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

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

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Rubi [A]  time = 0.51, 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} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [A]  time = 0.34, size = 764, normalized size = 1.79 \begin {gather*} \frac {-3 c^2 e^2 \left (2 a^2 e^2 \left (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\right )+20 a b e \left (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\right )+b^2 (-d) \left (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\right )\right )-2 c e^3 \left (a^3 e^3 \left (d^2+6 d e x+15 e^2 x^2\right )+3 a^2 b e^2 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+6 a b^2 e \left (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\right )+10 b^3 \left (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\right )\right )-e^4 \left (5 a^4 e^4+4 a^3 b e^3 (d+6 e x)+3 a^2 b^2 e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+2 a b^3 e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+b^4 \left (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\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 (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\right )-b \left (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\right )\right )+c^4 \left (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\right )}{30 e^9 (d+e x)^6} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x+c x^2\right )^4}{(d+e x)^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.40, size = 1191, normalized size = 2.80

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)^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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giac [B]  time = 0.21, size = 842, normalized size = 1.98 \begin {gather*} 2 \, {\left (14 \, c^{4} d^{2} - 14 \, b c^{3} d e + 3 \, b^{2} c^{2} e^{2} + 2 \, a c^{3} e^{2}\right )} e^{\left (-9\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{2} \, {\left (c^{4} x^{2} e^{7} - 14 \, c^{4} d x e^{6} + 8 \, b c^{3} x e^{7}\right )} e^{\left (-14\right )} + \frac {{\left (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 \, {\left (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}\right )} x^{5} - 4 \, a^{3} b d e^{7} + 15 \, {\left (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}\right )} x^{4} - 5 \, a^{4} e^{8} + 20 \, {\left (658 \, c^{4} d^{5} e^{3} - 910 \, b c^{3} d^{4} e^{4} + 330 \, b^{2} c^{2} d^{3} e^{5} + 220 \, 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}\right )} x^{3} + 15 \, {\left (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}\right )} x^{2} + 6 \, {\left (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}\right )} x\right )} e^{\left (-9\right )}}{30 \, {\left (x e + d\right )}^{6}} \end {gather*}

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

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maple [B]  time = 0.07, size = 1364, normalized size = 3.20

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)^7,x)

[Out]

-18/e^5/(e*x+d)^4*a*b^2*c*d^2+30/e^6/(e*x+d)^4*a*b*c^2*d^3-4/5/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^7-3/e^5/(e*x+d)^2*c^2*a^2-35/e^9/(e*x+d)^2*c^4*d^4-4/3/e^4/(e*x+d)^3*a*b^3+4/3/e^5/(e*x
+d)^3*b^4*d+56/3/e^9/(e*x+d)^3*c^4*d^5-4*c/e^6/(e*x+d)*b^3+56*c^4/e^9/(e*x+d)*d^3-1/6/e^5/(e*x+d)^6*d^4*b^4-1/
6/e^9/(e*x+d)^6*c^4*d^8+4*c^3/e^7*x*b-7*c^4/e^8*x*d+6*c^2/e^7*ln(e*x+d)*b^2+28*c^4/e^9*ln(e*x+d)*d^2-1/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*c^3/e^7*ln(e*x+d)*a
+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-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^7-1/6/e/(e*x+d)^6*a^4-1/2/e^5/(e*x+d)^2*b^4-40/e^6/(e*
x+d)^3*a*b*c^2*d^2+9/e^4/(e*x+d)^4*a^2*b*c*d+2/e^4/(e*x+d)^6*d^3*a^2*b*c-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^3-12/e^6/(e*x+d)^5*a*b*c^2*d^4+
30/e^6/(e*x+d)^2*a*b*c^2*d+16/e^5/(e*x+d)^3*a*b^2*c*d-28*c^3/e^8*ln(e*x+d)*b*d+24/5/e^5/(e*x+d)^5*a^2*c^2*d^3+
1/2*c^4*x^2/e^7+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^6-6/e^5/(e*x+d)^2*a*b^2*c-30/e^7/(e*
x+d)^2*a*c^3*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^3-4/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^3-140/3/e^8/(e*x+d)^3*b*c^3*d^4-9/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^5-12*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^2-12/5/e^4/(e*x+d)^5*a*b^3*
d^2+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^2-1/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+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

