3.1523 \(\int \frac {(b+2 c x) (a+b x+c x^2)^3}{(d+e x)^5} \, dx\)

Optimal. Leaf size=389 \[ -\frac {6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{e^8 (d+e x)}+\frac {3 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{2 e^8 (d+e x)^2}-\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^8 (d+e x)^3}-\frac {5 c (2 c d-b e) \log (d+e x) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8}+\frac {c^2 x \left (-c e (35 b d-6 a e)+9 b^2 e^2+30 c^2 d^2\right )}{e^7}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{4 e^8 (d+e x)^4}-\frac {c^3 x^2 (10 c d-7 b e)}{2 e^6}+\frac {2 c^4 x^3}{3 e^5} \]

[Out]

c^2*(30*c^2*d^2+9*b^2*e^2-c*e*(-6*a*e+35*b*d))*x/e^7-1/2*c^3*(-7*b*e+10*c*d)*x^2/e^6+2/3*c^4*x^3/e^5+1/4*(-b*e
+2*c*d)*(a*e^2-b*d*e+c*d^2)^3/e^8/(e*x+d)^4-1/3*(a*e^2-b*d*e+c*d^2)^2*(14*c^2*d^2+3*b^2*e^2-2*c*e*(-a*e+7*b*d)
)/e^8/(e*x+d)^3+3/2*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(7*c^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))/e^8/(e*x+d)^2+(-70
*c^4*d^4-b^4*e^4+4*b^2*c*e^3*(-3*a*e+5*b*d)+20*c^3*d^2*e*(-3*a*e+7*b*d)-6*c^2*e^2*(a^2*e^2-10*a*b*d*e+15*b^2*d
^2))/e^8/(e*x+d)-5*c*(-b*e+2*c*d)*(7*c^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))*ln(e*x+d)/e^8

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Rubi [A]  time = 0.48, antiderivative size = 389, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {771} \[ -\frac {6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{e^8 (d+e x)}+\frac {c^2 x \left (-c e (35 b d-6 a e)+9 b^2 e^2+30 c^2 d^2\right )}{e^7}+\frac {3 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{2 e^8 (d+e x)^2}-\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^8 (d+e x)^3}-\frac {5 c (2 c d-b e) \log (d+e x) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{4 e^8 (d+e x)^4}-\frac {c^3 x^2 (10 c d-7 b e)}{2 e^6}+\frac {2 c^4 x^3}{3 e^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*(30*c^2*d^2 + 9*b^2*e^2 - c*e*(35*b*d - 6*a*e))*x)/e^7 - (c^3*(10*c*d - 7*b*e)*x^2)/(2*e^6) + (2*c^4*x^3)
/(3*e^5) + ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(4*e^8*(d + e*x)^4) - ((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)))/(3*e^8*(d + e*x)^3) + (3*(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)))/(2*e^8*(d + e*x)^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))/(e^8*(d + e*x)) - (5*c*(2*c*
d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*Log[d + e*x])/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} \, dx &=\int \left (\frac {c^2 \left (30 c^2 d^2+9 b^2 e^2-c e (35 b d-6 a e)\right )}{e^7}-\frac {c^3 (10 c d-7 b e) x}{e^6}+\frac {2 c^4 x^2}{e^5}+\frac {(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^7 (d+e x)^5}+\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)^4}+\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 (d+e x)^3}+\frac {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 )}{e^7 (d+e x)^2}+\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 )}{e^7 (d+e x)}\right ) \, dx\\ &=\frac {c^2 \left (30 c^2 d^2+9 b^2 e^2-c e (35 b d-6 a e)\right ) x}{e^7}-\frac {c^3 (10 c d-7 b e) x^2}{2 e^6}+\frac {2 c^4 x^3}{3 e^5}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{4 e^8 (d+e x)^4}-\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 )}{3 e^8 (d+e x)^3}+\frac {3 (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 )}{2 e^8 (d+e x)^2}-\frac {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 )}{e^8 (d+e x)}-\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 ) \log (d+e x)}{e^8}\\ \end {align*}

