3.19.22 \(\int \frac {(a+b x+c x^2)^4}{(d+e x)^3} \, dx\)

Optimal. Leaf size=430 \[ \frac {(d+e x)^2 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{2 e^9}+\frac {c^2 (d+e x)^4 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{2 e^9}-\frac {4 c (d+e x)^3 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9}+\frac {2 \log (d+e x) \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9}-\frac {4 x (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8}+\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^9 (d+e x)}-\frac {\left (a e^2-b d e+c d^2\right )^4}{2 e^9 (d+e x)^2}-\frac {4 c^3 (d+e x)^5 (2 c d-b e)}{5 e^9}+\frac {c^4 (d+e x)^6}{6 e^9} \]

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Rubi [A]  time = 0.67, antiderivative size = 430, 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 {(d+e x)^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 )}{2 e^9}+\frac {c^2 (d+e x)^4 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{2 e^9}-\frac {4 c (d+e x)^3 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9}-\frac {4 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 \log (d+e x) \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^9}+\frac {4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^9 (d+e x)}-\frac {\left (a e^2-b d e+c d^2\right )^4}{2 e^9 (d+e x)^2}-\frac {4 c^3 (d+e x)^5 (2 c d-b e)}{5 e^9}+\frac {c^4 (d+e x)^6}{6 e^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^4/(d + e*x)^3,x]

[Out]

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

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Mathematica [A]  time = 0.20, size = 440, normalized size = 1.02 \begin {gather*} \frac {15 e^2 x^2 \left (6 c^2 e^2 \left (a^2 e^2-6 a b d e+6 b^2 d^2\right )-12 b^2 c e^3 (b d-a e)-8 c^3 d^2 e (5 b d-3 a e)+b^4 e^4+15 c^4 d^4\right )+30 e x \left (-6 c^2 d e^2 \left (3 a^2 e^2-12 a b d e+10 b^2 d^2\right )+12 b c e^3 \left (a^2 e^2-3 a b d e+2 b^2 d^2\right )+b^3 e^4 (4 a e-3 b d)+20 c^3 d^3 e (3 b d-2 a e)-21 c^4 d^5\right )+60 \log (d+e x) \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+15 c^2 e^4 x^4 \left (2 c e (a e-3 b d)+3 b^2 e^2+3 c^2 d^2\right )+20 c e^3 x^3 (b e-c d) \left (c e (6 a e-7 b d)+2 b^2 e^2+5 c^2 d^2\right )+\frac {120 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^3}{d+e x}-\frac {15 \left (e (a e-b d)+c d^2\right )^4}{(d+e x)^2}+6 c^3 e^5 x^5 (4 b e-3 c d)+5 c^4 e^6 x^6}{30 e^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^4/(d + e*x)^3,x]

[Out]

(30*e*(-21*c^4*d^5 + 20*c^3*d^3*e*(3*b*d - 2*a*e) + b^3*e^4*(-3*b*d + 4*a*e) + 12*b*c*e^3*(2*b^2*d^2 - 3*a*b*d
*e + a^2*e^2) - 6*c^2*d*e^2*(10*b^2*d^2 - 12*a*b*d*e + 3*a^2*e^2))*x + 15*e^2*(15*c^4*d^4 + b^4*e^4 - 8*c^3*d^
2*e*(5*b*d - 3*a*e) - 12*b^2*c*e^3*(b*d - a*e) + 6*c^2*e^2*(6*b^2*d^2 - 6*a*b*d*e + a^2*e^2))*x^2 + 20*c*e^3*(
-(c*d) + b*e)*(5*c^2*d^2 + 2*b^2*e^2 + c*e*(-7*b*d + 6*a*e))*x^3 + 15*c^2*e^4*(3*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(
-3*b*d + a*e))*x^4 + 6*c^3*e^5*(-3*c*d + 4*b*e)*x^5 + 5*c^4*e^6*x^6 - (15*(c*d^2 + e*(-(b*d) + a*e))^4)/(d + e
*x)^2 + (120*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e*x) + 60*(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*Log[d + e*x])/(30*e^9)

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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)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x + c*x^2)^4/(d + e*x)^3,x]

[Out]

IntegrateAlgebraic[(a + b*x + c*x^2)^4/(d + e*x)^3, x]

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fricas [B]  time = 0.42, size = 1218, normalized size = 2.83

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)^3,x, algorithm="fricas")

