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

Optimal. Leaf size=421 \[ \frac {2 (d+e x)^{5/2} \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{5 e^8}+\frac {2 c^2 (d+e x)^{9/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^8}-\frac {10 c (d+e x)^{7/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{7 e^8}-\frac {2 (d+e x)^{3/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8}+\frac {2 \sqrt {d+e x} \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^8 \sqrt {d+e x}}-\frac {14 c^3 (d+e x)^{11/2} (2 c d-b e)}{11 e^8}+\frac {4 c^4 (d+e x)^{13/2}}{13 e^8} \]

________________________________________________________________________________________

Rubi [A]  time = 0.24, antiderivative size = 421, 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)^{5/2} \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 )}{5 e^8}+\frac {2 c^2 (d+e x)^{9/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^8}-\frac {10 c (d+e x)^{7/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{7 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 ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8}+\frac {2 \sqrt {d+e x} \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 )}{e^8}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^8 \sqrt {d+e x}}-\frac {14 c^3 (d+e x)^{11/2} (2 c d-b e)}{11 e^8}+\frac {4 c^4 (d+e x)^{13/2}}{13 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.54, size = 594, normalized size = 1.41 \begin {gather*} \frac {2 \left (429 c^2 e^2 \left (42 a^2 e^2 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )+15 a b e \left (-128 d^4-64 d^3 e x+16 d^2 e^2 x^2-8 d e^3 x^3+5 e^4 x^4\right )+5 b^2 \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )\right )+429 c e^3 \left (70 a^3 e^3 (2 d+e x)+105 a^2 b e^2 \left (-8 d^2-4 d e x+e^2 x^2\right )+84 a b^2 e \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )-5 b^3 \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )\right )+3003 b e^4 \left (-5 a^3 e^3+15 a^2 b e^2 (2 d+e x)+5 a b^2 e \left (-8 d^2-4 d e x+e^2 x^2\right )+b^3 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )-65 c^3 e \left (7 b \left (1024 d^6+512 d^5 e x-128 d^4 e^2 x^2+64 d^3 e^3 x^3-40 d^2 e^4 x^4+28 d e^5 x^5-21 e^6 x^6\right )-22 a e \left (256 d^5+128 d^4 e x-32 d^3 e^2 x^2+16 d^2 e^3 x^3-10 d e^4 x^4+7 e^5 x^5\right )\right )+70 c^4 \left (2048 d^7+1024 d^6 e x-256 d^5 e^2 x^2+128 d^4 e^3 x^3-80 d^3 e^4 x^4+56 d^2 e^5 x^5-42 d e^6 x^6+33 e^7 x^7\right )\right )}{15015 e^8 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(70*c^4*(2048*d^7 + 1024*d^6*e*x - 256*d^5*e^2*x^2 + 128*d^4*e^3*x^3 - 80*d^3*e^4*x^4 + 56*d^2*e^5*x^5 - 42
*d*e^6*x^6 + 33*e^7*x^7) + 3003*b*e^4*(-5*a^3*e^3 + 15*a^2*b*e^2*(2*d + e*x) + 5*a*b^2*e*(-8*d^2 - 4*d*e*x + e
^2*x^2) + b^3*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3)) + 429*c*e^3*(70*a^3*e^3*(2*d + e*x) + 105*a^2*b*e^
2*(-8*d^2 - 4*d*e*x + e^2*x^2) + 84*a*b^2*e*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) - 5*b^3*(128*d^4 + 64
*d^3*e*x - 16*d^2*e^2*x^2 + 8*d*e^3*x^3 - 5*e^4*x^4)) + 429*c^2*e^2*(42*a^2*e^2*(16*d^3 + 8*d^2*e*x - 2*d*e^2*
x^2 + e^3*x^3) + 15*a*b*e*(-128*d^4 - 64*d^3*e*x + 16*d^2*e^2*x^2 - 8*d*e^3*x^3 + 5*e^4*x^4) + 5*b^2*(256*d^5
+ 128*d^4*e*x - 32*d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 10*d*e^4*x^4 + 7*e^5*x^5)) - 65*c^3*e*(-22*a*e*(256*d^5 + 12
8*d^4*e*x - 32*d^3*e^2*x^2 + 16*d^2*e^3*x^3 - 10*d*e^4*x^4 + 7*e^5*x^5) + 7*b*(1024*d^6 + 512*d^5*e*x - 128*d^
4*e^2*x^2 + 64*d^3*e^3*x^3 - 40*d^2*e^4*x^4 + 28*d*e^5*x^5 - 21*e^6*x^6))))/(15015*e^8*Sqrt[d + e*x])

