3.2155 \(\int \frac {(a+b x+c x^2)^4}{(d+e x)^5} \, dx\)
Optimal. Leaf size=426 \[ \frac {\log (d+e x) \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^9}-\frac {c x \left (-20 c^2 d e (3 b d-a e)+6 b c e^2 (5 b d-2 a e)-4 b^3 e^3+35 c^3 d^3\right )}{e^8}-\frac {\left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right ) \left (a e^2-b d e+c d^2\right )^2}{e^9 (d+e x)^2}+\frac {4 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right ) \left (a e^2-b d e+c d^2\right )}{e^9 (d+e x)}+\frac {c^2 x^2 \left (-4 c e (5 b d-a e)+6 b^2 e^2+15 c^2 d^2\right )}{2 e^7}-\frac {\left (a e^2-b d e+c d^2\right )^4}{4 e^9 (d+e x)^4}+\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^9 (d+e x)^3}-\frac {c^3 x^3 (5 c d-4 b e)}{3 e^6}+\frac {c^4 x^4}{4 e^5} \]
[Out]
-c*(35*c^3*d^3-4*b^3*e^3+6*b*c*e^2*(-2*a*e+5*b*d)-20*c^2*d*e*(-a*e+3*b*d))*x/e^8+1/2*c^2*(15*c^2*d^2+6*b^2*e^2
-4*c*e*(-a*e+5*b*d))*x^2/e^7-1/3*c^3*(-4*b*e+5*c*d)*x^3/e^6+1/4*c^4*x^4/e^5-1/4*(a*e^2-b*d*e+c*d^2)^4/e^9/(e*x
+d)^4+4/3*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^3/e^9/(e*x+d)^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^9/(e*x+d)^2+4*(-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^9/(e*x+
d)+(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))*ln(e*x+d)/e^9
________________________________________________________________________________________
Rubi [A] time = 0.63, antiderivative size = 426, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used =
{698} \[ \frac {\log (d+e x) \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^9}+\frac {c^2 x^2 \left (-4 c e (5 b d-a e)+6 b^2 e^2+15 c^2 d^2\right )}{2 e^7}-\frac {\left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right ) \left (a e^2-b d e+c d^2\right )^2}{e^9 (d+e x)^2}+\frac {4 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right ) \left (a e^2-b d e+c d^2\right )}{e^9 (d+e x)}-\frac {c x \left (-20 c^2 d e (3 b d-a e)+6 b c e^2 (5 b d-2 a e)-4 b^3 e^3+35 c^3 d^3\right )}{e^8}-\frac {\left (a e^2-b d e+c d^2\right )^4}{4 e^9 (d+e x)^4}+\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^9 (d+e x)^3}-\frac {c^3 x^3 (5 c d-4 b e)}{3 e^6}+\frac {c^4 x^4}{4 e^5} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*x + c*x^2)^4/(d + e*x)^5,x]
[Out]
-((c*(35*c^3*d^3 - 4*b^3*e^3 + 6*b*c*e^2*(5*b*d - 2*a*e) - 20*c^2*d*e*(3*b*d - a*e))*x)/e^8) + (c^2*(15*c^2*d^
2 + 6*b^2*e^2 - 4*c*e*(5*b*d - a*e))*x^2)/(2*e^7) - (c^3*(5*c*d - 4*b*e)*x^3)/(3*e^6) + (c^4*x^4)/(4*e^5) - (c
*d^2 - b*d*e + a*e^2)^4/(4*e^9*(d + e*x)^4) + (4*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(3*e^9*(d + e*x)^3)
