3.15.20 \(\int \frac {(b+2 c x) (a+b x+c x^2)^3}{(d+e x)^{5/2}} \, dx\)

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

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Rubi [A]  time = 0.23, 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)^{3/2} \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 )}{3 e^8}+\frac {6 c^2 (d+e x)^{7/2} \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{7 e^8}-\frac {2 c (d+e x)^{5/2} (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8}-\frac {6 \sqrt {d+e x} (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 \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 \sqrt {d+e x}}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8 (d+e x)^{3/2}}-\frac {14 c^3 (d+e x)^{9/2} (2 c d-b e)}{9 e^8}+\frac {4 c^4 (d+e x)^{11/2}}{11 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((b + 2*c*x)*(a + b*x + c*x^2)^3)/(d + e*x)^(5/2),x]

[Out]

(2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(3*e^8*(d + e*x)^(3/2)) - (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)))/(e^8*Sqrt[d + e*x]) - (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))*Sqrt[d + e*x])/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)^(3/2))/(3*e^8) - (2*
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)^(5/2))/e^8 + (6*c^2*(14*c^2*d^2 + 3*b^2*
e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^(7/2))/(7*e^8) - (14*c^3*(2*c*d - b*e)*(d + e*x)^(9/2))/(9*e^8) + (4*c^4*
(d + e*x)^(11/2))/(11*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)^{5/2}} \, 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)^{5/2}}+\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 (d+e x)^{3/2}}+\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 )}{e^7 \sqrt {d+e x}}+\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 ) \sqrt {d+e x}}{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)^{3/2}}{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)^{5/2}}{e^7}-\frac {7 c^3 (2 c d-b e) (d+e x)^{7/2}}{e^7}+\frac {2 c^4 (d+e x)^{9/2}}{e^7}\right ) \, dx\\ &=\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{3 e^8 (d+e x)^{3/2}}-\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 )}{e^8 \sqrt {d+e x}}-\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 ) \sqrt {d+e x}}{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)^{3/2}}{3 e^8}-\frac {2 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^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)^{7/2}}{7 e^8}-\frac {14 c^3 (2 c d-b e) (d+e x)^{9/2}}{9 e^8}+\frac {4 c^4 (d+e x)^{11/2}}{11 e^8}\\ \end {align*}

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Mathematica [A]  time = 0.59, size = 598, normalized size = 1.42 \begin {gather*} \frac {-198 c^2 e^2 \left (14 a^2 e^2 \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )-7 a b e \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )+3 b^2 \left (256 d^5+384 d^4 e x+96 d^3 e^2 x^2-16 d^2 e^3 x^3+6 d e^4 x^4-3 e^5 x^5\right )\right )+462 c e^3 \left (-2 a^3 e^3 (2 d+3 e x)+9 a^2 b e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+12 a b^2 e \left (-16 d^3-24 d^2 e x-6 d e^2 x^2+e^3 x^3\right )+b^3 \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )\right )-462 b e^4 \left (a^3 e^3+3 a^2 b e^2 (2 d+3 e x)-3 a b^2 e \left (8 d^2+12 d e x+3 e^2 x^2\right )+b^3 \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )\right )+22 c^3 e \left (7 b \left (1024 d^6+1536 d^5 e x+384 d^4 e^2 x^2-64 d^3 e^3 x^3+24 d^2 e^4 x^4-12 d e^5 x^5+7 e^6 x^6\right )-18 a e \left (256 d^5+384 d^4 e x+96 d^3 e^2 x^2-16 d^2 e^3 x^3+6 d e^4 x^4-3 e^5 x^5\right )\right )-28 c^4 \left (2048 d^7+3072 d^6 e x+768 d^5 e^2 x^2-128 d^4 e^3 x^3+48 d^3 e^4 x^4-24 d^2 e^5 x^5+14 d e^6 x^6-9 e^7 x^7\right )}{693 e^8 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((b + 2*c*x)*(a + b*x + c*x^2)^3)/(d + e*x)^(5/2),x]

[Out]

