3.22.39 \(\int \frac {(a+b x+c x^2)^3}{(d+e x)^4} \, dx\) [2139]

Optimal. Leaf size=251 \[ \frac {c \left (10 c^2 d^2+3 b^2 e^2-3 c e (4 b d-a e)\right ) x}{e^6}-\frac {c^2 (4 c d-3 b e) x^2}{2 e^5}+\frac {c^3 x^3}{3 e^4}-\frac {\left (c d^2-b d e+a e^2\right )^3}{3 e^7 (d+e x)^3}+\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{2 e^7 (d+e x)^2}-\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^7 (d+e x)}-\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) \log (d+e x)}{e^7} \]

[Out]

c*(10*c^2*d^2+3*b^2*e^2-3*c*e*(-a*e+4*b*d))*x/e^6-1/2*c^2*(-3*b*e+4*c*d)*x^2/e^5+1/3*c^3*x^3/e^4-1/3*(a*e^2-b*
d*e+c*d^2)^3/e^7/(e*x+d)^3+3/2*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^2/e^7/(e*x+d)^2-3*(a*e^2-b*d*e+c*d^2)*(5*c^2*d
^2+b^2*e^2-c*e*(-a*e+5*b*d))/e^7/(e*x+d)-(-b*e+2*c*d)*(10*c^2*d^2+b^2*e^2-2*c*e*(-3*a*e+5*b*d))*ln(e*x+d)/e^7

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Rubi [A]
time = 0.21, antiderivative size = 251, 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 = {712} \begin {gather*} -\frac {3 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right ) \left (a e^2-b d e+c d^2\right )}{e^7 (d+e x)}-\frac {(2 c d-b e) \log (d+e x) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7}+\frac {c x \left (-3 c e (4 b d-a e)+3 b^2 e^2+10 c^2 d^2\right )}{e^6}-\frac {\left (a e^2-b d e+c d^2\right )^3}{3 e^7 (d+e x)^3}+\frac {3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{2 e^7 (d+e x)^2}-\frac {c^2 x^2 (4 c d-3 b e)}{2 e^5}+\frac {c^3 x^3}{3 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(10*c^2*d^2 + 3*b^2*e^2 - 3*c*e*(4*b*d - a*e))*x)/e^6 - (c^2*(4*c*d - 3*b*e)*x^2)/(2*e^5) + (c^3*x^3)/(3*e^
4) - (c*d^2 - b*d*e + a*e^2)^3/(3*e^7*(d + e*x)^3) + (3*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2)/(2*e^7*(d + e
*x)^2) - (3*(c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))/(e^7*(d + e*x)) - ((2*c*d - b*e
)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*Log[d + e*x])/e^7

Rule 712

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

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Mathematica [A]
