Integrand size = 24, antiderivative size = 126 \[ \int (A+B x) (d+e x)^{7/2} \left (b x+c x^2\right ) \, dx=-\frac {2 d (B d-A e) (c d-b e) (d+e x)^{9/2}}{9 e^4}+\frac {2 (B d (3 c d-2 b e)-A e (2 c d-b e)) (d+e x)^{11/2}}{11 e^4}-\frac {2 (3 B c d-b B e-A c e) (d+e x)^{13/2}}{13 e^4}+\frac {2 B c (d+e x)^{15/2}}{15 e^4} \] Output:
-2/9*d*(-A*e+B*d)*(-b*e+c*d)*(e*x+d)^(9/2)/e^4+2/11*(B*d*(-2*b*e+3*c*d)-A* e*(-b*e+2*c*d))*(e*x+d)^(11/2)/e^4-2/13*(-A*c*e-B*b*e+3*B*c*d)*(e*x+d)^(13 /2)/e^4+2/15*B*c*(e*x+d)^(15/2)/e^4
Time = 0.17 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.90 \[ \int (A+B x) (d+e x)^{7/2} \left (b x+c x^2\right ) \, dx=\frac {2 (d+e x)^{9/2} \left (5 A e \left (13 b e (-2 d+9 e x)+c \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )+B \left (5 b e \left (8 d^2-36 d e x+99 e^2 x^2\right )+c \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )\right )\right )}{6435 e^4} \] Input:
Integrate[(A + B*x)*(d + e*x)^(7/2)*(b*x + c*x^2),x]
Output:
(2*(d + e*x)^(9/2)*(5*A*e*(13*b*e*(-2*d + 9*e*x) + c*(8*d^2 - 36*d*e*x + 9 9*e^2*x^2)) + B*(5*b*e*(8*d^2 - 36*d*e*x + 99*e^2*x^2) + c*(-16*d^3 + 72*d ^2*e*x - 198*d*e^2*x^2 + 429*e^3*x^3))))/(6435*e^4)
Time = 0.49 (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.
| \(\displaystyle \int (A+B x) \left (b x+c x^2\right ) (d+e x)^{7/2} \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {(d+e x)^{11/2} (A c e+b B e-3 B c d)}{e^3}+\frac {(d+e x)^{9/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{e^3}-\frac {d (d+e x)^{7/2} (B d-A e) (c d-b e)}{e^3}+\frac {B c (d+e x)^{13/2}}{e^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 (d+e x)^{13/2} (-A c e-b B e+3 B c d)}{13 e^4}+\frac {2 (d+e x)^{11/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{11 e^4}-\frac {2 d (d+e x)^{9/2} (B d-A e) (c d-b e)}{9 e^4}+\frac {2 B c (d+e x)^{15/2}}{15 e^4}\) |
Input:
Int[(A + B*x)*(d + e*x)^(7/2)*(b*x + c*x^2),x]
Output:
(-2*d*(B*d - A*e)*(c*d - b*e)*(d + e*x)^(9/2))/(9*e^4) + (2*(B*d*(3*c*d - 2*b*e) - A*e*(2*c*d - b*e))*(d + e*x)^(11/2))/(11*e^4) - (2*(3*B*c*d - b*B *e - A*c*e)*(d + e*x)^(13/2))/(13*e^4) + (2*B*c*(d + e*x)^(15/2))/(15*e^4)
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 = 1.98 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.73
| method | result | size |
| pseudoelliptic | \(-\frac {4 \left (e x +d \right )^{\frac {9}{2}} \left (-\frac {9 \left (\frac {11 B c \,x^{2}}{15}+\frac {11 \left (A c +B b \right ) x}{13}+A b \right ) x \,e^{3}}{2}+d \left (\frac {99 B c \,x^{2}}{65}+\frac {18 \left (A c +B b \right ) x}{13}+A b \right ) e^{2}-\frac {4 d^{2} \left (\frac {9}{5} B c x +A c +B b \right ) e}{13}+\frac {8 B c \,d^{3}}{65}\right )}{99 e^{4}}\) | \(92\) |
