3.14.21 \(\int \frac {(b+2 c x) (a+b x+c x^2)^2}{(d+e x)^5} \, dx\)

Optimal. Leaf size=227 \[ -\frac {4 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6 (d+e x)}+\frac {(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^6 (d+e x)^2}-\frac {2 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^6 (d+e x)^3}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{4 e^6 (d+e x)^4}-\frac {5 c^2 (2 c d-b e) \log (d+e x)}{e^6}+\frac {2 c^3 x}{e^5} \]

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Rubi [A]  time = 0.21, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {771} \begin {gather*} -\frac {4 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6 (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^6 (d+e x)^2}-\frac {2 \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 )}{3 e^6 (d+e x)^3}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{4 e^6 (d+e x)^4}-\frac {5 c^2 (2 c d-b e) \log (d+e x)}{e^6}+\frac {2 c^3 x}{e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((b + 2*c*x)*(a + b*x + c*x^2)^2)/(d + e*x)^5,x]

[Out]

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

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

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Mathematica [A]  time = 0.18, size = 292, normalized size = 1.29 \begin {gather*} -\frac {2 c e^2 \left (a^2 e^2 (d+4 e x)+3 a b e \left (d^2+4 d e x+6 e^2 x^2\right )+6 b^2 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )+b e^3 \left (3 a^2 e^2+2 a b e (d+4 e x)+b^2 \left (d^2+4 d e x+6 e^2 x^2\right )\right )+c^2 e \left (12 a e \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-5 b d \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )\right )+60 c^2 (d+e x)^4 (2 c d-b e) \log (d+e x)+2 c^3 \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )}{12 e^6 (d+e x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((b + 2*c*x)*(a + b*x + c*x^2)^2)/(d + e*x)^5,x]

[Out]

-1/12*(2*c^3*(77*d^5 + 248*d^4*e*x + 252*d^3*e^2*x^2 + 48*d^2*e^3*x^3 - 48*d*e^4*x^4 - 12*e^5*x^5) + b*e^3*(3*
a^2*e^2 + 2*a*b*e*(d + 4*e*x) + b^2*(d^2 + 4*d*e*x + 6*e^2*x^2)) + 2*c*e^2*(a^2*e^2*(d + 4*e*x) + 3*a*b*e*(d^2
+ 4*d*e*x + 6*e^2*x^2) + 6*b^2*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3)) + c^2*e*(12*a*e*(d^3 + 4*d^2*e*x
+ 6*d*e^2*x^2 + 4*e^3*x^3) - 5*b*d*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3)) + 60*c^2*(2*c*d - b*e)*
(d + e*x)^4*Log[d + e*x])/(e^6*(d + e*x)^4)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^2}{(d+e x)^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((b + 2*c*x)*(a + b*x + c*x^2)^2)/(d + e*x)^5,x]

[Out]

IntegrateAlgebraic[((b + 2*c*x)*(a + b*x + c*x^2)^2)/(d + e*x)^5, x]

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fricas [B]  time = 0.43, size = 458, normalized size = 2.02 \begin {gather*} \frac {24 \, c^{3} e^{5} x^{5} + 96 \, c^{3} d e^{4} x^{4} - 154 \, c^{3} d^{5} + 125 \, b c^{2} d^{4} e - 3 \, a^{2} b e^{5} - 12 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} - 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{4} - 48 \, {\left (2 \, c^{3} d^{2} e^{3} - 5 \, b c^{2} d e^{4} + {\left (b^{2} c + a c^{2}\right )} e^{5}\right )} x^{3} - 6 \, {\left (84 \, c^{3} d^{3} e^{2} - 90 \, b c^{2} d^{2} e^{3} + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{4} + {\left (b^{3} + 6 \, a b c\right )} e^{5}\right )} x^{2} - 4 \, {\left (124 \, c^{3} d^{4} e - 110 \, b c^{2} d^{3} e^{2} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} + {\left (b^{3} + 6 \, a b c\right )} d e^{4} + 2 \, {\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x - 60 \, {\left (2 \, c^{3} d^{5} - b c^{2} d^{4} e + {\left (2 \, c^{3} d e^{4} - b c^{2} e^{5}\right )} x^{4} + 4 \, {\left (2 \, c^{3} d^{2} e^{3} - b c^{2} d e^{4}\right )} x^{3} + 6 \, {\left (2 \, c^{3} d^{3} e^{2} - b c^{2} d^{2} e^{3}\right )} x^{2} + 4 \, {\left (2 \, c^{3} d^{4} e - b c^{2} d^{3} e^{2}\right )} x\right )} \log \left (e x + d\right )}{12 \, {\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)^2/(e*x+d)^5,x, algorithm="fricas")