time = 0.07, size = 260, normalized size = 1.04 \begin {gather*} \frac {6 c e \left (10 c^2 d^2+3 b^2 e^2+3 c e (-4 b d+a e)\right ) x+3 c^2 e^2 (-4 c d+3 b e) x^2+2 c^3 e^3 x^3-\frac {2 \left (c d^2+e (-b d+a e)\right )^3}{(d+e x)^3}+\frac {9 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )^2}{(d+e x)^2}-\frac {18 \left (5 c^3 d^4+b^2 e^3 (-b d+a e)+2 c^2 d^2 e (-5 b d+3 a e)+c e^2 \left (6 b^2 d^2-6 a b d e+a^2 e^2\right )\right )}{d+e x}-6 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2+2 c e (-5 b d+3 a e)\right ) \log (d+e x)}{6 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(6*c*e*(10*c^2*d^2 + 3*b^2*e^2 + 3*c*e*(-4*b*d + a*e))*x + 3*c^2*e^2*(-4*c*d + 3*b*e)*x^2 + 2*c^3*e^3*x^3 - (2
*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e*x)^3 + (9*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^2)/(d + e*x)^2 - (18*
(5*c^3*d^4 + b^2*e^3*(-(b*d) + a*e) + 2*c^2*d^2*e*(-5*b*d + 3*a*e) + c*e^2*(6*b^2*d^2 - 6*a*b*d*e + a^2*e^2)))
/(d + e*x) - 6*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 + 2*c*e*(-5*b*d + 3*a*e))*Log[d + e*x])/(6*e^7)

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Maple [A]
time = 0.87, size = 450, normalized size = 1.79

method	result	size
norman	\(\frac {-\frac {2 e^{6} a^{3}+3 a^{2} b d \,e^{5}+6 e^{4} d^{2} a^{2} c +6 a \,b^{2} d^{2} e^{4}-66 a b c \,d^{3} e^{3}+132 d^{4} e^{2} c^{2} a -11 b^{3} d^{3} e^{3}+132 b^{2} c \,d^{4} e^{2}-330 b \,c^{2} d^{5} e +220 d^{6} c^{3}}{6 e^{7}}+\frac {c^{3} x^{6}}{3 e}-\frac {3 \left (e^{4} a^{2} c +a \,b^{2} e^{4}-6 a b c d \,e^{3}+12 d^{2} e^{2} c^{2} a -b^{3} d \,e^{3}+12 b^{2} c \,d^{2} e^{2}-30 d^{3} e b \,c^{2}+20 d^{4} c^{3}\right ) x^{2}}{e^{5}}-\frac {3 \left (a^{2} b \,e^{5}+2 d \,e^{4} a^{2} c +2 a \,b^{2} d \,e^{4}-18 a b c \,d^{2} e^{3}+36 d^{3} e^{2} c^{2} a -3 b^{3} d^{2} e^{3}+36 b^{2} c \,d^{3} e^{2}-90 b \,c^{2} d^{4} e +60 d^{5} c^{3}\right ) x}{2 e^{6}}+\frac {c \left (6 a c \,e^{2}+6 b^{2} e^{2}-15 b c d e +10 c^{2} d^{2}\right ) x^{4}}{2 e^{3}}+\frac {c^{2} \left (3 b e -2 c d \right ) x^{5}}{2 e^{2}}}{\left (e x +d \right )^{3}}+\frac {\left (6 a b c \,e^{3}-12 d \,e^{2} c^{2} a +b^{3} e^{3}-12 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -20 c^{3} d^{3}\right ) \ln \left (e x +d \right )}{e^{7}}\)	\(441\)
default	\(\frac {c \left (\frac {1}{3} c^{2} e^{2} x^{3}+\frac {3}{2} b c \,e^{2} x^{2}-2 c^{2} d e \,x^{2}+3 a c \,e^{2} x +3 b^{2} e^{2} x -12 b c d e x +10 d^{2} c^{2} x \right )}{e^{6}}-\frac {e^{6} a^{3}-3 a^{2} b d \,e^{5}+3 e^{4} d^{2} a^{2} c +3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} c^{2} a -b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +d^{6} c^{3}}{3 e^{7} \left (e x +d \right )^{3}}-\frac {3 e^{4} a^{2} c +3 