3.22.41 \(\int \frac {(a+b x+c x^2)^3}{(d+e x)^6} \, dx\) [2141]

Optimal. Leaf size=256 \[ \frac {c^3 x}{e^6}-\frac {\left (c d^2-b d e+a e^2\right )^3}{5 e^7 (d+e x)^5}+\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{4 e^7 (d+e x)^4}-\frac {\left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^7 (d+e x)^3}+\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )}{2 e^7 (d+e x)^2}-\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^7 (d+e x)}-\frac {3 c^2 (2 c d-b e) \log (d+e x)}{e^7} \]

[Out]

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

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Rubi [A]
time = 0.17, antiderivative size = 256, 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 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (d+e x)}+\frac {(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^7 (d+e x)^2}-\frac {\left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{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}{4 e^7 (d+e x)^4}-\frac {\left (a e^2-b d e+c d^2\right )^3}{5 e^7 (d+e x)^5}-\frac {3 c^2 (2 c d-b e) \log (d+e x)}{e^7}+\frac {c^3 x}{e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^3/(d + e*x)^6,x]

[Out]

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

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Mathematica [A]
time = 0.17, size = 396, normalized size = 1.55 \begin {gather*} -\frac {2 c^3 \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )+e^3 \left (4 a^3 e^3+3 a^2 b e^2 (d+5 e x)+2 a b^2 e \left (d^2+5 d e x+10 e^2 x^2\right )+b^3 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )+2 c e^2 \left (a^2 e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+3 a b e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+6 b^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )+c^2 e \left (12 a e \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )-b d \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )\right )+60 c^2 (2 c d-b e) (d+e x)^5 \log (d+e x)}{20 e^7 (d+e x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^3/(d + e*x)^6,x]

[Out]

-1/20*(2*c^3*(87*d^6 + 375*d^5*e*x + 600*d^4*e^2*x^2 + 400*d^3*e^3*x^3 + 50*d^2*e^4*x^4 - 50*d*e^5*x^5 - 10*e^
6*x^6) + e^3*(4*a^3*e^3 + 3*a^2*b*e^2*(d + 5*e*x) + 2*a*b^2*e*(d^2 + 5*d*e*x + 10*e^2*x^2) + b^3*(d^3 + 5*d^2*
e*x + 10*d*e^2*x^2 + 10*e^3*x^3)) + 2*c*e^2*(a^2*e^2*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*a*b*e*(d^3 + 5*d^2*e*x +
10*d*e^2*x^2 + 10*e^3*x^3) + 6*b^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4)) + c^2*e*(12
*a*e*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4) - b*d*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2
*x^2 + 900*d*e^3*x^3 + 300*e^4*x^4)) + 60*c^2*(2*c*d - b*e)*(d + e*x)^5*Log[d + e*x])/(e^7*(d + e*x)^5)

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Maple [A]
time = 0.73, size = 453, normalized size = 1.77

