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

Optimal. Leaf size=427 \[ \frac {2 (d+e x)^{9/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{9 e^8}+\frac {6 c^2 (d+e x)^{13/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{13 e^8}-\frac {10 c (d+e x)^{11/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{11 e^8}-\frac {6 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{7 e^8}+\frac {2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^8}-\frac {2 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8}-\frac {14 c^3 (d+e x)^{15/2} (2 c d-b e)}{15 e^8}+\frac {4 c^4 (d+e x)^{17/2}}{17 e^8} \]

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Rubi [A]  time = 0.24, antiderivative size = 427, 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 {2 (d+e x)^{9/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{9 e^8}+\frac {6 c^2 (d+e x)^{13/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{13 e^8}-\frac {10 c (d+e x)^{11/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{11 e^8}-\frac {6 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{7 e^8}+\frac {2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^8}-\frac {2 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8}-\frac {14 c^3 (d+e x)^{15/2} (2 c d-b e)}{15 e^8}+\frac {4 c^4 (d+e x)^{17/2}}{17 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^(3/2))/(3*e^8) + (2*(c*d^2 - b*d*e + a*e^2)^2*(14*c^2*d^
2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^(5/2))/(5*e^8) - (6*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(7*c^
2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(7/2))/(7*e^8) + (2*(70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*
d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*(d + e*x)^(9/2))/(9
*e^8) - (10*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^(11/2))/(11*e^8) + (6*c^2*(1
4*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^(13/2))/(13*e^8) - (14*c^3*(2*c*d - b*e)*(d + e*x)^(15/
2))/(15*e^8) + (4*c^4*(d + e*x)^(17/2))/(17*e^8)

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 )^3 \, dx &=\int \left (\frac {(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}{e^7}+\frac {\left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{3/2}}{e^7}+\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (-7 c^2 d^2+7 b c d e-b^2 e^2-3 a c e^2\right ) (d+e x)^{5/2}}{e^7}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^{7/2}}{e^7}+\frac {5 c (2 c d-b e) \left (-7 c^2 d^2-b^2 e^2+c e (7 b d-3 a e)\right ) (d+e x)^{9/2}}{e^7}+\frac {3 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{11/2}}{e^7}-\frac {7 c^3 (2 c d-b e) (d+e x)^{13/2}}{e^7}+\frac {2 c^4 (d+e x)^{15/2}}{e^7}\right ) \, dx\\ &=-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3 (d+e x)^{3/2}}{3 e^8}+\frac {2 \left (c d^2-b d e+a e^2\right )^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{5/2}}{5 e^8}-\frac {6 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{7/2}}{7 e^8}+\frac {2 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (d+e x)^{9/2}}{9 e^8}-\frac {10 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (d+e x)^{11/2}}{11 e^8}+\frac {6 c^2 \left (14 c^2 d^2+3 b^2 e^2-2 c e (7 b d-a e)\right ) (d+e x)^{13/2}}{13 e^8}-\frac {14 c^3 (2 c d-b e) (d+e x)^{15/2}}{15 e^8}+\frac {4 c^4 (d+e x)^{17/2}}{17 e^8}\\ \end {align*}

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Mathematica [A]  time = 0.60, size = 601, normalized size = 1.41 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (-51 c^2 e^2 \left (286 a^2 e^2 \left (16 d^3-24 d^2 e x+30 d e^2 x^2-35 e^3 x^3\right )-65 a b 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 )+15 b^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 )+221 c e^3 \left (462 a^3 e^3 (3 e x-2 d)+297 a^2 b e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+132 a b^2 e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+5 b^3 \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 )+2431 b e^4 \left (105 a^3 e^3+63 a^2 b e^2 (3 e x-2 d)+9 a b^2 e \left (8 d^2-12 d e x+15 e^2 x^2\right )+b^3 \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )\right )+17 c^3 e \left (30 a e \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 )+7 b \left (1024 d^6-1536 d^5 e x+1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-2772 d e^5 x^5+3003 e^6 x^6\right )\right )-14 c^4 \left (2048 d^7-3072 d^6 e x+3840 d^5 e^2 x^2-4480 d^4 e^3 x^3+5040 d^3 e^4 x^4-5544 d^2 e^5 x^5+6006 d e^6 x^6-6435 e^7 x^7\right )\right )}{765765 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(-14*c^4*(2048*d^7 - 3072*d^6*e*x + 3840*d^5*e^2*x^2 - 4480*d^4*e^3*x^3 + 5040*d^3*e^4*x^4
- 5544*d^2*e^5*x^5 + 6006*d*e^6*x^6 - 6435*e^7*x^7) + 2431*b*e^4*(105*a^3*e^3 + 63*a^2*b*e^2*(-2*d + 3*e*x) +
9*a*b^2*e*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + b^3*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3)) + 221*c*e^
3*(462*a^3*e^3*(-2*d + 3*e*x) + 297*a^2*b*e^2*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + 132*a*b^2*e*(-16*d^3 + 24*d^2*
e*x - 30*d*e^2*x^2 + 35*e^3*x^3) + 5*b^3*(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)) - 51*c^2*e^2*(286*a^2*e^2*(16*d^3 - 24*d^2*e*x + 30*d*e^2*x^2 - 35*e^3*x^3) - 65*a*b*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) + 15*b^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)) + 17*c^3*e*(30*a*e*(-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) + 7*b*(1024*d^6 - 1536*d^5*e*x + 1920*d^4*e^2*x^2 - 2240*d^3*e^3*x^
3 + 2520*d^2*e^4*x^4 - 2772*d*e^5*x^5 + 3003*e^6*x^6))))/(765765*e^8)

