3.2157 \(\int \frac{(a+b x+c x^2)^4}{(d+e x)^7} \, dx\)

Optimal. Leaf size=426 \[ -\frac{6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{2 e^9 (d+e x)^2}+\frac{4 c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (d+e x)}+\frac{4 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9 (d+e x)^3}-\frac{\left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{2 e^9 (d+e x)^4}+\frac{2 c^2 \log (d+e x) \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9}+\frac{4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{5 e^9 (d+e x)^5}-\frac{\left (a e^2-b d e+c d^2\right )^4}{6 e^9 (d+e x)^6}-\frac{c^3 x (7 c d-4 b e)}{e^8}+\frac{c^4 x^2}{2 e^7} \]

[Out]

-((c^3*(7*c*d - 4*b*e)*x)/e^8) + (c^4*x^2)/(2*e^7) - (c*d^2 - b*d*e + a*e^2)^4/(6*e^9*(d + e*x)^6) + (4*(2*c*d
- b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(5*e^9*(d + e*x)^5) - ((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)))/(2*e^9*(d + e*x)^4) + (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)))/(3*e^9*(d + e*x)^3) - (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))/(2*e^9*(d + e*x)^2) + (4*c*(2*c*d - b*e)*(7*c^
2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e)))/(e^9*(d + e*x)) + (2*c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e)
)*Log[d + e*x])/e^9

________________________________________________________________________________________

Rubi [A]  time = 0.507144, antiderivative size = 426, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {698} \[ -\frac{6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{2 e^9 (d+e x)^2}+\frac{4 c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (d+e x)}+\frac{4 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9 (d+e x)^3}-\frac{\left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{2 e^9 (d+e x)^4}+\frac{2 c^2 \log (d+e x) \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9}+\frac{4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{5 e^9 (d+e x)^5}-\frac{\left (a e^2-b d e+c d^2\right )^4}{6 e^9 (d+e x)^6}-\frac{c^3 x (7 c d-4 b e)}{e^8}+\frac{c^4 x^2}{2 e^7} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^4/(d + e*x)^7,x]

[Out]

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

Mathematica [A]  time = 0.3502, size = 764, normalized size = 1.79 \[ \frac{-3 c^2 e^2 \left (2 a^2 e^2 \left (15 d^2 e^2 x^2+6 d^3 e x+d^4+20 d e^3 x^3+15 e^4 x^4\right )+20 a b e \left (15 d^3 e^2 x^2+20 d^2 e^3 x^3+6 d^4 e x+d^5+15 d e^4 x^4+6 e^5 x^5\right )+b^2 (-d) \left (1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+822 d^4 e x+147 d^5+1350 d e^4 x^4+360 e^5 x^5\right )\right )-2 c e^3 \left (3 a^2 b e^2 \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )+a^3 e^3 \left (d^2+6 d e x+15 e^2 x^2\right )+6 a b^2 e \left (15 d^2 e^2 x^2+6 d^3 e x+d^4+20 d e^3 x^3+15 e^4 x^4\right )+10 b^3 \left (15 d^3 e^2 x^2+20 d^2 e^3 x^3+6 d^4 e x+d^5+15 d e^4 x^4+6 e^5 x^5\right )\right )-e^4 \left (3 a^2 b^2 e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+4 a^3 b e^3 (d+6 e x)+5 a^4 e^4+2 a b^3 e \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )+b^4 \left (15 d^2 e^2 x^2+6 d^3 e x+d^4+20 d e^3 x^3+15 e^4 x^4\right )\right )+60 c^2 (d+e x)^6 \log (d+e x) \left (2 c e (a e-7 b d)+3 b^2 e^2+14 c^2 d^2\right )+2 c^3 e \left (a d e \left (1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+822 d^4 e x+147 d^5+1350 d e^4 x^4+360 e^5 x^5\right )-b \left (7725 d^5 e^2 x^2+8200 d^4 e^3 x^3+4050 d^3 e^4 x^4+360 d^2 e^5 x^5+3594 d^6 e x+669 d^7-360 d e^6 x^6-60 e^7 x^7\right )\right )+c^4 \left (10725 d^6 e^2 x^2+10100 d^5 e^3 x^3+3375 d^4 e^4 x^4-1170 d^3 e^5 x^5-1035 d^2 e^6 x^6+5298 d^7 e x+1023 d^8-120 d e^7 x^7+15 e^8 x^8\right )}{30 e^9 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^4/(d + e*x)^7,x]