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Mathematica [A]  time = 0.30, size = 614, normalized size = 1.58 \[ -\frac {3 c^2 e^2 \left (6 a^2 e^2 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-5 a b d e \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )+3 b^2 \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )\right )+c e^3 \left (2 a^3 e^3 (d+4 e x)+9 a^2 b e^2 \left (d^2+4 d e x+6 e^2 x^2\right )+36 a b^2 e \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-5 b^3 d \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )\right )+3 b e^4 \left (a^3 e^3+a^2 b e^2 (d+4 e x)+a b^2 e \left (d^2+4 d e x+6 e^2 x^2\right )+b^3 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )+60 c (d+e x)^4 (2 c d-b e) \log (d+e x) \left (c e (3 a e-7 b d)+b^2 e^2+7 c^2 d^2\right )-3 c^3 e \left (2 a e \left (-77 d^5-248 d^4 e x-252 d^3 e^2 x^2-48 d^2 e^3 x^3+48 d e^4 x^4+12 e^5 x^5\right )+7 b \left (57 d^6+168 d^5 e x+132 d^4 e^2 x^2-32 d^3 e^3 x^3-68 d^2 e^4 x^4-12 d e^5 x^5+2 e^6 x^6\right )\right )+2 c^4 \left (319 d^7+856 d^6 e x+444 d^5 e^2 x^2-544 d^4 e^3 x^3-556 d^3 e^4 x^4-84 d^2 e^5 x^5+14 d e^6 x^6-4 e^7 x^7\right )}{12 e^8 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*(2*c^4*(319*d^7 + 856*d^6*e*x + 444*d^5*e^2*x^2 - 544*d^4*e^3*x^3 - 556*d^3*e^4*x^4 - 84*d^2*e^5*x^5 + 1
4*d*e^6*x^6 - 4*e^7*x^7) + 3*b*e^4*(a^3*e^3 + a^2*b*e^2*(d + 4*e*x) + a*b^2*e*(d^2 + 4*d*e*x + 6*e^2*x^2) + b^
3*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3)) + c*e^3*(2*a^3*e^3*(d + 4*e*x) + 9*a^2*b*e^2*(d^2 + 4*d*e*x + 6
*e^2*x^2) + 36*a*b^2*e*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) - 5*b^3*d*(25*d^3 + 88*d^2*e*x + 108*d*e^2*
x^2 + 48*e^3*x^3)) + 3*c^2*e^2*(6*a^2*e^2*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) - 5*a*b*d*e*(25*d^3 + 88
*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3) + 3*b^2*(77*d^5 + 248*d^4*e*x + 252*d^3*e^2*x^2 + 48*d^2*e^3*x^3 - 48*d
*e^4*x^4 - 12*e^5*x^5)) - 3*c^3*e*(2*a*e*(-77*d^5 - 248*d^4*e*x - 252*d^3*e^2*x^2 - 48*d^2*e^3*x^3 + 48*d*e^4*
x^4 + 12*e^5*x^5) + 7*b*(57*d^6 + 168*d^5*e*x + 132*d^4*e^2*x^2 - 32*d^3*e^3*x^3 - 68*d^2*e^4*x^4 - 12*d*e^5*x
^5 + 2*e^6*x^6)) + 60*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)^4*Log[d + e*x])/(
e^8*(d + e*x)^4)

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fricas [B]  time = 0.79, size = 1017, normalized size = 2.61 \[ \frac {8 \, c^{4} e^{7} x^{7} - 638 \, c^{4} d^{7} + 1197 \, b c^{3} d^{6} e - 3 \, a^{3} b e^{7} - 231 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} + 125 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} - 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} - 14 \, {\left (2 \, c^{4} d e^{6} - 3 \, b c^{3} e^{7}\right )} x^{6} + 12 \, {\left (14 \, c^{4} d^{2} e^{5} - 21 \, b c^{3} d e^{6} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{7}\right )} x^{5} + 4 \, {\left (278 \, c^{4} d^{3} e^{4} - 357 \, b c^{3} d^{2} e^{5} + 36 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{6}\right )} x^{4} + 4 \, {\left (272 \, c^{4} d^{4} e^{3} - 168 \, b c^{3} d^{3} e^{4} - 36 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{5} + 60 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{6} - 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{7}\right )} x^{3} - 6 \, {\left (148 \, c^{4} d^{5} e^{2} - 462 \, b c^{3} d^{4} e^{3} + 126 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{4} - 90 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{5} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{6} + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{7}\right )} x^{2} - 4 \, {\left (428 \, c^{4} d^{6} e - 882 \, b c^{3} d^{5} e^{2} + 186 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{3} - 110 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{4} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{5} + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{6} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{7}\right )} x - 60 \, {\left (14 \, c^{4} d^{7} - 21 \, b c^{3} d^{6} e + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} - {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} + {\left (14 \, c^{4} d^{3} e^{4} - 21 \, b c^{3} d^{2} e^{5} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{6} - {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{7}\right )} x^{4} + 4 \, {\left (14 \, c^{4} d^{4} e^{3} - 21 \, b c^{3} d^{3} e^{4} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{5} - {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{6}\right )} x^{3} + 6 \, {\left (14 \, c^{4} d^{5} e^{2} - 21 \, b c^{3} d^{4} e^{3} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{4} - {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{5}\right )} x^{2} + 4 \, {\left (14 \, c^{4} d^{6} e - 21 \, b c^{3} d^{5} e^{2} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{3} - {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{4}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (e^{12} x^{4} + 4 \, d e^{11} x^{3} + 6 \, d^{2} e^{10} x^{2} + 4 \, d^{3} e^{9} x + d^{4} e^{8}\right )}} \]