________________________________________________________________________________________

IntegrateAlgebraic [B]  time = 0.34, size = 951, normalized size = 2.26 \begin {gather*} \frac {2 \left (30030 c^4 d^7-105105 b c^3 e d^6+210210 c^4 (d+e x) d^6+90090 a c^3 e^2 d^5+135135 b^2 c^2 e^2 d^5-210210 c^4 (d+e x)^2 d^5-630630 b c^3 e (d+e x) d^5-225225 a b c^2 e^3 d^4-75075 b^3 c e^3 d^4+210210 c^4 (d+e x)^3 d^4+525525 b c^3 e (d+e x)^2 d^4+450450 a c^3 e^2 (d+e x) d^4+675675 b^2 c^2 e^2 (d+e x) d^4+15015 b^4 e^4 d^3+90090 a^2 c^2 e^4 d^3+180180 a b^2 c e^4 d^3-150150 c^4 (d+e x)^4 d^3-420420 b c^3 e (d+e x)^3 d^3-300300 a c^3 e^2 (d+e x)^2 d^3-450450 b^2 c^2 e^2 (d+e x)^2 d^3-900900 a b c^2 e^3 (d+e x) d^3-300300 b^3 c e^3 (d+e x) d^3-45045 a b^3 e^5 d^2-135135 a^2 b c e^5 d^2+70070 c^4 (d+e x)^5 d^2+225225 b c^3 e (d+e x)^4 d^2+180180 a c^3 e^2 (d+e x)^3 d^2+270270 b^2 c^2 e^2 (d+e x)^3 d^2+450450 a b c^2 e^3 (d+e x)^2 d^2+150150 b^3 c e^3 (d+e x)^2 d^2+45045 b^4 e^4 (d+e x) d^2+270270 a^2 c^2 e^4 (d+e x) d^2+540540 a b^2 c e^4 (d+e x) d^2+45045 a^2 b^2 e^6 d+30030 a^3 c e^6 d-19110 c^4 (d+e x)^6 d-70070 b c^3 e (d+e x)^5 d-64350 a c^3 e^2 (d+e x)^4 d-96525 b^2 c^2 e^2 (d+e x)^4 d-180180 a b c^2 e^3 (d+e x)^3 d-60060 b^3 c e^3 (d+e x)^3 d-15015 b^4 e^4 (d+e x)^2 d-90090 a^2 c^2 e^4 (d+e x)^2 d-180180 a b^2 c e^4 (d+e x)^2 d-90090 a b^3 e^5 (d+e x) d-270270 a^2 b c e^5 (d+e x) d-15015 a^3 b e^7+2310 c^4 (d+e x)^7+9555 b c^3 e (d+e x)^6+10010 a c^3 e^2 (d+e x)^5+15015 b^2 c^2 e^2 (d+e x)^5+32175 a b c^2 e^3 (d+e x)^4+10725 b^3 c e^3 (d+e x)^4+3003 b^4 e^4 (d+e x)^3+18018 a^2 c^2 e^4 (d+e x)^3+36036 a b^2 c e^4 (d+e x)^3+15015 a b^3 e^5 (d+e x)^2+45045 a^2 b c e^5 (d+e x)^2+45045 a^2 b^2 e^6 (d+e x)+30030 a^3 c e^6 (d+e x)\right )}{15015 e^8 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(30030*c^4*d^7 - 105105*b*c^3*d^6*e + 135135*b^2*c^2*d^5*e^2 + 90090*a*c^3*d^5*e^2 - 75075*b^3*c*d^4*e^3 -
