3.15.11 \(\int (b+2 c x) \sqrt {d+e x} (a+b x+c x^2)^2 \, dx\)

Optimal. Leaf size=252 \[ \frac {8 c (d+e x)^{9/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{9 e^6}-\frac {2 (d+e x)^{7/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{7 e^6}+\frac {4 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^6}-\frac {2 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6}-\frac {10 c^2 (d+e x)^{11/2} (2 c d-b e)}{11 e^6}+\frac {4 c^3 (d+e x)^{13/2}}{13 e^6} \]

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Rubi [A]  time = 0.12, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {771} \begin {gather*} \frac {8 c (d+e x)^{9/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{9 e^6}-\frac {2 (d+e x)^{7/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{7 e^6}+\frac {4 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^6}-\frac {2 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6}-\frac {10 c^2 (d+e x)^{11/2} (2 c d-b e)}{11 e^6}+\frac {4 c^3 (d+e x)^{13/2}}{13 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(3/2))/(3*e^6) + (4*(c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 +
b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(5/2))/(5*e^6) - (2*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d
- 3*a*e))*(d + e*x)^(7/2))/(7*e^6) + (8*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(9/2))/(9*e^6)
- (10*c^2*(2*c*d - b*e)*(d + e*x)^(11/2))/(11*e^6) + (4*c^3*(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 (b+2 c x) \sqrt {d+e x} \left (a+b x+c x^2\right )^2 \, dx &=\int \left (\frac {(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}{e^5}+\frac {2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right ) (d+e x)^{3/2}}{e^5}+\frac {(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right ) (d+e x)^{5/2}}{e^5}+\frac {4 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{7/2}}{e^5}-\frac {5 c^2 (2 c d-b e) (d+e x)^{9/2}}{e^5}+\frac {2 c^3 (d+e x)^{11/2}}{e^5}\right ) \, dx\\ &=-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}{3 e^6}+\frac {4 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{5/2}}{5 e^6}-\frac {2 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{7/2}}{7 e^6}+\frac {8 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{9/2}}{9 e^6}-\frac {10 c^2 (2 c d-b e) (d+e x)^{11/2}}{11 e^6}+\frac {4 c^3 (d+e x)^{13/2}}{13 e^6}\\ \end {align*}

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Mathematica [A]  time = 0.36, size = 291, normalized size = 1.15 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (-286 c e^2 \left (21 a^2 e^2 (2 d-3 e x)-9 a b e \left (8 d^2-12 d e x+15 e^2 x^2\right )+b^2 \left (32 d^3-48 d^2 e x+60 d e^2 x^2-70 e^3 x^3\right )\right )+429 b e^3 \left (35 a^2 e^2+14 a b e (3 e x-2 d)+b^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )+13 c^2 e \left (44 a e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+5 b \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 )-10 c^3 \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 )}{45045 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(-10*c^3*(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) + 429*b*e^3*(35*a^2*e^2 + 14*a*b*e*(-2*d + 3*e*x) + b^2*(8*d^2 - 12*d*e*x + 15*e^2*x^2)) - 286*c*e^2*(
21*a^2*e^2*(2*d - 3*e*x) - 9*a*b*e*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + b^2*(32*d^3 - 48*d^2*e*x + 60*d*e^2*x^2 -
70*e^3*x^3)) + 13*c^2*e*(44*a*e*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3) + 5*b*(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))))/(45045*e^6)