(-28*c^4*(2048*d^7 + 3072*d^6*e*x + 768*d^5*e^2*x^2 - 128*d^4*e^3*x^3 + 48*d^3*e^4*x^4 - 24*d^2*e^5*x^5 + 14*d
*e^6*x^6 - 9*e^7*x^7) - 462*b*e^4*(a^3*e^3 + 3*a^2*b*e^2*(2*d + 3*e*x) - 3*a*b^2*e*(8*d^2 + 12*d*e*x + 3*e^2*x
^2) + b^3*(16*d^3 + 24*d^2*e*x + 6*d*e^2*x^2 - e^3*x^3)) + 462*c*e^3*(-2*a^3*e^3*(2*d + 3*e*x) + 9*a^2*b*e^2*(
8*d^2 + 12*d*e*x + 3*e^2*x^2) + 12*a*b^2*e*(-16*d^3 - 24*d^2*e*x - 6*d*e^2*x^2 + e^3*x^3) + b^3*(128*d^4 + 192
*d^3*e*x + 48*d^2*e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^4)) - 198*c^2*e^2*(14*a^2*e^2*(16*d^3 + 24*d^2*e*x + 6*d*e^2
*x^2 - e^3*x^3) - 7*a*b*e*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^4) + 3*b^2*(256*d^5
+ 384*d^4*e*x + 96*d^3*e^2*x^2 - 16*d^2*e^3*x^3 + 6*d*e^4*x^4 - 3*e^5*x^5)) + 22*c^3*e*(-18*a*e*(256*d^5 + 384
*d^4*e*x + 96*d^3*e^2*x^2 - 16*d^2*e^3*x^3 + 6*d*e^4*x^4 - 3*e^5*x^5) + 7*b*(1024*d^6 + 1536*d^5*e*x + 384*d^4
*e^2*x^2 - 64*d^3*e^3*x^3 + 24*d^2*e^4*x^4 - 12*d*e^5*x^5 + 7*e^6*x^6)))/(693*e^8*(d + e*x)^(3/2))

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IntegrateAlgebraic [B]  time = 0.35, size = 951, normalized size = 2.26 \begin {gather*} \frac {2 \left (462 c^4 d^7-1617 b c^3 e d^6-9702 c^4 (d+e x) d^6+1386 a c^3 e^2 d^5+2079 b^2 c^2 e^2 d^5-29106 c^4 (d+e x)^2 d^5+29106 b c^3 e (d+e x) d^5-3465 a b c^2 e^3 d^4-1155 b^3 c e^3 d^4+16170 c^4 (d+e x)^3 d^4+72765 b c^3 e (d+e x)^2 d^4-20790 a c^3 e^2 (d+e x) d^4-31185 b^2 c^2 e^2 (d+e x) d^4+231 b^4 e^4 d^3+1386 a^2 c^2 e^4 d^3+2772 a b^2 c e^4 d^3-9702 c^4 (d+e x)^4 d^3-32340 b c^3 e (d+e x)^3 d^3-41580 a c^3 e^2 (d+e x)^2 d^3-62370 b^2 c^2 e^2 (d+e x)^2 d^3+41580 a b c^2 e^3 (d+e x) d^3+13860 b^3 c e^3 (d+e x) d^3-693 a b^3 e^5 d^2-2079 a^2 b c e^5 d^2+4158 c^4 (d+e x)^5 d^2+14553 b c^3 e (d+e x)^4 d^2+13860 a c^3 e^2 (d+e x)^3 d^2+20790 b^2 c^2 e^2 (d+e x)^3 d^2+62370 a b c^2 e^3 (d+e x)^2 d^2+20790 b^3 c e^3 (d+e x)^2 d^2-2079 b^4 e^4 (d+e x) d^2-12474 a^2 c^2 e^4 (d+e x) d^2-24948 a b^2 c e^4 (d+e x) d^2+693 a^2 b^2 e^6 d+462 a^3 c e^6 d-1078 c^4 (d+e x)^6 d-4158 b c^3 e (d+e x)^5 d-4158 a c^3 e^2 (d+e x)^4 d-6237 b^2 c^2 e^2 (d+e x)^4 d-13860 a b c^2 e^3 (d+e x)^3 d-4620 b^3 c e^3 (d+e x)^3 d-2079 b^4 e^4 (d+e x)^2 d-12474 a^2 c^2 e^4 (d+e x)^2 d-24948 a b^2 c e^4 (d+e x)^2 d+4158 a b^3 e^5 (d+e x) d+12474 a^2 b c e^5 (d+e x) d-231 a^3 b e^7+126 c^4 (d+e x)^7+539 b c^3 e (d+e x)^6+594 a c^3 e^2 (d+e x)^5+891 b^2 c^2 e^2 (d+e x)^5+2079 a b c^2 e^3 (d+e x)^4+693 b^3 c e^3 (d+e x)^4+231 b^4 e^4 (d+e x)^3+1386 a^2 c^2 e^4 (d+e x)^3+2772 a b^2 c e^4 (d+e x)^3+2079 a b^3 e^5 (d+e x)^2+6237 a^2 b c e^5 (d+e x)^2-2079 a^2 b^2 e^6 (d+e x)-1386 a^3 c e^6 (d+e x)\right )}{693 e^8 (d+e x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((b + 2*c*x)*(a + b*x + c*x^2)^3)/(d + e*x)^(5/2),x]

