3.11.79 \(\int (A+B x) \sqrt {d+e x} (b x+c x^2)^2 \, dx\)

Optimal. Leaf size=267 \[ -\frac {2 (d+e x)^{9/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{9 e^6}+\frac {2 (d+e x)^{7/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{7 e^6}-\frac {2 d^2 (d+e x)^{3/2} (B d-A e) (c d-b e)^2}{3 e^6}-\frac {2 c (d+e x)^{11/2} (-A c e-2 b B e+5 B c d)}{11 e^6}+\frac {2 d (d+e x)^{5/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{5 e^6}+\frac {2 B c^2 (d+e x)^{13/2}}{13 e^6} \]

________________________________________________________________________________________

Rubi [A]  time = 0.15, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {771} \begin {gather*} -\frac {2 (d+e x)^{9/2} \left (2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{9 e^6}+\frac {2 (d+e x)^{7/2} \left (A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )\right )}{7 e^6}-\frac {2 d^2 (d+e x)^{3/2} (B d-A e) (c d-b e)^2}{3 e^6}-\frac {2 c (d+e x)^{11/2} (-A c e-2 b B e+5 B c d)}{11 e^6}+\frac {2 d (d+e x)^{5/2} (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{5 e^6}+\frac {2 B c^2 (d+e x)^{13/2}}{13 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*Sqrt[d + e*x]*(b*x + c*x^2)^2,x]

[Out]

(-2*d^2*(B*d - A*e)*(c*d - b*e)^2*(d + e*x)^(3/2))/(3*e^6) + (2*d*(c*d - b*e)*(B*d*(5*c*d - 3*b*e) - 2*A*e*(2*
c*d - b*e))*(d + e*x)^(5/2))/(5*e^6) + (2*(A*e*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*d*
e + 3*b^2*e^2))*(d + e*x)^(7/2))/(7*e^6) - (2*(2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2))*(
d + e*x)^(9/2))/(9*e^6) - (2*c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^(11/2))/(11*e^6) + (2*B*c^2*(d + e*x)^(13
/2))/(13*e^6)

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int (A+B x) \sqrt {d+e x} \left (b x+c x^2\right )^2 \, dx &=\int \left (-\frac {d^2 (B d-A e) (c d-b e)^2 \sqrt {d+e x}}{e^5}+\frac {d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) (d+e x)^{3/2}}{e^5}+\frac {\left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{5/2}}{e^5}+\frac {\left (-2 A c e (2 c d-b e)+B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{7/2}}{e^5}+\frac {c (-5 B c d+2 b B e+A c e) (d+e x)^{9/2}}{e^5}+\frac {B c^2 (d+e x)^{11/2}}{e^5}\right ) \, dx\\ &=-\frac {2 d^2 (B d-A e) (c d-b e)^2 (d+e x)^{3/2}}{3 e^6}+\frac {2 d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e)) (d+e x)^{5/2}}{5 e^6}+\frac {2 \left (A e \left (6 c^2 d^2-6 b c d e+b^2 e^2\right )-B d \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) (d+e x)^{7/2}}{7 e^6}-\frac {2 \left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2-8 b c d e+b^2 e^2\right )\right ) (d+e x)^{9/2}}{9 e^6}-\frac {2 c (5 B c d-2 b B e-A c e) (d+e x)^{11/2}}{11 e^6}+\frac {2 B c^2 (d+e x)^{13/2}}{13 e^6}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.18, size = 273, normalized size = 1.02 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (13 A e \left (33 b^2 e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+22 b c e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+c^2 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )+B \left (143 b^2 e^2 \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+26 b c e \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )-5 c^2 \left (256 d^5-384 d^4 e x+480 d^3 e^2 x^2-560 d^2 e^3 x^3+630 d e^4 x^4-693 e^5 x^5\right )\right )\right )}{45045 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*Sqrt[d + e*x]*(b*x + c*x^2)^2,x]

[Out]