[Out]

1/30*(5*c^4*e^8*x^8 + 225*c^4*d^8 - 780*b*c^3*d^7*e - 60*a^3*b*d*e^7 - 15*a^4*e^8 + 330*(3*b^2*c^2 + 2*a*c^3)*
d^6*e^2 - 540*(b^3*c + 3*a*b*c^2)*d^5*e^3 + 105*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 - 300*(a*b^3 + 3*a^2*b*
c)*d^3*e^5 + 90*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 - 8*(c^4*d*e^7 - 3*b*c^3*e^8)*x^7 + (14*c^4*d^2*e^6 - 42*b*c^3*d
*e^7 + 15*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 - 2*(14*c^4*d^3*e^5 - 42*b*c^3*d^2*e^6 + 15*(3*b^2*c^2 + 2*a*c^3)*d*e
^7 - 20*(b^3*c + 3*a*b*c^2)*e^8)*x^5 + 5*(14*c^4*d^4*e^4 - 42*b*c^3*d^3*e^5 + 15*(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 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*x^4 - 20*(14*c^4*d^5*e^3 - 42*b*c^3*d^
4*e^4 + 15*(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 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d
*e^7 - 6*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 - 15*(69*c^4*d^6*e^2 - 200*b*c^3*d^5*e^3 + 68*(3*b^2*c^2 + 2*a*c^3)*d^4*
e^4 - 84*(b^3*c + 3*a*b*c^2)*d^3*e^5 + 11*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 - 16*(a*b^3 + 3*a^2*b*c)*d*e^
7)*x^2 - 30*(13*c^4*d^7*e - 32*b*c^3*d^6*e^2 + 4*a^3*b*e^8 + 8*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 4*(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 + 8*(a*b^3 + 3*a^2*b*c)*d^2*e^6 - 4*(3*a^2*b^2 + 2*a^3
*c)*d*e^7)*x + 60*(14*c^4*d^8 - 42*b*c^3*d^7*e + 15*(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 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 - 6*(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 + (14*c^4*d^6*e^2 - 42*b*c^3*d^5*e^3 + 15*(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 + 3
*(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 + 2*(14
*c^4*d^7*e - 42*b*c^3*d^6*e^2 + 15*(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 + 3*(b^4 + 1
2*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)*log(e*x + d))
/(e^11*x^2 + 2*d*e^10*x + d^2*e^9)