[Out]

(2*(462*c^4*d^7 - 1617*b*c^3*d^6*e + 2079*b^2*c^2*d^5*e^2 + 1386*a*c^3*d^5*e^2 - 1155*b^3*c*d^4*e^3 - 3465*a*b
*c^2*d^4*e^3 + 231*b^4*d^3*e^4 + 2772*a*b^2*c*d^3*e^4 + 1386*a^2*c^2*d^3*e^4 - 693*a*b^3*d^2*e^5 - 2079*a^2*b*
c*d^2*e^5 + 693*a^2*b^2*d*e^6 + 462*a^3*c*d*e^6 - 231*a^3*b*e^7 - 9702*c^4*d^6*(d + e*x) + 29106*b*c^3*d^5*e*(
d + e*x) - 31185*b^2*c^2*d^4*e^2*(d + e*x) - 20790*a*c^3*d^4*e^2*(d + e*x) + 13860*b^3*c*d^3*e^3*(d + e*x) + 4
1580*a*b*c^2*d^3*e^3*(d + e*x) - 2079*b^4*d^2*e^4*(d + e*x) - 24948*a*b^2*c*d^2*e^4*(d + e*x) - 12474*a^2*c^2*
d^2*e^4*(d + e*x) + 4158*a*b^3*d*e^5*(d + e*x) + 12474*a^2*b*c*d*e^5*(d + e*x) - 2079*a^2*b^2*e^6*(d + e*x) -
1386*a^3*c*e^6*(d + e*x) - 29106*c^4*d^5*(d + e*x)^2 + 72765*b*c^3*d^4*e*(d + e*x)^2 - 62370*b^2*c^2*d^3*e^2*(
d + e*x)^2 - 41580*a*c^3*d^3*e^2*(d + e*x)^2 + 20790*b^3*c*d^2*e^3*(d + e*x)^2 + 62370*a*b*c^2*d^2*e^3*(d + e*
x)^2 - 2079*b^4*d*e^4*(d + e*x)^2 - 24948*a*b^2*c*d*e^4*(d + e*x)^2 - 12474*a^2*c^2*d*e^4*(d + e*x)^2 + 2079*a
*b^3*e^5*(d + e*x)^2 + 6237*a^2*b*c*e^5*(d + e*x)^2 + 16170*c^4*d^4*(d + e*x)^3 - 32340*b*c^3*d^3*e*(d + e*x)^
3 + 20790*b^2*c^2*d^2*e^2*(d + e*x)^3 + 13860*a*c^3*d^2*e^2*(d + e*x)^3 - 4620*b^3*c*d*e^3*(d + e*x)^3 - 13860
*a*b*c^2*d*e^3*(d + e*x)^3 + 231*b^4*e^4*(d + e*x)^3 + 2772*a*b^2*c*e^4*(d + e*x)^3 + 1386*a^2*c^2*e^4*(d + e*
x)^3 - 9702*c^4*d^3*(d + e*x)^4 + 14553*b*c^3*d^2*e*(d + e*x)^4 - 6237*b^2*c^2*d*e^2*(d + e*x)^4 - 4158*a*c^3*
d*e^2*(d + e*x)^4 + 693*b^3*c*e^3*(d + e*x)^4 + 2079*a*b*c^2*e^3*(d + e*x)^4 + 4158*c^4*d^2*(d + e*x)^5 - 4158
*b*c^3*d*e*(d + e*x)^5 + 891*b^2*c^2*e^2*(d + e*x)^5 + 594*a*c^3*e^2*(d + e*x)^5 - 1078*c^4*d*(d + e*x)^6 + 53
9*b*c^3*e*(d + e*x)^6 + 126*c^4*(d + e*x)^7))/(693*e^8*(d + e*x)^(3/2))