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IntegrateAlgebraic [B]  time = 0.34, size = 951, normalized size = 2.23 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (-510510 c^4 d^7+1786785 b c^3 e d^6+2144142 c^4 (d+e x) d^6-1531530 a c^3 e^2 d^5-2297295 b^2 c^2 e^2 d^5-4594590 c^4 (d+e x)^2 d^5-6432426 b c^3 e (d+e x) d^5+3828825 a b c^2 e^3 d^4+1276275 b^3 c e^3 d^4+5955950 c^4 (d+e x)^3 d^4+11486475 b c^3 e (d+e x)^2 d^4+4594590 a c^3 e^2 (d+e x) d^4+6891885 b^2 c^2 e^2 (d+e x) d^4-255255 b^4 e^4 d^3-1531530 a^2 c^2 e^4 d^3-3063060 a b^2 c e^4 d^3-4873050 c^4 (d+e x)^4 d^3-11911900 b c^3 e (d+e x)^3 d^3-6563700 a c^3 e^2 (d+e x)^2 d^3-9845550 b^2 c^2 e^2 (d+e x)^2 d^3-9189180 a b c^2 e^3 (d+e x) d^3-3063060 b^3 c e^3 (d+e x) d^3+765765 a b^3 e^5 d^2+2297295 a^2 b c e^5 d^2+2474010 c^4 (d+e x)^5 d^2+7309575 b c^3 e (d+e x)^4 d^2+5105100 a c^3 e^2 (d+e x)^3 d^2+7657650 b^2 c^2 e^2 (d+e x)^3 d^2+9845550 a b c^2 e^3 (d+e x)^2 d^2+3281850 b^3 c e^3 (d+e x)^2 d^2+459459 b^4 e^4 (d+e x) d^2+2756754 a^2 c^2 e^4 (d+e x) d^2+5513508 a b^2 c e^4 (d+e x) d^2-765765 a^2 b^2 e^6 d-510510 a^3 c e^6 d-714714 c^4 (d+e x)^6 d-2474010 b c^3 e (d+e x)^5 d-2088450 a c^3 e^2 (d+e x)^4 d-3132675 b^2 c^2 e^2 (d+e x)^4 d-5105100 a b c^2 e^3 (d+e x)^3 d-1701700 b^3 c e^3 (d+e x)^3 d-328185 b^4 e^4 (d+e x)^2 d-1969110 a^2 c^2 e^4 (d+e x)^2 d-3938220 a b^2 c e^4 (d+e x)^2 d-918918 a b^3 e^5 (d+e x) d-2756754 a^2 b c e^5 (d+e x) d+255255 a^3 b e^7+90090 c^4 (d+e x)^7+357357 b c^3 e (d+e x)^6+353430 a c^3 e^2 (d+e x)^5+530145 b^2 c^2 e^2 (d+e x)^5+1044225 a b c^2 e^3 (d+e x)^4+348075 b^3 c e^3 (d+e x)^4+85085 b^4 e^4 (d+e x)^3+510510 a^2 c^2 e^4 (d+e x)^3+1021020 a b^2 c e^4 (d+e x)^3+328185 a b^3 e^5 (d+e x)^2+984555 a^2 b c e^5 (d+e x)^2+459459 a^2 b^2 e^6 (d+e x)+306306 a^3 c e^6 (d+e x)\right )}{765765 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(-510510*c^4*d^7 + 1786785*b*c^3*d^6*e - 2297295*b^2*c^2*d^5*e^2 - 1531530*a*c^3*d^5*e^2 +
1276275*b^3*c*d^4*e^3 + 3828825*a*b*c^2*d^4*e^3 - 255255*b^4*d^3*e^4 - 3063060*a*b^2*c*d^3*e^4 - 1531530*a^2*c
^2*d^3*e^4 + 765765*a*b^3*d^2*e^5 + 2297295*a^2*b*c*d^2*e^5 - 765765*a^2*b^2*d*e^6 - 510510*a^3*c*d*e^6 + 2552
55*a^3*b*e^7 + 2144142*c^4*d^6*(d + e*x) - 6432426*b*c^3*d^5*e*(d + e*x) + 6891885*b^2*c^2*d^4*e^2*(d + e*x) +
4594590*a*c^3*d^4*e^2*(d + e*x) - 3063060*b^3*c*d^3*e^3*(d + e*x) - 9189180*a*b*c^2*d^3*e^3*(d + e*x) + 45945
9*b^4*d^2*e^4*(d + e*x) + 5513508*a*b^2*c*d^2*e^4*(d + e*x) + 2756754*a^2*c^2*d^2*e^4*(d + e*x) - 918918*a*b^3
*d*e^5*(d + e*x) - 2756754*a^2*b*c*d*e^5*(d + e*x) + 459459*a^2*b^2*e^6*(d + e*x) + 306306*a^3*c*e^6*(d + e*x)
- 4594590*c^4*d^5*(d + e*x)^2 + 11486475*b*c^3*d^4*e*(d + e*x)^2 - 9845550*b^2*c^2*d^3*e^2*(d + e*x)^2 - 6563
700*a*c^3*d^3*e^2*(d + e*x)^2 + 3281850*b^3*c*d^2*e^3*(d + e*x)^2 + 9845550*a*b*c^2*d^2*e^3*(d + e*x)^2 - 3281
85*b^4*d*e^4*(d + e*x)^2 - 3938220*a*b^2*c*d*e^4*(d + e*x)^2 - 1969110*a^2*c^2*d*e^4*(d + e*x)^2 + 328185*a*b^
3*e^5*(d + e*x)^2 + 984555*a^2*b*c*e^5*(d + e*x)^2 + 5955950*c^4*d^4*(d + e*x)^3 - 11911900*b*c^3*d^3*e*(d + e
*x)^3 + 7657650*b^2*c^2*d^2*e^2*(d + e*x)^3 + 5105100*a*c^3*d^2*e^2*(d + e*x)^3 - 1701700*b^3*c*d*e^3*(d + e*x
)^3 - 5105100*a*b*c^2*d*e^3*(d + e*x)^3 + 85085*b^4*e^4*(d + e*x)^3 + 1021020*a*b^2*c*e^4*(d + e*x)^3 + 510510
*a^2*c^2*e^4*(d + e*x)^3 - 4873050*c^4*d^3*(d + e*x)^4 + 7309575*b*c^3*d^2*e*(d + e*x)^4 - 3132675*b^2*c^2*d*e
^2*(d + e*x)^4 - 2088450*a*c^3*d*e^2*(d + e*x)^4 + 348075*b^3*c*e^3*(d + e*x)^4 + 1044225*a*b*c^2*e^3*(d + e*x
)^4 + 2474010*c^4*d^2*(d + e*x)^5 - 2474010*b*c^3*d*e*(d + e*x)^5 + 530145*b^2*c^2*e^2*(d + e*x)^5 + 353430*a*
c^3*e^2*(d + e*x)^5 - 714714*c^4*d*(d + e*x)^6 + 357357*b*c^3*e*(d + e*x)^6 + 90090*c^4*(d + e*x)^7))/(765765*
e^8)