225225*a*b*c^2*d^4*e^3 + 15015*b^4*d^3*e^4 + 180180*a*b^2*c*d^3*e^4 + 90090*a^2*c^2*d^3*e^4 - 45045*a*b^3*d^2*
e^5 - 135135*a^2*b*c*d^2*e^5 + 45045*a^2*b^2*d*e^6 + 30030*a^3*c*d*e^6 - 15015*a^3*b*e^7 + 210210*c^4*d^6*(d +
e*x) - 630630*b*c^3*d^5*e*(d + e*x) + 675675*b^2*c^2*d^4*e^2*(d + e*x) + 450450*a*c^3*d^4*e^2*(d + e*x) - 300
300*b^3*c*d^3*e^3*(d + e*x) - 900900*a*b*c^2*d^3*e^3*(d + e*x) + 45045*b^4*d^2*e^4*(d + e*x) + 540540*a*b^2*c*
d^2*e^4*(d + e*x) + 270270*a^2*c^2*d^2*e^4*(d + e*x) - 90090*a*b^3*d*e^5*(d + e*x) - 270270*a^2*b*c*d*e^5*(d +
e*x) + 45045*a^2*b^2*e^6*(d + e*x) + 30030*a^3*c*e^6*(d + e*x) - 210210*c^4*d^5*(d + e*x)^2 + 525525*b*c^3*d^
4*e*(d + e*x)^2 - 450450*b^2*c^2*d^3*e^2*(d + e*x)^2 - 300300*a*c^3*d^3*e^2*(d + e*x)^2 + 150150*b^3*c*d^2*e^3
*(d + e*x)^2 + 450450*a*b*c^2*d^2*e^3*(d + e*x)^2 - 15015*b^4*d*e^4*(d + e*x)^2 - 180180*a*b^2*c*d*e^4*(d + e*
x)^2 - 90090*a^2*c^2*d*e^4*(d + e*x)^2 + 15015*a*b^3*e^5*(d + e*x)^2 + 45045*a^2*b*c*e^5*(d + e*x)^2 + 210210*
c^4*d^4*(d + e*x)^3 - 420420*b*c^3*d^3*e*(d + e*x)^3 + 270270*b^2*c^2*d^2*e^2*(d + e*x)^3 + 180180*a*c^3*d^2*e
^2*(d + e*x)^3 - 60060*b^3*c*d*e^3*(d + e*x)^3 - 180180*a*b*c^2*d*e^3*(d + e*x)^3 + 3003*b^4*e^4*(d + e*x)^3 +
36036*a*b^2*c*e^4*(d + e*x)^3 + 18018*a^2*c^2*e^4*(d + e*x)^3 - 150150*c^4*d^3*(d + e*x)^4 + 225225*b*c^3*d^2
*e*(d + e*x)^4 - 96525*b^2*c^2*d*e^2*(d + e*x)^4 - 64350*a*c^3*d*e^2*(d + e*x)^4 + 10725*b^3*c*e^3*(d + e*x)^4
+ 32175*a*b*c^2*e^3*(d + e*x)^4 + 70070*c^4*d^2*(d + e*x)^5 - 70070*b*c^3*d*e*(d + e*x)^5 + 15015*b^2*c^2*e^2
*(d + e*x)^5 + 10010*a*c^3*e^2*(d + e*x)^5 - 19110*c^4*d*(d + e*x)^6 + 9555*b*c^3*e*(d + e*x)^6 + 2310*c^4*(d
+ e*x)^7))/(15015*e^8*Sqrt[d + e*x])