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giac [B]  time = 0.17, size = 890, normalized size = 2.07 \begin {gather*} 2 \, {\left (14 \, c^{4} d^{6} - 42 \, b c^{3} d^{5} e + 45 \, b^{2} c^{2} d^{4} e^{2} + 30 \, a c^{3} d^{4} e^{2} - 20 \, b^{3} c d^{3} e^{3} - 60 \, a b c^{2} d^{3} e^{3} + 3 \, b^{4} d^{2} e^{4} + 36 \, a b^{2} c d^{2} e^{4} + 18 \, a^{2} c^{2} d^{2} e^{4} - 6 \, a b^{3} d e^{5} - 18 \, a^{2} b c d e^{5} + 3 \, a^{2} b^{2} e^{6} + 2 \, a^{3} c e^{6}\right )} e^{\left (-9\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{30} \, {\left (5 \, c^{4} x^{6} e^{15} - 18 \, c^{4} d x^{5} e^{14} + 45 \, c^{4} d^{2} x^{4} e^{13} - 100 \, c^{4} d^{3} x^{3} e^{12} + 225 \, c^{4} d^{4} x^{2} e^{11} - 630 \, c^{4} d^{5} x e^{10} + 24 \, b c^{3} x^{5} e^{15} - 90 \, b c^{3} d x^{4} e^{14} + 240 \, b c^{3} d^{2} x^{3} e^{13} - 600 \, b c^{3} d^{3} x^{2} e^{12} + 1800 \, b c^{3} d^{4} x e^{11} + 45 \, b^{2} c^{2} x^{4} e^{15} + 30 \, a c^{3} x^{4} e^{15} - 180 \, b^{2} c^{2} d x^{3} e^{14} - 120 \, a c^{3} d x^{3} e^{14} + 540 \, b^{2} c^{2} d^{2} x^{2} e^{13} + 360 \, a c^{3} d^{2} x^{2} e^{13} - 1800 \, b^{2} c^{2} d^{3} x e^{12} - 1200 \, a c^{3} d^{3} x e^{12} + 40 \, b^{3} c x^{3} e^{15} + 120 \, a b c^{2} x^{3} e^{15} - 180 \, b^{3} c d x^{2} e^{14} - 540 \, a b c^{2} d x^{2} e^{14} + 720 \, b^{3} c d^{2} x e^{13} + 2160 \, a b c^{2} d^{2} x e^{13} + 15 \, b^{4} x^{2} e^{15} + 180 \, a b^{2} c x^{2} e^{15} + 90 \, a^{2} c^{2} x^{2} e^{15} - 90 \, b^{4} d x e^{14} - 1080 \, a b^{2} c d x e^{14} - 540 \, a^{2} c^{2} d x e^{14} + 120 \, a b^{3} x e^{15} + 360 \, a^{2} b c x e^{15}\right )} e^{\left (-18\right )} + \frac {{\left (15 \, c^{4} d^{8} - 52 \, b c^{3} d^{7} e + 66 \, b^{2} c^{2} d^{6} e^{2} + 44 \, a c^{3} d^{6} e^{2} - 36 \, b^{3} c d^{5} e^{3} - 108 \, a b c^{2} d^{5} e^{3} + 7 \, b^{4} d^{4} e^{4} + 84 \, a b^{2} c d^{4} e^{4} + 42 \, a^{2} c^{2} d^{4} e^{4} - 20 \, a b^{3} d^{3} e^{5} - 60 \, a^{2} b c d^{3} e^{5} + 18 \, a^{2} b^{2} d^{2} e^{6} + 12 \, a^{3} c d^{2} e^{6} - 4 \, a^{3} b d e^{7} - a^{4} e^{8} + 8 \, {\left (2 \, c^{4} d^{7} e - 7 \, b c^{3} d^{6} e^{2} + 9 \, b^{2} c^{2} d^{5} e^{3} + 6 \, a c^{3} d^{5} e^{3} - 5 \, b^{3} c d^{4} e^{4} - 15 \, 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} - 3 \, a b^{3} d^{2} e^{6} - 9 \, a^{2} b c d^{2} e^{6} + 3 \, a^{2} b^{2} d e^{7} + 2 \, a^{3} c d e^{7} - a^{3} b e^{8}\right )} x\right )} e^{\left (-9\right )}}{2 \, {\left (x e + d\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^4/(e*x+d)^3,x, algorithm="giac")

[Out]

2*(14*c^4*d^6 - 42*b*c^3*d^5*e + 45*b^2*c^2*d^4*e^2 + 30*a*c^3*d^4*e^2 - 20*b^3*c*d^3*e^3 - 60*a*b*c^2*d^3*e^3
+ 3*b^4*d^2*e^4 + 36*a*b^2*c*d^2*e^4 + 18*a^2*c^2*d^2*e^4 - 6*a*b^3*d*e^5 - 18*a^2*b*c*d*e^5 + 3*a^2*b^2*e^6
+ 2*a^3*c*e^6)*e^(-9)*log(abs(x*e + d)) + 1/30*(5*c^4*x^6*e^15 - 18*c^4*d*x^5*e^14 + 45*c^4*d^2*x^4*e^13 - 100
*c^4*d^3*x^3*e^12 + 225*c^4*d^4*x^2*e^11 - 630*c^4*d^5*x*e^10 + 24*b*c^3*x^5*e^15 - 90*b*c^3*d*x^4*e^14 + 240*
b*c^3*d^2*x^3*e^13 - 600*b*c^3*d^3*x^2*e^12 + 1800*b*c^3*d^4*x*e^11 + 45*b^2*c^2*x^4*e^15 + 30*a*c^3*x^4*e^15
- 180*b^2*c^2*d*x^3*e^14 - 120*a*c^3*d*x^3*e^14 + 540*b^2*c^2*d^2*x^2*e^13 + 360*a*c^3*d^2*x^2*e^13 - 1800*b^2
*c^2*d^3*x*e^12 - 1200*a*c^3*d^3*x*e^12 + 40*b^3*c*x^3*e^15 + 120*a*b*c^2*x^3*e^15 - 180*b^3*c*d*x^2*e^14 - 54
0*a*b*c^2*d*x^2*e^14 + 720*b^3*c*d^2*x*e^13 + 2160*a*b*c^2*d^2*x*e^13 + 15*b^4*x^2*e^15 + 180*a*b^2*c*x^2*e^15
+ 90*a^2*c^2*x^2*e^15 - 90*b^4*d*x*e^14 - 1080*a*b^2*c*d*x*e^14 - 540*a^2*c^2*d*x*e^14 + 120*a*b^3*x*e^15 + 3
60*a^2*b*c*x*e^15)*e^(-18) + 1/2*(15*c^4*d^8 - 52*b*c^3*d^7*e + 66*b^2*c^2*d^6*e^2 + 44*a*c^3*d^6*e^2 - 36*b^3
*c*d^5*e^3 - 108*a*b*c^2*d^5*e^3 + 7*b^4*d^4*e^4 + 84*a*b^2*c*d^4*e^4 + 42*a^2*c^2*d^4*e^4 - 20*a*b^3*d^3*e^5
- 60*a^2*b*c*d^3*e^5 + 18*a^2*b^2*d^2*e^6 + 12*a^3*c*d^2*e^6 - 4*a^3*b*d*e^7 - a^4*e^8 + 8*(2*c^4*d^7*e - 7*b*
c^3*d^6*e^2 + 9*b^2*c^2*d^5*e^3 + 6*a*c^3*d^5*e^3 - 5*b^3*c*d^4*e^4 - 15*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 - 3*a*b^3*d^2*e^6 - 9*a^2*b*c*d^2*e^6 + 3*a^2*b^2*d*e^7 + 2*a^3*c*d*e^7 - a^
3*b*e^8)*x)*e^(-9)/(x*e + d)^2