[Out]

1/12*(24*c^3*e^5*x^5 + 96*c^3*d*e^4*x^4 - 154*c^3*d^5 + 125*b*c^2*d^4*e - 3*a^2*b*e^5 - 12*(b^2*c + a*c^2)*d^3
*e^2 - (b^3 + 6*a*b*c)*d^2*e^3 - 2*(a*b^2 + a^2*c)*d*e^4 - 48*(2*c^3*d^2*e^3 - 5*b*c^2*d*e^4 + (b^2*c + a*c^2)
*e^5)*x^3 - 6*(84*c^3*d^3*e^2 - 90*b*c^2*d^2*e^3 + 12*(b^2*c + a*c^2)*d*e^4 + (b^3 + 6*a*b*c)*e^5)*x^2 - 4*(12
4*c^3*d^4*e - 110*b*c^2*d^3*e^2 + 12*(b^2*c + a*c^2)*d^2*e^3 + (b^3 + 6*a*b*c)*d*e^4 + 2*(a*b^2 + a^2*c)*e^5)*
x - 60*(2*c^3*d^5 - b*c^2*d^4*e + (2*c^3*d*e^4 - b*c^2*e^5)*x^4 + 4*(2*c^3*d^2*e^3 - b*c^2*d*e^4)*x^3 + 6*(2*c
^3*d^3*e^2 - b*c^2*d^2*e^3)*x^2 + 4*(2*c^3*d^4*e - b*c^2*d^3*e^2)*x)*log(e*x + d))/(e^10*x^4 + 4*d*e^9*x^3 + 6
*d^2*e^8*x^2 + 4*d^3*e^7*x + d^4*e^6)