________________________________________________________________________________________

fricas [A]  time = 0.41, size = 657, normalized size = 1.56 \begin {gather*} \frac {2 \, {\left (2310 \, c^{4} e^{7} x^{7} + 143360 \, c^{4} d^{7} - 465920 \, b c^{3} d^{6} e - 15015 \, a^{3} b e^{7} + 183040 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} - 274560 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} + 48048 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{4} - 120120 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{5} + 30030 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{6} - 735 \, {\left (4 \, c^{4} d e^{6} - 13 \, b c^{3} e^{7}\right )} x^{6} + 35 \, {\left (112 \, c^{4} d^{2} e^{5} - 364 \, b c^{3} d e^{6} + 143 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{7}\right )} x^{5} - 25 \, {\left (224 \, c^{4} d^{3} e^{4} - 728 \, b c^{3} d^{2} e^{5} + 286 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{6} - 429 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{7}\right )} x^{4} + {\left (8960 \, c^{4} d^{4} e^{3} - 29120 \, b c^{3} d^{3} e^{4} + 11440 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{5} - 17160 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{6} + 3003 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{7}\right )} x^{3} - {\left (17920 \, c^{4} d^{5} e^{2} - 58240 \, b c^{3} d^{4} e^{3} + 22880 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{4} - 34320 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{5} + 6006 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{6} - 15015 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{7}\right )} x^{2} + {\left (71680 \, c^{4} d^{6} e - 232960 \, b c^{3} d^{5} e^{2} + 91520 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{3} - 137280 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{4} + 24024 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{5} - 60060 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{6} + 15015 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{7}\right )} x\right )} \sqrt {e x + d}}{15015 \, {\left (e^{9} x + d e^{8}\right )}} \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)^(3/2),x, algorithm="fricas")

[Out]

2/15015*(2310*c^4*e^7*x^7 + 143360*c^4*d^7 - 465920*b*c^3*d^6*e - 15015*a^3*b*e^7 + 183040*(3*b^2*c^2 + 2*a*c^
3)*d^5*e^2 - 274560*(b^3*c + 3*a*b*c^2)*d^4*e^3 + 48048*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 - 120120*(a*b^3
+ 3*a^2*b*c)*d^2*e^5 + 30030*(3*a^2*b^2 + 2*a^3*c)*d*e^6 - 735*(4*c^4*d*e^6 - 13*b*c^3*e^7)*x^6 + 35*(112*c^4
*d^2*e^5 - 364*b*c^3*d*e^6 + 143*(3*b^2*c^2 + 2*a*c^3)*e^7)*x^5 - 25*(224*c^4*d^3*e^4 - 728*b*c^3*d^2*e^5 + 28
6*(3*b^2*c^2 + 2*a*c^3)*d*e^6 - 429*(b^3*c + 3*a*b*c^2)*e^7)*x^4 + (8960*c^4*d^4*e^3 - 29120*b*c^3*d^3*e^4 + 1
1440*(3*b^2*c^2 + 2*a*c^3)*d^2*e^5 - 17160*(b^3*c + 3*a*b*c^2)*d*e^6 + 3003*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^7
)*x^3 - (17920*c^4*d^5*e^2 - 58240*b*c^3*d^4*e^3 + 22880*(3*b^2*c^2 + 2*a*c^3)*d^3*e^4 - 34320*(b^3*c + 3*a*b*
c^2)*d^2*e^5 + 6006*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^6 - 15015*(a*b^3 + 3*a^2*b*c)*e^7)*x^2 + (71680*c^4*d^6
*e - 232960*b*c^3*d^5*e^2 + 91520*(3*b^2*c^2 + 2*a*c^3)*d^4*e^3 - 137280*(b^3*c + 3*a*b*c^2)*d^3*e^4 + 24024*(
b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^5 - 60060*(a*b^3 + 3*a^2*b*c)*d*e^6 + 15015*(3*a^2*b^2 + 2*a^3*c)*e^7)*x)*
sqrt(e*x + d)/(e^9*x + d*e^8)

