3.15.10 \(\int (b+2 c x) (d+e x)^{3/2} (a+b x+c x^2)^2 \, dx\)

Optimal. Leaf size=252 \[ \frac {8 c (d+e x)^{11/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{11 e^6}-\frac {2 (d+e x)^{9/2} (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{9 e^6}+\frac {4 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{7 e^6}-\frac {2 (d+e x)^{5/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^6}-\frac {10 c^2 (d+e x)^{13/2} (2 c d-b e)}{13 e^6}+\frac {4 c^3 (d+e x)^{15/2}}{15 e^6} \]

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Rubi [A]  time = 0.13, 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)^{11/2} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{11 e^6}-\frac {2 (d+e x)^{9/2} (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 )}{9 e^6}+\frac {4 (d+e x)^{7/2} \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 )}{7 e^6}-\frac {2 (d+e x)^{5/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^6}-\frac {10 c^2 (d+e x)^{13/2} (2 c d-b e)}{13 e^6}+\frac {4 c^3 (d+e x)^{15/2}}{15 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)*(d + e*x)^(3/2)*(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)^(5/2))/(5*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)^(7/2))/(7*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)^(9/2))/(9*e^6) + (8*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*(d + e*x)^(11/2))/(11*e^6
) - (10*c^2*(2*c*d - b*e)*(d + e*x)^(13/2))/(13*e^6) + (4*c^3*(d + e*x)^(15/2))/(15*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) (d+e x)^{3/2} \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 (d+e x)^{3/2}}{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)^{5/2}}{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)^{7/2}}{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)^{9/2}}{e^5}-\frac {5 c^2 (2 c d-b e) (d+e x)^{11/2}}{e^5}+\frac {2 c^3 (d+e x)^{13/2}}{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)^{5/2}}{5 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)^{7/2}}{7 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)^{9/2}}{9 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)^{11/2}}{11 e^6}-\frac {10 c^2 (2 c d-b e) (d+e x)^{13/2}}{13 e^6}+\frac {4 c^3 (d+e x)^{15/2}}{15 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)^{5/2} \left (-78 c e^2 \left (33 a^2 e^2 (2 d-5 e x)-11 a b e \left (8 d^2-20 d e x+35 e^2 x^2\right )+2 b^2 \left (16 d^3-40 d^2 e x+70 d e^2 x^2-105 e^3 x^3\right )\right )+143 b e^3 \left (63 a^2 e^2+18 a b e (5 e x-2 d)+b^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )+3 c^2 e \left (52 a e \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+5 b \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )+c^3 \left (-512 d^5+1280 d^4 e x-2240 d^3 e^2 x^2+3360 d^2 e^3 x^3-4620 d e^4 x^4+6006 e^5 x^5\right )\right )}{45045 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*c*x)*(d + e*x)^(3/2)*(a + b*x + c*x^2)^2,x]

[Out]

(2*(d + e*x)^(5/2)*(c^3*(-512*d^5 + 1280*d^4*e*x - 2240*d^3*e^2*x^2 + 3360*d^2*e^3*x^3 - 4620*d*e^4*x^4 + 6006
*e^5*x^5) + 143*b*e^3*(63*a^2*e^2 + 18*a*b*e*(-2*d + 5*e*x) + b^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2)) - 78*c*e^2*
(33*a^2*e^2*(2*d - 5*e*x) - 11*a*b*e*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 2*b^2*(16*d^3 - 40*d^2*e*x + 70*d*e^2*x
^2 - 105*e^3*x^3)) + 3*c^2*e*(52*a*e*(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3) + 5*b*(128*d^4 - 320*
d^3*e*x + 560*d^2*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4))))/(45045*e^6)

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IntegrateAlgebraic [A]  time = 0.19, size = 425, normalized size = 1.69 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (9009 a^2 b e^5+12870 a^2 c e^4 (d+e x)-18018 a^2 c d e^4+12870 a b^2 e^4 (d+e x)-18018 a b^2 d e^4+54054 a b c d^2 e^3-77220 a b c d e^3 (d+e x)+30030 a b c e^3 (d+e x)^2-36036 a c^2 d^3 e^2+77220 a c^2 d^2 e^2 (d+e x)-60060 a c^2 d e^2 (d+e x)^2+16380 a c^2 e^2 (d+e x)^3+9009 b^3 d^2 e^3-12870 b^3 d e^3 (d+e x)+5005 b^3 e^3 (d+e x)^2-36036 b^2 c d^3 e^2+77220 b^2 c d^2 e^2 (d+e x)-60060 b^2 c d e^2 (d+e x)^2+16380 b^2 c e^2 (d+e x)^3+45045 b c^2 d^4 e-128700 b c^2 d^3 e (d+e x)+150150 b c^2 d^2 e (d+e x)^2-81900 b c^2 d e (d+e x)^3+17325 b c^2 e (d+e x)^4-18018 c^3 d^5+64350 c^3 d^4 (d+e x)-100100 c^3 d^3 (d+e x)^2+81900 c^3 d^2 (d+e x)^3-34650 c^3 d (d+e x)^4+6006 c^3 (d+e x)^5\right )}{45045 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(b + 2*c*x)*(d + e*x)^(3/2)*(a + b*x + c*x^2)^2,x]