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fricas [A]  time = 0.42, size = 669, normalized size = 1.59 \begin {gather*} \frac {2 \, {\left (126 \, c^{4} e^{7} x^{7} - 28672 \, c^{4} d^{7} + 78848 \, b c^{3} d^{6} e - 231 \, a^{3} b e^{7} - 25344 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} + 29568 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} - 3696 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{4} + 5544 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{5} - 462 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{6} - 49 \, {\left (4 \, c^{4} d e^{6} - 11 \, b c^{3} e^{7}\right )} x^{6} + 3 \, {\left (112 \, c^{4} d^{2} e^{5} - 308 \, b c^{3} d e^{6} + 99 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{7}\right )} x^{5} - 3 \, {\left (224 \, c^{4} d^{3} e^{4} - 616 \, b c^{3} d^{2} e^{5} + 198 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{6} - 231 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{7}\right )} x^{4} + {\left (1792 \, c^{4} d^{4} e^{3} - 4928 \, b c^{3} d^{3} e^{4} + 1584 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{5} - 1848 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{6} + 231 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{7}\right )} x^{3} - 3 \, {\left (3584 \, c^{4} d^{5} e^{2} - 9856 \, b c^{3} d^{4} e^{3} + 3168 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{4} - 3696 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{5} + 462 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{6} - 693 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{7}\right )} x^{2} - 3 \, {\left (14336 \, c^{4} d^{6} e - 39424 \, b c^{3} d^{5} e^{2} + 12672 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{3} - 14784 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{4} + 1848 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{5} - 2772 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{6} + 231 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{7}\right )} x\right )} \sqrt {e x + d}}{693 \, {\left (e^{10} x^{2} + 2 \, d e^{9} x + d^{2} 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)^(5/2),x, algorithm="fricas")

[Out]

2/693*(126*c^4*e^7*x^7 - 28672*c^4*d^7 + 78848*b*c^3*d^6*e - 231*a^3*b*e^7 - 25344*(3*b^2*c^2 + 2*a*c^3)*d^5*e
^2 + 29568*(b^3*c + 3*a*b*c^2)*d^4*e^3 - 3696*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 + 5544*(a*b^3 + 3*a^2*b*c
)*d^2*e^5 - 462*(3*a^2*b^2 + 2*a^3*c)*d*e^6 - 49*(4*c^4*d*e^6 - 11*b*c^3*e^7)*x^6 + 3*(112*c^4*d^2*e^5 - 308*b
*c^3*d*e^6 + 99*(3*b^2*c^2 + 2*a*c^3)*e^7)*x^5 - 3*(224*c^4*d^3*e^4 - 616*b*c^3*d^2*e^5 + 198*(3*b^2*c^2 + 2*a
*c^3)*d*e^6 - 231*(b^3*c + 3*a*b*c^2)*e^7)*x^4 + (1792*c^4*d^4*e^3 - 4928*b*c^3*d^3*e^4 + 1584*(3*b^2*c^2 + 2*
a*c^3)*d^2*e^5 - 1848*(b^3*c + 3*a*b*c^2)*d*e^6 + 231*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^7)*x^3 - 3*(3584*c^4*d^
5*e^2 - 9856*b*c^3*d^4*e^3 + 3168*(3*b^2*c^2 + 2*a*c^3)*d^3*e^4 - 3696*(b^3*c + 3*a*b*c^2)*d^2*e^5 + 462*(b^4
+ 12*a*b^2*c + 6*a^2*c^2)*d*e^6 - 693*(a*b^3 + 3*a^2*b*c)*e^7)*x^2 - 3*(14336*c^4*d^6*e - 39424*b*c^3*d^5*e^2
+ 12672*(3*b^2*c^2 + 2*a*c^3)*d^4*e^3 - 14784*(b^3*c + 3*a*b*c^2)*d^3*e^4 + 1848*(b^4 + 12*a*b^2*c + 6*a^2*c^2
)*d^2*e^5 - 2772*(a*b^3 + 3*a^2*b*c)*d*e^6 + 231*(3*a^2*b^2 + 2*a^3*c)*e^7)*x)*sqrt(e*x + d)/(e^10*x^2 + 2*d*e
^9*x + d^2*e^8)