________________________________________________________________________________________

giac [B]  time = 0.31, size = 1000, normalized size = 2.38

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)^(3/2),x, algorithm="giac")

[Out]

2/15015*(2310*(x*e + d)^(13/2)*c^4*e^96 - 19110*(x*e + d)^(11/2)*c^4*d*e^96 + 70070*(x*e + d)^(9/2)*c^4*d^2*e^
96 - 150150*(x*e + d)^(7/2)*c^4*d^3*e^96 + 210210*(x*e + d)^(5/2)*c^4*d^4*e^96 - 210210*(x*e + d)^(3/2)*c^4*d^
5*e^96 + 210210*sqrt(x*e + d)*c^4*d^6*e^96 + 9555*(x*e + d)^(11/2)*b*c^3*e^97 - 70070*(x*e + d)^(9/2)*b*c^3*d*
e^97 + 225225*(x*e + d)^(7/2)*b*c^3*d^2*e^97 - 420420*(x*e + d)^(5/2)*b*c^3*d^3*e^97 + 525525*(x*e + d)^(3/2)*
b*c^3*d^4*e^97 - 630630*sqrt(x*e + d)*b*c^3*d^5*e^97 + 15015*(x*e + d)^(9/2)*b^2*c^2*e^98 + 10010*(x*e + d)^(9
/2)*a*c^3*e^98 - 96525*(x*e + d)^(7/2)*b^2*c^2*d*e^98 - 64350*(x*e + d)^(7/2)*a*c^3*d*e^98 + 270270*(x*e + d)^
(5/2)*b^2*c^2*d^2*e^98 + 180180*(x*e + d)^(5/2)*a*c^3*d^2*e^98 - 450450*(x*e + d)^(3/2)*b^2*c^2*d^3*e^98 - 300
300*(x*e + d)^(3/2)*a*c^3*d^3*e^98 + 675675*sqrt(x*e + d)*b^2*c^2*d^4*e^98 + 450450*sqrt(x*e + d)*a*c^3*d^4*e^
98 + 10725*(x*e + d)^(7/2)*b^3*c*e^99 + 32175*(x*e + d)^(7/2)*a*b*c^2*e^99 - 60060*(x*e + d)^(5/2)*b^3*c*d*e^9
9 - 180180*(x*e + d)^(5/2)*a*b*c^2*d*e^99 + 150150*(x*e + d)^(3/2)*b^3*c*d^2*e^99 + 450450*(x*e + d)^(3/2)*a*b
*c^2*d^2*e^99 - 300300*sqrt(x*e + d)*b^3*c*d^3*e^99 - 900900*sqrt(x*e + d)*a*b*c^2*d^3*e^99 + 3003*(x*e + d)^(
5/2)*b^4*e^100 + 36036*(x*e + d)^(5/2)*a*b^2*c*e^100 + 18018*(x*e + d)^(5/2)*a^2*c^2*e^100 - 15015*(x*e + d)^(
3/2)*b^4*d*e^100 - 180180*(x*e + d)^(3/2)*a*b^2*c*d*e^100 - 90090*(x*e + d)^(3/2)*a^2*c^2*d*e^100 + 45045*sqrt
(x*e + d)*b^4*d^2*e^100 + 540540*sqrt(x*e + d)*a*b^2*c*d^2*e^100 + 270270*sqrt(x*e + d)*a^2*c^2*d^2*e^100 + 15
015*(x*e + d)^(3/2)*a*b^3*e^101 + 45045*(x*e + d)^(3/2)*a^2*b*c*e^101 - 90090*sqrt(x*e + d)*a*b^3*d*e^101 - 27
0270*sqrt(x*e + d)*a^2*b*c*d*e^101 + 45045*sqrt(x*e + d)*a^2*b^2*e^102 + 30030*sqrt(x*e + d)*a^3*c*e^102)*e^(-
104) + 2*(2*c^4*d^7 - 7*b*c^3*d^6*e + 9*b^2*c^2*d^5*e^2 + 6*a*c^3*d^5*e^2 - 5*b^3*c*d^4*e^3 - 15*a*b*c^2*d^4*e
^3 + b^4*d^3*e^4 + 12*a*b^2*c*d^3*e^4 + 6*a^2*c^2*d^3*e^4 - 3*a*b^3*d^2*e^5 - 9*a^2*b*c*d^2*e^5 + 3*a^2*b^2*d*
e^6 + 2*a^3*c*d*e^6 - a^3*b*e^7)*e^(-8)/sqrt(x*e + d)