[Out]

(c^4*(1023*d^8 + 5298*d^7*e*x + 10725*d^6*e^2*x^2 + 10100*d^5*e^3*x^3 + 3375*d^4*e^4*x^4 - 1170*d^3*e^5*x^5 -
1035*d^2*e^6*x^6 - 120*d*e^7*x^7 + 15*e^8*x^8) - e^4*(5*a^4*e^4 + 4*a^3*b*e^3*(d + 6*e*x) + 3*a^2*b^2*e^2*(d^2
+ 6*d*e*x + 15*e^2*x^2) + 2*a*b^3*e*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + b^4*(d^4 + 6*d^3*e*x + 15
*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4)) - 2*c*e^3*(a^3*e^3*(d^2 + 6*d*e*x + 15*e^2*x^2) + 3*a^2*b*e^2*(d^3
+ 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + 6*a*b^2*e*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4
*x^4) + 10*b^3*(d^5 + 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5)) - 3*c^2*e^2*(2*
a^2*e^2*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4) + 20*a*b*e*(d^5 + 6*d^4*e*x + 15*d^3*e^
2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5) - b^2*d*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^
2*e^3*x^3 + 1350*d*e^4*x^4 + 360*e^5*x^5)) + 2*c^3*e*(a*d*e*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d
^2*e^3*x^3 + 1350*d*e^4*x^4 + 360*e^5*x^5) - b*(669*d^7 + 3594*d^6*e*x + 7725*d^5*e^2*x^2 + 8200*d^4*e^3*x^3 +
4050*d^3*e^4*x^4 + 360*d^2*e^5*x^5 - 360*d*e^6*x^6 - 60*e^7*x^7)) + 60*c^2*(14*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-
7*b*d + a*e))*(d + e*x)^6*Log[d + e*x])/(30*e^9*(d + e*x)^6)

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Maple [B]  time = 0.056, size = 1364, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^4/(e*x+d)^7,x)

[Out]