a \,b^{2} e^{4}-18 a b c d \,e^{3}+18 d^{2} e^{2} c^{2} a -3 b^{3} d \,e^{3}+18 b^{2} c \,d^{2} e^{2}-30 d^{3} e b \,c^{2}+15 d^{4} c^{3}}{e^{7} \left (e x +d \right )}+\frac {\left (6 a b c \,e^{3}-12 d \,e^{2} c^{2} a +b^{3} e^{3}-12 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -20 c^{3} d^{3}\right ) \ln \left (e x +d \right )}{e^{7}}-\frac {3 a^{2} b \,e^{5}-6 d \,e^{4} a^{2} c -6 a \,b^{2} d \,e^{4}+18 a b c \,d^{2} e^{3}-12 d^{3} e^{2} c^{2} a +3 b^{3} d^{2} e^{3}-12 b^{2} c \,d^{3} e^{2}+15 b \,c^{2} d^{4} e -6 d^{5} c^{3}}{2 e^{7} \left (e x +d \right )^{2}}\)	\(450\)
risch	\(\frac {c^{3} x^{3}}{3 e^{4}}+\frac {3 c^{2} b \,x^{2}}{2 e^{4}}-\frac {2 c^{3} d \,x^{2}}{e^{5}}+\frac {3 c^{2} a x}{e^{4}}+\frac {3 c \,b^{2} x}{e^{4}}-\frac {12 c^{2} b d x}{e^{5}}+\frac {10 c^{3} d^{2} x}{e^{6}}+\frac {\left (-3 a^{2} c \,e^{5}-3 a \,b^{2} e^{5}+18 a b c d \,e^{4}-18 e^{3} c^{2} d^{2} a +3 b^{3} d \,e^{4}-18 b^{2} c \,d^{2} e^{3}+30 d^{3} b \,c^{2} e^{2}-15 d^{4} e \,c^{3}\right ) x^{2}+\left (-\frac {3}{2} a^{2} b \,e^{5}-3 d \,e^{4} a^{2} c -3 a \,b^{2} d \,e^{4}+27 a b c \,d^{2} e^{3}-30 d^{3} e^{2} c^{2} a +\frac {9}{2} b^{3} d^{2} e^{3}-30 b^{2} c \,d^{3} e^{2}+\frac {105}{2} b \,c^{2} d^{4} e -27 d^{5} c^{3}\right ) x -\frac {2 e^{6} a^{3}+3 a^{2} b d \,e^{5}+6 e^{4} d^{2} a^{2} c +6 a \,b^{2} d^{2} e^{4}-66 a b c \,d^{3} e^{3}+78 d^{4} e^{2} c^{2} a -11 b^{3} d^{3} e^{3}+78 b^{2} c \,d^{4} e^{2}-141 b \,c^{2} d^{5} e +74 d^{6} c^{3}}{6 e}}{e^{6} \left (e x +d \right )^{3}}+\frac {6 \ln \left (e x +d \right ) a b c}{e^{4}}-\frac {12 \ln \left (e x +d \right ) d \,c^{2} a}{e^{5}}+\frac {\ln \left (e x +d \right ) b^{3}}{e^{4}}-\frac {12 \ln \left (e x +d \right ) b^{2} d c}{e^{5}}+\frac {30 \ln \left (e x +d \right ) b \,c^{2} d^{2}}{e^{6}}-\frac {20 \ln \left (e x +d \right ) c^{3} d^{3}}{e^{7}}\)	\(480\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c/e^6*(1/3*c^2*e^2*x^3+3/2*b*c*e^2*x^2-2*c^2*d*e*x^2+3*a*c*e^2*x+3*b^2*e^2*x-12*b*c*d*e*x+10*d^2*c^2*x)-1/3*(a
^3*e^6-3*a^2*b*d*e^5+3*a^2*c*d^2*e^4+3*a*b^2*d^2*e^4-6*a*b*c*d^3*e^3+3*a*c^2*d^4*e^2-b^3*d^3*e^3+3*b^2*c*d^4*e
^2-3*b*c^2*d^5*e+c^3*d^6)/e^7/(e*x+d)^3-(3*a^2*c*e^4+3*a*b^2*e^4-18*a*b*c*d*e^3+18*a*c^2*d^2*e^2-3*b^3*d*e^3+1
8*b^2*c*d^2*e^2-30*b*c^2*d^3*e+15*c^3*d^4)/e^7/(e*x+d)+1/e^7*(6*a*b*c*e^3-12*a*c^2*d*e^2+b^3*e^3-12*b^2*c*d*e^
2+30*b*c^2*d^2*e-20*c^3*d^3)*ln(e*x+d)-1/2*(3*a^2*b*e^5-6*a^2*c*d*e^4-6*a*b^2*d*e^4+18*a*b*c*d^2*e^3-12*a*c^2*