________________________________________________________________________________________

maple [B]  time = 0.06, size = 795, normalized size = 1.89 \begin {gather*} -\frac {2 \left (-2310 c^{4} e^{7} x^{7}-9555 b \,c^{3} e^{7} x^{6}+2940 c^{4} d \,e^{6} x^{6}-10010 a \,c^{3} e^{7} x^{5}-15015 b^{2} c^{2} e^{7} x^{5}+12740 b \,c^{3} d \,e^{6} x^{5}-3920 c^{4} d^{2} e^{5} x^{5}-32175 a b \,c^{2} e^{7} x^{4}+14300 a \,c^{3} d \,e^{6} x^{4}-10725 b^{3} c \,e^{7} x^{4}+21450 b^{2} c^{2} d \,e^{6} x^{4}-18200 b \,c^{3} d^{2} e^{5} x^{4}+5600 c^{4} d^{3} e^{4} x^{4}-18018 a^{2} c^{2} e^{7} x^{3}-36036 a \,b^{2} c \,e^{7} x^{3}+51480 a b \,c^{2} d \,e^{6} x^{3}-22880 a \,c^{3} d^{2} e^{5} x^{3}-3003 b^{4} e^{7} x^{3}+17160 b^{3} c d \,e^{6} x^{3}-34320 b^{2} c^{2} d^{2} e^{5} x^{3}+29120 b \,c^{3} d^{3} e^{4} x^{3}-8960 c^{4} d^{4} e^{3} x^{3}-45045 a^{2} b c \,e^{7} x^{2}+36036 a^{2} c^{2} d \,e^{6} x^{2}-15015 a \,b^{3} e^{7} x^{2}+72072 a \,b^{2} c d \,e^{6} x^{2}-102960 a b \,c^{2} d^{2} e^{5} x^{2}+45760 a \,c^{3} d^{3} e^{4} x^{2}+6006 b^{4} d \,e^{6} x^{2}-34320 b^{3} c \,d^{2} e^{5} x^{2}+68640 b^{2} c^{2} d^{3} e^{4} x^{2}-58240 b \,c^{3} d^{4} e^{3} x^{2}+17920 c^{4} d^{5} e^{2} x^{2}-30030 a^{3} c \,e^{7} x -45045 a^{2} b^{2} e^{7} x +180180 a^{2} b c d \,e^{6} x -144144 a^{2} c^{2} d^{2} e^{5} x +60060 a \,b^{3} d \,e^{6} x -288288 a \,b^{2} c \,d^{2} e^{5} x +411840 a b \,c^{2} d^{3} e^{4} x -183040 a \,c^{3} d^{4} e^{3} x -24024 b^{4} d^{2} e^{5} x +137280 b^{3} c \,d^{3} e^{4} x -274560 b^{2} c^{2} d^{4} e^{3} x +232960 b \,c^{3} d^{5} e^{2} x -71680 c^{4} d^{6} e x +15015 b \,a^{3} e^{7}-60060 a^{3} c d \,e^{6}-90090 a^{2} b^{2} d \,e^{6}+360360 a^{2} b c \,d^{2} e^{5}-288288 a^{2} c^{2} d^{3} e^{4}+120120 a \,b^{3} d^{2} e^{5}-576576 a \,b^{2} c \,d^{3} e^{4}+823680 a b \,c^{2} d^{4} e^{3}-366080 a \,c^{3} d^{5} e^{2}-48048 b^{4} d^{3} e^{4}+274560 b^{3} c \,d^{4} e^{3}-549120 b^{2} c^{2} d^{5} e^{2}+465920 b \,c^{3} d^{6} e -143360 c^{4} d^{7}\right )}{15015 \sqrt {e x +d}\, 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)^(3/2),x)