-140/3/e^8/(e*x+d)^3*b*c^3*d^4-6/e^5/(e*x+d)^2*a*b^2*c+28*c^4/e^9*ln(e*x+d)*d^2-1/6/e^5/(e*x+d)^6*d^4*b^4-1/6/
e^9/(e*x+d)^6*c^4*d^8-4/5/e^2/(e*x+d)^5*a^3*b+4/5/e^5/(e*x+d)^5*b^4*d^3+8/5/e^9/(e*x+d)^5*c^4*d^7-4*c/e^6/(e*x
+d)*b^3+56*c^4/e^9/(e*x+d)*d^3-1/e^3/(e*x+d)^4*a^3*c-3/2/e^3/(e*x+d)^4*a^2*b^2-3/2/e^5/(e*x+d)^4*b^4*d^2-7/e^9
/(e*x+d)^4*c^4*d^6-4/3/e^4/(e*x+d)^3*a*b^3-1/e^5/(e*x+d)^6*a^2*c^2*d^4+2/3/e^4/(e*x+d)^6*d^3*a*b^3-2/3/e^7/(e*
x+d)^6*a*c^3*d^6+2/3/e^6/(e*x+d)^6*d^5*b^3*c-1/e^7/(e*x+d)^6*d^6*b^2*c^2+2/3/e^8/(e*x+d)^6*b*c^3*d^7+8/5/e^3/(
e*x+d)^5*a^3*c*d+12/5/e^3/(e*x+d)^5*a^2*b^2*d+24/5/e^5/(e*x+d)^5*a^2*c^2*d^3-12/5/e^4/(e*x+d)^5*a*b^3*d^2+24/5
/e^7/(e*x+d)^5*a*c^3*d^5-4/e^6/(e*x+d)^5*b^3*c*d^4+36/5/e^7/(e*x+d)^5*b^2*c^2*d^5-28/5/e^8/(e*x+d)^5*b*c^3*d^6
+1/2*c^4*x^2/e^7-1/2/e^5/(e*x+d)^2*b^4-1/6/e/(e*x+d)^6*a^4+4/3/e^5/(e*x+d)^3*b^4*d+56/3/e^9/(e*x+d)^3*c^4*d^5-
3/e^5/(e*x+d)^2*c^2*a^2-35/e^9/(e*x+d)^2*c^4*d^4+4*c^3/e^7*ln(e*x+d)*a+6*c^2/e^7*ln(e*x+d)*b^2+4*c^3/e^7*x*b-7
*c^4/e^8*x*d-30/e^7/(e*x+d)^2*c^3*a*d^2+10/e^6/(e*x+d)^2*b^3*c*d-45/e^7/(e*x+d)^2*b^2*c^2*d^2+70/e^8/(e*x+d)^2
*b*c^3*d^3-9/e^5/(e*x+d)^4*a^2*c^2*d^2+3/e^4/(e*x+d)^4*a*b^3*d-15/e^7/(e*x+d)^4*a*c^3*d^4+10/e^6/(e*x+d)^4*b^3
*c*d^3-45/2/e^7/(e*x+d)^4*b^2*c^2*d^4+21/e^8/(e*x+d)^4*b*c^3*d^5-4/e^4/(e*x+d)^3*a^2*b*c+8/e^5/(e*x+d)^3*a^2*c
^2*d+80/3/e^7/(e*x+d)^3*a*c^3*d^3-40/3/e^6/(e*x+d)^3*b^3*c*d^2+40/e^7/(e*x+d)^3*b^2*c^2*d^3-28*c^3/e^8*ln(e*x+
d)*b*d+2/3/e^2/(e*x+d)^6*d*a^3*b-2/3/e^3/(e*x+d)^6*a^3*c*d^2-1/e^3/(e*x+d)^6*d^2*a^2*b^2-12*c^2/e^6/(e*x+d)*a*
b+24*c^3/e^7/(e*x+d)*a*d+36*c^2/e^7/(e*x+d)*b^2*d-84*c^3/e^8/(e*x+d)*b*d^2-2/e^5/(e*x+d)^6*d^4*a*b^2*c+2/e^6/(
e*x+d)^6*d^5*a*b*c^2-36/5/e^4/(e*x+d)^5*a^2*b*c*d^2+48/5/e^5/(e*x+d)^5*a*b^2*c*d^3+9/e^4/(e*x+d)^4*a^2*b*c*d-1
8/e^5/(e*x+d)^4*a*b^2*c*d^2-12/e^6/(e*x+d)^5*a*b*c^2*d^4+30/e^6/(e*x+d)^2*a*b*c^2*d+2/e^4/(e*x+d)^6*d^3*a^2*b*
c-40/e^6/(e*x+d)^3*a*b*c^2*d^2+30/e^6/(e*x+d)^4*a*b*c^2*d^3+16/e^5/(e*x+d)^3*a*b^2*c*d

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Maxima [B]  time = 1.14516, size = 1169, normalized size = 2.74 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d)^7,x, algorithm="maxima")

[Out]