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IntegrateAlgebraic [A]  time = 0.17, size = 425, normalized size = 1.69 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (15015 a^2 b e^5+18018 a^2 c e^4 (d+e x)-30030 a^2 c d e^4+18018 a b^2 e^4 (d+e x)-30030 a b^2 d e^4+90090 a b c d^2 e^3-108108 a b c d e^3 (d+e x)+38610 a b c e^3 (d+e x)^2-60060 a c^2 d^3 e^2+108108 a c^2 d^2 e^2 (d+e x)-77220 a c^2 d e^2 (d+e x)^2+20020 a c^2 e^2 (d+e x)^3+15015 b^3 d^2 e^3-18018 b^3 d e^3 (d+e x)+6435 b^3 e^3 (d+e x)^2-60060 b^2 c d^3 e^2+108108 b^2 c d^2 e^2 (d+e x)-77220 b^2 c d e^2 (d+e x)^2+20020 b^2 c e^2 (d+e x)^3+75075 b c^2 d^4 e-180180 b c^2 d^3 e (d+e x)+193050 b c^2 d^2 e (d+e x)^2-100100 b c^2 d e (d+e x)^3+20475 b c^2 e (d+e x)^4-30030 c^3 d^5+90090 c^3 d^4 (d+e x)-128700 c^3 d^3 (d+e x)^2+100100 c^3 d^2 (d+e x)^3-40950 c^3 d (d+e x)^4+6930 c^3 (d+e x)^5\right )}{45045 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(-30030*c^3*d^5 + 75075*b*c^2*d^4*e - 60060*b^2*c*d^3*e^2 - 60060*a*c^2*d^3*e^2 + 15015*b^3
*d^2*e^3 + 90090*a*b*c*d^2*e^3 - 30030*a*b^2*d*e^4 - 30030*a^2*c*d*e^4 + 15015*a^2*b*e^5 + 90090*c^3*d^4*(d +
e*x) - 180180*b*c^2*d^3*e*(d + e*x) + 108108*b^2*c*d^2*e^2*(d + e*x) + 108108*a*c^2*d^2*e^2*(d + e*x) - 18018*
b^3*d*e^3*(d + e*x) - 108108*a*b*c*d*e^3*(d + e*x) + 18018*a*b^2*e^4*(d + e*x) + 18018*a^2*c*e^4*(d + e*x) - 1
28700*c^3*d^3*(d + e*x)^2 + 193050*b*c^2*d^2*e*(d + e*x)^2 - 77220*b^2*c*d*e^2*(d + e*x)^2 - 77220*a*c^2*d*e^2
*(d + e*x)^2 + 6435*b^3*e^3*(d + e*x)^2 + 38610*a*b*c*e^3*(d + e*x)^2 + 100100*c^3*d^2*(d + e*x)^3 - 100100*b*
c^2*d*e*(d + e*x)^3 + 20020*b^2*c*e^2*(d + e*x)^3 + 20020*a*c^2*e^2*(d + e*x)^3 - 40950*c^3*d*(d + e*x)^4 + 20
475*b*c^2*e*(d + e*x)^4 + 6930*c^3*(d + e*x)^5))/(45045*e^6)