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giac [B]  time = 0.30, size = 972, normalized size = 2.31

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)^(5/2),x, algorithm="giac")

[Out]

2/693*(126*(x*e + d)^(11/2)*c^4*e^80 - 1078*(x*e + d)^(9/2)*c^4*d*e^80 + 4158*(x*e + d)^(7/2)*c^4*d^2*e^80 - 9
702*(x*e + d)^(5/2)*c^4*d^3*e^80 + 16170*(x*e + d)^(3/2)*c^4*d^4*e^80 - 29106*sqrt(x*e + d)*c^4*d^5*e^80 + 539
*(x*e + d)^(9/2)*b*c^3*e^81 - 4158*(x*e + d)^(7/2)*b*c^3*d*e^81 + 14553*(x*e + d)^(5/2)*b*c^3*d^2*e^81 - 32340
*(x*e + d)^(3/2)*b*c^3*d^3*e^81 + 72765*sqrt(x*e + d)*b*c^3*d^4*e^81 + 891*(x*e + d)^(7/2)*b^2*c^2*e^82 + 594*
(x*e + d)^(7/2)*a*c^3*e^82 - 6237*(x*e + d)^(5/2)*b^2*c^2*d*e^82 - 4158*(x*e + d)^(5/2)*a*c^3*d*e^82 + 20790*(
x*e + d)^(3/2)*b^2*c^2*d^2*e^82 + 13860*(x*e + d)^(3/2)*a*c^3*d^2*e^82 - 62370*sqrt(x*e + d)*b^2*c^2*d^3*e^82
- 41580*sqrt(x*e + d)*a*c^3*d^3*e^82 + 693*(x*e + d)^(5/2)*b^3*c*e^83 + 2079*(x*e + d)^(5/2)*a*b*c^2*e^83 - 46
20*(x*e + d)^(3/2)*b^3*c*d*e^83 - 13860*(x*e + d)^(3/2)*a*b*c^2*d*e^83 + 20790*sqrt(x*e + d)*b^3*c*d^2*e^83 +
62370*sqrt(x*e + d)*a*b*c^2*d^2*e^83 + 231*(x*e + d)^(3/2)*b^4*e^84 + 2772*(x*e + d)^(3/2)*a*b^2*c*e^84 + 1386
*(x*e + d)^(3/2)*a^2*c^2*e^84 - 2079*sqrt(x*e + d)*b^4*d*e^84 - 24948*sqrt(x*e + d)*a*b^2*c*d*e^84 - 12474*sqr
t(x*e + d)*a^2*c^2*d*e^84 + 2079*sqrt(x*e + d)*a*b^3*e^85 + 6237*sqrt(x*e + d)*a^2*b*c*e^85)*e^(-88) - 2/3*(42
*(x*e + d)*c^4*d^6 - 2*c^4*d^7 - 126*(x*e + d)*b*c^3*d^5*e + 7*b*c^3*d^6*e + 135*(x*e + d)*b^2*c^2*d^4*e^2 + 9
0*(x*e + d)*a*c^3*d^4*e^2 - 9*b^2*c^2*d^5*e^2 - 6*a*c^3*d^5*e^2 - 60*(x*e + d)*b^3*c*d^3*e^3 - 180*(x*e + d)*a
*b*c^2*d^3*e^3 + 5*b^3*c*d^4*e^3 + 15*a*b*c^2*d^4*e^3 + 9*(x*e + d)*b^4*d^2*e^4 + 108*(x*e + d)*a*b^2*c*d^2*e^
4 + 54*(x*e + d)*a^2*c^2*d^2*e^4 - b^4*d^3*e^4 - 12*a*b^2*c*d^3*e^4 - 6*a^2*c^2*d^3*e^4 - 18*(x*e + d)*a*b^3*d
*e^5 - 54*(x*e + d)*a^2*b*c*d*e^5 + 3*a*b^3*d^2*e^5 + 9*a^2*b*c*d^2*e^5 + 9*(x*e + d)*a^2*b^2*e^6 + 6*(x*e + d
)*a^3*c*e^6 - 3*a^2*b^2*d*e^6 - 2*a^3*c*d*e^6 + a^3*b*e^7)*e^(-8)/(x*e + d)^(3/2)