[Out]

(2*(d + e*x)^(5/2)*(-18018*c^3*d^5 + 45045*b*c^2*d^4*e - 36036*b^2*c*d^3*e^2 - 36036*a*c^2*d^3*e^2 + 9009*b^3*
d^2*e^3 + 54054*a*b*c*d^2*e^3 - 18018*a*b^2*d*e^4 - 18018*a^2*c*d*e^4 + 9009*a^2*b*e^5 + 64350*c^3*d^4*(d + e*
x) - 128700*b*c^2*d^3*e*(d + e*x) + 77220*b^2*c*d^2*e^2*(d + e*x) + 77220*a*c^2*d^2*e^2*(d + e*x) - 12870*b^3*
d*e^3*(d + e*x) - 77220*a*b*c*d*e^3*(d + e*x) + 12870*a*b^2*e^4*(d + e*x) + 12870*a^2*c*e^4*(d + e*x) - 100100
*c^3*d^3*(d + e*x)^2 + 150150*b*c^2*d^2*e*(d + e*x)^2 - 60060*b^2*c*d*e^2*(d + e*x)^2 - 60060*a*c^2*d*e^2*(d +
e*x)^2 + 5005*b^3*e^3*(d + e*x)^2 + 30030*a*b*c*e^3*(d + e*x)^2 + 81900*c^3*d^2*(d + e*x)^3 - 81900*b*c^2*d*e
*(d + e*x)^3 + 16380*b^2*c*e^2*(d + e*x)^3 + 16380*a*c^2*e^2*(d + e*x)^3 - 34650*c^3*d*(d + e*x)^4 + 17325*b*c
^2*e*(d + e*x)^4 + 6006*c^3*(d + e*x)^5))/(45045*e^6)

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fricas [B]  time = 0.40, size = 495, normalized size = 1.96 \begin {gather*} \frac {2 \, {\left (6006 \, c^{3} e^{7} x^{7} - 512 \, c^{3} d^{7} + 1920 \, b c^{2} d^{6} e + 9009 \, a^{2} b d^{2} e^{5} - 2496 \, {\left (b^{2} c + a c^{2}\right )} d^{5} e^{2} + 1144 \, {\left (b^{3} + 6 \, a b c\right )} d^{4} e^{3} - 5148 \, {\left (a b^{2} + a^{2} c\right )} d^{3} e^{4} + 231 \, {\left (32 \, c^{3} d e^{6} + 75 \, b c^{2} e^{7}\right )} x^{6} + 126 \, {\left (c^{3} d^{2} e^{5} + 175 \, b c^{2} d e^{6} + 130 \, {\left (b^{2} c + a c^{2}\right )} e^{7}\right )} x^{5} - 35 \, {\left (4 \, c^{3} d^{3} e^{4} - 15 \, b c^{2} d^{2} e^{5} - 624 \, {\left (b^{2} c + a c^{2}\right )} d e^{6} - 143 \, {\left (b^{3} + 6 \, a b c\right )} e^{7}\right )} x^{4} + 10 \, {\left (16 \, c^{3} d^{4} e^{3} - 60 \, b c^{2} d^{3} e^{4} + 78 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{5} + 715 \, {\left (b^{3} + 6 \, a b c\right )} d e^{6} + 1287 \, {\left (a b^{2} + a^{2} c\right )} e^{7}\right )} x^{3} - 3 \, {\left (64 \, c^{3} d^{5} e^{2} - 240 \, b c^{2} d^{4} e^{3} - 3003 \, a^{2} b e^{7} + 312 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{4} - 143 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{5} - 6864 \, {\left (a b^{2} + a^{2} c\right )} d e^{6}\right )} x^{2} + 2 \, {\left (128 \, c^{3} d^{6} e - 480 \, b c^{2} d^{5} e^{2} + 9009 \, a^{2} b d e^{6} + 624 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{3} - 286 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{4} + 1287 \, {\left (a b^{2} + a^{2} c\right )} d^{2} 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)*(e*x+d)^(3/2)*(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