| default | \(\frac {\frac {2 B c \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (A e -2 B d \right ) c +B \left (b e -c d \right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (-A e +B d \right ) d c +\left (A e -2 B d \right ) \left (b e -c d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (-A e +B d \right ) d \left (b e -c d \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{4}}\) | \(112\) |
| derivativedivides | \(\frac {\frac {2 B c \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (A e -2 B d \right ) c +B \left (b e -c d \right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (-\left (A e -B d \right ) d c +\left (A e -2 B d \right ) \left (b e -c d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}-\frac {2 \left (A e -B d \right ) d \left (b e -c d \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{4}}\) | \(113\) |
| gosper | \(-\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (-429 B c \,x^{3} e^{3}-495 A c \,e^{3} x^{2}-495 B b \,e^{3} x^{2}+198 B c d \,e^{2} x^{2}-585 A x b \,e^{3}+180 A x c d \,e^{2}+180 B x b d \,e^{2}-72 B x c \,d^{2} e +130 A b d \,e^{2}-40 A c \,d^{2} e -40 B b \,d^{2} e +16 B c \,d^{3}\right )}{6435 e^{4}}\) | \(121\) |
| orering | \(-\frac {2 \left (-429 B c \,x^{3} e^{3}-495 A c \,e^{3} x^{2}-495 B b \,e^{3} x^{2}+198 B c d \,e^{2} x^{2}-585 A x b \,e^{3}+180 A x c d \,e^{2}+180 B x b d \,e^{2}-72 B x c \,d^{2} e +130 A b d \,e^{2}-40 A c \,d^{2} e -40 B b \,d^{2} e +16 B c \,d^{3}\right ) \left (e x +d \right )^{\frac {9}{2}} \left (c \,x^{2}+b x \right )}{6435 e^{4} x \left (c x +b \right )}\) | \(140\) |
| trager | \(-\frac {2 \left (-429 B \,e^{7} c \,x^{7}-495 A c \,e^{7} x^{6}-495 B b \,e^{7} x^{6}-1518 B c d \,e^{6} x^{6}-585 A b \,e^{7} x^{5}-1800 A c d \,e^{6} x^{5}-1800 B b d \,e^{6} x^{5}-1854 B c \,d^{2} e^{5} x^{5}-2210 A b d \,e^{6} x^{4}-2290 A c \,d^{2} e^{5} x^{4}-2290 B b \,d^{2} e^{5} x^{4}-800 B c \,d^{3} e^{4} x^{4}-2990 A b \,d^{2} e^{5} x^{3}-1060 A c \,d^{3} e^{4} x^{3}-1060 B b \,d^{3} e^{4} x^{3}-5 B c \,d^{4} e^{3} x^{3}-1560 A b \,d^{3} e^{4} x^{2}-15 A c \,d^{4} e^{3} x^{2}-15 B b \,d^{4} e^{3} x^{2}+6 B c \,d^{5} e^{2} x^{2}-65 A b \,d^{4} e^{3} x +20 A c \,d^{5} e^{2} x +20 B b \,d^{5} e^{2} x -8 B c \,d^{6} e x +130 A b \,d^{5} e^{2}-40 A c \,d^{6} e -40 B b \,d^{6} e +16 B c \,d^{7}\right ) \sqrt {e x +d}}{6435 e^{4}}\) | \(329\) |