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giac [B]  time = 0.20, size = 525, normalized size = 2.31 \begin {gather*} 2 \, {\left (x e + d\right )} c^{3} e^{\left (-6\right )} + 5 \, {\left (2 \, c^{3} d - b c^{2} e\right )} e^{\left (-6\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - \frac {1}{12} \, {\left (\frac {240 \, c^{3} d^{2} e^{22}}{x e + d} - \frac {120 \, c^{3} d^{3} e^{22}}{{\left (x e + d\right )}^{2}} + \frac {40 \, c^{3} d^{4} e^{22}}{{\left (x e + d\right )}^{3}} - \frac {6 \, c^{3} d^{5} e^{22}}{{\left (x e + d\right )}^{4}} - \frac {240 \, b c^{2} d e^{23}}{x e + d} + \frac {180 \, b c^{2} d^{2} e^{23}}{{\left (x e + d\right )}^{2}} - \frac {80 \, b c^{2} d^{3} e^{23}}{{\left (x e + d\right )}^{3}} + \frac {15 \, b c^{2} d^{4} e^{23}}{{\left (x e + d\right )}^{4}} + \frac {48 \, b^{2} c e^{24}}{x e + d} + \frac {48 \, a c^{2} e^{24}}{x e + d} - \frac {72 \, b^{2} c d e^{24}}{{\left (x e + d\right )}^{2}} - \frac {72 \, a c^{2} d e^{24}}{{\left (x e + d\right )}^{2}} + \frac {48 \, b^{2} c d^{2} e^{24}}{{\left (x e + d\right )}^{3}} + \frac {48 \, a c^{2} d^{2} e^{24}}{{\left (x e + d\right )}^{3}} - \frac {12 \, b^{2} c d^{3} e^{24}}{{\left (x e + d\right )}^{4}} - \frac {12 \, a c^{2} d^{3} e^{24}}{{\left (x e + d\right )}^{4}} + \frac {6 \, b^{3} e^{25}}{{\left (x e + d\right )}^{2}} + \frac {36 \, a b c e^{25}}{{\left (x e + d\right )}^{2}} - \frac {8 \, b^{3} d e^{25}}{{\left (x e + d\right )}^{3}} - \frac {48 \, a b c d e^{25}}{{\left (x e + d\right )}^{3}} + \frac {3 \, b^{3} d^{2} e^{25}}{{\left (x e + d\right )}^{4}} + \frac {18 \, a b c d^{2} e^{25}}{{\left (x e + d\right )}^{4}} + \frac {8 \, a b^{2} e^{26}}{{\left (x e + d\right )}^{3}} + \frac {8 \, a^{2} c e^{26}}{{\left (x e + d\right )}^{3}} - \frac {6 \, a b^{2} d e^{26}}{{\left (x e + d\right )}^{4}} - \frac {6 \, a^{2} c d e^{26}}{{\left (x e + d\right )}^{4}} + \frac {3 \, a^{2} b e^{27}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-28\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)^2/(e*x+d)^5,x, algorithm="giac")

[Out]

2*(x*e + d)*c^3*e^(-6) + 5*(2*c^3*d - b*c^2*e)*e^(-6)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) - 1/12*(240*c^3*d^2
*e^22/(x*e + d) - 120*c^3*d^3*e^22/(x*e + d)^2 + 40*c^3*d^4*e^22/(x*e + d)^3 - 6*c^3*d^5*e^22/(x*e + d)^4 - 24
0*b*c^2*d*e^23/(x*e + d) + 180*b*c^2*d^2*e^23/(x*e + d)^2 - 80*b*c^2*d^3*e^23/(x*e + d)^3 + 15*b*c^2*d^4*e^23/
(x*e + d)^4 + 48*b^2*c*e^24/(x*e + d) + 48*a*c^2*e^24/(x*e + d) - 72*b^2*c*d*e^24/(x*e + d)^2 - 72*a*c^2*d*e^2
4/(x*e + d)^2 + 48*b^2*c*d^2*e^24/(x*e + d)^3 + 48*a*c^2*d^2*e^24/(x*e + d)^3 - 12*b^2*c*d^3*e^24/(x*e + d)^4
- 12*a*c^2*d^3*e^24/(x*e + d)^4 + 6*b^3*e^25/(x*e + d)^2 + 36*a*b*c*e^25/(x*e + d)^2 - 8*b^3*d*e^25/(x*e + d)^
3 - 48*a*b*c*d*e^25/(x*e + d)^3 + 3*b^3*d^2*e^25/(x*e + d)^4 + 18*a*b*c*d^2*e^25/(x*e + d)^4 + 8*a*b^2*e^26/(x
*e + d)^3 + 8*a^2*c*e^26/(x*e + d)^3 - 6*a*b^2*d*e^26/(x*e + d)^4 - 6*a^2*c*d*e^26/(x*e + d)^4 + 3*a^2*b*e^27/
(x*e + d)^4)*e^(-28)