- ((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^9*(d + e*x)^2) + (4*(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)))/(e^9*(d + e*x)) + ((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))*Log[d + e*x])/e^9
Rule 698
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^5} \, dx &=\int \left (\frac {c \left (-35 c^3 d^3+4 b^3 e^3-6 b c e^2 (5 b d-2 a e)+20 c^2 d e (3 b d-a e)\right )}{e^8}+\frac {c^2 \left (15 c^2 d^2+6 b^2 e^2-4 c e (5 b d-a e)\right ) x}{e^7}-\frac {c^3 (5 c d-4 b e) x^2}{e^6}+\frac {c^4 x^3}{e^5}+\frac {\left (c d^2-b d e+a e^2\right )^4}{e^8 (d+e x)^5}+\frac {4 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^3}{e^8 (d+e x)^4}+\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 (d+e x)^3}+\frac {4 (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^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)}\right ) \, dx\\ &=-\frac {c \left (35 c^3 d^3-4 b^3 e^3+6 b c e^2 (5 b d-2 a e)-20 c^2 d e (3 b d-a e)\right ) x}{e^8}+\frac {c^2 \left (15 c^2 d^2+6 b^2 e^2-4 c e (5 b d-a e)\right ) x^2}{2 e^7}-\frac {c^3 (5 c d-4 b e) x^3}{3 e^6}+\frac {c^4 x^4}{4 e^5}-\frac {\left (c d^2-b d e+a e^2\right )^4}{4 e^9 (d+e x)^4}+\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^3}{3 e^9 (d+e x)^3}-\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^9 (d+e x)^2}+\frac {4 (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 )}{e^9 (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 ) \log (d+e x)}{e^9}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.20, size = 430, normalized size = 1.01 \[ \frac {\frac {48 (2 c d-b e) \left (c e^2 \left (3 a^2 e^2-10 a b d e+8 b^2 d^2\right )+b^2 e^3 (a e-b d)-2 c^2 d^2 e (7 b d-5 a e)+7 c^3 d^4\right )}{d+e x}+12 \log (d+e x) \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )+12 c e x \left (20 c^2 d e (3 b d-a e)-6 b c e^2 (5 b d-2 a e)+4 b^3 e^3-35 c^3 d^3\right )+6 c^2 e^2 x^2 \left (4 c e (a e-5 b d)+6 b^2 e^2+15 c^2 d^2\right )-\frac {12 \left (2 c e (a e-7 b d)+3 b^2 e^2+14 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )^2}{(d+e x)^2}-\frac {3 \left (e (a e-b d)+c d^2\right )^4}{(d+e x)^4}+\frac {16 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^3}{(d+e x)^3}+4 c^3 e^3 x^3 (4 b e-5 c d)+3 c^4 e^4 x^4}{12 e^9} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x + c*x^2)^4/(d + e*x)^5,x]
[Out]
(12*c*e*(-35*c^3*d^3 + 4*b^3*e^3 - 6*b*c*e^2*(5*b*d - 2*a*e) + 20*c^2*d*e*(3*b*d - a*e))*x + 6*c^2*e^2*(15*c^2
*d^2 + 6*b^2*e^2 + 4*c*e*(-5*b*d + a*e))*x^2 + 4*c^3*e^3*(-5*c*d + 4*b*e)*x^3 + 3*c^4*e^4*x^4 - (3*(c*d^2 + e*