| risch | \(-\frac {2 \left (-429 B \,e^{7} c \,x^{7}-495 A c \,e^{7} x^{6}-495 B b \,e^{7} x^{6}-1518 B c d \,e^{6} x^{6}-585 A b \,e^{7} x^{5}-1800 A c d \,e^{6} x^{5}-1800 B b d \,e^{6} x^{5}-1854 B c \,d^{2} e^{5} x^{5}-2210 A b d \,e^{6} x^{4}-2290 A c \,d^{2} e^{5} x^{4}-2290 B b \,d^{2} e^{5} x^{4}-800 B c \,d^{3} e^{4} x^{4}-2990 A b \,d^{2} e^{5} x^{3}-1060 A c \,d^{3} e^{4} x^{3}-1060 B b \,d^{3} e^{4} x^{3}-5 B c \,d^{4} e^{3} x^{3}-1560 A b \,d^{3} e^{4} x^{2}-15 A c \,d^{4} e^{3} x^{2}-15 B b \,d^{4} e^{3} x^{2}+6 B c \,d^{5} e^{2} x^{2}-65 A b \,d^{4} e^{3} x +20 A c \,d^{5} e^{2} x +20 B b \,d^{5} e^{2} x -8 B c \,d^{6} e x +130 A b \,d^{5} e^{2}-40 A c \,d^{6} e -40 B b \,d^{6} e +16 B c \,d^{7}\right ) \sqrt {e x +d}}{6435 e^{4}}\) | \(329\) |
Input:
int((B*x+A)*(e*x+d)^(7/2)*(c*x^2+b*x),x,method=_RETURNVERBOSE)
Output:
-4/99*(e*x+d)^(9/2)*(-9/2*(11/15*B*c*x^2+11/13*(A*c+B*b)*x+A*b)*x*e^3+d*(9 9/65*B*c*x^2+18/13*(A*c+B*b)*x+A*b)*e^2-4/13*d^2*(9/5*B*c*x+A*c+B*b)*e+8/6 5*B*c*d^3)/e^4
Leaf count of result is larger than twice the leaf count of optimal. 271 vs. \(2 (110) = 220\).
Time = 0.08 (sec) , antiderivative size = 271, normalized size of antiderivative = 2.15 \[ \int (A+B x) (d+e x)^{7/2} \left (b x+c x^2\right ) \, dx=\frac {2 \, {\left (429 \, B c e^{7} x^{7} - 16 \, B c d^{7} - 130 \, A b d^{5} e^{2} + 40 \, {\left (B b + A c\right )} d^{6} e + 33 \, {\left (46 \, B c d e^{6} + 15 \, {\left (B b + A c\right )} e^{7}\right )} x^{6} + 9 \, {\left (206 \, B c d^{2} e^{5} + 65 \, A b e^{7} + 200 \, {\left (B b + A c\right )} d e^{6}\right )} x^{5} + 10 \, {\left (80 \, B c d^{3} e^{4} + 221 \, A b d e^{6} + 229 \, {\left (B b + A c\right )} d^{2} e^{5}\right )} x^{4} + 5 \, {\left (B c d^{4} e^{3} + 598 \, A b d^{2} e^{5} + 212 \, {\left (B b + A c\right )} d^{3} e^{4}\right )} x^{3} - 3 \, {\left (2 \, B c d^{5} e^{2} - 520 \, A b d^{3} e^{4} - 5 \, {\left (B b + A c\right )} d^{4} e^{3}\right )} x^{2} + {\left (8 \, B c d^{6} e + 65 \, A b d^{4} e^{3} - 20 \, {\left (B b + A c\right )} d^{5} e^{2}\right )} x\right )} \sqrt {e x + d}}{6435 \, e^{4}} \] Input:
integrate((B*x+A)*(e*x+d)^(7/2)*(c*x^2+b*x),x, algorithm="fricas")
Output:
2/6435*(429*B*c*e^7*x^7 - 16*B*c*d^7 - 130*A*b*d^5*e^2 + 40*(B*b + A*c)*d^ 6*e + 33*(46*B*c*d*e^6 + 15*(B*b + A*c)*e^7)*x^6 + 9*(206*B*c*d^2*e^5 + 65 *A*b*e^7 + 200*(B*b + A*c)*d*e^6)*x^5 + 10*(80*B*c*d^3*e^4 + 221*A*b*d*e^6 + 229*(B*b + A*c)*d^2*e^5)*x^4 + 5*(B*c*d^4*e^3 + 598*A*b*d^2*e^5 + 212*( B*b + A*c)*d^3*e^4)*x^3 - 3*(2*B*c*d^5*e^2 - 520*A*b*d^3*e^4 - 5*(B*b + A* c)*d^4*e^3)*x^2 + (8*B*c*d^6*e + 65*A*b*d^4*e^3 - 20*(B*b + A*c)*d^5*e^2)* x)*sqrt(e*x + d)/e^4
Leaf count of result is larger than twice the leaf count of optimal. 683 vs. \(2 (129) = 258\).