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maxima [B]  time = 1.43, size = 866, normalized size = 2.03 \begin {gather*} \frac {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 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{2} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5} e^{3} - {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} e^{4} - 2 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e^{5} - {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{6} + 120 \, {\left (14 \, c^{4} d^{3} e^{5} - 21 \, b c^{3} d^{2} e^{6} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{7} - {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{8}\right )} x^{5} + 15 \, {\left (490 \, c^{4} d^{4} e^{4} - 700 \, b c^{3} d^{3} e^{5} + 90 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{6} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{7} - {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{8}\right )} x^{4} + 20 \, {\left (658 \, c^{4} d^{5} e^{3} - 910 \, b c^{3} d^{4} e^{4} + 110 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{5} - 20 \, {\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} - 2 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{8}\right )} x^{3} + 15 \, {\left (798 \, c^{4} d^{6} e^{2} - 1078 \, b c^{3} d^{5} e^{3} + 125 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{4} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{5} - {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{6} - 2 \, {\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} + 6 \, {\left (918 \, c^{4} d^{7} e - 1218 \, b c^{3} d^{6} e^{2} - 4 \, a^{3} b e^{8} + 137 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{3} - 20 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{4} - {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{5} - 2 \, {\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}{30 \, {\left (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}\right )}} + \frac {c^{4} e x^{2} - 2 \, {\left (7 \, c^{4} d - 4 \, b c^{3} e\right )} x}{2 \, e^{8}} + \frac {2 \, {\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 )} \log \left (e x + d\right )}{e^{9}} \end {gather*}