d^3*e^2+3*b^3*d^2*e^3-12*b^2*c*d^3*e^2+15*b*c^2*d^4*e-6*c^3*d^5)/e^7/(e*x+d)^2

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Maxima [A]
time = 0.29, size = 433, normalized size = 1.73 \begin {gather*} -{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e - b^{3} e^{3} - 6 \, a b c e^{3} + 12 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d\right )} e^{\left (-7\right )} \log \left (x e + d\right ) + \frac {1}{6} \, {\left (2 \, c^{3} x^{3} e^{2} - 3 \, {\left (4 \, c^{3} d e - 3 \, b c^{2} e^{2}\right )} x^{2} + 6 \, {\left (10 \, c^{3} d^{2} - 12 \, b c^{2} d e + 3 \, b^{2} c e^{2} + 3 \, a c^{2} e^{2}\right )} x\right )} e^{\left (-6\right )} - \frac {74 \, c^{3} d^{6} - 141 \, b c^{2} d^{5} e + 78 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d^{4} + 3 \, a^{2} b d e^{5} - 11 \, {\left (b^{3} e^{3} + 6 \, a b c e^{3}\right )} d^{3} + 2 \, a^{3} e^{6} + 6 \, {\left (a b^{2} e^{4} + a^{2} c e^{4}\right )} d^{2} + 18 \, {\left (5 \, c^{3} d^{4} e^{2} - 10 \, b c^{2} d^{3} e^{3} + a b^{2} e^{6} + a^{2} c e^{6} + 6 \, {\left (b^{2} c e^{4} + a c^{2} e^{4}\right )} d^{2} - {\left (b^{3} e^{5} + 6 \, a b c e^{5}\right )} d\right )} x^{2} + 9 \, {\left (18 \, c^{3} d^{5} e - 35 \, b c^{2} d^{4} e^{2} + 20 \, {\left (b^{2} c e^{3} + a c^{2} e^{3}\right )} d^{3} + a^{2} b e^{6} - 3 \, {\left (b^{3} e^{4} + 6 \, a b c e^{4}\right )} d^{2} + 2 \, {\left (a b^{2} e^{5} + a^{2} c e^{5}\right )} d\right )} x}{6 \, {\left (x^{3} e^{10} + 3 \, d x^{2} e^{9} + 3 \, d^{2} x e^{8} + d^{3} e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(20*c^3*d^3 - 30*b*c^2*d^2*e - b^3*e^3 - 6*a*b*c*e^3 + 12*(b^2*c*e^2 + a*c^2*e^2)*d)*e^(-7)*log(x*e + d) + 1/
6*(2*c^3*x^3*e^2 - 3*(4*c^3*d*e - 3*b*c^2*e^2)*x^2 + 6*(10*c^3*d^2 - 12*b*c^2*d*e + 3*b^2*c*e^2 + 3*a*c^2*e^2)
*x)*e^(-6) - 1/6*(74*c^3*d^6 - 141*b*c^2*d^5*e + 78*(b^2*c*e^2 + a*c^2*e^2)*d^4 + 3*a^2*b*d*e^5 - 11*(b^3*e^3
+ 6*a*b*c*e^3)*d^3 + 2*a^3*e^6 + 6*(a*b^2*e^4 + a^2*c*e^4)*d^2 + 18*(5*c^3*d^4*e^2 - 10*b*c^2*d^3*e^3 + a*b^2*
e^6 + a^2*c*e^6 + 6*(b^2*c*e^4 + a*c^2*e^4)*d^2 - (b^3*e^5 + 6*a*b*c*e^5)*d)*x^2 + 9*(18*c^3*d^5*e - 35*b*c^2*
d^4*e^2 + 20*(b^2*c*e^3 + a*c^2*e^3)*d^3 + a^2*b*e^6 - 3*(b^3*e^4 + 6*a*b*c*e^4)*d^2 + 2*(a*b^2*e^5 + a^2*c*e^
5)*d)*x)/(x^3*e^10 + 3*d*x^2*e^9 + 3*d^2*x*e^8 + d^3*e^7)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 631 vs. \(2 (246) = 492\).