method	result	size
norman	\(\frac {\frac {c^{3} x^{6}}{e}-\frac {4 e^{6} a^{3}+3 a^{2} b d \,e^{5}+2 e^{4} d^{2} a^{2} c +2 a \,b^{2} d^{2} e^{4}+6 a b c \,d^{3} e^{3}+12 d^{4} e^{2} c^{2} a +b^{3} d^{3} e^{3}+12 b^{2} c \,d^{4} e^{2}-137 b \,c^{2} d^{5} e +274 d^{6} c^{3}}{20 e^{7}}-\frac {\left (3 e^{2} c^{2} a +3 b^{2} e^{2} c -15 d e b \,c^{2}+30 c^{3} d^{2}\right ) x^{4}}{e^{3}}-\frac {\left (6 a b c \,e^{3}+12 d \,e^{2} c^{2} a +b^{3} e^{3}+12 b^{2} d \,e^{2} c -90 b \,c^{2} d^{2} e +180 c^{3} d^{3}\right ) x^{3}}{2 e^{4}}-\frac {\left (2 e^{4} a^{2} c +2 a \,b^{2} e^{4}+6 a b c d \,e^{3}+12 d^{2} e^{2} c^{2} a +b^{3} d \,e^{3}+12 b^{2} c \,d^{2} e^{2}-110 d^{3} e b \,c^{2}+220 d^{4} c^{3}\right ) x^{2}}{2 e^{5}}-\frac {\left (3 a^{2} b \,e^{5}+2 d \,e^{4} a^{2} c +2 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}+12 d^{3} e^{2} c^{2} a +b^{3} d^{2} e^{3}+12 b^{2} c \,d^{3} e^{2}-125 b \,c^{2} d^{4} e +250 d^{5} c^{3}\right ) x}{4 e^{6}}}{\left (e x +d \right )^{5}}+\frac {3 c^{2} \left (b e -2 c d \right ) \ln \left (e x +d \right )}{e^{7}}\)	\(444\)
risch	\(\frac {c^{3} x}{e^{6}}+\frac {\left (-3 a \,c^{2} e^{5}-3 b^{2} c \,e^{5}+15 b \,c^{2} d \,e^{4}-15 c^{3} d^{2} e^{3}\right ) x^{4}-\frac {e^{2} \left (6 a b c \,e^{3}+12 d \,e^{2} c^{2} a +b^{3} e^{3}+12 b^{2} d \,e^{2} c -90 b \,c^{2} d^{2} e +100 c^{3} d^{3}\right ) x^{3}}{2}-\frac {e \left (2 e^{4} a^{2} c +2 a \,b^{2} e^{4}+6 a b c d \,e^{3}+12 d^{2} e^{2} c^{2} a +b^{3} d \,e^{3}+12 b^{2} c \,d^{2} e^{2}-110 d^{3} e b \,c^{2}+130 d^{4} c^{3}\right ) x^{2}}{2}+\left (-\frac {3}{4} a^{2} b \,e^{5}-\frac {1}{2} d \,e^{4} a^{2} c -\frac {1}{2} a \,b^{2} d \,e^{4}-\frac {3}{2} a b c \,d^{2} e^{3}-3 d^{3} e^{2} c^{2} a -\frac {1}{4} b^{3} d^{2} e^{3}-3 b^{2} c \,d^{3} e^{2}+\frac {125}{4} b \,c^{2} d^{4} e -\frac {77}{2} d^{5} c^{3}\right ) x -\frac {4 e^{6} a^{3}+3 a^{2} b d \,e^{5}+2 e^{4} d^{2} a^{2} c +2 a \,b^{2} d^{2} e^{4}+6 a b c \,d^{3} e^{3}+12 d^{4} e^{2} c^{2} a +b^{3} d^{3} e^{3}+12 b^{2} c \,d^{4} e^{2}-137 b \,c^{2} d^{5} e +174 d^{6} c^{3}}{20 e}}{e^{6} \left (e x +d \right )^{5}}+\frac {3 c^{2} \ln \left (e x +d \right ) b}{e^{6}}-\frac {6 c^{3} d \ln \left (e x +d \right )}{e^{7}}\)	\(449\)