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fricas [A]  time = 0.39, size = 399, normalized size = 1.58 \begin {gather*} \frac {2 \, {\left (6930 \, c^{3} e^{6} x^{6} - 2560 \, c^{3} d^{6} + 8320 \, b c^{2} d^{5} e + 15015 \, a^{2} b d e^{5} - 9152 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + 3432 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 12012 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 315 \, {\left (2 \, c^{3} d e^{5} + 65 \, b c^{2} e^{6}\right )} x^{5} - 35 \, {\left (20 \, c^{3} d^{2} e^{4} - 65 \, b c^{2} d e^{5} - 572 \, {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 5 \, {\left (160 \, c^{3} d^{3} e^{3} - 520 \, b c^{2} d^{2} e^{4} + 572 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} + 1287 \, {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} - 3 \, {\left (320 \, c^{3} d^{4} e^{2} - 1040 \, b c^{2} d^{3} e^{3} + 1144 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - 429 \, {\left (b^{3} + 6 \, a b c\right )} d e^{5} - 6006 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + {\left (1280 \, c^{3} d^{5} e - 4160 \, b c^{2} d^{4} e^{2} + 15015 \, a^{2} b e^{6} + 4576 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 1716 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 6006 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(6930*c^3*e^6*x^6 - 2560*c^3*d^6 + 8320*b*c^2*d^5*e + 15015*a^2*b*d*e^5 - 9152*(b^2*c + a*c^2)*d^4*e^2
+ 3432*(b^3 + 6*a*b*c)*d^3*e^3 - 12012*(a*b^2 + a^2*c)*d^2*e^4 + 315*(2*c^3*d*e^5 + 65*b*c^2*e^6)*x^5 - 35*(2
0*c^3*d^2*e^4 - 65*b*c^2*d*e^5 - 572*(b^2*c + a*c^2)*e^6)*x^4 + 5*(160*c^3*d^3*e^3 - 520*b*c^2*d^2*e^4 + 572*(
b^2*c + a*c^2)*d*e^5 + 1287*(b^3 + 6*a*b*c)*e^6)*x^3 - 3*(320*c^3*d^4*e^2 - 1040*b*c^2*d^3*e^3 + 1144*(b^2*c +
a*c^2)*d^2*e^4 - 429*(b^3 + 6*a*b*c)*d*e^5 - 6006*(a*b^2 + a^2*c)*e^6)*x^2 + (1280*c^3*d^5*e - 4160*b*c^2*d^4
*e^2 + 15015*a^2*b*e^6 + 4576*(b^2*c + a*c^2)*d^3*e^3 - 1716*(b^3 + 6*a*b*c)*d^2*e^4 + 6006*(a*b^2 + a^2*c)*d*
e^5)*x)*sqrt(e*x + d)/e^6

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giac [B]  time = 0.22, size = 966, normalized size = 3.83

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(30030*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a*b^2*d*e^(-1) + 30030*((x*e + d)^(3/2) - 3*sqrt(x*e + d)
*d)*a^2*c*d*e^(-1) + 3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*b^3*d*e^(-2) + 180
18*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*b*c*d*e^(-2) + 5148*(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^2*c*d*e^(-3) + 5148*(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*c^2*d*e^(-3) + 715*(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*c
^2*d*e^(-4) + 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)*c^3*d*e^(-5) + 6006*(3*(x*e + d)^(5/2) - 10*(x*e + d
)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*b^2*e^(-1) + 6006*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e
+ d)*d^2)*a^2*c*e^(-1) + 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^3*e^(-2) + 7722*(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*e^(-2) + 572*(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^2*c*e^(-3) + 572*(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*e^(-3) + 325*(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*e^(-4) + 30*(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)*c
^3*e^(-5) + 45045*sqrt(x*e + d)*a^2*b*d + 15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2*b)*e^(-1)

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maple [A]  time = 0.06, size = 359, normalized size = 1.42 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (6930 c^{3} e^{5} x^{5}+20475 b \,c^{2} e^{5} x^{4}-6300 c^{3} d \,e^{4} x^{4}+20020 a \,c^{2} e^{5} x^{3}+20020 b^{2} c \,e^{5} x^{3}-18200 b \,c^{2} d \,e^{4} x^{3}+5600 c^{3} d^{2} e^{3} x^{3}+38610 a b c \,e^{5} x^{2}-17160 a \,c^{2} d \,e^{4} x^{2}+6435 b^{3} e^{5} x^{2}-17160 b^{2} c d \,e^{4} x^{2}+15600 b \,c^{2} d^{2} e^{3} x^{2}-4800 c^{3} d^{3} e^{2} x^{2}+18018 a^{2} c \,e^{5} x +18018 a \,b^{2} e^{5} x -30888 a b c d \,e^{4} x +13728 a \,c^{2} d^{2} e^{3} x -5148 b^{3} d \,e^{4} x +13728 b^{2} c \,d^{2} e^{3} x -12480 b \,c^{2} d^{3} e^{2} x +3840 c^{3} d^{4} e x +15015 a^{2} b \,e^{5}-12012 a^{2} c d \,e^{4}-12012 a \,b^{2} d \,e^{4}+20592 a b c \,d^{2} e^{3}-9152 a \,c^{2} d^{3} e^{2}+3432 b^{3} d^{2} e^{3}-9152 b^{2} c \,d^{3} e^{2}+8320 b \,c^{2} d^{4} e -2560 c^{3} d^{5}\right )}{45045 e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(e*x+d)^(3/2)*(6930*c^3*e^5*x^5+20475*b*c^2*e^5*x^4-6300*c^3*d*e^4*x^4+20020*a*c^2*e^5*x^3+20020*b^2*c
*e^5*x^3-18200*b*c^2*d*e^4*x^3+5600*c^3*d^2*e^3*x^3+38610*a*b*c*e^5*x^2-17160*a*c^2*d*e^4*x^2+6435*b^3*e^5*x^2
-17160*b^2*c*d*e^4*x^2+15600*b*c^2*d^2*e^3*x^2-4800*c^3*d^3*e^2*x^2+18018*a^2*c*e^5*x+18018*a*b^2*e^5*x-30888*
a*b*c*d*e^4*x+13728*a*c^2*d^2*e^3*x-5148*b^3*d*e^4*x+13728*b^2*c*d^2*e^3*x-12480*b*c^2*d^3*e^2*x+3840*c^3*d^4*
e*x+15015*a^2*b*e^5-12012*a^2*c*d*e^4-12012*a*b^2*d*e^4+20592*a*b*c*d^2*e^3-9152*a*c^2*d^3*e^2+3432*b^3*d^2*e^
3-9152*b^2*c*d^3*e^2+8320*b*c^2*d^4*e-2560*c^3*d^5)/e^6