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maple [B]  time = 0.07, size = 1216, normalized size = 2.83

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)^3,x)

[Out]

-18/e^4*x^2*a*b*c^2*d-36/e^4/(e*x+d)*a^2*b*c*d^2+48/e^5/(e*x+d)*a*b^2*c*d^3-60/e^6/(e*x+d)*a*b*c^2*d^4+6/e^4/(
e*x+d)^2*a^2*b*c*d^3-6/e^5/(e*x+d)^2*a*b^2*c*d^4+6/e^6/(e*x+d)^2*a*b*c^2*d^5-36/e^4*a*b^2*c*d*x+72/e^5*ln(e*x+
d)*a*b^2*c*d^2-120/e^6*ln(e*x+d)*a*b*c^2*d^3+72/e^5*a*b*c^2*d^2*x-36/e^4*ln(e*x+d)*a^2*b*c*d-10/3/e^6*x^3*c^4*
d^3+3/e^3*x^2*a^2*c^2+4/5/e^3*x^5*b*c^3-3/5/e^4*x^5*c^4*d-4/e^2/(e*x+d)*a^3*b+4/e^5/(e*x+d)*b^4*d^3+8/e^9/(e*x
+d)*c^4*d^7-1/2/e^5/(e*x+d)^2*b^4*d^4-1/2/e^9/(e*x+d)^2*c^4*d^8+4/e^3*ln(e*x+d)*a^3*c+6/e^3*ln(e*x+d)*a^2*b^2+
6/e^5*ln(e*x+d)*b^4*d^2+28/e^9*ln(e*x+d)*c^4*d^6+1/e^3*x^4*a*c^3+15/2/e^7*x^2*c^4*d^4+4/e^3*a*b^3*x-3/e^4*b^4*
d*x+3/2/e^3*x^4*b^2*c^2+3/2/e^5*x^4*c^4*d^2+4/3/e^3*x^3*b^3*c-21/e^8*c^4*d^5*x-1/2/e/(e*x+d)^2*a^4+1/6/e^3*c^4
*x^6+1/2*b^4/e^3*x^2-84/e^8*ln(e*x+d)*b*c^3*d^5-28/e^8/(e*x+d)*b*c^3*d^6+2/e^2/(e*x+d)^2*d*a^3*b-40/e^6*ln(e*x
+d)*b^3*c*d^3+90/e^7*ln(e*x+d)*b^2*c^2*d^4-60/e^6*b^2*c^2*d^3*x+60/e^7*b*c^3*d^4*x+8/e^3/(e*x+d)*a^3*c*d+12/e^
3/(e*x+d)*a^2*b^2*d+24/e^5/(e*x+d)*a^2*c^2*d^3-12/e^4/(e*x+d)*a*b^3*d^2+24/e^7/(e*x+d)*a*c^3*d^5-20/e^6/(e*x+d
)*b^3*c*d^4+36/e^7/(e*x+d)*b^2*c^2*d^5-2/e^3/(e*x+d)^2*a^3*c*d^2-3/e^3/(e*x+d)^2*d^2*a^2*b^2-3/e^5/(e*x+d)^2*a
^2*c^2*d^4+2/e^4/(e*x+d)^2*a*b^3*d^3-2/e^7/(e*x+d)^2*a*c^3*d^6+2/e^6/(e*x+d)^2*b^3*c*d^5-3/e^7/(e*x+d)^2*b^2*c
^2*d^6+2/e^8/(e*x+d)^2*b*c^3*d^7+36/e^5*ln(e*x+d)*a^2*c^2*d^2-12/e^4*ln(e*x+d)*a*b^3*d+60/e^7*ln(e*x+d)*a*c^3*
d^4+6/e^3*x^2*a*b^2*c+12/e^5*x^2*a*c^3*d^2-6/e^4*x^2*b^3*c*d+18/e^5*x^2*b^2*c^2*d^2-20/e^6*x^2*b*c^3*d^3+12/e^
3*a^2*b*c*x-18/e^4*a^2*c^2*d*x-40/e^6*a*c^3*d^3*x+24/e^5*b^3*c*d^2*x-3/e^4*x^4*b*c^3*d+4/e^3*x^3*a*b*c^2-4/e^4
*x^3*a*c^3*d-6/e^4*x^3*b^2*c^2*d+8/e^5*x^3*b*c^3*d^2