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,x, algorithm="fricas")

[Out]

1/12*(8*c^4*e^7*x^7 - 638*c^4*d^7 + 1197*b*c^3*d^6*e - 3*a^3*b*e^7 - 231*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 + 125*(
b^3*c + 3*a*b*c^2)*d^4*e^3 - 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 - 14*(2*c^4*d*e^6 - 3*b*c^3*e^7)*x^6 + 12*(14*c^4*d^2*e^5 - 21*b*c^3*d*e^6 + 3*(3*b^2*c^
2 + 2*a*c^3)*e^7)*x^5 + 4*(278*c^4*d^3*e^4 - 357*b*c^3*d^2*e^5 + 36*(3*b^2*c^2 + 2*a*c^3)*d*e^6)*x^4 + 4*(272*
c^4*d^4*e^3 - 168*b*c^3*d^3*e^4 - 36*(3*b^2*c^2 + 2*a*c^3)*d^2*e^5 + 60*(b^3*c + 3*a*b*c^2)*d*e^6 - 3*(b^4 + 1
2*a*b^2*c + 6*a^2*c^2)*e^7)*x^3 - 6*(148*c^4*d^5*e^2 - 462*b*c^3*d^4*e^3 + 126*(3*b^2*c^2 + 2*a*c^3)*d^3*e^4 -
90*(b^3*c + 3*a*b*c^2)*d^2*e^5 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^6 + 3*(a*b^3 + 3*a^2*b*c)*e^7)*x^2 - 4*
(428*c^4*d^6*e - 882*b*c^3*d^5*e^2 + 186*(3*b^2*c^2 + 2*a*c^3)*d^4*e^3 - 110*(b^3*c + 3*a*b*c^2)*d^3*e^4 + 3*(
b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^5 + 3*(a*b^3 + 3*a^2*b*c)*d*e^6 + (3*a^2*b^2 + 2*a^3*c)*e^7)*x - 60*(14*c^
4*d^7 - 21*b*c^3*d^6*e + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - (b^3*c + 3*a*b*c^2)*d^4*e^3 + (14*c^4*d^3*e^4 - 21*
b*c^3*d^2*e^5 + 3*(3*b^2*c^2 + 2*a*c^3)*d*e^6 - (b^3*c + 3*a*b*c^2)*e^7)*x^4 + 4*(14*c^4*d^4*e^3 - 21*b*c^3*d^
3*e^4 + 3*(3*b^2*c^2 + 2*a*c^3)*d^2*e^5 - (b^3*c + 3*a*b*c^2)*d*e^6)*x^3 + 6*(14*c^4*d^5*e^2 - 21*b*c^3*d^4*e^
3 + 3*(3*b^2*c^2 + 2*a*c^3)*d^3*e^4 - (b^3*c + 3*a*b*c^2)*d^2*e^5)*x^2 + 4*(14*c^4*d^6*e - 21*b*c^3*d^5*e^2 +
3*(3*b^2*c^2 + 2*a*c^3)*d^4*e^3 - (b^3*c + 3*a*b*c^2)*d^3*e^4)*x)*log(e*x + d))/(e^12*x^4 + 4*d*e^11*x^3 + 6*d
^2*e^10*x^2 + 4*d^3*e^9*x + d^4*e^8)

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giac [B]  time = 0.22, size = 1057, normalized size = 2.72 \[ \text {result too large to display} \]

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,x, algorithm="giac")

[Out]