(-(b*d) + a*e))^4)/(d + e*x)^4 + (16*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e*x)^3 - (12*(14*c^2*d^2
+ 3*b^2*e^2 + 2*c*e*(-7*b*d + a*e))*(c*d^2 + e*(-(b*d) + a*e))^2)/(d + e*x)^2 + (48*(2*c*d - b*e)*(7*c^3*d^4
- 2*c^2*d^2*e*(7*b*d - 5*a*e) + b^2*e^3*(-(b*d) + a*e) + c*e^2*(8*b^2*d^2 - 10*a*b*d*e + 3*a^2*e^2)))/(d + e*x
) + 12*(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))*Log[d + e*x])/(12*e^9)
________________________________________________________________________________________
fricas [B] time = 0.96, size = 1303, normalized size = 3.06 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x^2+b*x+a)^4/(e*x+d)^5,x, algorithm="fricas")
[Out]
1/12*(3*c^4*e^8*x^8 + 533*c^4*d^8 - 1276*b*c^3*d^7*e - 4*a^3*b*d*e^7 - 3*a^4*e^8 + 342*(3*b^2*c^2 + 2*a*c^3)*d
^6*e^2 - 308*(b^3*c + 3*a*b*c^2)*d^5*e^3 + 25*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 - 12*(a*b^3 + 3*a^2*b*c)*
d^3*e^5 - 2*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 - 8*(c^4*d*e^7 - 2*b*c^3*e^8)*x^7 + 4*(7*c^4*d^2*e^6 - 14*b*c^3*d*e^
7 + 3*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 - 24*(7*c^4*d^3*e^5 - 14*b*c^3*d^2*e^6 + 3*(3*b^2*c^2 + 2*a*c^3)*d*e^7 -
2*(b^3*c + 3*a*b*c^2)*e^8)*x^5 - (1217*c^4*d^4*e^4 - 2224*b*c^3*d^3*e^5 + 408*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 -
192*(b^3*c + 3*a*b*c^2)*d*e^7)*x^4 - 4*(377*c^4*d^5*e^3 - 544*b*c^3*d^4*e^4 + 48*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5
+ 48*(b^3*c + 3*a*b*c^2)*d^2*e^6 - 12*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^7 + 12*(a*b^3 + 3*a^2*b*c)*e^8)*x^3
+ 6*(43*c^4*d^6*e^2 - 296*b*c^3*d^5*e^3 + 132*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 168*(b^3*c + 3*a*b*c^2)*d^3*e^5
+ 18*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 - 12*(a*b^3 + 3*a^2*b*c)*d*e^7 - 2*(3*a^2*b^2 + 2*a^3*c)*e^8)*x^2
+ 4*(323*c^4*d^7*e - 856*b*c^3*d^6*e^2 - 4*a^3*b*e^8 + 252*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 248*(b^3*c + 3*a*b*
c^2)*d^4*e^4 + 22*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 12*(a*b^3 + 3*a^2*b*c)*d^2*e^6 - 2*(3*a^2*b^2 + 2*a
^3*c)*d*e^7)*x + 12*(70*c^4*d^8 - 140*b*c^3*d^7*e + 30*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 20*(b^3*c + 3*a*b*c^2)*
d^5*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + (70*c^4*d^4*e^4 - 140*b*c^3*d^3*e^5 + 30*(3*b^2*c^2 + 2*a*c
^3)*d^2*e^6 - 20*(b^3*c + 3*a*b*c^2)*d*e^7 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*x^4 + 4*(70*c^4*d^5*e^3 - 140
*b*c^3*d^4*e^4 + 30*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 20*(b^3*c + 3*a*b*c^2)*d^2*e^6 + (b^4 + 12*a*b^2*c + 6*a^2