time = 2.43, size = 631, normalized size = 2.51 \begin {gather*} -\frac {74 \, c^{3} d^{6} - {\left (2 \, c^{3} x^{6} + 9 \, b c^{2} x^{5} + 18 \, {\left (b^{2} c + a c^{2}\right )} x^{4} - 9 \, a^{2} b x - 2 \, a^{3} - 18 \, {\left (a b^{2} + a^{2} c\right )} x^{2}\right )} e^{6} + 3 \, {\left (2 \, c^{3} d x^{5} + 15 \, b c^{2} d x^{4} - 18 \, {\left (b^{2} c + a c^{2}\right )} d x^{3} + a^{2} b d - 6 \, {\left (b^{3} + 6 \, a b c\right )} d x^{2} + 6 \, {\left (a b^{2} + a^{2} c\right )} d x\right )} e^{5} - 3 \, {\left (10 \, c^{3} d^{2} x^{4} - 63 \, b c^{2} d^{2} x^{3} - 18 \, {\left (b^{2} c + a c^{2}\right )} d^{2} x^{2} + 9 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} x - 2 \, {\left (a b^{2} + a^{2} c\right )} d^{2}\right )} e^{4} - {\left (146 \, c^{3} d^{3} x^{3} - 27 \, b c^{2} d^{3} x^{2} - 162 \, {\left (b^{2} c + a c^{2}\right )} d^{3} x + 11 \, {\left (b^{3} + 6 \, a b c\right )} d^{3}\right )} e^{3} - 3 \, {\left (26 \, c^{3} d^{4} x^{2} + 81 \, b c^{2} d^{4} x - 26 \, {\left (b^{2} c + a c^{2}\right )} d^{4}\right )} e^{2} + 3 \, {\left (34 \, c^{3} d^{5} x - 47 \, b c^{2} d^{5}\right )} e + 6 \, {\left (20 \, c^{3} d^{6} - {\left (b^{3} + 6 \, a b c\right )} x^{3} e^{6} + 3 \, {\left (4 \, {\left (b^{2} c + a c^{2}\right )} d x^{3} - {\left (b^{3} + 6 \, a b c\right )} d x^{2}\right )} e^{5} - 3 \, {\left (10 \, b c^{2} d^{2} x^{3} - 12 \, {\left (b^{2} c + a c^{2}\right )} d^{2} x^{2} + {\left (b^{3} + 6 \, a b c\right )} d^{2} x\right )} e^{4} + {\left (20 \, c^{3} d^{3} x^{3} - 90 \, b c^{2} d^{3} x^{2} + 36 \, {\left (b^{2} c + a c^{2}\right )} d^{3} x - {\left (b^{3} + 6 \, a b c\right )} d^{3}\right )} e^{3} + 6 \, {\left (10 \, c^{3} d^{4} x^{2} - 15 \, b c^{2} d^{4} x + 2 \, {\left (b^{2} c + a c^{2}\right )} d^{4}\right )} e^{2} + 30 \, {\left (2 \, c^{3} d^{5} x - b c^{2} d^{5}\right )} e\right )} \log \left (x e + d\right )}{6 \, {\left (x^{3} e^{10} + 3 \, d x^{2} e^{9} + 3 \, d^{2} x e^{8} + d^{3} e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(74*c^3*d^6 - (2*c^3*x^6 + 9*b*c^2*x^5 + 18*(b^2*c + a*c^2)*x^4 - 9*a^2*b*x - 2*a^3 - 18*(a*b^2 + a^2*c)*
x^2)*e^6 + 3*(2*c^3*d*x^5 + 15*b*c^2*d*x^4 - 18*(b^2*c + a*c^2)*d*x^3 + a^2*b*d - 6*(b^3 + 6*a*b*c)*d*x^2 + 6*
(a*b^2 + a^2*c)*d*x)*e^5 - 3*(10*c^3*d^2*x^4 - 63*b*c^2*d^2*x^3 - 18*(b^2*c + a*c^2)*d^2*x^2 + 9*(b^3 + 6*a*b*
c)*d^2*x - 2*(a*b^2 + a^2*c)*d^2)*e^4 - (146*c^3*d^3*x^3 - 27*b*c^2*d^3*x^2 - 162*(b^2*c + a*c^2)*d^3*x + 11*(
b^3 + 6*a*b*c)*d^3)*e^3 - 3*(26*c^3*d^4*x^2 + 81*b*c^2*d^4*x - 26*(b^2*c + a*c^2)*d^4)*e^2 + 3*(34*c^3*d^5*x -
47*b*c^2*d^5)*e + 6*(20*c^3*d^6 - (b^3 + 6*a*b*c)*x^3*e^6 + 3*(4*(b^2*c + a*c^2)*d*x^3 - (b^3 + 6*a*b*c)*d*x^
2)*e^5 - 3*(10*b*c^2*d^2*x^3 - 12*(b^2*c + a*c^2)*d^2*x^2 + (b^3 + 6*a*b*c)*d^2*x)*e^4 + (20*c^3*d^3*x^3 - 90*
b*c^2*d^3*x^2 + 36*(b^2*c + a*c^2)*d^3*x - (b^3 + 6*a*b*c)*d^3)*e^3 + 6*(10*c^3*d^4*x^2 - 15*b*c^2*d^4*x + 2*(
b^2*c + a*c^2)*d^4)*e^2 + 30*(2*c^3*d^5*x - b*c^2*d^5)*e)*log(x*e + d))/(x^3*e^10 + 3*d*x^2*e^9 + 3*d^2*x*e^8
+ d^3*e^7)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (243) = 486\).