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maxima [A]  time = 0.54, size = 308, normalized size = 1.22 \begin {gather*} \frac {2 \, {\left (6930 \, {\left (e x + d\right )}^{\frac {13}{2}} c^{3} - 20475 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 20020 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + {\left (b^{2} c + a c^{2}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 6435 \, {\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} - {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 18018 \, {\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d e^{3} + {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 15015 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{4}\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((2*c*x+b)*(c*x^2+b*x+a)^2*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/45045*(6930*(e*x + d)^(13/2)*c^3 - 20475*(2*c^3*d - b*c^2*e)*(e*x + d)^(11/2) + 20020*(5*c^3*d^2 - 5*b*c^2*d
*e + (b^2*c + a*c^2)*e^2)*(e*x + d)^(9/2) - 6435*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*(b^2*c + a*c^2)*d*e^2 - (b^
3 + 6*a*b*c)*e^3)*(e*x + d)^(7/2) + 18018*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*(b^2*c + a*c^2)*d^2*e^2 - (b^3 + 6*a
*b*c)*d*e^3 + (a*b^2 + a^2*c)*e^4)*(e*x + d)^(5/2) - 15015*(2*c^3*d^5 - 5*b*c^2*d^4*e - a^2*b*e^5 + 4*(b^2*c +
a*c^2)*d^3*e^2 - (b^3 + 6*a*b*c)*d^2*e^3 + 2*(a*b^2 + a^2*c)*d*e^4)*(e*x + d)^(3/2))/e^6