1/30*(1023*c^4*d^8 - 1338*b*c^3*d^7*e - 4*a^3*b*d*e^7 - 5*a^4*e^8 + 147*(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 - 2*(a*b^3 + 3*a^2*b*c)*d^3*e^5 - (3*a^2*b^2
+ 2*a^3*c)*d^2*e^6 + 120*(14*c^4*d^3*e^5 - 21*b*c^3*d^2*e^6 + 3*(3*b^2*c^2 + 2*a*c^3)*d*e^7 - (b^3*c + 3*a*b*
c^2)*e^8)*x^5 + 15*(490*c^4*d^4*e^4 - 700*b*c^3*d^3*e^5 + 90*(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 + 20*(658*c^4*d^5*e^3 - 910*b*c^3*d^4*e^4 + 110*(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 - 2*(a*b^3 + 3*a
^2*b*c)*e^8)*x^3 + 15*(798*c^4*d^6*e^2 - 1078*b*c^3*d^5*e^3 + 125*(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 - 2*(a*b^3 + 3*a^2*b*c)*d*e^7 - (3*a^2*b^2 + 2*a^3
*c)*e^8)*x^2 + 6*(918*c^4*d^7*e - 1218*b*c^3*d^6*e^2 - 4*a^3*b*e^8 + 137*(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 - 2*(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^15*x^6 + 6*d*e^14*x^5 + 15*d^2*e^13*x^4 + 20*d^3*e^12*x^3 + 15*d^4*e^11*x^2 + 6*d^5*
e^10*x + d^6*e^9) + 1/2*(c^4*e*x^2 - 2*(7*c^4*d - 4*b*c^3*e)*x)/e^8 + 2*(14*c^4*d^2 - 14*b*c^3*d*e + (3*b^2*c^
2 + 2*a*c^3)*e^2)*log(e*x + d)/e^9

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Fricas [B]  time = 1.76318, size = 2498, normalized size = 5.86 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d)^7,x, algorithm="fricas")

[Out]

1/30*(15*c^4*e^8*x^8 + 1023*c^4*d^8 - 1338*b*c^3*d^7*e - 4*a^3*b*d*e^7 - 5*a^4*e^8 + 147*(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 - 2*(a*b^3 + 3*a^2*b*c)*d^3
*e^5 - (3*a^2*b^2 + 2*a^3*c)*d^2*e^6 - 120*(c^4*d*e^7 - b*c^3*e^8)*x^7 - 45*(23*c^4*d^2*e^6 - 16*b*c^3*d*e^7)*
x^6 - 30*(39*c^4*d^3*e^5 + 24*b*c^3*d^2*e^6 - 12*(3*b^2*c^2 + 2*a*c^3)*d*e^7 + 4*(b^3*c + 3*a*b*c^2)*e^8)*x^5
+ 15*(225*c^4*d^4*e^4 - 540*b*c^3*d^3*e^5 + 90*(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 + 20*(505*c^4*d^5*e^3 - 820*b*c^3*d^4*e^4 + 110*(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 - 2*(a*b^3 + 3*a^2*b*c)*e^8)*x
^3 + 15*(715*c^4*d^6*e^2 - 1030*b*c^3*d^5*e^3 + 125*(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 - 2*(a*b^3 + 3*a^2*b*c)*d*e^7 - (3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 +
6*(883*c^4*d^7*e - 1198*b*c^3*d^6*e^2 - 4*a^3*b*e^8 + 137*(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 - 2*(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 + 60*(14*c^4*d^8 - 14*b*c^3*d^7*e + (3*b^2*c^2 + 2*a*c^3)*d^6*e^2 + (14*c^4*d^2*e^6 - 14*b*c^3*d*e^7 +
(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 + 6*(14*c^4*d^3*e^5 - 14*b*c^3*d^2*e^6 + (3*b^2*c^2 + 2*a*c^3)*d*e^7)*x^5 + 15
*(14*c^4*d^4*e^4 - 14*b*c^3*d^3*e^5 + (3*b^2*c^2 + 2*a*c^3)*d^2*e^6)*x^4 + 20*(14*c^4*d^5*e^3 - 14*b*c^3*d^4*e
^4 + (3*b^2*c^2 + 2*a*c^3)*d^3*e^5)*x^3 + 15*(14*c^4*d^6*e^2 - 14*b*c^3*d^5*e^3 + (3*b^2*c^2 + 2*a*c^3)*d^4*e^
4)*x^2 + 6*(14*c^4*d^7*e - 14*b*c^3*d^6*e^2 + (3*b^2*c^2 + 2*a*c^3)*d^5*e^3)*x)*log(e*x + d))/(e^15*x^6 + 6*d*
e^14*x^5 + 15*d^2*e^13*x^4 + 20*d^3*e^12*x^3 + 15*d^4*e^11*x^2 + 6*d^5*e^10*x + d^6*e^9)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**4/(e*x+d)**7,x)