time = 65.99, size = 502, normalized size = 2.00 \begin {gather*} \frac {c^{3} x^{3}}{3 e^{4}} + x^{2} \cdot \left (\frac {3 b c^{2}}{2 e^{4}} - \frac {2 c^{3} d}{e^{5}}\right ) + x \left (\frac {3 a c^{2}}{e^{4}} + \frac {3 b^{2} c}{e^{4}} - \frac {12 b c^{2} d}{e^{5}} + \frac {10 c^{3} d^{2}}{e^{6}}\right ) + \frac {- 2 a^{3} e^{6} - 3 a^{2} b d e^{5} - 6 a^{2} c d^{2} e^{4} - 6 a b^{2} d^{2} e^{4} + 66 a b c d^{3} e^{3} - 78 a c^{2} d^{4} e^{2} + 11 b^{3} d^{3} e^{3} - 78 b^{2} c d^{4} e^{2} + 141 b c^{2} d^{5} e - 74 c^{3} d^{6} + x^{2} \left (- 18 a^{2} c e^{6} - 18 a b^{2} e^{6} + 108 a b c d e^{5} - 108 a c^{2} d^{2} e^{4} + 18 b^{3} d e^{5} - 108 b^{2} c d^{2} e^{4} + 180 b c^{2} d^{3} e^{3} - 90 c^{3} d^{4} e^{2}\right ) + x \left (- 9 a^{2} b e^{6} - 18 a^{2} c d e^{5} - 18 a b^{2} d e^{5} + 162 a b c d^{2} e^{4} - 180 a c^{2} d^{3} e^{3} + 27 b^{3} d^{2} e^{4} - 180 b^{2} c d^{3} e^{3} + 315 b c^{2} d^{4} e^{2} - 162 c^{3} d^{5} e\right )}{6 d^{3} e^{7} + 18 d^{2} e^{8} x + 18 d e^{9} x^{2} + 6 e^{10} x^{3}} + \frac {\left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right ) \log {\left (d + e x \right )}}{e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**3*x**3/(3*e**4) + x**2*(3*b*c**2/(2*e**4) - 2*c**3*d/e**5) + x*(3*a*c**2/e**4 + 3*b**2*c/e**4 - 12*b*c**2*d
/e**5 + 10*c**3*d**2/e**6) + (-2*a**3*e**6 - 3*a**2*b*d*e**5 - 6*a**2*c*d**2*e**4 - 6*a*b**2*d**2*e**4 + 66*a*
b*c*d**3*e**3 - 78*a*c**2*d**4*e**2 + 11*b**3*d**3*e**3 - 78*b**2*c*d**4*e**2 + 141*b*c**2*d**5*e - 74*c**3*d*
*6 + x**2*(-18*a**2*c*e**6 - 18*a*b**2*e**6 + 108*a*b*c*d*e**5 - 108*a*c**2*d**2*e**4 + 18*b**3*d*e**5 - 108*b
**2*c*d**2*e**4 + 180*b*c**2*d**3*e**3 - 90*c**3*d**4*e**2) + x*(-9*a**2*b*e**6 - 18*a**2*c*d*e**5 - 18*a*b**2
*d*e**5 + 162*a*b*c*d**2*e**4 - 180*a*c**2*d**3*e**3 + 27*b**3*d**2*e**4 - 180*b**2*c*d**3*e**3 + 315*b*c**2*d
**4*e**2 - 162*c**3*d**5*e))/(6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) + (b*e - 2*c*d)*(6
*a*c*e**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2)*log(d + e*x)/e**7