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mupad [B]  time = 1.92, size = 267, normalized size = 1.06 \begin {gather*} \frac {{\left (d+e\,x\right )}^{5/2}\,\left (4\,a^2\,c\,e^4+4\,a\,b^2\,e^4-24\,a\,b\,c\,d\,e^3+24\,a\,c^2\,d^2\,e^2-4\,b^3\,d\,e^3+24\,b^2\,c\,d^2\,e^2-40\,b\,c^2\,d^3\,e+20\,c^3\,d^4\right )}{5\,e^6}+\frac {4\,c^3\,{\left (d+e\,x\right )}^{13/2}}{13\,e^6}-\frac {\left (20\,c^3\,d-10\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^6}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (8\,b^2\,c\,e^2-40\,b\,c^2\,d\,e+40\,c^3\,d^2+8\,a\,c^2\,e^2\right )}{9\,e^6}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{7\,e^6}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{3/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{3\,e^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d + e*x)^(5/2)*(20*c^3*d^4 + 4*a*b^2*e^4 + 4*a^2*c*e^4 - 4*b^3*d*e^3 + 24*a*c^2*d^2*e^2 + 24*b^2*c*d^2*e^2 -
40*b*c^2*d^3*e - 24*a*b*c*d*e^3))/(5*e^6) + (4*c^3*(d + e*x)^(13/2))/(13*e^6) - ((20*c^3*d - 10*b*c^2*e)*(d +
e*x)^(11/2))/(11*e^6) + ((d + e*x)^(9/2)*(40*c^3*d^2 + 8*a*c^2*e^2 + 8*b^2*c*e^2 - 40*b*c^2*d*e))/(9*e^6) + (
2*(b*e - 2*c*d)*(d + e*x)^(7/2)*(b^2*e^2 + 10*c^2*d^2 + 6*a*c*e^2 - 10*b*c*d*e))/(7*e^6) + (2*(b*e - 2*c*d)*(d
+ e*x)^(3/2)*(a*e^2 + c*d^2 - b*d*e)^2)/(3*e^6)

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sympy [A]  time = 8.78, size = 405, normalized size = 1.61 \begin {gather*} \frac {2 \left (\frac {2 c^{3} \left (d + e x\right )^{\frac {13}{2}}}{13 e^{5}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \left (5 b c^{2} e - 10 c^{3} d\right )}{11 e^{5}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (4 a c^{2} e^{2} + 4 b^{2} c e^{2} - 20 b c^{2} d e + 20 c^{3} d^{2}\right )}{9 e^{5}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (6 a b c e^{3} - 12 a c^{2} d e^{2} + b^{3} e^{3} - 12 b^{2} c d e^{2} + 30 b c^{2} d^{2} e - 20 c^{3} d^{3}\right )}{7 e^{5}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (2 a^{2} c e^{4} + 2 a b^{2} e^{4} - 12 a b c d e^{3} + 12 a c^{2} d^{2} e^{2} - 2 b^{3} d e^{3} + 12 b^{2} c d^{2} e^{2} - 20 b c^{2} d^{3} e + 10 c^{3} d^{4}\right )}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a^{2} b e^{5} - 2 a^{2} c d e^{4} - 2 a b^{2} d e^{4} + 6 a b c d^{2} e^{3} - 4 a c^{2} d^{3} e^{2} + b^{3} d^{2} e^{3} - 4 b^{2} c d^{3} e^{2} + 5 b c^{2} d^{4} e - 2 c^{3} d^{5}\right )}{3 e^{5}}\right )}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(2*c**3*(d + e*x)**(13/2)/(13*e**5) + (d + e*x)**(11/2)*(5*b*c**2*e - 10*c**3*d)/(11*e**5) + (d + e*x)**(9/2
)*(4*a*c**2*e**2 + 4*b**2*c*e**2 - 20*b*c**2*d*e + 20*c**3*d**2)/(9*e**5) + (d + e*x)**(7/2)*(6*a*b*c*e**3 - 1
2*a*c**2*d*e**2 + b**3*e**3 - 12*b**2*c*d*e**2 + 30*b*c**2*d**2*e - 20*c**3*d**3)/(7*e**5) + (d + e*x)**(5/2)*
(2*a**2*c*e**4 + 2*a*b**2*e**4 - 12*a*b*c*d*e**3 + 12*a*c**2*d**2*e**2 - 2*b**3*d*e**3 + 12*b**2*c*d**2*e**2 -
20*b*c**2*d**3*e + 10*c**3*d**4)/(5*e**5) + (d + e*x)**(3/2)*(a**2*b*e**5 - 2*a**2*c*d*e**4 - 2*a*b**2*d*e**4
+ 6*a*b*c*d**2*e**3 - 4*a*c**2*d**3*e**2 + b**3*d**2*e**3 - 4*b**2*c*d**3*e**2 + 5*b*c**2*d**4*e - 2*c**3*d**
5)/(3*e**5))/e

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