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maxima [A]  time = 1.39, size = 819, normalized size = 1.90 \begin {gather*} \frac {15 \, c^{4} d^{8} - 52 \, b c^{3} d^{7} e - 4 \, a^{3} b d e^{7} - a^{4} e^{8} + 22 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{6} e^{2} - 36 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{5} e^{3} + 7 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{4} e^{4} - 20 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} d^{3} e^{5} + 6 \, {\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d^{2} e^{6} + 8 \, {\left (2 \, c^{4} d^{7} e - 7 \, b c^{3} d^{6} e^{2} - a^{3} b e^{8} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{3} - 5 \, {\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} - 3 \, {\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}{2 \, {\left (e^{11} x^{2} + 2 \, d e^{10} x + d^{2} e^{9}\right )}} + \frac {5 \, c^{4} e^{5} x^{6} - 6 \, {\left (3 \, c^{4} d e^{4} - 4 \, b c^{3} e^{5}\right )} x^{5} + 15 \, {\left (3 \, c^{4} d^{2} e^{3} - 6 \, b c^{3} d e^{4} + {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{5}\right )} x^{4} - 20 \, {\left (5 \, c^{4} d^{3} e^{2} - 12 \, b c^{3} d^{2} e^{3} + 3 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{4} - 2 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} e^{5}\right )} x^{3} + 15 \, {\left (15 \, c^{4} d^{4} e - 40 \, b c^{3} d^{3} e^{2} + 12 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{3} - 12 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{4} + {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{5}\right )} x^{2} - 30 \, {\left (21 \, c^{4} d^{5} - 60 \, b c^{3} d^{4} e + 20 \, {\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{2} - 24 \, {\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{3} + 3 \, {\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{4} - 4 \, {\left (a b^{3} + 3 \, a^{2} b c\right )} e^{5}\right )} x}{30 \, e^{8}} + \frac {2 \, {\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 )} \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)^3,x, algorithm="maxima")

[Out]