[Out]

-2/15015/(e*x+d)^(1/2)*(-2310*c^4*e^7*x^7-9555*b*c^3*e^7*x^6+2940*c^4*d*e^6*x^6-10010*a*c^3*e^7*x^5-15015*b^2*
c^2*e^7*x^5+12740*b*c^3*d*e^6*x^5-3920*c^4*d^2*e^5*x^5-32175*a*b*c^2*e^7*x^4+14300*a*c^3*d*e^6*x^4-10725*b^3*c
*e^7*x^4+21450*b^2*c^2*d*e^6*x^4-18200*b*c^3*d^2*e^5*x^4+5600*c^4*d^3*e^4*x^4-18018*a^2*c^2*e^7*x^3-36036*a*b^
2*c*e^7*x^3+51480*a*b*c^2*d*e^6*x^3-22880*a*c^3*d^2*e^5*x^3-3003*b^4*e^7*x^3+17160*b^3*c*d*e^6*x^3-34320*b^2*c
^2*d^2*e^5*x^3+29120*b*c^3*d^3*e^4*x^3-8960*c^4*d^4*e^3*x^3-45045*a^2*b*c*e^7*x^2+36036*a^2*c^2*d*e^6*x^2-1501
5*a*b^3*e^7*x^2+72072*a*b^2*c*d*e^6*x^2-102960*a*b*c^2*d^2*e^5*x^2+45760*a*c^3*d^3*e^4*x^2+6006*b^4*d*e^6*x^2-
34320*b^3*c*d^2*e^5*x^2+68640*b^2*c^2*d^3*e^4*x^2-58240*b*c^3*d^4*e^3*x^2+17920*c^4*d^5*e^2*x^2-30030*a^3*c*e^
7*x-45045*a^2*b^2*e^7*x+180180*a^2*b*c*d*e^6*x-144144*a^2*c^2*d^2*e^5*x+60060*a*b^3*d*e^6*x-288288*a*b^2*c*d^2
*e^5*x+411840*a*b*c^2*d^3*e^4*x-183040*a*c^3*d^4*e^3*x-24024*b^4*d^2*e^5*x+137280*b^3*c*d^3*e^4*x-274560*b^2*c
^2*d^4*e^3*x+232960*b*c^3*d^5*e^2*x-71680*c^4*d^6*e*x+15015*a^3*b*e^7-60060*a^3*c*d*e^6-90090*a^2*b^2*d*e^6+36
0360*a^2*b*c*d^2*e^5-288288*a^2*c^2*d^3*e^4+120120*a*b^3*d^2*e^5-576576*a*b^2*c*d^3*e^4+823680*a*b*c^2*d^4*e^3
-366080*a*c^3*d^5*e^2-48048*b^4*d^3*e^4+274560*b^3*c*d^4*e^3-549120*b^2*c^2*d^5*e^2+465920*b*c^3*d^6*e-143360*
c^4*d^7)/e^8

________________________________________________________________________________________

maxima [A]  time = 0.54, size = 653, normalized size = 1.55 \begin {gather*} \frac {2 \, {\left (\frac {2310 \, {\left (e x + d\right )}^{\frac {13}{2}} c^{4} - 9555 \, {\left (2 \, c^{4} d - b c^{3} e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 5005 \, {\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 {9}{2}} - 10725 \, {\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 {7}{2}} + 3003 \, {\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 {5}{2}} - 15015 \, {\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 {3}{2}} + 15015 \, {\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 )} \sqrt {e x + d}}{e^{7}} + \frac {15015 \, {\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 )}}{\sqrt {e x + d} e^{7}}\right )}}{15015 \, 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)^(3/2),x, algorithm="maxima")