(2*(d + e*x)^(3/2)*(13*A*e*(33*b^2*e^2*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + 22*b*c*e*(-16*d^3 + 24*d^2*e*x - 30*d
*e^2*x^2 + 35*e^3*x^3) + c^2*(128*d^4 - 192*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4)) + B*(143
*b^2*e^2*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3) + 26*b*c*e*(128*d^4 - 192*d^3*e*x + 240*d^2*e^2*x^
2 - 280*d*e^3*x^3 + 315*e^4*x^4) - 5*c^2*(256*d^5 - 384*d^4*e*x + 480*d^3*e^2*x^2 - 560*d^2*e^3*x^3 + 630*d*e^
4*x^4 - 693*e^5*x^5))))/(45045*e^6)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.17, size = 399, normalized size = 1.49 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (15015 A b^2 d^2 e^3-18018 A b^2 d e^3 (d+e x)+6435 A b^2 e^3 (d+e x)^2-30030 A b c d^3 e^2+54054 A b c d^2 e^2 (d+e x)-38610 A b c d e^2 (d+e x)^2+10010 A b c e^2 (d+e x)^3+15015 A c^2 d^4 e-36036 A c^2 d^3 e (d+e x)+38610 A c^2 d^2 e (d+e x)^2-20020 A c^2 d e (d+e x)^3+4095 A c^2 e (d+e x)^4-15015 b^2 B d^3 e^2+27027 b^2 B d^2 e^2 (d+e x)-19305 b^2 B d e^2 (d+e x)^2+5005 b^2 B e^2 (d+e x)^3+30030 b B c d^4 e-72072 b B c d^3 e (d+e x)+77220 b B c d^2 e (d+e x)^2-40040 b B c d e (d+e x)^3+8190 b B c e (d+e x)^4-15015 B c^2 d^5+45045 B c^2 d^4 (d+e x)-64350 B c^2 d^3 (d+e x)^2+50050 B c^2 d^2 (d+e x)^3-20475 B c^2 d (d+e x)^4+3465 B c^2 (d+e x)^5\right )}{45045 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x)*Sqrt[d + e*x]*(b*x + c*x^2)^2,x]

[Out]

(2*(d + e*x)^(3/2)*(-15015*B*c^2*d^5 + 30030*b*B*c*d^4*e + 15015*A*c^2*d^4*e - 15015*b^2*B*d^3*e^2 - 30030*A*b
*c*d^3*e^2 + 15015*A*b^2*d^2*e^3 + 45045*B*c^2*d^4*(d + e*x) - 72072*b*B*c*d^3*e*(d + e*x) - 36036*A*c^2*d^3*e
*(d + e*x) + 27027*b^2*B*d^2*e^2*(d + e*x) + 54054*A*b*c*d^2*e^2*(d + e*x) - 18018*A*b^2*d*e^3*(d + e*x) - 643
50*B*c^2*d^3*(d + e*x)^2 + 77220*b*B*c*d^2*e*(d + e*x)^2 + 38610*A*c^2*d^2*e*(d + e*x)^2 - 19305*b^2*B*d*e^2*(
d + e*x)^2 - 38610*A*b*c*d*e^2*(d + e*x)^2 + 6435*A*b^2*e^3*(d + e*x)^2 + 50050*B*c^2*d^2*(d + e*x)^3 - 40040*
b*B*c*d*e*(d + e*x)^3 - 20020*A*c^2*d*e*(d + e*x)^3 + 5005*b^2*B*e^2*(d + e*x)^3 + 10010*A*b*c*e^2*(d + e*x)^3
- 20475*B*c^2*d*(d + e*x)^4 + 8190*b*B*c*e*(d + e*x)^4 + 4095*A*c^2*e*(d + e*x)^4 + 3465*B*c^2*(d + e*x)^5))/
(45045*e^6)

________________________________________________________________________________________

fricas [A]  time = 0.43, size = 357, normalized size = 1.34 \begin {gather*} \frac {2 \, {\left (3465 \, B c^{2} e^{6} x^{6} - 1280 \, B c^{2} d^{6} + 3432 \, A b^{2} d^{3} e^{3} + 1664 \, {\left (2 \, B b c + A c^{2}\right )} d^{5} e - 2288 \, {\left (B b^{2} + 2 \, A b c\right )} d^{4} e^{2} + 315 \, {\left (B c^{2} d e^{5} + 13 \, {\left (2 \, B b c + A c^{2}\right )} e^{6}\right )} x^{5} - 35 \, {\left (10 \, B c^{2} d^{2} e^{4} - 13 \, {\left (2 \, B b c + A c^{2}\right )} d e^{5} - 143 \, {\left (B b^{2} + 2 \, A b c\right )} e^{6}\right )} x^{4} + 5 \, {\left (80 \, B c^{2} d^{3} e^{3} + 1287 \, A b^{2} e^{6} - 104 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{4} + 143 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{5}\right )} x^{3} - 3 \, {\left (160 \, B c^{2} d^{4} e^{2} - 429 \, A b^{2} d e^{5} - 208 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{3} + 286 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{4}\right )} x^{2} + 4 \, {\left (160 \, B c^{2} d^{5} e - 429 \, A b^{2} d^{2} e^{4} - 208 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e^{2} + 286 \, {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{3}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(1/2)*(c*x^2+b*x)^2,x, algorithm="fricas")