1/2*(15*c^4*d^8 - 52*b*c^3*d^7*e - 4*a^3*b*d*e^7 - a^4*e^8 + 22*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 36*(b^3*c + 3*
a*b*c^2)*d^5*e^3 + 7*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 - 20*(a*b^3 + 3*a^2*b*c)*d^3*e^5 + 6*(3*a^2*b^2 +
2*a^3*c)*d^2*e^6 + 8*(2*c^4*d^7*e - 7*b*c^3*d^6*e^2 - a^3*b*e^8 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 5*(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 - 3*(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^11*x^2 + 2*d*e^10*x + d^2*e^9) + 1/30*(5*c^4*e^5*x^6 - 6*(3*c^4*d*e^4 - 4*b*c^3*e^5)*x^5 +
15*(3*c^4*d^2*e^3 - 6*b*c^3*d*e^4 + (3*b^2*c^2 + 2*a*c^3)*e^5)*x^4 - 20*(5*c^4*d^3*e^2 - 12*b*c^3*d^2*e^3 + 3
*(3*b^2*c^2 + 2*a*c^3)*d*e^4 - 2*(b^3*c + 3*a*b*c^2)*e^5)*x^3 + 15*(15*c^4*d^4*e - 40*b*c^3*d^3*e^2 + 12*(3*b^
2*c^2 + 2*a*c^3)*d^2*e^3 - 12*(b^3*c + 3*a*b*c^2)*d*e^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^5)*x^2 - 30*(21*c^4
*d^5 - 60*b*c^3*d^4*e + 20*(3*b^2*c^2 + 2*a*c^3)*d^3*e^2 - 24*(b^3*c + 3*a*b*c^2)*d^2*e^3 + 3*(b^4 + 12*a*b^2*
c + 6*a^2*c^2)*d*e^4 - 4*(a*b^3 + 3*a^2*b*c)*e^5)*x)/e^8 + 2*(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)*log(e*x + d)/e^9

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mupad [B]  time = 0.82, size = 1444, normalized size = 3.36

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^4/(d + e*x)^3,x)

[Out]

(x*(8*c^4*d^7 - 4*a^3*b*e^7 + 4*b^4*d^3*e^4 - 12*a*b^3*d^2*e^5 + 12*a^2*b^2*d*e^6 + 24*a*c^3*d^5*e^2 - 20*b^3*
c*d^4*e^3 + 24*a^2*c^2*d^3*e^4 + 36*b^2*c^2*d^5*e^2 + 8*a^3*c*d*e^6 - 28*b*c^3*d^6*e - 60*a*b*c^2*d^4*e^3 + 48
*a*b^2*c*d^3*e^4 - 36*a^2*b*c*d^2*e^5) + (15*c^4*d^8 - a^4*e^8 + 7*b^4*d^4*e^4 - 20*a*b^3*d^3*e^5 + 44*a*c^3*d
^6*e^2 + 12*a^3*c*d^2*e^6 - 36*b^3*c*d^5*e^3 + 18*a^2*b^2*d^2*e^6 + 42*a^2*c^2*d^4*e^4 + 66*b^2*c^2*d^6*e^2 -
4*a^3*b*d*e^7 - 52*b*c^3*d^7*e - 108*a*b*c^2*d^5*e^3 + 84*a*b^2*c*d^4*e^4 - 60*a^2*b*c*d^3*e^5)/(2*e))/(d^2*e^
8 + e^10*x^2 + 2*d*e^9*x) + x^5*((4*b*c^3)/(5*e^3) - (3*c^4*d)/(5*e^4)) - x^3*((c^4*d^3)/(3*e^6) + (d^2*((4*b*
c^3)/e^3 - (3*c^4*d)/e^4))/e^2 - (d*((3*d*((4*b*c^3)/e^3 - (3*c^4*d)/e^4))/e - (4*a*c^3 + 6*b^2*c^2)/e^3 + (3*
c^4*d^2)/e^5))/e - (4*b*c*(3*a*c + b^2))/(3*e^3)) + x*((d^3*((3*d*((4*b*c^3)/e^3 - (3*c^4*d)/e^4))/e - (4*a*c^
3 + 6*b^2*c^2)/e^3 + (3*c^4*d^2)/e^5))/e^3 - (3*d*((b^4 + 6*a^2*c^2 + 12*a*b^2*c)/e^3 + (3*d^2*((3*d*((4*b*c^3
)/e^3 - (3*c^4*d)/e^4))/e - (4*a*c^3 + 6*b^2*c^2)/e^3 + (3*c^4*d^2)/e^5))/e^2 + (3*d*((c^4*d^3)/e^6 + (3*d^2*(
(4*b*c^3)/e^3 - (3*c^4*d)/e^4))/e^2 - (3*d*((3*d*((4*b*c^3)/e^3 - (3*c^4*d)/e^4))/e - (4*a*c^3 + 6*b^2*c^2)/e^
3 + (3*c^4*d^2)/e^5))/e - (4*b*c*(3*a*c + b^2))/e^3))/e - (d^3*((4*b*c^3)/e^3 - (3*c^4*d)/e^4))/e^3))/e + (3*d
^2*((c^4*d^3)/e^6 + (3*d^2*((4*b*c^3)/e^3 - (3*c^4*d)/e^4))/e^2 - (3*d*((3*d*((4*b*c^3)/e^3 - (3*c^4*d)/e^4))/
e - (4*a*c^3 + 6*b^2*c^2)/e^3 + (3*c^4*d^2)/e^5))/e - (4*b*c*(3*a*c + b^2))/e^3))/e^2 + (4*a*b*(3*a*c + b^2))/
e^3) + x^2*((b^4 + 6*a^2*c^2 + 12*a*b^2*c)/(2*e^3) + (3*d^2*((3*d*((4*b*c^3)/e^3 - (3*c^4*d)/e^4))/e - (4*a*c^
3 + 6*b^2*c^2)/e^3 + (3*c^4*d^2)/e^5))/(2*e^2) + (3*d*((c^4*d^3)/e^6 + (3*d^2*((4*b*c^3)/e^3 - (3*c^4*d)/e^4))
/e^2 - (3*d*((3*d*((4*b*c^3)/e^3 - (3*c^4*d)/e^4))/e - (4*a*c^3 + 6*b^2*c^2)/e^3 + (3*c^4*d^2)/e^5))/e - (4*b*
c*(3*a*c + b^2))/e^3))/(2*e) - (d^3*((4*b*c^3)/e^3 - (3*c^4*d)/e^4))/(2*e^3)) - x^4*((3*d*((4*b*c^3)/e^3 - (3*
c^4*d)/e^4))/(4*e) - (4*a*c^3 + 6*b^2*c^2)/(4*e^3) + (3*c^4*d^2)/(4*e^5)) + (log(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))/e^9
+ (c^4*x^6)/(6*e^3)