Time = 0.72 (sec) , antiderivative size = 683, normalized size of antiderivative = 5.42 \[ \int (A+B x) (d+e x)^{7/2} \left (b x+c x^2\right ) \, dx=\begin {cases} - \frac {4 A b d^{5} \sqrt {d + e x}}{99 e^{2}} + \frac {2 A b d^{4} x \sqrt {d + e x}}{99 e} + \frac {16 A b d^{3} x^{2} \sqrt {d + e x}}{33} + \frac {92 A b d^{2} e x^{3} \sqrt {d + e x}}{99} + \frac {68 A b d e^{2} x^{4} \sqrt {d + e x}}{99} + \frac {2 A b e^{3} x^{5} \sqrt {d + e x}}{11} + \frac {16 A c d^{6} \sqrt {d + e x}}{1287 e^{3}} - \frac {8 A c d^{5} x \sqrt {d + e x}}{1287 e^{2}} + \frac {2 A c d^{4} x^{2} \sqrt {d + e x}}{429 e} + \frac {424 A c d^{3} x^{3} \sqrt {d + e x}}{1287} + \frac {916 A c d^{2} e x^{4} \sqrt {d + e x}}{1287} + \frac {80 A c d e^{2} x^{5} \sqrt {d + e x}}{143} + \frac {2 A c e^{3} x^{6} \sqrt {d + e x}}{13} + \frac {16 B b d^{6} \sqrt {d + e x}}{1287 e^{3}} - \frac {8 B b d^{5} x \sqrt {d + e x}}{1287 e^{2}} + \frac {2 B b d^{4} x^{2} \sqrt {d + e x}}{429 e} + \frac {424 B b d^{3} x^{3} \sqrt {d + e x}}{1287} + \frac {916 B b d^{2} e x^{4} \sqrt {d + e x}}{1287} + \frac {80 B b d e^{2} x^{5} \sqrt {d + e x}}{143} + \frac {2 B b e^{3} x^{6} \sqrt {d + e x}}{13} - \frac {32 B c d^{7} \sqrt {d + e x}}{6435 e^{4}} + \frac {16 B c d^{6} x \sqrt {d + e x}}{6435 e^{3}} - \frac {4 B c d^{5} x^{2} \sqrt {d + e x}}{2145 e^{2}} + \frac {2 B c d^{4} x^{3} \sqrt {d + e x}}{1287 e} + \frac {320 B c d^{3} x^{4} \sqrt {d + e x}}{1287} + \frac {412 B c d^{2} e x^{5} \sqrt {d + e x}}{715} + \frac {92 B c d e^{2} x^{6} \sqrt {d + e x}}{195} + \frac {2 B c e^{3} x^{7} \sqrt {d + e x}}{15} & \text {for}\: e \neq 0 \\d^{\frac {7}{2}} \left (\frac {A b x^{2}}{2} + \frac {A c x^{3}}{3} + \frac {B b x^{3}}{3} + \frac {B c x^{4}}{4}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((B*x+A)*(e*x+d)**(7/2)*(c*x**2+b*x),x)
Output:
Piecewise((-4*A*b*d**5*sqrt(d + e*x)/(99*e**2) + 2*A*b*d**4*x*sqrt(d + e*x )/(99*e) + 16*A*b*d**3*x**2*sqrt(d + e*x)/33 + 92*A*b*d**2*e*x**3*sqrt(d + e*x)/99 + 68*A*b*d*e**2*x**4*sqrt(d + e*x)/99 + 2*A*b*e**3*x**5*sqrt(d + e*x)/11 + 16*A*c*d**6*sqrt(d + e*x)/(1287*e**3) - 8*A*c*d**5*x*sqrt(d + e* x)/(1287*e**2) + 2*A*c*d**4*x**2*sqrt(d + e*x)/(429*e) + 424*A*c*d**3*x**3 *sqrt(d + e*x)/1287 + 916*A*c*d**2*e*x**4*sqrt(d + e*x)/1287 + 80*A*c*d*e* *2*x**5*sqrt(d + e*x)/143 + 2*A*c*e**3*x**6*sqrt(d + e*x)/13 + 16*B*b*d**6 *sqrt(d + e*x)/(1287*e**3) - 8*B*b*d**5*x*sqrt(d + e*x)/(1287*e**2) + 2*B* b*d**4*x**2*sqrt(d + e*x)/(429*e) + 424*B*b*d**3*x**3*sqrt(d + e*x)/1287 + 916*B*b*d**2*e*x**4*sqrt(d + e*x)/1287 + 80*B*b*d*e**2*x**5*sqrt(d + e*x) /143 + 2*B*b*e**3*x**6*sqrt(d + e*x)/13 - 32*B*c*d**7*sqrt(d + e*x)/(6435* e**4) + 16*B*c*d**6*x*sqrt(d + e*x)/(6435*e**3) - 4*B*c*d**5*x**2*sqrt(d + e*x)/(2145*e**2) + 2*B*c*d**4*x**3*sqrt(d + e*x)/(1287*e) + 320*B*c*d**3* x**4*sqrt(d + e*x)/1287 + 412*B*c*d**2*e*x**5*sqrt(d + e*x)/715 + 92*B*c*d *e**2*x**6*sqrt(d + e*x)/195 + 2*B*c*e**3*x**7*sqrt(d + e*x)/15, Ne(e, 0)) , (d**(7/2)*(A*b*x**2/2 + A*c*x**3/3 + B*b*x**3/3 + B*c*x**4/4), True))
Time = 0.03 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.89 \[ \int (A+B x) (d+e x)^{7/2} \left (b x+c x^2\right ) \, dx=\frac {2 \, {\left (429 \, {\left (e x + d\right )}^{\frac {15}{2}} B c - 495 \, {\left (3 \, B c d - {\left (B b + A c\right )} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 585 \, {\left (3 \, B c d^{2} + A b e^{2} - 2 \, {\left (B b + A c\right )} d e\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 715 \, {\left (B c d^{3} + A b d e^{2} - {\left (B b + A c\right )} d^{2} e\right )} {\left (e x + d\right )}^{\frac {9}{2}}\right )}}{6435 \, e^{4}} \] Input:
integrate((B*x+A)*(e*x+d)^(7/2)*(c*x^2+b*x),x, algorithm="maxima")
Output:
2/6435*(429*(e*x + d)^(15/2)*B*c - 495*(3*B*c*d - (B*b + A*c)*e)*(e*x + d) ^(13/2) + 585*(3*B*c*d^2 + A*b*e^2 - 2*(B*b + A*c)*d*e)*(e*x + d)^(11/2) - 715*(B*c*d^3 + A*b*d*e^2 - (B*b + A*c)*d^2*e)*(e*x + d)^(9/2))/e^4
Leaf count of result is larger than twice the leaf count of optimal. 1304 vs. \(2 (110) = 220\).
Time = 0.25 (sec) , antiderivative size = 1304, normalized size of antiderivative = 10.35 \[ \int (A+B x) (d+e x)^{7/2} \left (b x+c x^2\right ) \, dx=\text {Too large to display} \] Input:
integrate((B*x+A)*(e*x+d)^(7/2)*(c*x^2+b*x),x, algorithm="giac")
Output:
2/45045*(15015*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*A*b*d^4/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^4/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^4/e^2 + 12012*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*A*b*d^3/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^4/e^3 + 5148*(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^3/e^2 + 5148*(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^3/e^2 + 7722*(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^2/e + 572*(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^ 3/e^3 + 858*(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^2/e^2 + 8 58*(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*c*d^2/e^2 + 572*(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*d/e + 390*(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*c*d^2/e^3 + ...