1/6*(4*c^4 - 21*(2*c^4*d*e - b*c^3*e^2)*e^(-1)/(x*e + d) + 18*(14*c^4*d^2*e^2 - 14*b*c^3*d*e^3 + 3*b^2*c^2*e^4
+ 2*a*c^3*e^4)*e^(-2)/(x*e + d)^2)*(x*e + d)^3*e^(-8) + 5*(14*c^4*d^3 - 21*b*c^3*d^2*e + 9*b^2*c^2*d*e^2 + 6*
a*c^3*d*e^2 - b^3*c*e^3 - 3*a*b*c^2*e^3)*e^(-8)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) - 1/12*(840*c^4*d^4*e^36/
(x*e + d) - 252*c^4*d^5*e^36/(x*e + d)^2 + 56*c^4*d^6*e^36/(x*e + d)^3 - 6*c^4*d^7*e^36/(x*e + d)^4 - 1680*b*c
^3*d^3*e^37/(x*e + d) + 630*b*c^3*d^4*e^37/(x*e + d)^2 - 168*b*c^3*d^5*e^37/(x*e + d)^3 + 21*b*c^3*d^6*e^37/(x
*e + d)^4 + 1080*b^2*c^2*d^2*e^38/(x*e + d) + 720*a*c^3*d^2*e^38/(x*e + d) - 540*b^2*c^2*d^3*e^38/(x*e + d)^2
- 360*a*c^3*d^3*e^38/(x*e + d)^2 + 180*b^2*c^2*d^4*e^38/(x*e + d)^3 + 120*a*c^3*d^4*e^38/(x*e + d)^3 - 27*b^2*
c^2*d^5*e^38/(x*e + d)^4 - 18*a*c^3*d^5*e^38/(x*e + d)^4 - 240*b^3*c*d*e^39/(x*e + d) - 720*a*b*c^2*d*e^39/(x*
e + d) + 180*b^3*c*d^2*e^39/(x*e + d)^2 + 540*a*b*c^2*d^2*e^39/(x*e + d)^2 - 80*b^3*c*d^3*e^39/(x*e + d)^3 - 2
40*a*b*c^2*d^3*e^39/(x*e + d)^3 + 15*b^3*c*d^4*e^39/(x*e + d)^4 + 45*a*b*c^2*d^4*e^39/(x*e + d)^4 + 12*b^4*e^4
0/(x*e + d) + 144*a*b^2*c*e^40/(x*e + d) + 72*a^2*c^2*e^40/(x*e + d) - 18*b^4*d*e^40/(x*e + d)^2 - 216*a*b^2*c
*d*e^40/(x*e + d)^2 - 108*a^2*c^2*d*e^40/(x*e + d)^2 + 12*b^4*d^2*e^40/(x*e + d)^3 + 144*a*b^2*c*d^2*e^40/(x*e
+ d)^3 + 72*a^2*c^2*d^2*e^40/(x*e + d)^3 - 3*b^4*d^3*e^40/(x*e + d)^4 - 36*a*b^2*c*d^3*e^40/(x*e + d)^4 - 18*
a^2*c^2*d^3*e^40/(x*e + d)^4 + 18*a*b^3*e^41/(x*e + d)^2 + 54*a^2*b*c*e^41/(x*e + d)^2 - 24*a*b^3*d*e^41/(x*e
+ d)^3 - 72*a^2*b*c*d*e^41/(x*e + d)^3 + 9*a*b^3*d^2*e^41/(x*e + d)^4 + 27*a^2*b*c*d^2*e^41/(x*e + d)^4 + 12*a
^2*b^2*e^42/(x*e + d)^3 + 8*a^3*c*e^42/(x*e + d)^3 - 9*a^2*b^2*d*e^42/(x*e + d)^4 - 6*a^3*c*d*e^42/(x*e + d)^4
+ 3*a^3*b*e^43/(x*e + d)^4)*e^(-44)

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maple [B]  time = 0.06, size = 1056, normalized size = 2.71 \[ \text {result too large to display} \]

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,x)

[Out]