default	\(\frac {c^{3} x}{e^{6}}-\frac {3 e^{4} a^{2} c +3 a \,b^{2} e^{4}-18 a b c d \,e^{3}+18 d^{2} e^{2} c^{2} a -3 b^{3} d \,e^{3}+18 b^{2} c \,d^{2} e^{2}-30 d^{3} e b \,c^{2}+15 d^{4} c^{3}}{3 e^{7} \left (e x +d \right )^{3}}-\frac {3 c \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right )}{e^{7} \left (e x +d \right )}+\frac {3 c^{2} \left (b e -2 c d \right ) \ln \left (e x +d \right )}{e^{7}}-\frac {6 a b c \,e^{3}-12 d \,e^{2} c^{2} a +b^{3} e^{3}-12 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -20 c^{3} d^{3}}{2 e^{7} \left (e x +d \right )^{2}}-\frac {e^{6} a^{3}-3 a^{2} b d \,e^{5}+3 e^{4} d^{2} a^{2} c +3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} c^{2} a -b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +d^{6} c^{3}}{5 e^{7} \left (e x +d \right )^{5}}-\frac {3 a^{2} b \,e^{5}-6 d \,e^{4} a^{2} c -6 a \,b^{2} d \,e^{4}+18 a b c \,d^{2} e^{3}-12 d^{3} e^{2} c^{2} a +3 b^{3} d^{2} e^{3}-12 b^{2} c \,d^{3} e^{2}+15 b \,c^{2} d^{4} e -6 d^{5} c^{3}}{4 e^{7} \left (e x +d \right )^{4}}\)	\(453\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^3/(e*x+d)^6,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, size = 446, normalized size = 1.74 \begin {gather*} c^{3} x e^{\left (-6\right )} - 3 \, {\left (2 \, c^{3} d - b c^{2} e\right )} e^{\left (-7\right )} \log \left (x e + d\right ) - \frac {174 \, c^{3} d^{6} - 137 \, b c^{2} d^{5} e + 12 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d^{4} + 60 \, {\left (5 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} + b^{2} c e^{6} + a c^{2} e^{6}\right )} x^{4} + 3 \, a^{2} b d e^{5} + {\left (b^{3} e^{3} + 6 \, a b c e^{3}\right )} d^{3} + 10 \, {\left (100 \, c^{3} d^{3} e^{3} - 90 \, b c^{2} d^{2} e^{4} + b^{3} e^{6} + 6 \, a b c e^{6} + 12 \, {\left (b^{2} c e^{5} + a c^{2} e^{5}\right )} d\right )} x^{3} + 4 \, a^{3} e^{6} + 2 \, {\left (a b^{2} e^{4} + a^{2} c e^{4}\right )} d^{2} + 10 \, {\left (130 \, c^{3} d^{4} e^{2} - 110 \, b c^{2} d^{3} e^{3} + 2 \, a b^{2} e^{6} + 2 \, a^{2} c e^{6} + 12 \, {\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} + 5 \, {\left (154 \, c^{3} d^{5} e - 125 \, b c^{2} d^{4} e^{2} + 12 \, {\left (b^{2} c e^{3} + a c^{2} e^{3}\right )} d^{3} + 3 \, a^{2} b e^{6} + {\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}{20 \, {\left (x^{5} e^{12} + 5 \, d x^{4} e^{11} + 10 \, d^{2} x^{3} e^{10} + 10 \, d^{3} x^{2} e^{9} + 5 \, d^{4} x e^{8} + d^{5} 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)^6,x, algorithm="maxima")