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Giac [A]
time = 1.60, size = 424, normalized size = 1.69 \begin {gather*} -{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} + 12 \, a c^{2} d e^{2} - b^{3} e^{3} - 6 \, a b c e^{3}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{6} \, {\left (2 \, c^{3} x^{3} e^{8} - 12 \, c^{3} d x^{2} e^{7} + 60 \, c^{3} d^{2} x e^{6} + 9 \, b c^{2} x^{2} e^{8} - 72 \, b c^{2} d x e^{7} + 18 \, b^{2} c x e^{8} + 18 \, a c^{2} x e^{8}\right )} e^{\left (-12\right )} - \frac {{\left (74 \, c^{3} d^{6} - 141 \, b c^{2} d^{5} e + 78 \, b^{2} c d^{4} e^{2} + 78 \, a c^{2} d^{4} e^{2} - 11 \, b^{3} d^{3} e^{3} - 66 \, a b c d^{3} e^{3} + 6 \, a b^{2} d^{2} e^{4} + 6 \, a^{2} c d^{2} e^{4} + 3 \, a^{2} b d e^{5} + 2 \, a^{3} e^{6} + 18 \, {\left (5 \, c^{3} d^{4} e^{2} - 10 \, b c^{2} d^{3} e^{3} + 6 \, b^{2} c d^{2} e^{4} + 6 \, a c^{2} d^{2} e^{4} - b^{3} d e^{5} - 6 \, a b c d e^{5} + a b^{2} e^{6} + a^{2} c e^{6}\right )} x^{2} + 9 \, {\left (18 \, c^{3} d^{5} e - 35 \, b c^{2} d^{4} e^{2} + 20 \, b^{2} c d^{3} e^{3} + 20 \, a c^{2} d^{3} e^{3} - 3 \, b^{3} d^{2} e^{4} - 18 \, a b c d^{2} e^{4} + 2 \, a b^{2} d e^{5} + 2 \, a^{2} c d e^{5} + a^{2} b e^{6}\right )} x\right )} e^{\left (-7\right )}}{6 \, {\left (x e + d\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 + 12*a*c^2*d*e^2 - b^3*e^3 - 6*a*b*c*e^3)*e^(-7)*log(abs(x*e +
d)) + 1/6*(2*c^3*x^3*e^8 - 12*c^3*d*x^2*e^7 + 60*c^3*d^2*x*e^6 + 9*b*c^2*x^2*e^8 - 72*b*c^2*d*x*e^7 + 18*b^2*c
*x*e^8 + 18*a*c^2*x*e^8)*e^(-12) - 1/6*(74*c^3*d^6 - 141*b*c^2*d^5*e + 78*b^2*c*d^4*e^2 + 78*a*c^2*d^4*e^2 - 1
1*b^3*d^3*e^3 - 66*a*b*c*d^3*e^3 + 6*a*b^2*d^2*e^4 + 6*a^2*c*d^2*e^4 + 3*a^2*b*d*e^5 + 2*a^3*e^6 + 18*(5*c^3*d
^4*e^2 - 10*b*c^2*d^3*e^3 + 6*b^2*c*d^2*e^4 + 6*a*c^2*d^2*e^4 - b^3*d*e^5 - 6*a*b*c*d*e^5 + a*b^2*e^6 + a^2*c*
e^6)*x^2 + 9*(18*c^3*d^5*e - 35*b*c^2*d^4*e^2 + 20*b^2*c*d^3*e^3 + 20*a*c^2*d^3*e^3 - 3*b^3*d^2*e^4 - 18*a*b*c
*d^2*e^4 + 2*a*b^2*d*e^5 + 2*a^2*c*d*e^5 + a^2*b*e^6)*x)*e^(-7)/(x*e + d)^3