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fricas [B]  time = 0.42, size = 802, normalized size = 1.88 \begin {gather*} \frac {2 \, {\left (90090 \, c^{4} e^{8} x^{8} - 28672 \, c^{4} d^{8} + 121856 \, b c^{3} d^{7} e + 255255 \, a^{3} b d e^{7} - 65280 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{2} + 141440 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5} e^{3} - 38896 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} e^{4} + 175032 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e^{5} - 102102 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{6} + 3003 \, {\left (2 \, c^{4} d e^{7} + 119 \, b c^{3} e^{8}\right )} x^{7} - 231 \, {\left (28 \, c^{4} d^{2} e^{6} - 119 \, b c^{3} d e^{7} - 765 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{8}\right )} x^{6} + 63 \, {\left (112 \, c^{4} d^{3} e^{5} - 476 \, b c^{3} d^{2} e^{6} + 255 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{7} + 5525 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{8}\right )} x^{5} - 35 \, {\left (224 \, c^{4} d^{4} e^{4} - 952 \, b c^{3} d^{3} e^{5} + 510 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{6} - 1105 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{7} - 2431 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{8}\right )} x^{4} + 5 \, {\left (1792 \, c^{4} d^{5} e^{3} - 7616 \, b c^{3} d^{4} e^{4} + 4080 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{5} - 8840 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{6} + 2431 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{7} + 65637 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{8}\right )} x^{3} - 3 \, {\left (3584 \, c^{4} d^{6} e^{2} - 15232 \, b c^{3} d^{5} e^{3} + 8160 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{4} - 17680 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{5} + 4862 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{6} - 21879 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{7} - 51051 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{8}\right )} x^{2} + {\left (14336 \, c^{4} d^{7} e - 60928 \, b c^{3} d^{6} e^{2} + 255255 \, a^{3} b e^{8} + 32640 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{3} - 70720 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{4} + 19448 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{5} - 87516 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{6} + 51051 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{7}\right )} x\right )} \sqrt {e x + d}}{765765 \, e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)^3*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/765765*(90090*c^4*e^8*x^8 - 28672*c^4*d^8 + 121856*b*c^3*d^7*e + 255255*a^3*b*d*e^7 - 65280*(3*b^2*c^2 + 2*a
*c^3)*d^6*e^2 + 141440*(b^3*c + 3*a*b*c^2)*d^5*e^3 - 38896*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 175032*(a*
b^3 + 3*a^2*b*c)*d^3*e^5 - 102102*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 + 3003*(2*c^4*d*e^7 + 119*b*c^3*e^8)*x^7 - 231
*(28*c^4*d^2*e^6 - 119*b*c^3*d*e^7 - 765*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 + 63*(112*c^4*d^3*e^5 - 476*b*c^3*d^2*
e^6 + 255*(3*b^2*c^2 + 2*a*c^3)*d*e^7 + 5525*(b^3*c + 3*a*b*c^2)*e^8)*x^5 - 35*(224*c^4*d^4*e^4 - 952*b*c^3*d^
3*e^5 + 510*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 - 1105*(b^3*c + 3*a*b*c^2)*d*e^7 - 2431*(b^4 + 12*a*b^2*c + 6*a^2*c^
2)*e^8)*x^4 + 5*(1792*c^4*d^5*e^3 - 7616*b*c^3*d^4*e^4 + 4080*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 8840*(b^3*c + 3*
a*b*c^2)*d^2*e^6 + 2431*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^7 + 65637*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 - 3*(3584*c^
4*d^6*e^2 - 15232*b*c^3*d^5*e^3 + 8160*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 17680*(b^3*c + 3*a*b*c^2)*d^3*e^5 + 486
2*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 - 21879*(a*b^3 + 3*a^2*b*c)*d*e^7 - 51051*(3*a^2*b^2 + 2*a^3*c)*e^8)*
x^2 + (14336*c^4*d^7*e - 60928*b*c^3*d^6*e^2 + 255255*a^3*b*e^8 + 32640*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 70720*
(b^3*c + 3*a*b*c^2)*d^4*e^4 + 19448*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 87516*(a*b^3 + 3*a^2*b*c)*d^2*e^6
+ 51051*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*x)*sqrt(e*x + d)/e^8