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maple [B]  time = 0.06, size = 795, normalized size = 1.89 \begin {gather*} -\frac {2 \left (-126 c^{4} e^{7} x^{7}-539 b \,c^{3} e^{7} x^{6}+196 c^{4} d \,e^{6} x^{6}-594 a \,c^{3} e^{7} x^{5}-891 b^{2} c^{2} e^{7} x^{5}+924 b \,c^{3} d \,e^{6} x^{5}-336 c^{4} d^{2} e^{5} x^{5}-2079 a b \,c^{2} e^{7} x^{4}+1188 a \,c^{3} d \,e^{6} x^{4}-693 b^{3} c \,e^{7} x^{4}+1782 b^{2} c^{2} d \,e^{6} x^{4}-1848 b \,c^{3} d^{2} e^{5} x^{4}+672 c^{4} d^{3} e^{4} x^{4}-1386 a^{2} c^{2} e^{7} x^{3}-2772 a \,b^{2} c \,e^{7} x^{3}+5544 a b \,c^{2} d \,e^{6} x^{3}-3168 a \,c^{3} d^{2} e^{5} x^{3}-231 b^{4} e^{7} x^{3}+1848 b^{3} c d \,e^{6} x^{3}-4752 b^{2} c^{2} d^{2} e^{5} x^{3}+4928 b \,c^{3} d^{3} e^{4} x^{3}-1792 c^{4} d^{4} e^{3} x^{3}-6237 a^{2} b c \,e^{7} x^{2}+8316 a^{2} c^{2} d \,e^{6} x^{2}-2079 a \,b^{3} e^{7} x^{2}+16632 a \,b^{2} c d \,e^{6} x^{2}-33264 a b \,c^{2} d^{2} e^{5} x^{2}+19008 a \,c^{3} d^{3} e^{4} x^{2}+1386 b^{4} d \,e^{6} x^{2}-11088 b^{3} c \,d^{2} e^{5} x^{2}+28512 b^{2} c^{2} d^{3} e^{4} x^{2}-29568 b \,c^{3} d^{4} e^{3} x^{2}+10752 c^{4} d^{5} e^{2} x^{2}+1386 a^{3} c \,e^{7} x +2079 a^{2} b^{2} e^{7} x -24948 a^{2} b c d \,e^{6} x +33264 a^{2} c^{2} d^{2} e^{5} x -8316 a \,b^{3} d \,e^{6} x +66528 a \,b^{2} c \,d^{2} e^{5} x -133056 a b \,c^{2} d^{3} e^{4} x +76032 a \,c^{3} d^{4} e^{3} x +5544 b^{4} d^{2} e^{5} x -44352 b^{3} c \,d^{3} e^{4} x +114048 b^{2} c^{2} d^{4} e^{3} x -118272 b \,c^{3} d^{5} e^{2} x +43008 c^{4} d^{6} e x +231 b \,a^{3} e^{7}+924 a^{3} c d \,e^{6}+1386 a^{2} b^{2} d \,e^{6}-16632 a^{2} b c \,d^{2} e^{5}+22176 a^{2} c^{2} d^{3} e^{4}-5544 a \,b^{3} d^{2} e^{5}+44352 a \,b^{2} c \,d^{3} e^{4}-88704 a b \,c^{2} d^{4} e^{3}+50688 a \,c^{3} d^{5} e^{2}+3696 b^{4} d^{3} e^{4}-29568 b^{3} c \,d^{4} e^{3}+76032 b^{2} c^{2} d^{5} e^{2}-78848 b \,c^{3} d^{6} e +28672 c^{4} d^{7}\right )}{693 \left (e x +d \right )^{\frac {3}{2}} 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)^(5/2),x)

[Out]