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maple [B]  time = 0.06, size = 507, normalized size = 2.23 \begin {gather*} -\frac {a^{2} b}{4 \left (e x +d \right )^{4} e}+\frac {a^{2} c d}{2 \left (e x +d \right )^{4} e^{2}}+\frac {a \,b^{2} d}{2 \left (e x +d \right )^{4} e^{2}}-\frac {3 a b c \,d^{2}}{2 \left (e x +d \right )^{4} e^{3}}+\frac {a \,c^{2} d^{3}}{\left (e x +d \right )^{4} e^{4}}-\frac {b^{3} d^{2}}{4 \left (e x +d \right )^{4} e^{3}}+\frac {b^{2} c \,d^{3}}{\left (e x +d \right )^{4} e^{4}}-\frac {5 b \,c^{2} d^{4}}{4 \left (e x +d \right )^{4} e^{5}}+\frac {c^{3} d^{5}}{2 \left (e x +d \right )^{4} e^{6}}-\frac {2 a^{2} c}{3 \left (e x +d \right )^{3} e^{2}}-\frac {2 a \,b^{2}}{3 \left (e x +d \right )^{3} e^{2}}+\frac {4 a b c d}{\left (e x +d \right )^{3} e^{3}}-\frac {4 a \,c^{2} d^{2}}{\left (e x +d \right )^{3} e^{4}}+\frac {2 b^{3} d}{3 \left (e x +d \right )^{3} e^{3}}-\frac {4 b^{2} c \,d^{2}}{\left (e x +d \right )^{3} e^{4}}+\frac {20 b \,c^{2} d^{3}}{3 \left (e x +d \right )^{3} e^{5}}-\frac {10 c^{3} d^{4}}{3 \left (e x +d \right )^{3} e^{6}}-\frac {3 a b c}{\left (e x +d \right )^{2} e^{3}}+\frac {6 a \,c^{2} d}{\left (e x +d \right )^{2} e^{4}}-\frac {b^{3}}{2 \left (e x +d \right )^{2} e^{3}}+\frac {6 b^{2} c d}{\left (e x +d \right )^{2} e^{4}}-\frac {15 b \,c^{2} d^{2}}{\left (e x +d \right )^{2} e^{5}}+\frac {10 c^{3} d^{3}}{\left (e x +d \right )^{2} e^{6}}-\frac {4 a \,c^{2}}{\left (e x +d \right ) e^{4}}-\frac {4 b^{2} c}{\left (e x +d \right ) e^{4}}+\frac {20 b \,c^{2} d}{\left (e x +d \right ) e^{5}}+\frac {5 b \,c^{2} \ln \left (e x +d \right )}{e^{5}}-\frac {20 c^{3} d^{2}}{\left (e x +d \right ) e^{6}}-\frac {10 c^{3} d \ln \left (e x +d \right )}{e^{6}}+\frac {2 c^{3} x}{e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(c*x^2+b*x+a)^2/(e*x+d)^5,x)

[Out]

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

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maxima [A]  time = 0.66, size = 338, normalized size = 1.49 \begin {gather*} -\frac {154 \, c^{3} d^{5} - 125 \, b c^{2} d^{4} e + 3 \, a^{2} b e^{5} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} + {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{4} + 48 \, {\left (5 \, c^{3} d^{2} e^{3} - 5 \, b c^{2} d e^{4} + {\left (b^{2} c + a c^{2}\right )} e^{5}\right )} x^{3} + 6 \, {\left (100 \, c^{3} d^{3} e^{2} - 90 \, b c^{2} d^{2} e^{3} + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{4} + {\left (b^{3} + 6 \, a b c\right )} e^{5}\right )} x^{2} + 4 \, {\left (130 \, c^{3} d^{4} e - 110 \, b c^{2} d^{3} e^{2} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} + {\left (b^{3} + 6 \, a b c\right )} d e^{4} + 2 \, {\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x}{12 \, {\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} + \frac {2 \, c^{3} x}{e^{5}} - \frac {5 \, {\left (2 \, c^{3} d - b c^{2} e\right )} \log \left (e x + d\right )}{e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x+a)^2/(e*x+d)^5,x, algorithm="maxima")