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giac [B]  time = 0.28, size = 1876, normalized size = 4.39

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)^3*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/765765*(765765*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^2*b^2*d*e^(-1) + 510510*((x*e + d)^(3/2) - 3*sqrt(x*e
+ d)*d)*a^3*c*d*e^(-1) + 153153*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*b^3*d*e^(
-2) + 459459*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*b*c*d*e^(-2) + 21879*(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^4*d*e^(-3) + 262548*(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*c*d*e^(-3) + 131
274*(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^2*d*e^(-3
) + 12155*(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 + 31
5*sqrt(x*e + d)*d^4)*b^3*c*d*e^(-4) + 36465*(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^2*d*e^(-4) + 9945*(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^2*d*e^(-5) + 6630*(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^3*d*e^(-5) + 1785*(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^3*d*e^(-6) + 238*(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^4*d*e^(-7) + 153153*(3*(x*e + d)^(5
/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*b^2*e^(-1) + 102102*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(
3/2)*d + 15*sqrt(x*e + d)*d^2)*a^3*c*e^(-1) + 65637*(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^3*e^(-2) + 196911*(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*b*c*e^(-2) + 2431*(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^4*e^(-3) + 29172*(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^2*c*e^(
-3) + 14586*(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^2*c^2*e^(-3) + 5525*(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^3*c*e^(-4) + 16575*(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*b*c^2*e^(-4) + 2295*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 50
05*(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^2*c^2*e^(-5) + 1530*(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)*a*c^3*e^(-5) + 833*(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*s
qrt(x*e + d)*d^7)*b*c^3*e^(-6) + 14*(6435*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)*d + 235620*(x*e + d)^(13/2
)*d^2 - 556920*(x*e + d)^(11/2)*d^3 + 850850*(x*e + d)^(9/2)*d^4 - 875160*(x*e + d)^(7/2)*d^5 + 612612*(x*e +
d)^(5/2)*d^6 - 291720*(x*e + d)^(3/2)*d^7 + 109395*sqrt(x*e + d)*d^8)*c^4*e^(-7) + 765765*sqrt(x*e + d)*a^3*b*
d + 255255*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^3*b)*e^(-1)