[Out]

c^3*x*e^(-6) - 3*(2*c^3*d - b*c^2*e)*e^(-7)*log(x*e + d) - 1/20*(174*c^3*d^6 - 137*b*c^2*d^5*e + 12*(b^2*c*e^2
+ a*c^2*e^2)*d^4 + 60*(5*c^3*d^2*e^4 - 5*b*c^2*d*e^5 + b^2*c*e^6 + a*c^2*e^6)*x^4 + 3*a^2*b*d*e^5 + (b^3*e^3
+ 6*a*b*c*e^3)*d^3 + 10*(100*c^3*d^3*e^3 - 90*b*c^2*d^2*e^4 + b^3*e^6 + 6*a*b*c*e^6 + 12*(b^2*c*e^5 + a*c^2*e^
5)*d)*x^3 + 4*a^3*e^6 + 2*(a*b^2*e^4 + a^2*c*e^4)*d^2 + 10*(130*c^3*d^4*e^2 - 110*b*c^2*d^3*e^3 + 2*a*b^2*e^6
+ 2*a^2*c*e^6 + 12*(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 + 5*(154*c^3*d^5*e - 125*b*c^2
*d^4*e^2 + 12*(b^2*c*e^3 + a*c^2*e^3)*d^3 + 3*a^2*b*e^6 + (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^5*e^12 + 5*d*x^4*e^11 + 10*d^2*x^3*e^10 + 10*d^3*x^2*e^9 + 5*d^4*x*e^8 + d^5*e^7)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 563 vs. \(2 (255) = 510\).
time = 2.95, size = 563, normalized size = 2.20 \begin {gather*} -\frac {174 \, c^{3} d^{6} - {\left (20 \, c^{3} x^{6} - 60 \, {\left (b^{2} c + a c^{2}\right )} x^{4} - 15 \, a^{2} b x - 10 \, {\left (b^{3} + 6 \, a b c\right )} x^{3} - 4 \, a^{3} - 20 \, {\left (a b^{2} + a^{2} c\right )} x^{2}\right )} e^{6} - {\left (100 \, c^{3} d x^{5} + 300 \, b c^{2} d x^{4} - 120 \, {\left (b^{2} c + a c^{2}\right )} d x^{3} - 3 \, a^{2} b d - 10 \, {\left (b^{3} + 6 \, a b c\right )} d x^{2} - 10 \, {\left (a b^{2} + a^{2} c\right )} d x\right )} e^{5} + {\left (100 \, c^{3} d^{2} x^{4} - 900 \, b c^{2} d^{2} x^{3} + 120 \, {\left (b^{2} c + a c^{2}\right )} d^{2} x^{2} + 5 \, {\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 (800 \, c^{3} d^{3} x^{3} - 1100 \, b c^{2} d^{3} x^{2} + 60 \, {\left (b^{2} c + a c^{2}\right )} d^{3} x + {\left (b^{3} + 6 \, a b c\right )} d^{3}\right )} e^{3} + {\left (1200 \, c^{3} d^{4} x^{2} - 625 \, b c^{2} d^{4} x + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{4}\right )} e^{2} + {\left (750 \, c^{3} d^{5} x - 137 \, b c^{2} d^{5}\right )} e + 60 \, {\left (2 \, c^{3} d^{6} - b c^{2} x^{5} e^{6} + {\left (2 \, c^{3} d x^{5} - 5 \, b c^{2} d x^{4}\right )} e^{5} + 10 \, {\left (c^{3} d^{2} x^{4} - b c^{2} d^{2} x^{3}\right )} e^{4} + 10 \, {\left (2 \, c^{3} d^{3} x^{3} - b c^{2} d^{3} x^{2}\right )} e^{3} + 5 \, {\left (4 \, c^{3} d^{4} x^{2} - b c^{2} d^{4} x\right )} e^{2} + {\left (10 \, c^{3} d^{5} x - b c^{2} d^{5}\right )} e\right )} \log \left (x e + d\right )}{20 \, {\left (x^{5} e^{12} + 5 \, d x^{4} e^{11} + 10 \, d^{2} x^{3} e^{10} + 10 \, d^{3} x^{2} e^{9} + 5 \, d^{4} x e^{8} + d^{5} 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)^6,x, algorithm="fricas")

[Out]

-1/20*(174*c^3*d^6 - (20*c^3*x^6 - 60*(b^2*c + a*c^2)*x^4 - 15*a^2*b*x - 10*(b^3 + 6*a*b*c)*x^3 - 4*a^3 - 20*(
a*b^2 + a^2*c)*x^2)*e^6 - (100*c^3*d*x^5 + 300*b*c^2*d*x^4 - 120*(b^2*c + a*c^2)*d*x^3 - 3*a^2*b*d - 10*(b^3 +
6*a*b*c)*d*x^2 - 10*(a*b^2 + a^2*c)*d*x)*e^5 + (100*c^3*d^2*x^4 - 900*b*c^2*d^2*x^3 + 120*(b^2*c + a*c^2)*d^2
*x^2 + 5*(b^3 + 6*a*b*c)*d^2*x + 2*(a*b^2 + a^2*c)*d^2)*e^4 + (800*c^3*d^3*x^3 - 1100*b*c^2*d^3*x^2 + 60*(b^2*
c + a*c^2)*d^3*x + (b^3 + 6*a*b*c)*d^3)*e^3 + (1200*c^3*d^4*x^2 - 625*b*c^2*d^4*x + 12*(b^2*c + a*c^2)*d^4)*e^
2 + (750*c^3*d^5*x - 137*b*c^2*d^5)*e + 60*(2*c^3*d^6 - b*c^2*x^5*e^6 + (2*c^3*d*x^5 - 5*b*c^2*d*x^4)*e^5 + 10
*(c^3*d^2*x^4 - b*c^2*d^2*x^3)*e^4 + 10*(2*c^3*d^3*x^3 - b*c^2*d^3*x^2)*e^3 + 5*(4*c^3*d^4*x^2 - b*c^2*d^4*x)*
e^2 + (10*c^3*d^5*x - b*c^2*d^5)*e)*log(x*e + d))/(x^5*e^12 + 5*d*x^4*e^11 + 10*d^2*x^3*e^10 + 10*d^3*x^2*e^9
+ 5*d^4*x*e^8 + d^5*e^7)