[Out]

2/45045*(3465*B*c^2*e^6*x^6 - 1280*B*c^2*d^6 + 3432*A*b^2*d^3*e^3 + 1664*(2*B*b*c + A*c^2)*d^5*e - 2288*(B*b^2
+ 2*A*b*c)*d^4*e^2 + 315*(B*c^2*d*e^5 + 13*(2*B*b*c + A*c^2)*e^6)*x^5 - 35*(10*B*c^2*d^2*e^4 - 13*(2*B*b*c +
A*c^2)*d*e^5 - 143*(B*b^2 + 2*A*b*c)*e^6)*x^4 + 5*(80*B*c^2*d^3*e^3 + 1287*A*b^2*e^6 - 104*(2*B*b*c + A*c^2)*d
^2*e^4 + 143*(B*b^2 + 2*A*b*c)*d*e^5)*x^3 - 3*(160*B*c^2*d^4*e^2 - 429*A*b^2*d*e^5 - 208*(2*B*b*c + A*c^2)*d^3
*e^3 + 286*(B*b^2 + 2*A*b*c)*d^2*e^4)*x^2 + 4*(160*B*c^2*d^5*e - 429*A*b^2*d^2*e^4 - 208*(2*B*b*c + A*c^2)*d^4
*e^2 + 286*(B*b^2 + 2*A*b*c)*d^3*e^3)*x)*sqrt(e*x + d)/e^6

________________________________________________________________________________________

giac [B]  time = 0.21, size = 835, normalized size = 3.13

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(1/2)*(c*x^2+b*x)^2,x, algorithm="giac")

[Out]

2/45045*(3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*b^2*d*e^(-2) + 1287*(5*(x*e
+ d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*B*b^2*d*e^(-3) + 2574*(5*(x
*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*b*c*d*e^(-3) + 286*(35
*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e +
d)*d^4)*B*b*c*d*e^(-4) + 143*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e
+ d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*A*c^2*d*e^(-4) + 65*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990
*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*B*c^2*d*e^
(-5) + 1287*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*b^2*e
^(-2) + 143*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 +
315*sqrt(x*e + d)*d^4)*B*b^2*e^(-3) + 286*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^
2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*A*b*c*e^(-3) + 130*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(
9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5
)*B*b*c*e^(-4) + 65*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5
/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*A*c^2*e^(-4) + 15*(231*(x*e + d)^(13/2) - 1638*(x*
e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e +
d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*B*c^2*e^(-5))*e^(-1)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 341, normalized size = 1.28 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3465 B \,c^{2} x^{5} e^{5}+4095 A \,c^{2} e^{5} x^{4}+8190 B b c \,e^{5} x^{4}-3150 B \,c^{2} d \,e^{4} x^{4}+10010 A b c \,e^{5} x^{3}-3640 A \,c^{2} d \,e^{4} x^{3}+5005 B \,b^{2} e^{5} x^{3}-7280 B b c d \,e^{4} x^{3}+2800 B \,c^{2} d^{2} e^{3} x^{3}+6435 A \,b^{2} e^{5} x^{2}-8580 A b c d \,e^{4} x^{2}+3120 A \,c^{2} d^{2} e^{3} x^{2}-4290 B \,b^{2} d \,e^{4} x^{2}+6240 B b c \,d^{2} e^{3} x^{2}-2400 B \,c^{2} d^{3} e^{2} x^{2}-5148 A \,b^{2} d \,e^{4} x +6864 A b c \,d^{2} e^{3} x -2496 A \,c^{2} d^{3} e^{2} x +3432 B \,b^{2} d^{2} e^{3} x -4992 B b c \,d^{3} e^{2} x +1920 B \,c^{2} d^{4} e x +3432 A \,b^{2} d^{2} e^{3}-4576 A b c \,d^{3} e^{2}+1664 A \,c^{2} d^{4} e -2288 B \,b^{2} d^{3} e^{2}+3328 B b c \,d^{4} e -1280 B \,c^{2} d^{5}\right )}{45045 e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(1/2)*(c*x^2+b*x)^2,x)