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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mupad [B]  time = 0.81, size = 955, normalized size = 2.24 \begin {gather*} x\,\left (\frac {4\,b\,c^3}{e^7}-\frac {7\,c^4\,d}{e^8}\right )-\frac {x^3\,\left (4\,a^2\,b\,c\,e^7+4\,a^2\,c^2\,d\,e^6+\frac {4\,a\,b^3\,e^7}{3}+8\,a\,b^2\,c\,d\,e^6+40\,a\,b\,c^2\,d^2\,e^5-\frac {440\,a\,c^3\,d^3\,e^4}{3}+\frac {2\,b^4\,d\,e^6}{3}+\frac {40\,b^3\,c\,d^2\,e^5}{3}-220\,b^2\,c^2\,d^3\,e^4+\frac {1820\,b\,c^3\,d^4\,e^3}{3}-\frac {1316\,c^4\,d^5\,e^2}{3}\right )+x\,\left (\frac {4\,a^3\,b\,e^7}{5}+\frac {2\,a^3\,c\,d\,e^6}{5}+\frac {3\,a^2\,b^2\,d\,e^6}{5}+\frac {6\,a^2\,b\,c\,d^2\,e^5}{5}+\frac {6\,a^2\,c^2\,d^3\,e^4}{5}+\frac {2\,a\,b^3\,d^2\,e^5}{5}+\frac {12\,a\,b^2\,c\,d^3\,e^4}{5}+12\,a\,b\,c^2\,d^4\,e^3-\frac {274\,a\,c^3\,d^5\,e^2}{5}+\frac {b^4\,d^3\,e^4}{5}+4\,b^3\,c\,d^4\,e^3-\frac {411\,b^2\,c^2\,d^5\,e^2}{5}+\frac {1218\,b\,c^3\,d^6\,e}{5}-\frac {918\,c^4\,d^7}{5}\right )+x^4\,\left (3\,a^2\,c^2\,e^7+6\,a\,b^2\,c\,e^7+30\,a\,b\,c^2\,d\,e^6-90\,a\,c^3\,d^2\,e^5+\frac {b^4\,e^7}{2}+10\,b^3\,c\,d\,e^6-135\,b^2\,c^2\,d^2\,e^5+350\,b\,c^3\,d^3\,e^4-245\,c^4\,d^4\,e^3\right )+x^5\,\left (4\,b^3\,c\,e^7-36\,b^2\,c^2\,d\,e^6+84\,b\,c^3\,d^2\,e^5+12\,a\,b\,c^2\,e^7-56\,c^4\,d^3\,e^4-24\,a\,c^3\,d\,e^6\right )+\frac {5\,a^4\,e^8+4\,a^3\,b\,d\,e^7+2\,a^3\,c\,d^2\,e^6+3\,a^2\,b^2\,d^2\,e^6+6\,a^2\,b\,c\,d^3\,e^5+6\,a^2\,c^2\,d^4\,e^4+2\,a\,b^3\,d^3\,e^5+12\,a\,b^2\,c\,d^4\,e^4+60\,a\,b\,c^2\,d^5\,e^3-294\,a\,c^3\,d^6\,e^2+b^4\,d^4\,e^4+20\,b^3\,c\,d^5\,e^3-441\,b^2\,c^2\,d^6\,e^2+1338\,b\,c^3\,d^7\,e-1023\,c^4\,d^8}{30\,e}+x^2\,\left (a^3\,c\,e^7+\frac {3\,a^2\,b^2\,e^7}{2}+3\,a^2\,b\,c\,d\,e^6+3\,a^2\,c^2\,d^2\,e^5+a\,b^3\,d\,e^6+6\,a\,b^2\,c\,d^2\,e^5+30\,a\,b\,c^2\,d^3\,e^4-125\,a\,c^3\,d^4\,e^3+\frac {b^4\,d^2\,e^5}{2}+10\,b^3\,c\,d^3\,e^4-\frac {375\,b^2\,c^2\,d^4\,e^3}{2}+539\,b\,c^3\,d^5\,e^2-399\,c^4\,d^6\,e\right )}{d^6\,e^8+6\,d^5\,e^9\,x+15\,d^4\,e^{10}\,x^2+20\,d^3\,e^{11}\,x^3+15\,d^2\,e^{12}\,x^4+6\,d\,e^{13}\,x^5+e^{14}\,x^6}+\frac {\ln \left (d+e\,x\right )\,\left (6\,b^2\,c^2\,e^2-28\,b\,c^3\,d\,e+28\,c^4\,d^2+4\,a\,c^3\,e^2\right )}{e^9}+\frac {c^4\,x^2}{2\,e^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*((4*b*c^3)/e^7 - (7*c^4*d)/e^8) - (x^3*((4*a*b^3*e^7)/3 + (2*b^4*d*e^6)/3 - (1316*c^4*d^5*e^2)/3 - (440*a*c^
3*d^3*e^4)/3 + 4*a^2*c^2*d*e^6 + (1820*b*c^3*d^4*e^3)/3 + (40*b^3*c*d^2*e^5)/3 - 220*b^2*c^2*d^3*e^4 + 4*a^2*b
*c*e^7 + 8*a*b^2*c*d*e^6 + 40*a*b*c^2*d^2*e^5) + x*((4*a^3*b*e^7)/5 - (918*c^4*d^7)/5 + (b^4*d^3*e^4)/5 + (2*a
*b^3*d^2*e^5)/5 + (3*a^2*b^2*d*e^6)/5 - (274*a*c^3*d^5*e^2)/5 + 4*b^3*c*d^4*e^3 + (6*a^2*c^2*d^3*e^4)/5 - (411
*b^2*c^2*d^5*e^2)/5 + (2*a^3*c*d*e^6)/5 + (1218*b*c^3*d^6*e)/5 + 12*a*b*c^2*d^4*e^3 + (12*a*b^2*c*d^3*e^4)/5 +
(6*a^2*b*c*d^2*e^5)/5) + x^4*((b^4*e^7)/2 + 3*a^2*c^2*e^7 - 245*c^4*d^4*e^3 - 90*a*c^3*d^2*e^5 + 350*b*c^3*d^
3*e^4 - 135*b^2*c^2*d^2*e^5 + 6*a*b^2*c*e^7 + 10*b^3*c*d*e^6 + 30*a*b*c^2*d*e^6) + x^5*(4*b^3*c*e^7 - 56*c^4*d
^3*e^4 + 84*b*c^3*d^2*e^5 - 36*b^2*c^2*d*e^6 + 12*a*b*c^2*e^7 - 24*a*c^3*d*e^6) + (5*a^4*e^8 - 1023*c^4*d^8 +
b^4*d^4*e^4 + 2*a*b^3*d^3*e^5 - 294*a*c^3*d^6*e^2 + 2*a^3*c*d^2*e^6 + 20*b^3*c*d^5*e^3 + 3*a^2*b^2*d^2*e^6 + 6
*a^2*c^2*d^4*e^4 - 441*b^2*c^2*d^6*e^2 + 4*a^3*b*d*e^7 + 1338*b*c^3*d^7*e + 60*a*b*c^2*d^5*e^3 + 12*a*b^2*c*d^
4*e^4 + 6*a^2*b*c*d^3*e^5)/(30*e) + x^2*(a^3*c*e^7 - 399*c^4*d^6*e + (3*a^2*b^2*e^7)/2 + (b^4*d^2*e^5)/2 - 125
*a*c^3*d^4*e^3 + 539*b*c^3*d^5*e^2 + 10*b^3*c*d^3*e^4 + 3*a^2*c^2*d^2*e^5 - (375*b^2*c^2*d^4*e^3)/2 + a*b^3*d*
e^6 + 3*a^2*b*c*d*e^6 + 30*a*b*c^2*d^3*e^4 + 6*a*b^2*c*d^2*e^5))/(d^6*e^8 + e^14*x^6 + 6*d^5*e^9*x + 6*d*e^13*
x^5 + 15*d^4*e^10*x^2 + 20*d^3*e^11*x^3 + 15*d^2*e^12*x^4) + (log(d + e*x)*(28*c^4*d^2 + 4*a*c^3*e^2 + 6*b^2*c
^2*e^2 - 28*b*c^3*d*e))/e^9 + (c^4*x^2)/(2*e^7)

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

[Out]

Timed out

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