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Sympy [F(-1)] Timed out
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)**3/(e*x+d)**6,x)

[Out]

Timed out

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Giac [A]
time = 1.22, size = 415, normalized size = 1.62 \begin {gather*} c^{3} x e^{\left (-6\right )} - 3 \, {\left (2 \, c^{3} d - b c^{2} e\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) - \frac {{\left (174 \, c^{3} d^{6} - 137 \, b c^{2} d^{5} e + 12 \, b^{2} c d^{4} e^{2} + 12 \, a c^{2} d^{4} e^{2} + b^{3} d^{3} e^{3} + 6 \, a b c d^{3} e^{3} + 2 \, a b^{2} d^{2} e^{4} + 2 \, a^{2} c d^{2} e^{4} + 60 \, {\left (5 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} + b^{2} c e^{6} + a c^{2} e^{6}\right )} x^{4} + 3 \, a^{2} b d e^{5} + 10 \, {\left (100 \, c^{3} d^{3} e^{3} - 90 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} + 12 \, a c^{2} d e^{5} + b^{3} e^{6} + 6 \, a b c e^{6}\right )} x^{3} + 4 \, a^{3} e^{6} + 10 \, {\left (130 \, c^{3} d^{4} e^{2} - 110 \, b c^{2} d^{3} e^{3} + 12 \, b^{2} c d^{2} e^{4} + 12 \, a c^{2} d^{2} e^{4} + b^{3} d e^{5} + 6 \, a b c d e^{5} + 2 \, a b^{2} e^{6} + 2 \, a^{2} c e^{6}\right )} x^{2} + 5 \, {\left (154 \, c^{3} d^{5} e - 125 \, b c^{2} d^{4} e^{2} + 12 \, b^{2} c d^{3} e^{3} + 12 \, a c^{2} d^{3} e^{3} + b^{3} d^{2} e^{4} + 6 \, a b c d^{2} e^{4} + 2 \, a b^{2} d e^{5} + 2 \, a^{2} c d e^{5} + 3 \, a^{2} b e^{6}\right )} x\right )} e^{\left (-7\right )}}{20 \, {\left (x e + d\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(e*x+d)^6,x, algorithm="giac")

[Out]

c^3*x*e^(-6) - 3*(2*c^3*d - b*c^2*e)*e^(-7)*log(abs(x*e + d)) - 1/20*(174*c^3*d^6 - 137*b*c^2*d^5*e + 12*b^2*c
*d^4*e^2 + 12*a*c^2*d^4*e^2 + b^3*d^3*e^3 + 6*a*b*c*d^3*e^3 + 2*a*b^2*d^2*e^4 + 2*a^2*c*d^2*e^4 + 60*(5*c^3*d^
2*e^4 - 5*b*c^2*d*e^5 + b^2*c*e^6 + a*c^2*e^6)*x^4 + 3*a^2*b*d*e^5 + 10*(100*c^3*d^3*e^3 - 90*b*c^2*d^2*e^4 +
12*b^2*c*d*e^5 + 12*a*c^2*d*e^5 + b^3*e^6 + 6*a*b*c*e^6)*x^3 + 4*a^3*e^6 + 10*(130*c^3*d^4*e^2 - 110*b*c^2*d^3
*e^3 + 12*b^2*c*d^2*e^4 + 12*a*c^2*d^2*e^4 + b^3*d*e^5 + 6*a*b*c*d*e^5 + 2*a*b^2*e^6 + 2*a^2*c*e^6)*x^2 + 5*(1
54*c^3*d^5*e - 125*b*c^2*d^4*e^2 + 12*b^2*c*d^3*e^3 + 12*a*c^2*d^3*e^3 + b^3*d^2*e^4 + 6*a*b*c*d^2*e^4 + 2*a*b
^2*d*e^5 + 2*a^2*c*d*e^5 + 3*a^2*b*e^6)*x)*e^(-7)/(x*e + d)^5