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maple [B]  time = 0.05, size = 795, normalized size = 1.86 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (90090 c^{4} e^{7} x^{7}+357357 b \,c^{3} e^{7} x^{6}-84084 c^{4} d \,e^{6} x^{6}+353430 a \,c^{3} e^{7} x^{5}+530145 b^{2} c^{2} e^{7} x^{5}-329868 b \,c^{3} d \,e^{6} x^{5}+77616 c^{4} d^{2} e^{5} x^{5}+1044225 a b \,c^{2} e^{7} x^{4}-321300 a \,c^{3} d \,e^{6} x^{4}+348075 b^{3} c \,e^{7} x^{4}-481950 b^{2} c^{2} d \,e^{6} x^{4}+299880 b \,c^{3} d^{2} e^{5} x^{4}-70560 c^{4} d^{3} e^{4} x^{4}+510510 a^{2} c^{2} e^{7} x^{3}+1021020 a \,b^{2} c \,e^{7} x^{3}-928200 a b \,c^{2} d \,e^{6} x^{3}+285600 a \,c^{3} d^{2} e^{5} x^{3}+85085 b^{4} e^{7} x^{3}-309400 b^{3} c d \,e^{6} x^{3}+428400 b^{2} c^{2} d^{2} e^{5} x^{3}-266560 b \,c^{3} d^{3} e^{4} x^{3}+62720 c^{4} d^{4} e^{3} x^{3}+984555 a^{2} b c \,e^{7} x^{2}-437580 a^{2} c^{2} d \,e^{6} x^{2}+328185 a \,b^{3} e^{7} x^{2}-875160 a \,b^{2} c d \,e^{6} x^{2}+795600 a b \,c^{2} d^{2} e^{5} x^{2}-244800 a \,c^{3} d^{3} e^{4} x^{2}-72930 b^{4} d \,e^{6} x^{2}+265200 b^{3} c \,d^{2} e^{5} x^{2}-367200 b^{2} c^{2} d^{3} e^{4} x^{2}+228480 b \,c^{3} d^{4} e^{3} x^{2}-53760 c^{4} d^{5} e^{2} x^{2}+306306 a^{3} c \,e^{7} x +459459 a^{2} b^{2} e^{7} x -787644 a^{2} b c d \,e^{6} x +350064 a^{2} c^{2} d^{2} e^{5} x -262548 a \,b^{3} d \,e^{6} x +700128 a \,b^{2} c \,d^{2} e^{5} x -636480 a b \,c^{2} d^{3} e^{4} x +195840 a \,c^{3} d^{4} e^{3} x +58344 b^{4} d^{2} e^{5} x -212160 b^{3} c \,d^{3} e^{4} x +293760 b^{2} c^{2} d^{4} e^{3} x -182784 b \,c^{3} d^{5} e^{2} x +43008 c^{4} d^{6} e x +255255 b \,a^{3} e^{7}-204204 a^{3} c d \,e^{6}-306306 a^{2} b^{2} d \,e^{6}+525096 a^{2} b c \,d^{2} e^{5}-233376 a^{2} c^{2} d^{3} e^{4}+175032 a \,b^{3} d^{2} e^{5}-466752 a \,b^{2} c \,d^{3} e^{4}+424320 a b \,c^{2} d^{4} e^{3}-130560 a \,c^{3} d^{5} e^{2}-38896 b^{4} d^{3} e^{4}+141440 b^{3} c \,d^{4} e^{3}-195840 b^{2} c^{2} d^{5} e^{2}+121856 b \,c^{3} d^{6} e -28672 c^{4} d^{7}\right )}{765765 e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(c*x^2+b*x+a)^3*(e*x+d)^(1/2),x)

[Out]

