{3} - 3 \, {\left (6 \, B c d^{4} e^{2} - 715 \, A b d^{2} e^{4} - 13 \, {\left (B b + A c\right )} d^{3} e^{3}\right )} x^{2} + {\left (24 \, B c d^{5} e + 143 \, A b d^{3} e^{3} - 52 \, {\left (B b + A c\right )} d^{4} e^{2}\right )} x\right )} \sqrt {e x + d}}{9009 \, e^{4}} \]
2/9009*(693*B*c*e^6*x^6 - 48*B*c*d^6 - 286*A*b*d^4*e^2 + 104*(B*b + A*c)*d ^5*e + 63*(27*B*c*d*e^5 + 13*(B*b + A*c)*e^6)*x^5 + 7*(159*B*c*d^2*e^4 + 1 43*A*b*e^6 + 299*(B*b + A*c)*d*e^5)*x^4 + (15*B*c*d^3*e^3 + 2717*A*b*d*e^5 + 1469*(B*b + A*c)*d^2*e^4)*x^3 - 3*(6*B*c*d^4*e^2 - 715*A*b*d^2*e^4 - 13 *(B*b + A*c)*d^3*e^3)*x^2 + (24*B*c*d^5*e + 143*A*b*d^3*e^3 - 52*(B*b + A* c)*d^4*e^2)*x)*sqrt(e*x + d)/e^4
Leaf count of result is larger than twice the leaf count of optimal. 581 vs. \(2 (129) = 258\).
Time = 0.43 (sec) , antiderivative size = 581, normalized size of antiderivative = 4.61 \[ \int (A+B x) (d+e x)^{5/2} \left (b x+c x^2\right ) \, dx=\begin {cases} - \frac {4 A b d^{4} \sqrt {d + e x}}{63 e^{2}} + \frac {2 A b d^{3} x \sqrt {d + e x}}{63 e} + \frac {10 A b d^{2} x^{2} \sqrt {d + e x}}{21} + \frac {38 A b d e x^{3} \sqrt {d + e x}}{63} + \frac {2 A b e^{2} x^{4} \sqrt {d + e x}}{9} + \frac {16 A c d^{5} \sqrt {d + e x}}{693 e^{3}} - \frac {8 A c d^{4} x \sqrt {d + e x}}{693 e^{2}} + \frac {2 A c d^{3} x^{2} \sqrt {d + e x}}{231 e} + \frac {226 A c d^{2} x^{3} \sqrt {d + e x}}{693} + \frac {46 A c d e x^{4} \sqrt {d + e x}}{99} + \frac {2 A c e^{2} x^{5} \sqrt {d + e x}}{11} + \frac {16 B b d^{5} \sqrt {d + e x}}{693 e^{3}} - \frac {8 B b d^{4} x \sqrt {d + e x}}{693 e^{2}} + \frac {2 B b d^{3} x^{2} \sqrt {d + e x}}{231 e} + \frac {226 B b d^{2} x^{3} \sqrt {d + e x}}{693} + \frac {46 B b d e x^{4} \sqrt {d + e x}}{99} + \frac {2 B b e^{2} x^{5} \sqrt {d + e x}}{11} - \frac {32 B c d^{6} \sqrt {d + e x}}{3003 e^{4}} + \frac {16 B c d^{5} x \sqrt {d + e x}}{3003 e^{3}} - \frac {4 B c d^{4} x^{2} \sqrt {d + e x}}{1001 e^{2}} + \frac {10 B c d^{3} x^{3} \sqrt {d + e x}}{3003 e} + \frac {106 B c d^{2} x^{4} \sqrt {d + e x}}{429} + \frac {54 B c d e x^{5} \sqrt {d + e x}}{143} + \frac {2 B c e^{2} x^{6} \sqrt {d + e x}}{13} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (\frac {A b x^{2}}{2} + \frac {A c x^{3}}{3} + \frac {B b x^{3}}{3} + \frac {B c x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]
Piecewise((-4*A*b*d**4*sqrt(d + e*x)/(63*e**2) + 2*A*b*d**3*x*sqrt(d + e*x )/(63*e) + 10*A*b*d**2*x**2*sqrt(d + e*x)/21 + 38*A*b*d*e*x**3*sqrt(d + e* x)/63 + 2*A*b*e**2*x**4*sqrt(d + e*x)/9 + 16*A*c*d**5*sqrt(d + e*x)/(693*e **3) - 8*A*c*d**4*x*sqrt(d + e*x)/(693*e**2) + 2*A*c*d**3*x**2*sqrt(d + e* x)/(231*e) + 226*A*c*d**2*x**3*sqrt(d + e*x)/693 + 46*A*c*d*e*x**4*sqrt(d + e*x)/99 + 2*A*c*e**2*x**5*sqrt(d + e*x)/11 + 16*B*b*d**5*sqrt(d + e*x)/( 693*e**3) - 8*B*b*d**4*x*sqrt(d + e*x)/(693*e**2) + 2*B*b*d**3*x**2*sqrt(d + e*x)/(231*e) + 226*B*b*d**2*x**3*sqrt(d + e*x)/693 + 46*B*b*d*e*x**4*sq rt(d + e*x)/99 + 2*B*b*e**2*x**5*sqrt(d + e*x)/11 - 32*B*c*d**6*sqrt(d + e *x)/(3003*e**4) + 16*B*c*d**5*x*sqrt(d + e*x)/(3003*e**3) - 4*B*c*d**4*x** 2*sqrt(d + e*x)/(1001*e**2) + 10*B*c*d**3*x**3*sqrt(d + e*x)/(3003*e) + 10 6*B*c*d**2*x**4*sqrt(d + e*x)/429 + 54*B*c*d*e*x**5*sqrt(d + e*x)/143 + 2* B*c*e**2*x**6*sqrt(d + e*x)/13, Ne(e, 0)), (d**(5/2)*(A*b*x**2/2 + A*c*x** 3/3 + B*b*x**3/3 + B*c*x**4/4), True))
Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.89 \[ \int (A+B x) (d+e x)^{5/2} \left (b x+c x^2\right ) \, dx=\frac {2 \, {\left (693 \, {\left (e x + d\right )}^{\frac {13}{2}} B c - 819 \, {\left (3 \, B c d - {\left (B b + A c\right )} e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 1001 \, {\left (3 \, B c d^{2} + A b e^{2} - 2 \, {\left (B b + A c\right )} d e\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 1287 \, {\left (B c d^{3} + A b d e^{2} - {\left (B b + A c\right )} d^{2} e\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{9009 \, e^{4}} \]
2/9009*(693*(e*x + d)^(13/2)*B*c - 819*(3*B*c*d - (B*b + A*c)*e)*(e*x + d) ^(11/2) + 1001*(3*B*c*d^2 + A*b*e^2 - 2*(B*b + A*c)*d*e)*(e*x + d)^(9/2) - 1287*(B*c*d^3 + A*b*d*e^2 - (B*b + A*c)*d^2*e)*(e*x + d)^(7/2))/e^4
Leaf count of result is larger than twice the leaf count of optimal. 944 vs. \(2 (110) = 220\).
Time = 0.28 (sec) , antiderivative size = 944, normalized size of antiderivative = 7.49 \[ \int (A+B x) (d+e x)^{5/2} \left (b x+c x^2\right ) \, dx=\text {Too large to display} \]
2/45045*(15015*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*A*b*d^3/e + 3003*(3*( e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*B*b*d^3/e^2 + 3003*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A *c*d^3/e^2 + 9009*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*b*d^2/e + 1287*(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*c*d^3/e^3 + 3861*(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*b*d^2/e^2 + 3861*(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*c*d^2/e^2 + 3861*(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*d/e + 429*(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)*B*c*d^2/e ^3 + 429*(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)*B*b*d/e^2 + 429*(3 5*(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*c*d/e^2 + 143*(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/e + 195*(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 + 11 55*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*B*c*d/e^3 + 65*(63*(e*x...
Time = 10.32 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.88 \[ \int (A+B x) (d+e x)^{5/2} \left (b x+c x^2\right ) \, dx=\frac {{\left (d+e\,x\right )}^{9/2}\,\left (2\,A\,b\,e^2+6\,B\,c\,d^2-4\,A\,c\,d\,e-4\,B\,b\,d\,e\right )}{9\,e^4}+\frac {{\left (d+e\,x\right )}^{11/2}\,\left (2\,A\,c\,e+2\,B\,b\,e-6\,B\,c\,d\right )}{11\,e^4}+\frac {2\,B\,c\,{\left (d+e\,x\right )}^{13/2}}{13\,e^4}-\frac {2\,d\,\left (A\,e-B\,d\right )\,\left (b\,e-c\,d\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^4} \]