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Mupad [B]
time = 0.14, size = 484, normalized size = 1.93 \begin {gather*} x^2\,\left (\frac {3\,b\,c^2}{2\,e^4}-\frac {2\,c^3\,d}{e^5}\right )-x\,\left (\frac {4\,d\,\left (\frac {3\,b\,c^2}{e^4}-\frac {4\,c^3\,d}{e^5}\right )}{e}+\frac {6\,c^3\,d^2}{e^6}-\frac {3\,c\,\left (b^2+a\,c\right )}{e^4}\right )-\frac {x\,\left (\frac {3\,a^2\,b\,e^5}{2}+3\,a^2\,c\,d\,e^4+3\,a\,b^2\,d\,e^4-27\,a\,b\,c\,d^2\,e^3+30\,a\,c^2\,d^3\,e^2-\frac {9\,b^3\,d^2\,e^3}{2}+30\,b^2\,c\,d^3\,e^2-\frac {105\,b\,c^2\,d^4\,e}{2}+27\,c^3\,d^5\right )+\frac {2\,a^3\,e^6+3\,a^2\,b\,d\,e^5+6\,a^2\,c\,d^2\,e^4+6\,a\,b^2\,d^2\,e^4-66\,a\,b\,c\,d^3\,e^3+78\,a\,c^2\,d^4\,e^2-11\,b^3\,d^3\,e^3+78\,b^2\,c\,d^4\,e^2-141\,b\,c^2\,d^5\,e+74\,c^3\,d^6}{6\,e}+x^2\,\left (3\,a^2\,c\,e^5+3\,a\,b^2\,e^5-18\,a\,b\,c\,d\,e^4+18\,a\,c^2\,d^2\,e^3-3\,b^3\,d\,e^4+18\,b^2\,c\,d^2\,e^3-30\,b\,c^2\,d^3\,e^2+15\,c^3\,d^4\,e\right )}{d^3\,e^6+3\,d^2\,e^7\,x+3\,d\,e^8\,x^2+e^9\,x^3}+\frac {\ln \left (d+e\,x\right )\,\left (b^3\,e^3-12\,b^2\,c\,d\,e^2+30\,b\,c^2\,d^2\,e+6\,a\,b\,c\,e^3-20\,c^3\,d^3-12\,a\,c^2\,d\,e^2\right )}{e^7}+\frac {c^3\,x^3}{3\,e^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*((3*b*c^2)/(2*e^4) - (2*c^3*d)/e^5) - x*((4*d*((3*b*c^2)/e^4 - (4*c^3*d)/e^5))/e + (6*c^3*d^2)/e^6 - (3*c*
(a*c + b^2))/e^4) - (x*(27*c^3*d^5 + (3*a^2*b*e^5)/2 - (9*b^3*d^2*e^3)/2 + 30*a*c^2*d^3*e^2 + 30*b^2*c*d^3*e^2
+ 3*a*b^2*d*e^4 + 3*a^2*c*d*e^4 - (105*b*c^2*d^4*e)/2 - 27*a*b*c*d^2*e^3) + (2*a^3*e^6 + 74*c^3*d^6 - 11*b^3*
d^3*e^3 + 6*a*b^2*d^2*e^4 + 78*a*c^2*d^4*e^2 + 6*a^2*c*d^2*e^4 + 78*b^2*c*d^4*e^2 + 3*a^2*b*d*e^5 - 141*b*c^2*
d^5*e - 66*a*b*c*d^3*e^3)/(6*e) + x^2*(3*a*b^2*e^5 + 3*a^2*c*e^5 - 3*b^3*d*e^4 + 15*c^3*d^4*e + 18*a*c^2*d^2*e
^3 - 30*b*c^2*d^3*e^2 + 18*b^2*c*d^2*e^3 - 18*a*b*c*d*e^4))/(d^3*e^6 + e^9*x^3 + 3*d^2*e^7*x + 3*d*e^8*x^2) +
(log(d + e*x)*(b^3*e^3 - 20*c^3*d^3 + 6*a*b*c*e^3 - 12*a*c^2*d*e^2 + 30*b*c^2*d^2*e - 12*b^2*c*d*e^2))/e^7 + (
c^3*x^3)/(3*e^4)

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