*c^2)*d*e^7)*x^3 + 6*(70*c^4*d^6*e^2 - 140*b*c^3*d^5*e^3 + 30*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 20*(b^3*c + 3*a*
b*c^2)*d^3*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6)*x^2 + 4*(70*c^4*d^7*e - 140*b*c^3*d^6*e^2 + 30*(3*b^2
*c^2 + 2*a*c^3)*d^5*e^3 - 20*(b^3*c + 3*a*b*c^2)*d^4*e^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5)*x)*log(e*x
+ d))/(e^13*x^4 + 4*d*e^12*x^3 + 6*d^2*e^11*x^2 + 4*d^3*e^10*x + d^4*e^9)
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giac [B] time = 0.22, size = 1283, normalized size = 3.01 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x^2+b*x+a)^4/(e*x+d)^5,x, algorithm="giac")
[Out]
1/12*(3*c^4 - 16*(2*c^4*d*e - b*c^3*e^2)*e^(-1)/(x*e + d) + 12*(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 - 48*(14*c^4*d^3*e^3 - 21*b*c^3*d^2*e^4 + 9*b^2*c^2*d*e^5 + 6*a*c^3*d*e^5
- b^3*c*e^6 - 3*a*b*c^2*e^6)*e^(-3)/(x*e + d)^3)*(x*e + d)^4*e^(-9) - (70*c^4*d^4 - 140*b*c^3*d^3*e + 90*b^2*c
^2*d^2*e^2 + 60*a*c^3*d^2*e^2 - 20*b^3*c*d*e^3 - 60*a*b*c^2*d*e^3 + b^4*e^4 + 12*a*b^2*c*e^4 + 6*a^2*c^2*e^4)*
e^(-9)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) + 1/12*(672*c^4*d^5*e^43/(x*e + d) - 168*c^4*d^6*e^43/(x*e + d)^2
+ 32*c^4*d^7*e^43/(x*e + d)^3 - 3*c^4*d^8*e^43/(x*e + d)^4 - 1680*b*c^3*d^4*e^44/(x*e + d) + 504*b*c^3*d^5*e^4
4/(x*e + d)^2 - 112*b*c^3*d^6*e^44/(x*e + d)^3 + 12*b*c^3*d^7*e^44/(x*e + d)^4 + 1440*b^2*c^2*d^3*e^45/(x*e +
d) + 960*a*c^3*d^3*e^45/(x*e + d) - 540*b^2*c^2*d^4*e^45/(x*e + d)^2 - 360*a*c^3*d^4*e^45/(x*e + d)^2 + 144*b^
2*c^2*d^5*e^45/(x*e + d)^3 + 96*a*c^3*d^5*e^45/(x*e + d)^3 - 18*b^2*c^2*d^6*e^45/(x*e + d)^4 - 12*a*c^3*d^6*e^
45/(x*e + d)^4 - 480*b^3*c*d^2*e^46/(x*e + d) - 1440*a*b*c^2*d^2*e^46/(x*e + d) + 240*b^3*c*d^3*e^46/(x*e + d)
^2 + 720*a*b*c^2*d^3*e^46/(x*e + d)^2 - 80*b^3*c*d^4*e^46/(x*e + d)^3 - 240*a*b*c^2*d^4*e^46/(x*e + d)^3 + 12*
b^3*c*d^5*e^46/(x*e + d)^4 + 36*a*b*c^2*d^5*e^46/(x*e + d)^4 + 48*b^4*d*e^47/(x*e + d) + 576*a*b^2*c*d*e^47/(x
*e + d) + 288*a^2*c^2*d*e^47/(x*e + d) - 36*b^4*d^2*e^47/(x*e + d)^2 - 432*a*b^2*c*d^2*e^47/(x*e + d)^2 - 216*
a^2*c^2*d^2*e^47/(x*e + d)^2 + 16*b^4*d^3*e^47/(x*e + d)^3 + 192*a*b^2*c*d^3*e^47/(x*e + d)^3 + 96*a^2*c^2*d^3
*e^47/(x*e + d)^3 - 3*b^4*d^4*e^47/(x*e + d)^4 - 36*a*b^2*c*d^4*e^47/(x*e + d)^4 - 18*a^2*c^2*d^4*e^47/(x*e +
d)^4 - 48*a*b^3*e^48/(x*e + d) - 144*a^2*b*c*e^48/(x*e + d) + 72*a*b^3*d*e^48/(x*e + d)^2 + 216*a^2*b*c*d*e^48
/(x*e + d)^2 - 48*a*b^3*d^2*e^48/(x*e + d)^3 - 144*a^2*b*c*d^2*e^48/(x*e + d)^3 + 12*a*b^3*d^3*e^48/(x*e + d)^