[Out]

2/45045*(e*x+d)^(3/2)*(3465*B*c^2*e^5*x^5+4095*A*c^2*e^5*x^4+8190*B*b*c*e^5*x^4-3150*B*c^2*d*e^4*x^4+10010*A*b
*c*e^5*x^3-3640*A*c^2*d*e^4*x^3+5005*B*b^2*e^5*x^3-7280*B*b*c*d*e^4*x^3+2800*B*c^2*d^2*e^3*x^3+6435*A*b^2*e^5*
x^2-8580*A*b*c*d*e^4*x^2+3120*A*c^2*d^2*e^3*x^2-4290*B*b^2*d*e^4*x^2+6240*B*b*c*d^2*e^3*x^2-2400*B*c^2*d^3*e^2
*x^2-5148*A*b^2*d*e^4*x+6864*A*b*c*d^2*e^3*x-2496*A*c^2*d^3*e^2*x+3432*B*b^2*d^2*e^3*x-4992*B*b*c*d^3*e^2*x+19
20*B*c^2*d^4*e*x+3432*A*b^2*d^2*e^3-4576*A*b*c*d^3*e^2+1664*A*c^2*d^4*e-2288*B*b^2*d^3*e^2+3328*B*b*c*d^4*e-12
80*B*c^2*d^5)/e^6

________________________________________________________________________________________

maxima [A]  time = 0.51, size = 291, normalized size = 1.09 \begin {gather*} \frac {2 \, {\left (3465 \, {\left (e x + d\right )}^{\frac {13}{2}} B c^{2} - 4095 \, {\left (5 \, B c^{2} d - {\left (2 \, B b c + A c^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 5005 \, {\left (10 \, B c^{2} d^{2} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d e + {\left (B b^{2} + 2 \, A b c\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 6435 \, {\left (10 \, B c^{2} d^{3} - A b^{2} e^{3} - 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 9009 \, {\left (5 \, B c^{2} d^{4} - 2 \, A b^{2} d e^{3} - 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \, {\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 15015 \, {\left (B c^{2} d^{5} - A b^{2} d^{2} e^{3} - {\left (2 \, B b c + A c^{2}\right )} d^{4} e + {\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{45045 \, e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(1/2)*(c*x^2+b*x)^2,x, algorithm="maxima")

[Out]

2/45045*(3465*(e*x + d)^(13/2)*B*c^2 - 4095*(5*B*c^2*d - (2*B*b*c + A*c^2)*e)*(e*x + d)^(11/2) + 5005*(10*B*c^
2*d^2 - 4*(2*B*b*c + A*c^2)*d*e + (B*b^2 + 2*A*b*c)*e^2)*(e*x + d)^(9/2) - 6435*(10*B*c^2*d^3 - A*b^2*e^3 - 6*
(2*B*b*c + A*c^2)*d^2*e + 3*(B*b^2 + 2*A*b*c)*d*e^2)*(e*x + d)^(7/2) + 9009*(5*B*c^2*d^4 - 2*A*b^2*d*e^3 - 4*(
2*B*b*c + A*c^2)*d^3*e + 3*(B*b^2 + 2*A*b*c)*d^2*e^2)*(e*x + d)^(5/2) - 15015*(B*c^2*d^5 - A*b^2*d^2*e^3 - (2*
B*b*c + A*c^2)*d^4*e + (B*b^2 + 2*A*b*c)*d^3*e^2)*(e*x + d)^(3/2))/e^6

________________________________________________________________________________________

mupad [B]  time = 1.53, size = 254, normalized size = 0.95 \begin {gather*} \frac {{\left (d+e\,x\right )}^{11/2}\,\left (2\,A\,c^2\,e-10\,B\,c^2\,d+4\,B\,b\,c\,e\right )}{11\,e^6}+\frac {{\left (d+e\,x\right )}^{7/2}\,\left (-6\,B\,b^2\,d\,e^2+2\,A\,b^2\,e^3+24\,B\,b\,c\,d^2\,e-12\,A\,b\,c\,d\,e^2-20\,B\,c^2\,d^3+12\,A\,c^2\,d^2\,e\right )}{7\,e^6}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (2\,B\,b^2\,e^2-16\,B\,b\,c\,d\,e+4\,A\,b\,c\,e^2+20\,B\,c^2\,d^2-8\,A\,c^2\,d\,e\right )}{9\,e^6}+\frac {2\,B\,c^2\,{\left (d+e\,x\right )}^{13/2}}{13\,e^6}-\frac {2\,d\,\left (b\,e-c\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (2\,A\,b\,e^2+5\,B\,c\,d^2-4\,A\,c\,d\,e-3\,B\,b\,d\,e\right )}{5\,e^6}+\frac {2\,d^2\,\left (A\,e-B\,d\right )\,{\left (b\,e-c\,d\right )}^2\,{\left (d+e\,x\right )}^{3/2}}{3\,e^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^2*(A + B*x)*(d + e*x)^(1/2),x)