2/765765*(e*x+d)^(3/2)*(90090*c^4*e^7*x^7+357357*b*c^3*e^7*x^6-84084*c^4*d*e^6*x^6+353430*a*c^3*e^7*x^5+530145
*b^2*c^2*e^7*x^5-329868*b*c^3*d*e^6*x^5+77616*c^4*d^2*e^5*x^5+1044225*a*b*c^2*e^7*x^4-321300*a*c^3*d*e^6*x^4+3
48075*b^3*c*e^7*x^4-481950*b^2*c^2*d*e^6*x^4+299880*b*c^3*d^2*e^5*x^4-70560*c^4*d^3*e^4*x^4+510510*a^2*c^2*e^7
*x^3+1021020*a*b^2*c*e^7*x^3-928200*a*b*c^2*d*e^6*x^3+285600*a*c^3*d^2*e^5*x^3+85085*b^4*e^7*x^3-309400*b^3*c*
d*e^6*x^3+428400*b^2*c^2*d^2*e^5*x^3-266560*b*c^3*d^3*e^4*x^3+62720*c^4*d^4*e^3*x^3+984555*a^2*b*c*e^7*x^2-437
580*a^2*c^2*d*e^6*x^2+328185*a*b^3*e^7*x^2-875160*a*b^2*c*d*e^6*x^2+795600*a*b*c^2*d^2*e^5*x^2-244800*a*c^3*d^
3*e^4*x^2-72930*b^4*d*e^6*x^2+265200*b^3*c*d^2*e^5*x^2-367200*b^2*c^2*d^3*e^4*x^2+228480*b*c^3*d^4*e^3*x^2-537
60*c^4*d^5*e^2*x^2+306306*a^3*c*e^7*x+459459*a^2*b^2*e^7*x-787644*a^2*b*c*d*e^6*x+350064*a^2*c^2*d^2*e^5*x-262
548*a*b^3*d*e^6*x+700128*a*b^2*c*d^2*e^5*x-636480*a*b*c^2*d^3*e^4*x+195840*a*c^3*d^4*e^3*x+58344*b^4*d^2*e^5*x
-212160*b^3*c*d^3*e^4*x+293760*b^2*c^2*d^4*e^3*x-182784*b*c^3*d^5*e^2*x+43008*c^4*d^6*e*x+255255*a^3*b*e^7-204
204*a^3*c*d*e^6-306306*a^2*b^2*d*e^6+525096*a^2*b*c*d^2*e^5-233376*a^2*c^2*d^3*e^4+175032*a*b^3*d^2*e^5-466752
*a*b^2*c*d^3*e^4+424320*a*b*c^2*d^4*e^3-130560*a*c^3*d^5*e^2-38896*b^4*d^3*e^4+141440*b^3*c*d^4*e^3-195840*b^2
*c^2*d^5*e^2+121856*b*c^3*d^6*e-28672*c^4*d^7)/e^8

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maxima [A]  time = 0.65, size = 645, normalized size = 1.51 \begin {gather*} \frac {2 \, {\left (90090 \, {\left (e x + d\right )}^{\frac {17}{2}} c^{4} - 357357 \, {\left (2 \, c^{4} d - b c^{3} e\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 176715 \, {\left (14 \, c^{4} d^{2} - 14 \, b c^{3} d e + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 348075 \, {\left (14 \, c^{4} d^{3} - 21 \, b c^{3} d^{2} e + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{2} - {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 85085 \, {\left (70 \, c^{4} d^{4} - 140 \, b c^{3} d^{3} e + 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{2} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 328185 \, {\left (14 \, c^{4} d^{5} - 35 \, b c^{3} d^{4} e + 10 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{2} - 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{4} - {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{5}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 153153 \, {\left (14 \, c^{4} d^{6} - 42 \, b c^{3} d^{5} e + 15 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{2} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{3} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{4} - 6 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{5} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{6}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 255255 \, {\left (2 \, c^{4} d^{7} - 7 \, b c^{3} d^{6} e - a^{3} b e^{7} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} - 5 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{4} - 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{5} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{6}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{765765 \, e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)^3*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/765765*(90090*(e*x + d)^(17/2)*c^4 - 357357*(2*c^4*d - b*c^3*e)*(e*x + d)^(15/2) + 176715*(14*c^4*d^2 - 14*b
*c^3*d*e + (3*b^2*c^2 + 2*a*c^3)*e^2)*(e*x + d)^(13/2) - 348075*(14*c^4*d^3 - 21*b*c^3*d^2*e + 3*(3*b^2*c^2 +
2*a*c^3)*d*e^2 - (b^3*c + 3*a*b*c^2)*e^3)*(e*x + d)^(11/2) + 85085*(70*c^4*d^4 - 140*b*c^3*d^3*e + 30*(3*b^2*c
^2 + 2*a*c^3)*d^2*e^2 - 20*(b^3*c + 3*a*b*c^2)*d*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^4)*(e*x + d)^(9/2) - 3
28185*(14*c^4*d^5 - 35*b*c^3*d^4*e + 10*(3*b^2*c^2 + 2*a*c^3)*d^3*e^2 - 10*(b^3*c + 3*a*b*c^2)*d^2*e^3 + (b^4
+ 12*a*b^2*c + 6*a^2*c^2)*d*e^4 - (a*b^3 + 3*a^2*b*c)*e^5)*(e*x + d)^(7/2) + 153153*(14*c^4*d^6 - 42*b*c^3*d^5
*e + 15*(3*b^2*c^2 + 2*a*c^3)*d^4*e^2 - 20*(b^3*c + 3*a*b*c^2)*d^3*e^3 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*
e^4 - 6*(a*b^3 + 3*a^2*b*c)*d*e^5 + (3*a^2*b^2 + 2*a^3*c)*e^6)*(e*x + d)^(5/2) - 255255*(2*c^4*d^7 - 7*b*c^3*d
^6*e - a^3*b*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 5*(b^3*c + 3*a*b*c^2)*d^4*e^3 + (b^4 + 12*a*b^2*c + 6*a^2
*c^2)*d^3*e^4 - 3*(a*b^3 + 3*a^2*b*c)*d^2*e^5 + (3*a^2*b^2 + 2*a^3*c)*d*e^6)*(e*x + d)^(3/2))/e^8