4 + 36*a^2*b*c*d^3*e^48/(x*e + d)^4 - 36*a^2*b^2*e^49/(x*e + d)^2 - 24*a^3*c*e^49/(x*e + d)^2 + 48*a^2*b^2*d*e
^49/(x*e + d)^3 + 32*a^3*c*d*e^49/(x*e + d)^3 - 18*a^2*b^2*d^2*e^49/(x*e + d)^4 - 12*a^3*c*d^2*e^49/(x*e + d)^
4 - 16*a^3*b*e^50/(x*e + d)^3 + 12*a^3*b*d*e^50/(x*e + d)^4 - 3*a^4*e^51/(x*e + d)^4)*e^(-52)
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maple [B] time = 0.07, size = 1306, normalized size = 3.07 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c*x^2+b*x+a)^4/(e*x+d)^5,x)
[Out]
48/e^5/(e*x+d)*a*b^2*c*d-120/e^6/(e*x+d)*a*b*c^2*d^2+18/e^4/(e*x+d)^2*a^2*b*c*d-36/e^5/(e*x+d)^2*a*b^2*c*d^2+1
5/2*c^4/e^7*x^2*d^2+4*c/e^5*b^3*x-35*c^4/e^8*d^3*x-4/e^4/(e*x+d)*a*b^3+4/e^5/(e*x+d)*b^4*d+56/e^9/(e*x+d)*c^4*
d^5-2/e^3/(e*x+d)^2*a^3*c-3/e^3/(e*x+d)^2*a^2*b^2-3/e^5/(e*x+d)^2*b^4*d^2-14/e^9/(e*x+d)^2*c^4*d^6+4/3*c^3/e^5
*x^3*b-5/3*c^4/e^6*x^3*d+2*c^3/e^5*x^2*a+3*c^2/e^5*x^2*b^2-4/3/e^2/(e*x+d)^3*a^3*b-1/4/e^5/(e*x+d)^4*d^4*b^4-1
/4/e^9/(e*x+d)^4*c^4*d^8+6/e^5*ln(e*x+d)*c^2*a^2+70/e^9*ln(e*x+d)*c^4*d^4+4/3/e^5/(e*x+d)^3*b^4*d^3+8/3/e^9/(e
*x+d)^3*c^4*d^7-10*c^3/e^6*x^2*b*d+12/e^7/(e*x+d)^3*b^2*c^2*d^5+3/e^6/(e*x+d)^4*a*b*c^2*d^5+3/e^4/(e*x+d)^4*d^
3*a^2*b*c-60/e^6*ln(e*x+d)*a*b*c^2*d+60/e^6/(e*x+d)^2*a*b*c^2*d^3-12/e^4/(e*x+d)^3*a^2*b*c*d^2+16/e^5/(e*x+d)^
3*a*b^2*c*d^3-20/e^6/(e*x+d)^3*a*b*c^2*d^4-1/4/e/(e*x+d)^4*a^4-45/e^7/(e*x+d)^2*b^2*c^2*d^4+42/e^8/(e*x+d)^2*b
*c^3*d^5+8/3/e^3/(e*x+d)^3*a^3*c*d+4/e^3/(e*x+d)^3*a^2*b^2*d+8/e^5/(e*x+d)^3*a^2*c^2*d^3-4/e^4/(e*x+d)^3*a*b^3
*d^2+8/e^7/(e*x+d)^3*a*c^3*d^5-20/3/e^6/(e*x+d)^3*b^3*c*d^4-20*c^3/e^6*a*d*x+12*c^2/e^5*a*b*x-3/e^5/(e*x+d)^4*
d^4*a*b^2*c+b^4/e^5*ln(e*x+d)+1/4*c^4*x^4/e^5-28/3/e^8/(e*x+d)^3*b*c^3*d^6+1/e^2/(e*x+d)^4*d*a^3*b-1/e^3/(e*x+
d)^4*a^3*c*d^2-3/2/e^3/(e*x+d)^4*d^2*a^2*b^2-3/2/e^5/(e*x+d)^4*a^2*c^2*d^4+1/e^4/(e*x+d)^4*d^3*a*b^3-1/e^7/(e*
x+d)^4*a*c^3*d^6+1/e^6/(e*x+d)^4*b^3*c*d^5-3/2/e^7/(e*x+d)^4*b^2*c^2*d^6+1/e^8/(e*x+d)^4*b*c^3*d^7-30*c^2/e^6*
b^2*d*x+60*c^3/e^7*b*d^2*x-12/e^4/(e*x+d)*a^2*b*c+24/e^5/(e*x+d)*a^2*c^2*d+80/e^7/(e*x+d)*a*c^3*d^3-40/e^6/(e*
x+d)*b^3*c*d^2+120/e^7/(e*x+d)*b^2*c^2*d^3-140/e^8/(e*x+d)*b*c^3*d^4-18/e^5/(e*x+d)^2*a^2*c^2*d^2+6/e^4/(e*x+d
)^2*a*b^3*d-30/e^7/(e*x+d)^2*a*c^3*d^4+20/e^6/(e*x+d)^2*b^3*c*d^3+12/e^5*ln(e*x+d)*a*b^2*c+60/e^7*ln(e*x+d)*a*
c^3*d^2-20/e^6*ln(e*x+d)*b^3*c*d+90/e^7*ln(e*x+d)*b^2*c^2*d^2-140/e^8*ln(e*x+d)*b*c^3*d^3