[Out]

-1/12*(154*c^3*d^5 - 125*b*c^2*d^4*e + 3*a^2*b*e^5 + 12*(b^2*c + a*c^2)*d^3*e^2 + (b^3 + 6*a*b*c)*d^2*e^3 + 2*
(a*b^2 + a^2*c)*d*e^4 + 48*(5*c^3*d^2*e^3 - 5*b*c^2*d*e^4 + (b^2*c + a*c^2)*e^5)*x^3 + 6*(100*c^3*d^3*e^2 - 90
*b*c^2*d^2*e^3 + 12*(b^2*c + a*c^2)*d*e^4 + (b^3 + 6*a*b*c)*e^5)*x^2 + 4*(130*c^3*d^4*e - 110*b*c^2*d^3*e^2 +
12*(b^2*c + a*c^2)*d^2*e^3 + (b^3 + 6*a*b*c)*d*e^4 + 2*(a*b^2 + a^2*c)*e^5)*x)/(e^10*x^4 + 4*d*e^9*x^3 + 6*d^2
*e^8*x^2 + 4*d^3*e^7*x + d^4*e^6) + 2*c^3*x/e^5 - 5*(2*c^3*d - b*c^2*e)*log(e*x + d)/e^6

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mupad [B]  time = 1.92, size = 369, normalized size = 1.63 \begin {gather*} \frac {2\,c^3\,x}{e^5}-\frac {x\,\left (\frac {2\,a^2\,c\,e^4}{3}+\frac {2\,a\,b^2\,e^4}{3}+2\,a\,b\,c\,d\,e^3+4\,a\,c^2\,d^2\,e^2+\frac {b^3\,d\,e^3}{3}+4\,b^2\,c\,d^2\,e^2-\frac {110\,b\,c^2\,d^3\,e}{3}+\frac {130\,c^3\,d^4}{3}\right )+x^2\,\left (\frac {b^3\,e^4}{2}+6\,b^2\,c\,d\,e^3-45\,b\,c^2\,d^2\,e^2+3\,a\,b\,c\,e^4+50\,c^3\,d^3\,e+6\,a\,c^2\,d\,e^3\right )+x^3\,\left (4\,b^2\,c\,e^4-20\,b\,c^2\,d\,e^3+20\,c^3\,d^2\,e^2+4\,a\,c^2\,e^4\right )+\frac {3\,a^2\,b\,e^5+2\,a^2\,c\,d\,e^4+2\,a\,b^2\,d\,e^4+6\,a\,b\,c\,d^2\,e^3+12\,a\,c^2\,d^3\,e^2+b^3\,d^2\,e^3+12\,b^2\,c\,d^3\,e^2-125\,b\,c^2\,d^4\,e+154\,c^3\,d^5}{12\,e}}{d^4\,e^5+4\,d^3\,e^6\,x+6\,d^2\,e^7\,x^2+4\,d\,e^8\,x^3+e^9\,x^4}-\frac {\ln \left (d+e\,x\right )\,\left (10\,c^3\,d-5\,b\,c^2\,e\right )}{e^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b + 2*c*x)*(a + b*x + c*x^2)^2)/(d + e*x)^5,x)

[Out]