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mupad [B]  time = 0.13, size = 444, normalized size = 1.04 \begin {gather*} \frac {{\left (d+e\,x\right )}^{13/2}\,\left (18\,b^2\,c^2\,e^2-84\,b\,c^3\,d\,e+84\,c^4\,d^2+12\,a\,c^3\,e^2\right )}{13\,e^8}+\frac {4\,c^4\,{\left (d+e\,x\right )}^{17/2}}{17\,e^8}-\frac {\left (28\,c^4\,d-14\,b\,c^3\,e\right )\,{\left (d+e\,x\right )}^{15/2}}{15\,e^8}+\frac {{\left (d+e\,x\right )}^{9/2}\,\left (12\,a^2\,c^2\,e^4+24\,a\,b^2\,c\,e^4-120\,a\,b\,c^2\,d\,e^3+120\,a\,c^3\,d^2\,e^2+2\,b^4\,e^4-40\,b^3\,c\,d\,e^3+180\,b^2\,c^2\,d^2\,e^2-280\,b\,c^3\,d^3\,e+140\,c^4\,d^4\right )}{9\,e^8}+\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 )}^3}{3\,e^8}+\frac {6\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,\left (3\,a^2\,c\,e^4+a\,b^2\,e^4-10\,a\,b\,c\,d\,e^3+10\,a\,c^2\,d^2\,e^2-b^3\,d\,e^3+8\,b^2\,c\,d^2\,e^2-14\,b\,c^2\,d^3\,e+7\,c^3\,d^4\right )}{7\,e^8}+\frac {2\,{\left (d+e\,x\right )}^{5/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2\,\left (3\,b^2\,e^2-14\,b\,c\,d\,e+14\,c^2\,d^2+2\,a\,c\,e^2\right )}{5\,e^8}+\frac {10\,c\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{11/2}\,\left (b^2\,e^2-7\,b\,c\,d\,e+7\,c^2\,d^2+3\,a\,c\,e^2\right )}{11\,e^8} \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)^3,x)

[Out]

((d + e*x)^(13/2)*(84*c^4*d^2 + 12*a*c^3*e^2 + 18*b^2*c^2*e^2 - 84*b*c^3*d*e))/(13*e^8) + (4*c^4*(d + e*x)^(17
/2))/(17*e^8) - ((28*c^4*d - 14*b*c^3*e)*(d + e*x)^(15/2))/(15*e^8) + ((d + e*x)^(9/2)*(2*b^4*e^4 + 140*c^4*d^
4 + 12*a^2*c^2*e^4 + 120*a*c^3*d^2*e^2 + 180*b^2*c^2*d^2*e^2 + 24*a*b^2*c*e^4 - 280*b*c^3*d^3*e - 40*b^3*c*d*e
^3 - 120*a*b*c^2*d*e^3))/(9*e^8) + (2*(b*e - 2*c*d)*(d + e*x)^(3/2)*(a*e^2 + c*d^2 - b*d*e)^3)/(3*e^8) + (6*(b
*e - 2*c*d)*(d + e*x)^(7/2)*(7*c^3*d^4 + a*b^2*e^4 + 3*a^2*c*e^4 - b^3*d*e^3 + 10*a*c^2*d^2*e^2 + 8*b^2*c*d^2*
e^2 - 14*b*c^2*d^3*e - 10*a*b*c*d*e^3))/(7*e^8) + (2*(d + e*x)^(5/2)*(a*e^2 + c*d^2 - b*d*e)^2*(3*b^2*e^2 + 14
*c^2*d^2 + 2*a*c*e^2 - 14*b*c*d*e))/(5*e^8) + (10*c*(b*e - 2*c*d)*(d + e*x)^(11/2)*(b^2*e^2 + 7*c^2*d^2 + 3*a*
c*e^2 - 7*b*c*d*e))/(11*e^8)