Time = 0.09 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.88 \[ \int (A+B x) (d+e x)^{7/2} \left (b x+c x^2\right ) \, dx=\frac {{\left (d+e\,x\right )}^{11/2}\,\left (2\,A\,b\,e^2+6\,B\,c\,d^2-4\,A\,c\,d\,e-4\,B\,b\,d\,e\right )}{11\,e^4}+\frac {{\left (d+e\,x\right )}^{13/2}\,\left (2\,A\,c\,e+2\,B\,b\,e-6\,B\,c\,d\right )}{13\,e^4}+\frac {2\,B\,c\,{\left (d+e\,x\right )}^{15/2}}{15\,e^4}-\frac {2\,d\,\left (A\,e-B\,d\right )\,\left (b\,e-c\,d\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^4} \] Input:
int((b*x + c*x^2)*(A + B*x)*(d + e*x)^(7/2),x)
Output:
((d + e*x)^(11/2)*(2*A*b*e^2 + 6*B*c*d^2 - 4*A*c*d*e - 4*B*b*d*e))/(11*e^4 ) + ((d + e*x)^(13/2)*(2*A*c*e + 2*B*b*e - 6*B*c*d))/(13*e^4) + (2*B*c*(d + e*x)^(15/2))/(15*e^4) - (2*d*(A*e - B*d)*(b*e - c*d)*(d + e*x)^(9/2))/(9 *e^4)
Time = 0.25 (sec) , antiderivative size = 334, normalized size of antiderivative = 2.65 \[ \int (A+B x) (d+e x)^{7/2} \left (b x+c x^2\right ) \, dx=\frac {2 \sqrt {e x +d}\, \left (429 b c \,e^{7} x^{7}+495 a c \,e^{7} x^{6}+495 b^{2} e^{7} x^{6}+1518 b c d \,e^{6} x^{6}+585 a b \,e^{7} x^{5}+1800 a c d \,e^{6} x^{5}+1800 b^{2} d \,e^{6} x^{5}+1854 b c \,d^{2} e^{5} x^{5}+2210 a b d \,e^{6} x^{4}+2290 a c \,d^{2} e^{5} x^{4}+2290 b^{2} d^{2} e^{5} x^{4}+800 b c \,d^{3} e^{4} x^{4}+2990 a b \,d^{2} e^{5} x^{3}+1060 a c \,d^{3} e^{4} x^{3}+1060 b^{2} d^{3} e^{4} x^{3}+5 b c \,d^{4} e^{3} x^{3}+1560 a b \,d^{3} e^{4} x^{2}+15 a c \,d^{4} e^{3} x^{2}+15 b^{2} d^{4} e^{3} x^{2}-6 b c \,d^{5} e^{2} x^{2}+65 a b \,d^{4} e^{3} x -20 a c \,d^{5} e^{2} x -20 b^{2} d^{5} e^{2} x +8 b c \,d^{6} e x -130 a b \,d^{5} e^{2}+40 a c \,d^{6} e +40 b^{2} d^{6} e -16 b c \,d^{7}\right )}{6435 e^{4}} \] Input:
int((B*x+A)*(e*x+d)^(7/2)*(c*x^2+b*x),x)
Output:
(2*sqrt(d + e*x)*( - 130*a*b*d**5*e**2 + 65*a*b*d**4*e**3*x + 1560*a*b*d** 3*e**4*x**2 + 2990*a*b*d**2*e**5*x**3 + 2210*a*b*d*e**6*x**4 + 585*a*b*e** 7*x**5 + 40*a*c*d**6*e - 20*a*c*d**5*e**2*x + 15*a*c*d**4*e**3*x**2 + 1060 *a*c*d**3*e**4*x**3 + 2290*a*c*d**2*e**5*x**4 + 1800*a*c*d*e**6*x**5 + 495 *a*c*e**7*x**6 + 40*b**2*d**6*e - 20*b**2*d**5*e**2*x + 15*b**2*d**4*e**3* x**2 + 1060*b**2*d**3*e**4*x**3 + 2290*b**2*d**2*e**5*x**4 + 1800*b**2*d*e **6*x**5 + 495*b**2*e**7*x**6 - 16*b*c*d**7 + 8*b*c*d**6*e*x - 6*b*c*d**5* e**2*x**2 + 5*b*c*d**4*e**3*x**3 + 800*b*c*d**3*e**4*x**4 + 1854*b*c*d**2* e**5*x**5 + 1518*b*c*d*e**6*x**6 + 429*b*c*e**7*x**7))/(6435*e**4)