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sympy [B]  time = 19.58, size = 906, normalized size = 2.11 \begin {gather*} \frac {c^{4} x^{6}}{6 e^{3}} + x^{5} \left (\frac {4 b c^{3}}{5 e^{3}} - \frac {3 c^{4} d}{5 e^{4}}\right ) + x^{4} \left (\frac {a c^{3}}{e^{3}} + \frac {3 b^{2} c^{2}}{2 e^{3}} - \frac {3 b c^{3} d}{e^{4}} + \frac {3 c^{4} d^{2}}{2 e^{5}}\right ) + x^{3} \left (\frac {4 a b c^{2}}{e^{3}} - \frac {4 a c^{3} d}{e^{4}} + \frac {4 b^{3} c}{3 e^{3}} - \frac {6 b^{2} c^{2} d}{e^{4}} + \frac {8 b c^{3} d^{2}}{e^{5}} - \frac {10 c^{4} d^{3}}{3 e^{6}}\right ) + x^{2} \left (\frac {3 a^{2} c^{2}}{e^{3}} + \frac {6 a b^{2} c}{e^{3}} - \frac {18 a b c^{2} d}{e^{4}} + \frac {12 a c^{3} d^{2}}{e^{5}} + \frac {b^{4}}{2 e^{3}} - \frac {6 b^{3} c d}{e^{4}} + \frac {18 b^{2} c^{2} d^{2}}{e^{5}} - \frac {20 b c^{3} d^{3}}{e^{6}} + \frac {15 c^{4} d^{4}}{2 e^{7}}\right ) + x \left (\frac {12 a^{2} b c}{e^{3}} - \frac {18 a^{2} c^{2} d}{e^{4}} + \frac {4 a b^{3}}{e^{3}} - \frac {36 a b^{2} c d}{e^{4}} + \frac {72 a b c^{2} d^{2}}{e^{5}} - \frac {40 a c^{3} d^{3}}{e^{6}} - \frac {3 b^{4} d}{e^{4}} + \frac {24 b^{3} c d^{2}}{e^{5}} - \frac {60 b^{2} c^{2} d^{3}}{e^{6}} + \frac {60 b c^{3} d^{4}}{e^{7}} - \frac {21 c^{4} d^{5}}{e^{8}}\right ) + \frac {- a^{4} e^{8} - 4 a^{3} b d e^{7} + 12 a^{3} c d^{2} e^{6} + 18 a^{2} b^{2} d^{2} e^{6} - 60 a^{2} b c d^{3} e^{5} + 42 a^{2} c^{2} d^{4} e^{4} - 20 a b^{3} d^{3} e^{5} + 84 a b^{2} c d^{4} e^{4} - 108 a b c^{2} d^{5} e^{3} + 44 a c^{3} d^{6} e^{2} + 7 b^{4} d^{4} e^{4} - 36 b^{3} c d^{5} e^{3} + 66 b^{2} c^{2} d^{6} e^{2} - 52 b c^{3} d^{7} e + 15 c^{4} d^{8} + x \left (- 8 a^{3} b e^{8} + 16 a^{3} c d e^{7} + 24 a^{2} b^{2} d e^{7} - 72 a^{2} b c d^{2} e^{6} + 48 a^{2} c^{2} d^{3} e^{5} - 24 a b^{3} d^{2} e^{6} + 96 a b^{2} c d^{3} e^{5} - 120 a b c^{2} d^{4} e^{4} + 48 a c^{3} d^{5} e^{3} + 8 b^{4} d^{3} e^{5} - 40 b^{3} c d^{4} e^{4} + 72 b^{2} c^{2} d^{5} e^{3} - 56 b c^{3} d^{6} e^{2} + 16 c^{4} d^{7} e\right )}{2 d^{2} e^{9} + 4 d e^{10} x + 2 e^{11} x^{2}} + \frac {2 \left (a e^{2} - b d e + c d^{2}\right )^{2} \left (2 a c e^{2} + 3 b^{2} e^{2} - 14 b c d e + 14 c^{2} d^{2}\right ) \log {\left (d + e x \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)**3,x)

