3.1609 \(\int (b+2 c x) \sqrt{d+e x} (a+b x+c x^2)^3 \, dx\)

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.238388, antiderivative size = 427, normalized size of antiderivative = 1., 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} \[ \frac{2 (d+e x)^{9/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{9 e^8}+\frac{6 c^2 (d+e x)^{13/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{13 e^8}-\frac{10 c (d+e x)^{11/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{11 e^8}-\frac{6 (d+e x)^{7/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{7 e^8}+\frac{2 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^8}-\frac{2 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8}-\frac{14 c^3 (d+e x)^{15/2} (2 c d-b e)}{15 e^8}+\frac{4 c^4 (d+e x)^{17/2}}{17 e^8} \]

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*}

Mathematica [A]  time = 0.632712, size = 601, normalized size = 1.41 \[ \frac{2 (d+e x)^{3/2} \left (-51 c^2 e^2 \left (286 a^2 e^2 \left (-24 d^2 e x+16 d^3+30 d e^2 x^2-35 e^3 x^3\right )-65 a b e \left (240 d^2 e^2 x^2-192 d^3 e x+128 d^4-280 d e^3 x^3+315 e^4 x^4\right )+15 b^2 \left (480 d^3 e^2 x^2-560 d^2 e^3 x^3-384 d^4 e x+256 d^5+630 d e^4 x^4-693 e^5 x^5\right )\right )+221 c e^3 \left (297 a^2 b e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+462 a^3 e^3 (3 e x-2 d)+132 a b^2 e \left (24 d^2 e x-16 d^3-30 d e^2 x^2+35 e^3 x^3\right )+5 b^3 \left (240 d^2 e^2 x^2-192 d^3 e x+128 d^4-280 d e^3 x^3+315 e^4 x^4\right )\right )+2431 b e^4 \left (63 a^2 b e^2 (3 e x-2 d)+105 a^3 e^3+9 a b^2 e \left (8 d^2-12 d e x+15 e^2 x^2\right )+b^3 \left (24 d^2 e x-16 d^3-30 d e^2 x^2+35 e^3 x^3\right )\right )+17 c^3 e \left (30 a e \left (-480 d^3 e^2 x^2+560 d^2 e^3 x^3+384 d^4 e x-256 d^5-630 d e^4 x^4+693 e^5 x^5\right )+7 b \left (1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-1536 d^5 e x+1024 d^6-2772 d e^5 x^5+3003 e^6 x^6\right )\right )-14 c^4 \left (3840 d^5 e^2 x^2-4480 d^4 e^3 x^3+5040 d^3 e^4 x^4-5544 d^2 e^5 x^5-3072 d^6 e x+2048 d^7+6006 d e^6 x^6-6435 e^7 x^7\right )\right )}{765765 e^8} \]

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)

________________________________________________________________________________________

Maple [B]  time = 0.006, size = 795, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}

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

________________________________________________________________________________________

Maxima [A]  time = 1.04133, size = 871, normalized size = 2.04 \begin{align*} \text{result too large to display} \end{align*}

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

________________________________________________________________________________________

Fricas [B]  time = 1.38657, size = 1863, normalized size = 4.36 \begin{align*} \text{result too large to display} \end{align*}

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

________________________________________________________________________________________

Sympy [A]  time = 12.3477, size = 843, normalized size = 1.97 \begin{align*} \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{align*}

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

________________________________________________________________________________________

Giac [B]  time = 1.29283, size = 1141, normalized size = 2.67 \begin{align*} \text{result too large to display} \end{align*}

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*(153153*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^2*b^2*e^(-1) + 102102*(3*(x*e + d)^(5/2) - 5*(x*e
+ d)^(3/2)*d)*a^3*c*e^(-1) + 21879*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a*b^3
*e^(-2) + 65637*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^2*b*c*e^(-2) + 2431*(35
*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*b^4*e^(-3) + 291
72*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*a*b^2*c*e^
(-3) + 14586*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*
a^2*c^2*e^(-3) + 1105*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e +
d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*b^3*c*e^(-4) + 3315*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d +
2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*a*b*c^2*e^(-4) + 765*(693*(x*e
+ d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d
)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*b^2*c^2*e^(-5) + 510*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d +
10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*
a*c^3*e^(-5) + 119*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 100100*(x*
e + d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*b*c^3*e^
(-6) + 14*(6435*(x*e + d)^(17/2) - 51051*(x*e + d)^(15/2)*d + 176715*(x*e + d)^(13/2)*d^2 - 348075*(x*e + d)^(
11/2)*d^3 + 425425*(x*e + d)^(9/2)*d^4 - 328185*(x*e + d)^(7/2)*d^5 + 153153*(x*e + d)^(5/2)*d^6 - 36465*(x*e
+ d)^(3/2)*d^7)*c^4*e^(-7) + 255255*(x*e + d)^(3/2)*a^3*b)*e^(-1)