[Out]

Timed out

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Giac [B]  time = 1.12314, size = 1137, normalized size = 2.67 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d)^7,x, algorithm="giac")

[Out]

2*(14*c^4*d^2 - 14*b*c^3*d*e + 3*b^2*c^2*e^2 + 2*a*c^3*e^2)*e^(-9)*log(abs(x*e + d)) + 1/2*(c^4*x^2*e^7 - 14*c
^4*d*x*e^6 + 8*b*c^3*x*e^7)*e^(-14) + 1/30*(1023*c^4*d^8 - 1338*b*c^3*d^7*e + 441*b^2*c^2*d^6*e^2 + 294*a*c^3*
d^6*e^2 - 20*b^3*c*d^5*e^3 - 60*a*b*c^2*d^5*e^3 - b^4*d^4*e^4 - 12*a*b^2*c*d^4*e^4 - 6*a^2*c^2*d^4*e^4 - 2*a*b
^3*d^3*e^5 - 6*a^2*b*c*d^3*e^5 - 3*a^2*b^2*d^2*e^6 - 2*a^3*c*d^2*e^6 + 120*(14*c^4*d^3*e^5 - 21*b*c^3*d^2*e^6
+ 9*b^2*c^2*d*e^7 + 6*a*c^3*d*e^7 - b^3*c*e^8 - 3*a*b*c^2*e^8)*x^5 - 4*a^3*b*d*e^7 + 15*(490*c^4*d^4*e^4 - 700
*b*c^3*d^3*e^5 + 270*b^2*c^2*d^2*e^6 + 180*a*c^3*d^2*e^6 - 20*b^3*c*d*e^7 - 60*a*b*c^2*d*e^7 - b^4*e^8 - 12*a*
b^2*c*e^8 - 6*a^2*c^2*e^8)*x^4 - 5*a^4*e^8 + 20*(658*c^4*d^5*e^3 - 910*b*c^3*d^4*e^4 + 330*b^2*c^2*d^3*e^5 + 2
20*a*c^3*d^3*e^5 - 20*b^3*c*d^2*e^6 - 60*a*b*c^2*d^2*e^6 - b^4*d*e^7 - 12*a*b^2*c*d*e^7 - 6*a^2*c^2*d*e^7 - 2*
a*b^3*e^8 - 6*a^2*b*c*e^8)*x^3 + 15*(798*c^4*d^6*e^2 - 1078*b*c^3*d^5*e^3 + 375*b^2*c^2*d^4*e^4 + 250*a*c^3*d^
4*e^4 - 20*b^3*c*d^3*e^5 - 60*a*b*c^2*d^3*e^5 - b^4*d^2*e^6 - 12*a*b^2*c*d^2*e^6 - 6*a^2*c^2*d^2*e^6 - 2*a*b^3
*d*e^7 - 6*a^2*b*c*d*e^7 - 3*a^2*b^2*e^8 - 2*a^3*c*e^8)*x^2 + 6*(918*c^4*d^7*e - 1218*b*c^3*d^6*e^2 + 411*b^2*
c^2*d^5*e^3 + 274*a*c^3*d^5*e^3 - 20*b^3*c*d^4*e^4 - 60*a*b*c^2*d^4*e^4 - b^4*d^3*e^5 - 12*a*b^2*c*d^3*e^5 - 6
*a^2*c^2*d^3*e^5 - 2*a*b^3*d^2*e^6 - 6*a^2*b*c*d^2*e^6 - 3*a^2*b^2*d*e^7 - 2*a^3*c*d*e^7 - 4*a^3*b*e^8)*x)*e^(
-9)/(x*e + d)^6