[Out]

c**4*x**6/(6*e**3) + x**5*(4*b*c**3/(5*e**3) - 3*c**4*d/(5*e**4)) + x**4*(a*c**3/e**3 + 3*b**2*c**2/(2*e**3) -
3*b*c**3*d/e**4 + 3*c**4*d**2/(2*e**5)) + x**3*(4*a*b*c**2/e**3 - 4*a*c**3*d/e**4 + 4*b**3*c/(3*e**3) - 6*b**
2*c**2*d/e**4 + 8*b*c**3*d**2/e**5 - 10*c**4*d**3/(3*e**6)) + x**2*(3*a**2*c**2/e**3 + 6*a*b**2*c/e**3 - 18*a*
b*c**2*d/e**4 + 12*a*c**3*d**2/e**5 + b**4/(2*e**3) - 6*b**3*c*d/e**4 + 18*b**2*c**2*d**2/e**5 - 20*b*c**3*d**
3/e**6 + 15*c**4*d**4/(2*e**7)) + x*(12*a**2*b*c/e**3 - 18*a**2*c**2*d/e**4 + 4*a*b**3/e**3 - 36*a*b**2*c*d/e*
*4 + 72*a*b*c**2*d**2/e**5 - 40*a*c**3*d**3/e**6 - 3*b**4*d/e**4 + 24*b**3*c*d**2/e**5 - 60*b**2*c**2*d**3/e**
6 + 60*b*c**3*d**4/e**7 - 21*c**4*d**5/e**8) + (-a**4*e**8 - 4*a**3*b*d*e**7 + 12*a**3*c*d**2*e**6 + 18*a**2*b
**2*d**2*e**6 - 60*a**2*b*c*d**3*e**5 + 42*a**2*c**2*d**4*e**4 - 20*a*b**3*d**3*e**5 + 84*a*b**2*c*d**4*e**4 -
108*a*b*c**2*d**5*e**3 + 44*a*c**3*d**6*e**2 + 7*b**4*d**4*e**4 - 36*b**3*c*d**5*e**3 + 66*b**2*c**2*d**6*e**
2 - 52*b*c**3*d**7*e + 15*c**4*d**8 + x*(-8*a**3*b*e**8 + 16*a**3*c*d*e**7 + 24*a**2*b**2*d*e**7 - 72*a**2*b*c
*d**2*e**6 + 48*a**2*c**2*d**3*e**5 - 24*a*b**3*d**2*e**6 + 96*a*b**2*c*d**3*e**5 - 120*a*b*c**2*d**4*e**4 + 4
8*a*c**3*d**5*e**3 + 8*b**4*d**3*e**5 - 40*b**3*c*d**4*e**4 + 72*b**2*c**2*d**5*e**3 - 56*b*c**3*d**6*e**2 + 1
6*c**4*d**7*e))/(2*d**2*e**9 + 4*d*e**10*x + 2*e**11*x**2) + 2*(a*e**2 - b*d*e + c*d**2)**2*(2*a*c*e**2 + 3*b*
*2*e**2 - 14*b*c*d*e + 14*c**2*d**2)*log(d + e*x)/e**9

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