(2*c^3*x)/e^5 - (x*((130*c^3*d^4)/3 + (2*a*b^2*e^4)/3 + (2*a^2*c*e^4)/3 + (b^3*d*e^3)/3 + 4*a*c^2*d^2*e^2 + 4*
b^2*c*d^2*e^2 - (110*b*c^2*d^3*e)/3 + 2*a*b*c*d*e^3) + x^2*((b^3*e^4)/2 + 50*c^3*d^3*e - 45*b*c^2*d^2*e^2 + 3*
a*b*c*e^4 + 6*a*c^2*d*e^3 + 6*b^2*c*d*e^3) + x^3*(4*a*c^2*e^4 + 4*b^2*c*e^4 + 20*c^3*d^2*e^2 - 20*b*c^2*d*e^3)
+ (154*c^3*d^5 + 3*a^2*b*e^5 + b^3*d^2*e^3 + 12*a*c^2*d^3*e^2 + 12*b^2*c*d^3*e^2 + 2*a*b^2*d*e^4 + 2*a^2*c*d*
e^4 - 125*b*c^2*d^4*e + 6*a*b*c*d^2*e^3)/(12*e))/(d^4*e^5 + e^9*x^4 + 4*d^3*e^6*x + 4*d*e^8*x^3 + 6*d^2*e^7*x^
2) - (log(d + e*x)*(10*c^3*d - 5*b*c^2*e))/e^6

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sympy [A]  time = 58.01, size = 401, normalized size = 1.77 \begin {gather*} \frac {2 c^{3} x}{e^{5}} + \frac {5 c^{2} \left (b e - 2 c d\right ) \log {\left (d + e x \right )}}{e^{6}} + \frac {- 3 a^{2} b e^{5} - 2 a^{2} c d e^{4} - 2 a b^{2} d e^{4} - 6 a b c d^{2} e^{3} - 12 a c^{2} d^{3} e^{2} - b^{3} d^{2} e^{3} - 12 b^{2} c d^{3} e^{2} + 125 b c^{2} d^{4} e - 154 c^{3} d^{5} + x^{3} \left (- 48 a c^{2} e^{5} - 48 b^{2} c e^{5} + 240 b c^{2} d e^{4} - 240 c^{3} d^{2} e^{3}\right ) + x^{2} \left (- 36 a b c e^{5} - 72 a c^{2} d e^{4} - 6 b^{3} e^{5} - 72 b^{2} c d e^{4} + 540 b c^{2} d^{2} e^{3} - 600 c^{3} d^{3} e^{2}\right ) + x \left (- 8 a^{2} c e^{5} - 8 a b^{2} e^{5} - 24 a b c d e^{4} - 48 a c^{2} d^{2} e^{3} - 4 b^{3} d e^{4} - 48 b^{2} c d^{2} e^{3} + 440 b c^{2} d^{3} e^{2} - 520 c^{3} d^{4} e\right )}{12 d^{4} e^{6} + 48 d^{3} e^{7} x + 72 d^{2} e^{8} x^{2} + 48 d e^{9} x^{3} + 12 e^{10} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x**2+b*x+a)**2/(e*x+d)**5,x)

[Out]

2*c**3*x/e**5 + 5*c**2*(b*e - 2*c*d)*log(d + e*x)/e**6 + (-3*a**2*b*e**5 - 2*a**2*c*d*e**4 - 2*a*b**2*d*e**4 -
6*a*b*c*d**2*e**3 - 12*a*c**2*d**3*e**2 - b**3*d**2*e**3 - 12*b**2*c*d**3*e**2 + 125*b*c**2*d**4*e - 154*c**3
*d**5 + x**3*(-48*a*c**2*e**5 - 48*b**2*c*e**5 + 240*b*c**2*d*e**4 - 240*c**3*d**2*e**3) + x**2*(-36*a*b*c*e**
5 - 72*a*c**2*d*e**4 - 6*b**3*e**5 - 72*b**2*c*d*e**4 + 540*b*c**2*d**2*e**3 - 600*c**3*d**3*e**2) + x*(-8*a**
2*c*e**5 - 8*a*b**2*e**5 - 24*a*b*c*d*e**4 - 48*a*c**2*d**2*e**3 - 4*b**3*d*e**4 - 48*b**2*c*d**2*e**3 + 440*b
*c**2*d**3*e**2 - 520*c**3*d**4*e))/(12*d**4*e**6 + 48*d**3*e**7*x + 72*d**2*e**8*x**2 + 48*d*e**9*x**3 + 12*e
**10*x**4)

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