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maxima [B] time = 1.30, size = 843, normalized size = 1.98 \[ \frac {533 \, c^{4} d^{8} - 1276 \, b c^{3} d^{7} e - 4 \, a^{3} b d e^{7} - 3 \, a^{4} e^{8} + 342 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{2} - 308 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5} e^{3} + 25 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} e^{4} - 12 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e^{5} - 2 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{6} + 48 \, {\left (14 \, c^{4} d^{5} e^{3} - 35 \, b c^{3} d^{4} e^{4} + 10 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{5} - 10 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{6} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{7} - {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{8}\right )} x^{3} + 12 \, {\left (154 \, c^{4} d^{6} e^{2} - 378 \, b c^{3} d^{5} e^{3} + 105 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{4} - 100 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{5} + 9 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{6} - 6 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{7} - {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{8}\right )} x^{2} + 8 \, {\left (214 \, c^{4} d^{7} e - 518 \, b c^{3} d^{6} e^{2} - 2 \, a^{3} b e^{8} + 141 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{3} - 130 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{4} + 11 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{5} - 6 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{6} - {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{7}\right )} x}{12 \, {\left (e^{13} x^{4} + 4 \, d e^{12} x^{3} + 6 \, d^{2} e^{11} x^{2} + 4 \, d^{3} e^{10} x + d^{4} e^{9}\right )}} + \frac {3 \, c^{4} e^{3} x^{4} - 4 \, {\left (5 \, c^{4} d e^{2} - 4 \, b c^{3} e^{3}\right )} x^{3} + 6 \, {\left (15 \, c^{4} d^{2} e - 20 \, b c^{3} d e^{2} + 2 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{3}\right )} x^{2} - 12 \, {\left (35 \, c^{4} d^{3} - 60 \, b c^{3} d^{2} e + 10 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{2} - 4 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{3}\right )} x}{12 \, e^{8}} + \frac {{\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 )} \log \left (e x + d\right )}{e^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x^2+b*x+a)^4/(e*x+d)^5,x, algorithm="maxima")
[Out]
1/12*(533*c^4*d^8 - 1276*b*c^3*d^7*e - 4*a^3*b*d*e^7 - 3*a^4*e^8 + 342*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 308*(b^
3*c + 3*a*b*c^2)*d^5*e^3 + 25*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 - 12*(a*b^3 + 3*a^2*b*c)*d^3*e^5 - 2*(3*a
^2*b^2 + 2*a^3*c)*d^2*e^6 + 48*(14*c^4*d^5*e^3 - 35*b*c^3*d^4*e^4 + 10*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 10*(b^3