3/e^4/(e*x+d)^4*d^3*a*b^2*c+18/e^4/(e*x+d)^2*a*b^2*c*d-45/e^5/(e*x+d)^2*a*b*c^2*d^2+20/e^5/(e*x+d)^3*a*b*c^2*d
^3+6/e^3/(e*x+d)^3*a^2*b*c*d-12/e^4/(e*x+d)^3*a*b^2*c*d^2+60/e^5/(e*x+d)*a*b*c^2*d-9/4/e^3/(e*x+d)^4*d^2*a^2*b
*c+2/3*c^4*x^3/e^5-15/4/e^5/(e*x+d)^4*d^4*a*b*c^2-35*c^3/e^6*x*b*d-12/e^4/(e*x+d)*a*b^2*c+3/2/e^4/(e*x+d)^2*b^
4*d+21/e^8/(e*x+d)^2*c^4*d^5-1/4/e/(e*x+d)^4*a^3*b+1/4/e^4/(e*x+d)^4*d^3*b^4+1/2/e^8/(e*x+d)^4*c^4*d^7+5*c/e^5
*ln(e*x+d)*b^3-70*c^4/e^8*ln(e*x+d)*d^3-1/e^4/(e*x+d)*b^4+30*c^4/e^7*x*d^2-6/e^4/(e*x+d)*c^2*a^2-70/e^8/(e*x+d
)*c^4*d^4-2/3/e^2/(e*x+d)^3*a^3*c-1/e^2/(e*x+d)^3*a^2*b^2-1/e^4/(e*x+d)^3*b^4*d^2-14/3/e^8/(e*x+d)^3*c^4*d^6+7
/2*c^3/e^5*x^2*b-5*c^4/e^6*x^2*d+6*c^3/e^5*x*a+9*c^2/e^5*x*b^2-3/2/e^3/(e*x+d)^2*a*b^3+140/e^7/(e*x+d)*b*c^3*d
^3+15*c^2/e^5*ln(e*x+d)*a*b-30*c^3/e^6*ln(e*x+d)*a*d-45*c^2/e^6*ln(e*x+d)*b^2*d+105*c^3/e^7*ln(e*x+d)*b*d^2+14
/e^7/(e*x+d)^3*b*c^3*d^5-9/2/e^3/(e*x+d)^2*a^2*b*c-60/e^6/(e*x+d)*a*c^3*d^2+20/e^5/(e*x+d)*b^3*c*d-90/e^6/(e*x
+d)*b^2*c^2*d^2+9/e^4/(e*x+d)^2*a^2*c^2*d+30/e^6/(e*x+d)^2*a*c^3*d^3-15/e^5/(e*x+d)^2*b^3*c*d^2+45/e^6/(e*x+d)
^2*b^2*c^2*d^3-105/2/e^7/(e*x+d)^2*b*c^3*d^4+1/2/e^2/(e*x+d)^4*a^3*c*d+3/4/e^2/(e*x+d)^4*d*a^2*b^2+3/2/e^4/(e*
x+d)^4*a^2*c^2*d^3-3/4/e^3/(e*x+d)^4*d^2*a*b^3+3/2/e^6/(e*x+d)^4*a*c^3*d^5-5/4/e^5/(e*x+d)^4*d^4*b^3*c+9/4/e^6
/(e*x+d)^4*b^2*c^2*d^5-7/4/e^7/(e*x+d)^4*b*c^3*d^6+20/3/e^5/(e*x+d)^3*b^3*c*d^3-15/e^6/(e*x+d)^3*b^2*c^2*d^4-6
/e^4/(e*x+d)^3*a^2*c^2*d^2+2/e^3/(e*x+d)^3*a*b^3*d-10/e^6/(e*x+d)^3*a*c^3*d^4

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maxima [A]  time = 0.66, size = 679, normalized size = 1.75 \[ -\frac {638 \, c^{4} d^{7} - 1197 \, b c^{3} d^{6} e + 3 \, a^{3} b e^{7} + 231 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} - 125 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} + 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} + 12 \, {\left (70 \, c^{4} d^{4} e^{3} - 140 \, b c^{3} d^{3} e^{4} + 30 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{5} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{6} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{7}\right )} x^{3} + 18 \, {\left (126 \, c^{4} d^{5} e^{2} - 245 \, b c^{3} d^{4} e^{3} + 50 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{4} - 30 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{5} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{6} + {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{7}\right )} x^{2} + 4 \, {\left (518 \, c^{4} d^{6} e - 987 \, b c^{3} d^{5} e^{2} + 195 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{3} - 110 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{4} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{5} + 3 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{6} + {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{7}\right )} x}{12 \, {\left (e^{12} x^{4} + 4 \, d e^{11} x^{3} + 6 \, d^{2} e^{10} x^{2} + 4 \, d^{3} e^{9} x + d^{4} e^{8}\right )}} + \frac {4 \, c^{4} e^{2} x^{3} - 3 \, {\left (10 \, c^{4} d e - 7 \, b c^{3} e^{2}\right )} x^{2} + 6 \, {\left (30 \, c^{4} d^{2} - 35 \, b c^{3} d e + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{2}\right )} x}{6 \, e^{7}} - \frac {5 \, {\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 )} \log \left (e x + d\right )}{e^{8}} \]