2/45045*(6006*c^3*e^7*x^7 - 512*c^3*d^7 + 1920*b*c^2*d^6*e + 9009*a^2*b*d^2*e^5 - 2496*(b^2*c + a*c^2)*d^5*e^2
+ 1144*(b^3 + 6*a*b*c)*d^4*e^3 - 5148*(a*b^2 + a^2*c)*d^3*e^4 + 231*(32*c^3*d*e^6 + 75*b*c^2*e^7)*x^6 + 126*(
c^3*d^2*e^5 + 175*b*c^2*d*e^6 + 130*(b^2*c + a*c^2)*e^7)*x^5 - 35*(4*c^3*d^3*e^4 - 15*b*c^2*d^2*e^5 - 624*(b^2
*c + a*c^2)*d*e^6 - 143*(b^3 + 6*a*b*c)*e^7)*x^4 + 10*(16*c^3*d^4*e^3 - 60*b*c^2*d^3*e^4 + 78*(b^2*c + a*c^2)*
d^2*e^5 + 715*(b^3 + 6*a*b*c)*d*e^6 + 1287*(a*b^2 + a^2*c)*e^7)*x^3 - 3*(64*c^3*d^5*e^2 - 240*b*c^2*d^4*e^3 -
3003*a^2*b*e^7 + 312*(b^2*c + a*c^2)*d^3*e^4 - 143*(b^3 + 6*a*b*c)*d^2*e^5 - 6864*(a*b^2 + a^2*c)*d*e^6)*x^2 +
2*(128*c^3*d^6*e - 480*b*c^2*d^5*e^2 + 9009*a^2*b*d*e^6 + 624*(b^2*c + a*c^2)*d^4*e^3 - 286*(b^3 + 6*a*b*c)*d
^3*e^4 + 1287*(a*b^2 + a^2*c)*d^2*e^5)*x)*sqrt(e*x + d)/e^6

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giac [B]  time = 0.29, size = 1648, normalized size = 6.54

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^(3/2)*(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

2/45045*(30030*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a*b^2*d^2*e^(-1) + 30030*((x*e + d)^(3/2) - 3*sqrt(x*e +
d)*d)*a^2*c*d^2*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^2*e^(-2)
+ 18018*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*b*c*d^2*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^2*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^2*e^(-3) + 715*(3
5*(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^2*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^2*e^(-5) + 12012*(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^(-1) + 12012*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3
/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*c*d*e^(-1) + 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)*b^3*d*e^(-2) + 15444*(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^(-2) + 1144*(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*d*e^(-3) + 1144*(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
^(-3) + 650*(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^(-4) + 60*(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*d*e^(-5) + 45045*sqrt(x*e + d)*a^2*b*d^2 + 30030*((x*e + d)^(3/2) - 3*sq
rt(x*e + d)*d)*a^2*b*d + 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^2*e^(-1) + 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^2*c*e^(-1) + 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^3*e^(-2) + 858*(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^(-2) + 260*(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^2*c*e^(-3) + 260*(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^(-3) + 75*(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^(-4) + 14*(429*(x*e + d)^(15/2) - 3465*(x*e
+ d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*
e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*c^3*e^(-5) + 3003*(3*(x*e + d)^(5/2) -
10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*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 {5}{2}} \left (6006 c^{3} e^{5} x^{5}+17325 b \,c^{2} e^{5} x^{4}-4620 c^{3} d \,e^{4} x^{4}+16380 a \,c^{2} e^{5} x^{3}+16380 b^{2} c \,e^{5} x^{3}-12600 b \,c^{2} d \,e^{4} x^{3}+3360 c^{3} d^{2} e^{3} x^{3}+30030 a b c \,e^{5} x^{2}-10920 a \,c^{2} d \,e^{4} x^{2}+5005 b^{3} e^{5} x^{2}-10920 b^{2} c d \,e^{4} x^{2}+8400 b \,c^{2} d^{2} e^{3} x^{2}-2240 c^{3} d^{3} e^{2} x^{2}+12870 a^{2} c \,e^{5} x +12870 a \,b^{2} e^{5} x -17160 a b c d \,e^{4} x +6240 a \,c^{2} d^{2} e^{3} x -2860 b^{3} d \,e^{4} x +6240 b^{2} c \,d^{2} e^{3} x -4800 b \,c^{2} d^{3} e^{2} x +1280 c^{3} d^{4} e x +9009 a^{2} b \,e^{5}-5148 a^{2} c d \,e^{4}-5148 a \,b^{2} d \,e^{4}+6864 a b c \,d^{2} e^{3}-2496 a \,c^{2} d^{3} e^{2}+1144 b^{3} d^{2} e^{3}-2496 b^{2} c \,d^{3} e^{2}+1920 b \,c^{2} d^{4} e -512 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)*(e*x+d)^(3/2)*(c*x^2+b*x+a)^2,x)

[Out]