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sympy [A]  time = 13.41, size = 843, normalized size = 1.97 \begin {gather*} \frac {2 \left (\frac {2 c^{4} \left (d + e x\right )^{\frac {17}{2}}}{17 e^{7}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \left (7 b c^{3} e - 14 c^{4} d\right )}{15 e^{7}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \left (6 a c^{3} e^{2} + 9 b^{2} c^{2} e^{2} - 42 b c^{3} d e + 42 c^{4} d^{2}\right )}{13 e^{7}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \left (15 a b c^{2} e^{3} - 30 a c^{3} d e^{2} + 5 b^{3} c e^{3} - 45 b^{2} c^{2} d e^{2} + 105 b c^{3} d^{2} e - 70 c^{4} d^{3}\right )}{11 e^{7}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (6 a^{2} c^{2} e^{4} + 12 a b^{2} c e^{4} - 60 a b c^{2} d e^{3} + 60 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 20 b^{3} c d e^{3} + 90 b^{2} c^{2} d^{2} e^{2} - 140 b c^{3} d^{3} e + 70 c^{4} d^{4}\right )}{9 e^{7}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (9 a^{2} b c e^{5} - 18 a^{2} c^{2} d e^{4} + 3 a b^{3} e^{5} - 36 a b^{2} c d e^{4} + 90 a b c^{2} d^{2} e^{3} - 60 a c^{3} d^{3} e^{2} - 3 b^{4} d e^{4} + 30 b^{3} c d^{2} e^{3} - 90 b^{2} c^{2} d^{3} e^{2} + 105 b c^{3} d^{4} e - 42 c^{4} d^{5}\right )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (2 a^{3} c e^{6} + 3 a^{2} b^{2} e^{6} - 18 a^{2} b c d e^{5} + 18 a^{2} c^{2} d^{2} e^{4} - 6 a b^{3} d e^{5} + 36 a b^{2} c d^{2} e^{4} - 60 a b c^{2} d^{3} e^{3} + 30 a c^{3} d^{4} e^{2} + 3 b^{4} d^{2} e^{4} - 20 b^{3} c d^{3} e^{3} + 45 b^{2} c^{2} d^{4} e^{2} - 42 b c^{3} d^{5} e + 14 c^{4} d^{6}\right )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a^{3} b e^{7} - 2 a^{3} c d e^{6} - 3 a^{2} b^{2} d e^{6} + 9 a^{2} b c d^{2} e^{5} - 6 a^{2} c^{2} d^{3} e^{4} + 3 a b^{3} d^{2} e^{5} - 12 a b^{2} c d^{3} e^{4} + 15 a b c^{2} d^{4} e^{3} - 6 a c^{3} d^{5} e^{2} - b^{4} d^{3} e^{4} + 5 b^{3} c d^{4} e^{3} - 9 b^{2} c^{2} d^{5} e^{2} + 7 b c^{3} d^{6} e - 2 c^{4} d^{7}\right )}{3 e^{7}}\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)**3*(e*x+d)**(1/2),x)

[Out]

2*(2*c**4*(d + e*x)**(17/2)/(17*e**7) + (d + e*x)**(15/2)*(7*b*c**3*e - 14*c**4*d)/(15*e**7) + (d + e*x)**(13/
2)*(6*a*c**3*e**2 + 9*b**2*c**2*e**2 - 42*b*c**3*d*e + 42*c**4*d**2)/(13*e**7) + (d + e*x)**(11/2)*(15*a*b*c**
2*e**3 - 30*a*c**3*d*e**2 + 5*b**3*c*e**3 - 45*b**2*c**2*d*e**2 + 105*b*c**3*d**2*e - 70*c**4*d**3)/(11*e**7)
+ (d + e*x)**(9/2)*(6*a**2*c**2*e**4 + 12*a*b**2*c*e**4 - 60*a*b*c**2*d*e**3 + 60*a*c**3*d**2*e**2 + b**4*e**4
- 20*b**3*c*d*e**3 + 90*b**2*c**2*d**2*e**2 - 140*b*c**3*d**3*e + 70*c**4*d**4)/(9*e**7) + (d + e*x)**(7/2)*(
9*a**2*b*c*e**5 - 18*a**2*c**2*d*e**4 + 3*a*b**3*e**5 - 36*a*b**2*c*d*e**4 + 90*a*b*c**2*d**2*e**3 - 60*a*c**3
*d**3*e**2 - 3*b**4*d*e**4 + 30*b**3*c*d**2*e**3 - 90*b**2*c**2*d**3*e**2 + 105*b*c**3*d**4*e - 42*c**4*d**5)/
(7*e**7) + (d + e*x)**(5/2)*(2*a**3*c*e**6 + 3*a**2*b**2*e**6 - 18*a**2*b*c*d*e**5 + 18*a**2*c**2*d**2*e**4 -
6*a*b**3*d*e**5 + 36*a*b**2*c*d**2*e**4 - 60*a*b*c**2*d**3*e**3 + 30*a*c**3*d**4*e**2 + 3*b**4*d**2*e**4 - 20*
b**3*c*d**3*e**3 + 45*b**2*c**2*d**4*e**2 - 42*b*c**3*d**5*e + 14*c**4*d**6)/(5*e**7) + (d + e*x)**(3/2)*(a**3
*b*e**7 - 2*a**3*c*d*e**6 - 3*a**2*b**2*d*e**6 + 9*a**2*b*c*d**2*e**5 - 6*a**2*c**2*d**3*e**4 + 3*a*b**3*d**2*
e**5 - 12*a*b**2*c*d**3*e**4 + 15*a*b*c**2*d**4*e**3 - 6*a*c**3*d**5*e**2 - b**4*d**3*e**4 + 5*b**3*c*d**4*e**
3 - 9*b**2*c**2*d**5*e**2 + 7*b*c**3*d**6*e - 2*c**4*d**7)/(3*e**7))/e

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