[Out]

((d + e*x)^(11/2)*(2*A*c^2*e - 10*B*c^2*d + 4*B*b*c*e))/(11*e^6) + ((d + e*x)^(7/2)*(2*A*b^2*e^3 - 20*B*c^2*d^
3 + 12*A*c^2*d^2*e - 6*B*b^2*d*e^2 - 12*A*b*c*d*e^2 + 24*B*b*c*d^2*e))/(7*e^6) + ((d + e*x)^(9/2)*(2*B*b^2*e^2
+ 20*B*c^2*d^2 + 4*A*b*c*e^2 - 8*A*c^2*d*e - 16*B*b*c*d*e))/(9*e^6) + (2*B*c^2*(d + e*x)^(13/2))/(13*e^6) - (
2*d*(b*e - c*d)*(d + e*x)^(5/2)*(2*A*b*e^2 + 5*B*c*d^2 - 4*A*c*d*e - 3*B*b*d*e))/(5*e^6) + (2*d^2*(A*e - B*d)*
(b*e - c*d)^2*(d + e*x)^(3/2))/(3*e^6)

________________________________________________________________________________________

sympy [A]  time = 7.01, size = 377, normalized size = 1.41 \begin {gather*} \frac {2 \left (\frac {B c^{2} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{5}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \left (A c^{2} e + 2 B b c e - 5 B c^{2} d\right )}{11 e^{5}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (2 A b c e^{2} - 4 A c^{2} d e + B b^{2} e^{2} - 8 B b c d e + 10 B c^{2} d^{2}\right )}{9 e^{5}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (A b^{2} e^{3} - 6 A b c d e^{2} + 6 A c^{2} d^{2} e - 3 B b^{2} d e^{2} + 12 B b c d^{2} e - 10 B c^{2} d^{3}\right )}{7 e^{5}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (- 2 A b^{2} d e^{3} + 6 A b c d^{2} e^{2} - 4 A c^{2} d^{3} e + 3 B b^{2} d^{2} e^{2} - 8 B b c d^{3} e + 5 B c^{2} d^{4}\right )}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (A b^{2} d^{2} e^{3} - 2 A b c d^{3} e^{2} + A c^{2} d^{4} e - B b^{2} d^{3} e^{2} + 2 B b c d^{4} e - B c^{2} d^{5}\right )}{3 e^{5}}\right )}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**(1/2)*(c*x**2+b*x)**2,x)

[Out]

2*(B*c**2*(d + e*x)**(13/2)/(13*e**5) + (d + e*x)**(11/2)*(A*c**2*e + 2*B*b*c*e - 5*B*c**2*d)/(11*e**5) + (d +
e*x)**(9/2)*(2*A*b*c*e**2 - 4*A*c**2*d*e + B*b**2*e**2 - 8*B*b*c*d*e + 10*B*c**2*d**2)/(9*e**5) + (d + e*x)**
(7/2)*(A*b**2*e**3 - 6*A*b*c*d*e**2 + 6*A*c**2*d**2*e - 3*B*b**2*d*e**2 + 12*B*b*c*d**2*e - 10*B*c**2*d**3)/(7
*e**5) + (d + e*x)**(5/2)*(-2*A*b**2*d*e**3 + 6*A*b*c*d**2*e**2 - 4*A*c**2*d**3*e + 3*B*b**2*d**2*e**2 - 8*B*b
*c*d**3*e + 5*B*c**2*d**4)/(5*e**5) + (d + e*x)**(3/2)*(A*b**2*d**2*e**3 - 2*A*b*c*d**3*e**2 + A*c**2*d**4*e -
B*b**2*d**3*e**2 + 2*B*b*c*d**4*e - B*c**2*d**5)/(3*e**5))/e

________________________________________________________________________________________