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,x, algorithm="maxima")

[Out]

-1/12*(638*c^4*d^7 - 1197*b*c^3*d^6*e + 3*a^3*b*e^7 + 231*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 125*(b^3*c + 3*a*b*c
^2)*d^4*e^3 + 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 + 12*(70*c^4*d^4*e^3 - 140*b*c^3*d^3*e^4 + 30*(3*b^2*c^2 + 2*a*c^3)*d^2*e^5 - 20*(b^3*c + 3*a*b*c^2)*d*
e^6 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^7)*x^3 + 18*(126*c^4*d^5*e^2 - 245*b*c^3*d^4*e^3 + 50*(3*b^2*c^2 + 2*a*
c^3)*d^3*e^4 - 30*(b^3*c + 3*a*b*c^2)*d^2*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^6 + (a*b^3 + 3*a^2*b*c)*e^7
)*x^2 + 4*(518*c^4*d^6*e - 987*b*c^3*d^5*e^2 + 195*(3*b^2*c^2 + 2*a*c^3)*d^4*e^3 - 110*(b^3*c + 3*a*b*c^2)*d^3
*e^4 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^5 + 3*(a*b^3 + 3*a^2*b*c)*d*e^6 + (3*a^2*b^2 + 2*a^3*c)*e^7)*x)/
(e^12*x^4 + 4*d*e^11*x^3 + 6*d^2*e^10*x^2 + 4*d^3*e^9*x + d^4*e^8) + 1/6*(4*c^4*e^2*x^3 - 3*(10*c^4*d*e - 7*b*
c^3*e^2)*x^2 + 6*(30*c^4*d^2 - 35*b*c^3*d*e + 3*(3*b^2*c^2 + 2*a*c^3)*e^2)*x)/e^7 - 5*(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)*log(e*x + d)/e^8

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mupad [B]  time = 1.88, size = 763, normalized size = 1.96 \[ x^2\,\left (\frac {7\,b\,c^3}{2\,e^5}-\frac {5\,c^4\,d}{e^6}\right )-x\,\left (\frac {5\,d\,\left (\frac {7\,b\,c^3}{e^5}-\frac {10\,c^4\,d}{e^6}\right )}{e}-\frac {9\,b^2\,c^2+6\,a\,c^3}{e^5}+\frac {20\,c^4\,d^2}{e^7}\right )-\frac {x^2\,\left (\frac {9\,a^2\,b\,c\,e^6}{2}+9\,a^2\,c^2\,d\,e^5+\frac {3\,a\,b^3\,e^6}{2}+18\,a\,b^2\,c\,d\,e^5-135\,a\,b\,c^2\,d^2\,e^4+150\,a\,c^3\,d^3\,e^3+\frac {3\,b^4\,d\,e^5}{2}-45\,b^3\,c\,d^2\,e^4+225\,b^2\,c^2\,d^3\,e^3-\frac {735\,b\,c^3\,d^4\,e^2}{2}+189\,c^4\,d^5\,e\right )+x^3\,\left (6\,a^2\,c^2\,e^6+12\,a\,b^2\,c\,e^6-60\,a\,b\,c^2\,d\,e^5+60\,a\,c^3\,d^2\,e^4+b^4\,e^6-20\,b^3\,c\,d\,e^5+90\,b^2\,c^2\,d^2\,e^4-140\,b\,c^3\,d^3\,e^3+70\,c^4\,d^4\,e^2\right )+x\,\left (\frac {2\,a^3\,c\,e^6}{3}+a^2\,b^2\,e^6+3\,a^2\,b\,c\,d\,e^5+6\,a^2\,c^2\,d^2\,e^4+a\,b^3\,d\,e^5+12\,a\,b^2\,c\,d^2\,e^4-110\,a\,b\,c^2\,d^3\,e^3+130\,a\,c^3\,d^4\,e^2+b^4\,d^2\,e^4-\frac {110\,b^3\,c\,d^3\,e^3}{3}+195\,b^2\,c^2\,d^4\,e^2-329\,b\,c^3\,d^5\,e+\frac {518\,c^4\,d^6}{3}\right )+\frac {3\,a^3\,b\,e^7+2\,a^3\,c\,d\,e^6+3\,a^2\,b^2\,d\,e^6+9\,a^2\,b\,c\,d^2\,e^5+18\,a^2\,c^2\,d^3\,e^4+3\,a\,b^3\,d^2\,e^5+36\,a\,b^2\,c\,d^3\,e^4-375\,a\,b\,c^2\,d^4\,e^3+462\,a\,c^3\,d^5\,e^2+3\,b^4\,d^3\,e^4-125\,b^3\,c\,d^4\,e^3+693\,b^2\,c^2\,d^5\,e^2-1197\,b\,c^3\,d^6\,e+638\,c^4\,d^7}{12\,e}}{d^4\,e^7+4\,d^3\,e^8\,x+6\,d^2\,e^9\,x^2+4\,d\,e^{10}\,x^3+e^{11}\,x^4}-\frac {\ln \left (d+e\,x\right )\,\left (-5\,b^3\,c\,e^3+45\,b^2\,c^2\,d\,e^2-105\,b\,c^3\,d^2\,e-15\,a\,b\,c^2\,e^3+70\,c^4\,d^3+30\,a\,c^3\,d\,e^2\right )}{e^8}+\frac {2\,c^4\,x^3}{3\,e^5} \]