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Mupad [B]
time = 0.80, size = 493, normalized size = 1.93 \begin {gather*} \frac {c^3\,x}{e^6}-\frac {x\,\left (\frac {3\,a^2\,b\,e^5}{4}+\frac {a^2\,c\,d\,e^4}{2}+\frac {a\,b^2\,d\,e^4}{2}+\frac {3\,a\,b\,c\,d^2\,e^3}{2}+3\,a\,c^2\,d^3\,e^2+\frac {b^3\,d^2\,e^3}{4}+3\,b^2\,c\,d^3\,e^2-\frac {125\,b\,c^2\,d^4\,e}{4}+\frac {77\,c^3\,d^5}{2}\right )+x^4\,\left (3\,b^2\,c\,e^5-15\,b\,c^2\,d\,e^4+15\,c^3\,d^2\,e^3+3\,a\,c^2\,e^5\right )+\frac {4\,a^3\,e^6+3\,a^2\,b\,d\,e^5+2\,a^2\,c\,d^2\,e^4+2\,a\,b^2\,d^2\,e^4+6\,a\,b\,c\,d^3\,e^3+12\,a\,c^2\,d^4\,e^2+b^3\,d^3\,e^3+12\,b^2\,c\,d^4\,e^2-137\,b\,c^2\,d^5\,e+174\,c^3\,d^6}{20\,e}+x^2\,\left (a^2\,c\,e^5+a\,b^2\,e^5+3\,a\,b\,c\,d\,e^4+6\,a\,c^2\,d^2\,e^3+\frac {b^3\,d\,e^4}{2}+6\,b^2\,c\,d^2\,e^3-55\,b\,c^2\,d^3\,e^2+65\,c^3\,d^4\,e\right )+x^3\,\left (\frac {b^3\,e^5}{2}+6\,b^2\,c\,d\,e^4-45\,b\,c^2\,d^2\,e^3+3\,a\,b\,c\,e^5+50\,c^3\,d^3\,e^2+6\,a\,c^2\,d\,e^4\right )}{d^5\,e^6+5\,d^4\,e^7\,x+10\,d^3\,e^8\,x^2+10\,d^2\,e^9\,x^3+5\,d\,e^{10}\,x^4+e^{11}\,x^5}-\frac {\ln \left (d+e\,x\right )\,\left (6\,c^3\,d-3\,b\,c^2\,e\right )}{e^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^3/(d + e*x)^6,x)

[Out]

(c^3*x)/e^6 - (x*((77*c^3*d^5)/2 + (3*a^2*b*e^5)/4 + (b^3*d^2*e^3)/4 + 3*a*c^2*d^3*e^2 + 3*b^2*c*d^3*e^2 + (a*
b^2*d*e^4)/2 + (a^2*c*d*e^4)/2 - (125*b*c^2*d^4*e)/4 + (3*a*b*c*d^2*e^3)/2) + x^4*(3*a*c^2*e^5 + 3*b^2*c*e^5 +
15*c^3*d^2*e^3 - 15*b*c^2*d*e^4) + (4*a^3*e^6 + 174*c^3*d^6 + b^3*d^3*e^3 + 2*a*b^2*d^2*e^4 + 12*a*c^2*d^4*e^
2 + 2*a^2*c*d^2*e^4 + 12*b^2*c*d^4*e^2 + 3*a^2*b*d*e^5 - 137*b*c^2*d^5*e + 6*a*b*c*d^3*e^3)/(20*e) + x^2*(a*b^
2*e^5 + a^2*c*e^5 + (b^3*d*e^4)/2 + 65*c^3*d^4*e + 6*a*c^2*d^2*e^3 - 55*b*c^2*d^3*e^2 + 6*b^2*c*d^2*e^3 + 3*a*
b*c*d*e^4) + x^3*((b^3*e^5)/2 + 50*c^3*d^3*e^2 - 45*b*c^2*d^2*e^3 + 3*a*b*c*e^5 + 6*a*c^2*d*e^4 + 6*b^2*c*d*e^
4))/(d^5*e^6 + e^11*x^5 + 5*d^4*e^7*x + 5*d*e^10*x^4 + 10*d^3*e^8*x^2 + 10*d^2*e^9*x^3) - (log(d + e*x)*(6*c^3
*d - 3*b*c^2*e))/e^7

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