-2/693/(e*x+d)^(3/2)*(-126*c^4*e^7*x^7-539*b*c^3*e^7*x^6+196*c^4*d*e^6*x^6-594*a*c^3*e^7*x^5-891*b^2*c^2*e^7*x
^5+924*b*c^3*d*e^6*x^5-336*c^4*d^2*e^5*x^5-2079*a*b*c^2*e^7*x^4+1188*a*c^3*d*e^6*x^4-693*b^3*c*e^7*x^4+1782*b^
2*c^2*d*e^6*x^4-1848*b*c^3*d^2*e^5*x^4+672*c^4*d^3*e^4*x^4-1386*a^2*c^2*e^7*x^3-2772*a*b^2*c*e^7*x^3+5544*a*b*
c^2*d*e^6*x^3-3168*a*c^3*d^2*e^5*x^3-231*b^4*e^7*x^3+1848*b^3*c*d*e^6*x^3-4752*b^2*c^2*d^2*e^5*x^3+4928*b*c^3*
d^3*e^4*x^3-1792*c^4*d^4*e^3*x^3-6237*a^2*b*c*e^7*x^2+8316*a^2*c^2*d*e^6*x^2-2079*a*b^3*e^7*x^2+16632*a*b^2*c*
d*e^6*x^2-33264*a*b*c^2*d^2*e^5*x^2+19008*a*c^3*d^3*e^4*x^2+1386*b^4*d*e^6*x^2-11088*b^3*c*d^2*e^5*x^2+28512*b
^2*c^2*d^3*e^4*x^2-29568*b*c^3*d^4*e^3*x^2+10752*c^4*d^5*e^2*x^2+1386*a^3*c*e^7*x+2079*a^2*b^2*e^7*x-24948*a^2
*b*c*d*e^6*x+33264*a^2*c^2*d^2*e^5*x-8316*a*b^3*d*e^6*x+66528*a*b^2*c*d^2*e^5*x-133056*a*b*c^2*d^3*e^4*x+76032
*a*c^3*d^4*e^3*x+5544*b^4*d^2*e^5*x-44352*b^3*c*d^3*e^4*x+114048*b^2*c^2*d^4*e^3*x-118272*b*c^3*d^5*e^2*x+4300
8*c^4*d^6*e*x+231*a^3*b*e^7+924*a^3*c*d*e^6+1386*a^2*b^2*d*e^6-16632*a^2*b*c*d^2*e^5+22176*a^2*c^2*d^3*e^4-554
4*a*b^3*d^2*e^5+44352*a*b^2*c*d^3*e^4-88704*a*b*c^2*d^4*e^3+50688*a*c^3*d^5*e^2+3696*b^4*d^3*e^4-29568*b^3*c*d
^4*e^3+76032*b^2*c^2*d^5*e^2-78848*b*c^3*d^6*e+28672*c^4*d^7)/e^8

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maxima [A]  time = 0.67, size = 651, normalized size = 1.55 \begin {gather*} \frac {2 \, {\left (\frac {126 \, {\left (e x + d\right )}^{\frac {11}{2}} c^{4} - 539 \, {\left (2 \, c^{4} d - b c^{3} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 297 \, {\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 {7}{2}} - 693 \, {\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 {5}{2}} + 231 \, {\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 {3}{2}} - 2079 \, {\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 )} \sqrt {e x + d}}{e^{7}} + \frac {231 \, {\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} - 3 \, {\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 )}\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{7}}\right )}}{693 \, 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)^(5/2),x, algorithm="maxima")

[Out]

2/693*((126*(e*x + d)^(11/2)*c^4 - 539*(2*c^4*d - b*c^3*e)*(e*x + d)^(9/2) + 297*(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)^(7/2) - 693*(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)^(5/2) + 231*(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)^(3/2) - 2079*(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)*sqrt(e*x + d))/e^7 + 231*(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 - 3*(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))/((e*x + d)^(3/2)*e^7))/e