*c + 3*a*b*c^2)*d^2*e^6 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^7 - (a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 12*(154*c^4*d^
6*e^2 - 378*b*c^3*d^5*e^3 + 105*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 100*(b^3*c + 3*a*b*c^2)*d^3*e^5 + 9*(b^4 + 12*
a*b^2*c + 6*a^2*c^2)*d^2*e^6 - 6*(a*b^3 + 3*a^2*b*c)*d*e^7 - (3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 + 8*(214*c^4*d^7*e
- 518*b*c^3*d^6*e^2 - 2*a^3*b*e^8 + 141*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 130*(b^3*c + 3*a*b*c^2)*d^4*e^4 + 11*
(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 6*(a*b^3 + 3*a^2*b*c)*d^2*e^6 - (3*a^2*b^2 + 2*a^3*c)*d*e^7)*x)/(e^13
*x^4 + 4*d*e^12*x^3 + 6*d^2*e^11*x^2 + 4*d^3*e^10*x + d^4*e^9) + 1/12*(3*c^4*e^3*x^4 - 4*(5*c^4*d*e^2 - 4*b*c^
3*e^3)*x^3 + 6*(15*c^4*d^2*e - 20*b*c^3*d*e^2 + 2*(3*b^2*c^2 + 2*a*c^3)*e^3)*x^2 - 12*(35*c^4*d^3 - 60*b*c^3*d
^2*e + 10*(3*b^2*c^2 + 2*a*c^3)*d*e^2 - 4*(b^3*c + 3*a*b*c^2)*e^3)*x)/e^8 + (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)*log(e*x +
d)/e^9
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mupad [B] time = 0.81, size = 1005, normalized size = 2.36 \[ x^3\,\left (\frac {4\,b\,c^3}{3\,e^5}-\frac {5\,c^4\,d}{3\,e^6}\right )-\frac {x\,\left (\frac {4\,a^3\,b\,e^7}{3}+\frac {4\,a^3\,c\,d\,e^6}{3}+2\,a^2\,b^2\,d\,e^6+12\,a^2\,b\,c\,d^2\,e^5-44\,a^2\,c^2\,d^3\,e^4+4\,a\,b^3\,d^2\,e^5-88\,a\,b^2\,c\,d^3\,e^4+260\,a\,b\,c^2\,d^4\,e^3-188\,a\,c^3\,d^5\,e^2-\frac {22\,b^4\,d^3\,e^4}{3}+\frac {260\,b^3\,c\,d^4\,e^3}{3}-282\,b^2\,c^2\,d^5\,e^2+\frac {1036\,b\,c^3\,d^6\,e}{3}-\frac {428\,c^4\,d^7}{3}\right )-x^3\,\left (-12\,a^2\,b\,c\,e^7+24\,a^2\,c^2\,d\,e^6-4\,a\,b^3\,e^7+48\,a\,b^2\,c\,d\,e^6-120\,a\,b\,c^2\,d^2\,e^5+80\,a\,c^3\,d^3\,e^4+4\,b^4\,d\,e^6-40\,b^3\,c\,d^2\,e^5+120\,b^2\,c^2\,d^3\,e^4-140\,b\,c^3\,d^4\,e^3+56\,c^4\,d^5\,e^2\right )+\frac {3\,a^4\,e^8+4\,a^3\,b\,d\,e^7+4\,a^3\,c\,d^2\,e^6+6\,a^2\,b^2\,d^2\,e^6+36\,a^2\,b\,c\,d^3\,e^5-150\,a^2\,c^2\,d^4\,e^4+12\,a\,b^3\,d^3\,e^5-300\,a\,b^2\,c\,d^4\,e^4+924\,a\,b\,c^2\,d^5\,e^3-684\,a\,c^3\,d^6\,e^2-25\,b^4\,d^4\,e^4+308\,b^3\,c\,d^5\,e^3-1026\,b^2\,c^2\,d^6\,e^2+1276\,b\,c^3\,d^7\,e-533\,c^4\,d^8}{12\,e}+x^2\,\left (2\,a^3\,c\,e^7+3\,a^2\,b^2\,e^7+18\,a^2\,b\,c\,d\,e^6-54\,a^2\,c^2\,d^2\,e^5+6\,a\,b^3\,d\,e^6-108\,a\,b^2\,c\,d^2\,e^5+300\,a\,b\,c^2\,d^3\,e^4-210\,a\,c^3\,d^4\,e^3-9\,b^4\,d^2\,e^5+100\,b^3\,c\,d^3\,e^4-315\,b^2\,c^2\,d^4\,e^3+378\,b\,c^3\,d^5\,e^2-154\,c^4\,d^6\,e\right )}{d^4\,e^8+4\,d^3\,e^9\,x+6\,d^2\,e^{10}\,x^2+4\,d\,e^{11}\,x^3+e^{12}\,x^4}-x^2\,\left (\frac {5\,d\,\left (\frac {4\,b\,c^3}{e^5}-\frac {5\,c^4\,d}{e^6}\right )}{2\,e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{2\,e^5}+\frac {5\,c^4\,d^2}{e^7}\right )-x\,\left (\frac {10\,c^4\,d^3}{e^8}+\frac {10\,d^2\,\left (\frac {4\,b\,c^3}{e^5}-\frac {5\,c^4\,d}{e^6}\right )}{e^2}-\frac {5\,d\,\left (\frac {5\,d\,\left (\frac {4\,b\,c^3}{e^5}-\frac {5\,c^4\,d}{e^6}\right )}{e}-\frac {6\,b^2\,c^2+4\,a\,c^3}{e^5}+\frac {10\,c^4\,d^2}{e^7}\right )}{e}-\frac {4\,b\,c\,\left (b^2+3\,a\,c\right )}{e^5}\right )+\frac {c^4\,x^4}{4\,e^5}+\frac {\ln \left (d+e\,x\right )\,\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 )}{e^9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*x + c*x^2)^4/(d + e*x)^5,x)
[Out]
x^3*((4*b*c^3)/(3*e^5) - (5*c^4*d)/(3*e^6)) - (x*((4*a^3*b*e^7)/3 - (428*c^4*d^7)/3 - (22*b^4*d^3*e^4)/3 + 4*a
*b^3*d^2*e^5 + 2*a^2*b^2*d*e^6 - 188*a*c^3*d^5*e^2 + (260*b^3*c*d^4*e^3)/3 - 44*a^2*c^2*d^3*e^4 - 282*b^2*c^2*
d^5*e^2 + (4*a^3*c*d*e^6)/3 + (1036*b*c^3*d^6*e)/3 + 260*a*b*c^2*d^4*e^3 - 88*a*b^2*c*d^3*e^4 + 12*a^2*b*c*d^2
*e^5) - x^3*(4*b^4*d*e^6 - 4*a*b^3*e^7 + 56*c^4*d^5*e^2 + 80*a*c^3*d^3*e^4 + 24*a^2*c^2*d*e^6 - 140*b*c^3*d^4*
e^3 - 40*b^3*c*d^2*e^5 + 120*b^2*c^2*d^3*e^4 - 12*a^2*b*c*e^7 + 48*a*b^2*c*d*e^6 - 120*a*b*c^2*d^2*e^5) + (3*a
^4*e^8 - 533*c^4*d^8 - 25*b^4*d^4*e^4 + 12*a*b^3*d^3*e^5 - 684*a*c^3*d^6*e^2 + 4*a^3*c*d^2*e^6 + 308*b^3*c*d^5
*e^3 + 6*a^2*b^2*d^2*e^6 - 150*a^2*c^2*d^4*e^4 - 1026*b^2*c^2*d^6*e^2 + 4*a^3*b*d*e^7 + 1276*b*c^3*d^7*e + 924
*a*b*c^2*d^5*e^3 - 300*a*b^2*c*d^4*e^4 + 36*a^2*b*c*d^3*e^5)/(12*e) + x^2*(2*a^3*c*e^7 - 154*c^4*d^6*e + 3*a^2
*b^2*e^7 - 9*b^4*d^2*e^5 - 210*a*c^3*d^4*e^3 + 378*b*c^3*d^5*e^2 + 100*b^3*c*d^3*e^4 - 54*a^2*c^2*d^2*e^5 - 31
5*b^2*c^2*d^4*e^3 + 6*a*b^3*d*e^6 + 18*a^2*b*c*d*e^6 + 300*a*b*c^2*d^3*e^4 - 108*a*b^2*c*d^2*e^5))/(d^4*e^8 +
e^12*x^4 + 4*d^3*e^9*x + 4*d*e^11*x^3 + 6*d^2*e^10*x^2) - x^2*((5*d*((4*b*c^3)/e^5 - (5*c^4*d)/e^6))/(2*e) - (
4*a*c^3 + 6*b^2*c^2)/(2*e^5) + (5*c^4*d^2)/e^7) - x*((10*c^4*d^3)/e^8 + (10*d^2*((4*b*c^3)/e^5 - (5*c^4*d)/e^6
))/e^2 - (5*d*((5*d*((4*b*c^3)/e^5 - (5*c^4*d)/e^6))/e - (4*a*c^3 + 6*b^2*c^2)/e^5 + (10*c^4*d^2)/e^7))/e - (4
*b*c*(3*a*c + b^2))/e^5) + (c^4*x^4)/(4*e^5) + (log(d + e*x)*(b^4*e^4 + 70*c^4*d^4 + 6*a^2*c^2*e^4 + 60*a*c^3*
d^2*e^2 + 90*b^2*c^2*d^2*e^2 + 12*a*b^2*c*e^4 - 140*b*c^3*d^3*e - 20*b^3*c*d*e^3 - 60*a*b*c^2*d*e^3))/e^9
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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((c*x**2+b*x+a)**4/(e*x+d)**5,x)
[Out]
Timed out
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