2/45045*(e*x+d)^(5/2)*(6006*c^3*e^5*x^5+17325*b*c^2*e^5*x^4-4620*c^3*d*e^4*x^4+16380*a*c^2*e^5*x^3+16380*b^2*c
*e^5*x^3-12600*b*c^2*d*e^4*x^3+3360*c^3*d^2*e^3*x^3+30030*a*b*c*e^5*x^2-10920*a*c^2*d*e^4*x^2+5005*b^3*e^5*x^2
-10920*b^2*c*d*e^4*x^2+8400*b*c^2*d^2*e^3*x^2-2240*c^3*d^3*e^2*x^2+12870*a^2*c*e^5*x+12870*a*b^2*e^5*x-17160*a
*b*c*d*e^4*x+6240*a*c^2*d^2*e^3*x-2860*b^3*d*e^4*x+6240*b^2*c*d^2*e^3*x-4800*b*c^2*d^3*e^2*x+1280*c^3*d^4*e*x+
9009*a^2*b*e^5-5148*a^2*c*d*e^4-5148*a*b^2*d*e^4+6864*a*b*c*d^2*e^3-2496*a*c^2*d^3*e^2+1144*b^3*d^2*e^3-2496*b
^2*c*d^3*e^2+1920*b*c^2*d^4*e-512*c^3*d^5)/e^6

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maxima [A]  time = 0.53, size = 308, normalized size = 1.22 \begin {gather*} \frac {2 \, {\left (6006 \, {\left (e x + d\right )}^{\frac {15}{2}} c^{3} - 17325 \, {\left (2 \, c^{3} d - b c^{2} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 16380 \, {\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 {11}{2}} - 5005 \, {\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 {9}{2}} + 12870 \, {\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 {7}{2}} - 9009 \, {\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 {5}{2}}\right )}}{45045 \, e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^(3/2)*(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

2/45045*(6006*(e*x + d)^(15/2)*c^3 - 17325*(2*c^3*d - b*c^2*e)*(e*x + d)^(13/2) + 16380*(5*c^3*d^2 - 5*b*c^2*d
*e + (b^2*c + a*c^2)*e^2)*(e*x + d)^(11/2) - 5005*(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)^(9/2) + 12870*(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)^(7/2) - 9009*(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)^(5/2))/e^6

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mupad [B]  time = 0.07, size = 267, normalized size = 1.06 \begin {gather*} \frac {{\left (d+e\,x\right )}^{7/2}\,\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 )}{7\,e^6}+\frac {4\,c^3\,{\left (d+e\,x\right )}^{15/2}}{15\,e^6}-\frac {\left (20\,c^3\,d-10\,b\,c^2\,e\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^6}+\frac {{\left (d+e\,x\right )}^{11/2}\,\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 )}{11\,e^6}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{9/2}\,\left (b^2\,e^2-10\,b\,c\,d\,e+10\,c^2\,d^2+6\,a\,c\,e^2\right )}{9\,e^6}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{5\,e^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + 2*c*x)*(d + e*x)^(3/2)*(a + b*x + c*x^2)^2,x)

[Out]

((d + e*x)^(7/2)*(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))/(7*e^6) + (4*c^3*(d + e*x)^(15/2))/(15*e^6) - ((20*c^3*d - 10*b*c^2*e)*(d +
e*x)^(13/2))/(13*e^6) + ((d + e*x)^(11/2)*(40*c^3*d^2 + 8*a*c^2*e^2 + 8*b^2*c*e^2 - 40*b*c^2*d*e))/(11*e^6) +
(2*(b*e - 2*c*d)*(d + e*x)^(9/2)*(b^2*e^2 + 10*c^2*d^2 + 6*a*c*e^2 - 10*b*c*d*e))/(9*e^6) + (2*(b*e - 2*c*d)*
(d + e*x)^(5/2)*(a*e^2 + c*d^2 - b*d*e)^2)/(5*e^6)

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sympy [A]  time = 43.46, size = 1093, normalized size = 4.34

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)**(3/2)*(c*x**2+b*x+a)**2,x)

[Out]

a**2*b*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*a**2*b*(-d*(d + e*x)**(3/2)/3
+ (d + e*x)**(5/2)/5)/e + 4*a**2*c*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 4*a**2*c*(d**2*(d + e
*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 4*a*b**2*d*(-d*(d + e*x)**(3/2)/3 + (d + e*
x)**(5/2)/5)/e**2 + 4*a*b**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 12
*a*b*c*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 12*a*b*c*(-d**3*(d + e
*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 8*a*c**2*d*(-d*
*3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4 + 8*a*c*
*2*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 +
(d + e*x)**(11/2)/11)/e**4 + 2*b**3*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)
/e**3 + 2*b**3*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/
2)/9)/e**3 + 8*b**2*c*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d +
e*x)**(9/2)/9)/e**4 + 8*b**2*c*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/
7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4 + 10*b*c**2*d*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e
*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**5 + 10*b*c**2*(-d
**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d
*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**5 + 4*c**3*d*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2
) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)
/e**6 + 4*c**3*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d
+ e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**6

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