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mupad [B]  time = 1.96, size = 677, normalized size = 1.61 \begin {gather*} \frac {{\left (d+e\,x\right )}^{7/2}\,\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 )}{7\,e^8}+\frac {4\,c^4\,{\left (d+e\,x\right )}^{11/2}}{11\,e^8}-\frac {\left (28\,c^4\,d-14\,b\,c^3\,e\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^8}+\frac {\frac {4\,c^4\,d^7}{3}-\left (d+e\,x\right )\,\left (4\,a^3\,c\,e^6+6\,a^2\,b^2\,e^6-36\,a^2\,b\,c\,d\,e^5+36\,a^2\,c^2\,d^2\,e^4-12\,a\,b^3\,d\,e^5+72\,a\,b^2\,c\,d^2\,e^4-120\,a\,b\,c^2\,d^3\,e^3+60\,a\,c^3\,d^4\,e^2+6\,b^4\,d^2\,e^4-40\,b^3\,c\,d^3\,e^3+90\,b^2\,c^2\,d^4\,e^2-84\,b\,c^3\,d^5\,e+28\,c^4\,d^6\right )-\frac {2\,a^3\,b\,e^7}{3}+\frac {2\,b^4\,d^3\,e^4}{3}-2\,a\,b^3\,d^2\,e^5+2\,a^2\,b^2\,d\,e^6+4\,a\,c^3\,d^5\,e^2-\frac {10\,b^3\,c\,d^4\,e^3}{3}+4\,a^2\,c^2\,d^3\,e^4+6\,b^2\,c^2\,d^5\,e^2+\frac {4\,a^3\,c\,d\,e^6}{3}-\frac {14\,b\,c^3\,d^6\,e}{3}-10\,a\,b\,c^2\,d^4\,e^3+8\,a\,b^2\,c\,d^3\,e^4-6\,a^2\,b\,c\,d^2\,e^5}{e^8\,{\left (d+e\,x\right )}^{3/2}}+\frac {{\left (d+e\,x\right )}^{3/2}\,\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 )}{3\,e^8}+\frac {6\,\left (b\,e-2\,c\,d\right )\,\sqrt {d+e\,x}\,\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\,c\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{5/2}\,\left (b^2\,e^2-7\,b\,c\,d\,e+7\,c^2\,d^2+3\,a\,c\,e^2\right )}{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)^(5/2),x)

[Out]

((d + e*x)^(7/2)*(84*c^4*d^2 + 12*a*c^3*e^2 + 18*b^2*c^2*e^2 - 84*b*c^3*d*e))/(7*e^8) + (4*c^4*(d + e*x)^(11/2
))/(11*e^8) - ((28*c^4*d - 14*b*c^3*e)*(d + e*x)^(9/2))/(9*e^8) + ((4*c^4*d^7)/3 - (d + e*x)*(28*c^4*d^6 + 4*a
^3*c*e^6 + 6*a^2*b^2*e^6 + 6*b^4*d^2*e^4 + 60*a*c^3*d^4*e^2 - 40*b^3*c*d^3*e^3 + 36*a^2*c^2*d^2*e^4 + 90*b^2*c
^2*d^4*e^2 - 12*a*b^3*d*e^5 - 84*b*c^3*d^5*e - 36*a^2*b*c*d*e^5 - 120*a*b*c^2*d^3*e^3 + 72*a*b^2*c*d^2*e^4) -
(2*a^3*b*e^7)/3 + (2*b^4*d^3*e^4)/3 - 2*a*b^3*d^2*e^5 + 2*a^2*b^2*d*e^6 + 4*a*c^3*d^5*e^2 - (10*b^3*c*d^4*e^3)
/3 + 4*a^2*c^2*d^3*e^4 + 6*b^2*c^2*d^5*e^2 + (4*a^3*c*d*e^6)/3 - (14*b*c^3*d^6*e)/3 - 10*a*b*c^2*d^4*e^3 + 8*a
*b^2*c*d^3*e^4 - 6*a^2*b*c*d^2*e^5)/(e^8*(d + e*x)^(3/2)) + ((d + e*x)^(3/2)*(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))/(3*e^8) + (6*(b*e - 2*c*d)*(d + e*x)^(1/2)*(7*c^3*d^4 + a*b^2*e^4 + 3*a^2*c*e^4 - b^3*d*e^3 + 1
0*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*c*(b*e - 2*c*d)*(d + e*x)^(5/2)
*(b^2*e^2 + 7*c^2*d^2 + 3*a*c*e^2 - 7*b*c*d*e))/e^8

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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)**(5/2),x)

[Out]

Timed out

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