[Out]

2/15015*((2310*(e*x + d)^(13/2)*c^4 - 9555*(2*c^4*d - b*c^3*e)*(e*x + d)^(11/2) + 5005*(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)^(9/2) - 10725*(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)^(7/2) + 3003*(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)^(5/2) - 15015*(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)^(3/2) + 15015*(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)*sqrt(e*x + d))/e^7 + 15015*(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)/(sqrt(e*x + d)*e^7))/e

________________________________________________________________________________________

mupad [B]  time = 1.94, size = 581, normalized size = 1.38 \begin {gather*} \frac {{\left (d+e\,x\right )}^{9/2}\,\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 )}{9\,e^8}+\frac {4\,c^4\,{\left (d+e\,x\right )}^{13/2}}{13\,e^8}-\frac {\left (28\,c^4\,d-14\,b\,c^3\,e\right )\,{\left (d+e\,x\right )}^{11/2}}{11\,e^8}+\frac {-2\,a^3\,b\,e^7+4\,a^3\,c\,d\,e^6+6\,a^2\,b^2\,d\,e^6-18\,a^2\,b\,c\,d^2\,e^5+12\,a^2\,c^2\,d^3\,e^4-6\,a\,b^3\,d^2\,e^5+24\,a\,b^2\,c\,d^3\,e^4-30\,a\,b\,c^2\,d^4\,e^3+12\,a\,c^3\,d^5\,e^2+2\,b^4\,d^3\,e^4-10\,b^3\,c\,d^4\,e^3+18\,b^2\,c^2\,d^5\,e^2-14\,b\,c^3\,d^6\,e+4\,c^4\,d^7}{e^8\,\sqrt {d+e\,x}}+\frac {{\left (d+e\,x\right )}^{5/2}\,\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 )}{5\,e^8}+\frac {2\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{3/2}\,\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 )}{e^8}+\frac {2\,\sqrt {d+e\,x}\,{\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 )}{e^8}+\frac {10\,c\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,\left (b^2\,e^2-7\,b\,c\,d\,e+7\,c^2\,d^2+3\,a\,c\,e^2\right )}{7\,e^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d + e*x)^(9/2)*(84*c^4*d^2 + 12*a*c^3*e^2 + 18*b^2*c^2*e^2 - 84*b*c^3*d*e))/(9*e^8) + (4*c^4*(d + e*x)^(13/2
))/(13*e^8) - ((28*c^4*d - 14*b*c^3*e)*(d + e*x)^(11/2))/(11*e^8) + (4*c^4*d^7 - 2*a^3*b*e^7 + 2*b^4*d^3*e^4 -
6*a*b^3*d^2*e^5 + 6*a^2*b^2*d*e^6 + 12*a*c^3*d^5*e^2 - 10*b^3*c*d^4*e^3 + 12*a^2*c^2*d^3*e^4 + 18*b^2*c^2*d^5
*e^2 + 4*a^3*c*d*e^6 - 14*b*c^3*d^6*e - 30*a*b*c^2*d^4*e^3 + 24*a*b^2*c*d^3*e^4 - 18*a^2*b*c*d^2*e^5)/(e^8*(d
+ e*x)^(1/2)) + ((d + e*x)^(5/2)*(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))/(5*e^8) + (2*(b*e - 2*c*d)*(d
+ e*x)^(3/2)*(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))/e^8 + (2*(d + e*x)^(1/2)*(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))/e^8 + (10*c*(b*e - 2*c*d)*(d + e*x)^(7/2)*(b^2*e^2 + 7*c^2*d^2 + 3*a*c*e^2 - 7*b*c*d*e))/(7*
e^8)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________