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,x)

[Out]

x^2*((7*b*c^3)/(2*e^5) - (5*c^4*d)/e^6) - x*((5*d*((7*b*c^3)/e^5 - (10*c^4*d)/e^6))/e - (6*a*c^3 + 9*b^2*c^2)/
e^5 + (20*c^4*d^2)/e^7) - (x^2*((3*a*b^3*e^6)/2 + (3*b^4*d*e^5)/2 + 189*c^4*d^5*e + 150*a*c^3*d^3*e^3 + 9*a^2*
c^2*d*e^5 - (735*b*c^3*d^4*e^2)/2 - 45*b^3*c*d^2*e^4 + 225*b^2*c^2*d^3*e^3 + (9*a^2*b*c*e^6)/2 + 18*a*b^2*c*d*
e^5 - 135*a*b*c^2*d^2*e^4) + x^3*(b^4*e^6 + 6*a^2*c^2*e^6 + 70*c^4*d^4*e^2 + 60*a*c^3*d^2*e^4 - 140*b*c^3*d^3*
e^3 + 90*b^2*c^2*d^2*e^4 + 12*a*b^2*c*e^6 - 20*b^3*c*d*e^5 - 60*a*b*c^2*d*e^5) + x*((518*c^4*d^6)/3 + (2*a^3*c
*e^6)/3 + a^2*b^2*e^6 + b^4*d^2*e^4 + 130*a*c^3*d^4*e^2 - (110*b^3*c*d^3*e^3)/3 + 6*a^2*c^2*d^2*e^4 + 195*b^2*
c^2*d^4*e^2 + a*b^3*d*e^5 - 329*b*c^3*d^5*e + 3*a^2*b*c*d*e^5 - 110*a*b*c^2*d^3*e^3 + 12*a*b^2*c*d^2*e^4) + (6
38*c^4*d^7 + 3*a^3*b*e^7 + 3*b^4*d^3*e^4 + 3*a*b^3*d^2*e^5 + 3*a^2*b^2*d*e^6 + 462*a*c^3*d^5*e^2 - 125*b^3*c*d
^4*e^3 + 18*a^2*c^2*d^3*e^4 + 693*b^2*c^2*d^5*e^2 + 2*a^3*c*d*e^6 - 1197*b*c^3*d^6*e - 375*a*b*c^2*d^4*e^3 + 3
6*a*b^2*c*d^3*e^4 + 9*a^2*b*c*d^2*e^5)/(12*e))/(d^4*e^7 + e^11*x^4 + 4*d^3*e^8*x + 4*d*e^10*x^3 + 6*d^2*e^9*x^
2) - (log(d + e*x)*(70*c^4*d^3 - 5*b^3*c*e^3 + 45*b^2*c^2*d*e^2 - 15*a*b*c^2*e^3 + 30*a*c^3*d*e^2 - 105*b*c^3*
d^2*e))/e^8 + (2*c^4*x^3)/(3*e^5)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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,x)

[Out]

Timed out

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