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

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

________________________________________________________________________________________

Rubi [A]  time = 0.32, 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 = {698} \begin {gather*} \frac {c (d+e x)^3 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}-\frac {(d+e x)^2 (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}+\frac {3 x \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^6}-\frac {\left (a e^2-b d e+c d^2\right )^3}{e^7 (d+e x)}-\frac {3 (2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^2}{e^7}-\frac {3 c^2 (d+e x)^4 (2 c d-b e)}{4 e^7}+\frac {c^3 (d+e x)^5}{5 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

________________________________________________________________________________________

Mathematica [A]  time = 0.10, size = 255, normalized size = 1.00 \begin {gather*} \frac {20 e x \left (3 c e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )+b^2 e^3 (3 a e-2 b d)+3 c^2 d^2 e (3 a e-4 b d)+5 c^3 d^4\right )+10 e^2 x^2 (b e-c d) \left (c e (6 a e-5 b d)+b^2 e^2+4 c^2 d^2\right )+20 c e^3 x^3 \left (c e (a e-2 b d)+b^2 e^2+c^2 d^2\right )-\frac {20 \left (e (a e-b d)+c d^2\right )^3}{d+e x}-60 (2 c d-b e) \log (d+e x) \left (e (a e-b d)+c d^2\right )^2+5 c^2 e^4 x^4 (3 b e-2 c d)+4 c^3 e^5 x^5}{20 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(20*e*(5*c^3*d^4 + 3*c^2*d^2*e*(-4*b*d + 3*a*e) + b^2*e^3*(-2*b*d + 3*a*e) + 3*c*e^2*(3*b^2*d^2 - 4*a*b*d*e +
a^2*e^2))*x + 10*e^2*(-(c*d) + b*e)*(4*c^2*d^2 + b^2*e^2 + c*e*(-5*b*d + 6*a*e))*x^2 + 20*c*e^3*(c^2*d^2 + b^2
*e^2 + c*e*(-2*b*d + a*e))*x^3 + 5*c^2*e^4*(-2*c*d + 3*b*e)*x^4 + 4*c^3*e^5*x^5 - (20*(c*d^2 + e*(-(b*d) + a*e
))^3)/(d + e*x) - 60*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^2*Log[d + e*x])/(20*e^7)

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.40, size = 580, normalized size = 2.27 \begin {gather*} \frac {4 \, c^{3} e^{6} x^{6} - 20 \, c^{3} d^{6} + 60 \, b c^{2} d^{5} e + 60 \, a^{2} b d e^{5} - 20 \, a^{3} e^{6} - 60 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + 20 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 60 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 3 \, {\left (2 \, c^{3} d e^{5} - 5 \, b c^{2} e^{6}\right )} x^{5} + 5 \, {\left (2 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} + 4 \, {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} - 10 \, {\left (2 \, c^{3} d^{3} e^{3} - 5 \, b c^{2} d^{2} e^{4} + 4 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} - {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 30 \, {\left (2 \, c^{3} d^{4} e^{2} - 5 \, b c^{2} d^{3} e^{3} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - {\left (b^{3} + 6 \, a b c\right )} d e^{5} + 2 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 20 \, {\left (5 \, c^{3} d^{5} e - 12 \, b c^{2} d^{4} e^{2} + 9 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 2 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 3 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x - 60 \, {\left (2 \, c^{3} d^{6} - 5 \, b c^{2} d^{5} e - a^{2} b d e^{5} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 2 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + {\left (2 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} - a^{2} b e^{6} + 4 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x\right )} \log \left (e x + d\right )}{20 \, {\left (e^{8} x + d 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)^2,x, algorithm="fricas")

[Out]

1/20*(4*c^3*e^6*x^6 - 20*c^3*d^6 + 60*b*c^2*d^5*e + 60*a^2*b*d*e^5 - 20*a^3*e^6 - 60*(b^2*c + a*c^2)*d^4*e^2 +
20*(b^3 + 6*a*b*c)*d^3*e^3 - 60*(a*b^2 + a^2*c)*d^2*e^4 - 3*(2*c^3*d*e^5 - 5*b*c^2*e^6)*x^5 + 5*(2*c^3*d^2*e^
4 - 5*b*c^2*d*e^5 + 4*(b^2*c + a*c^2)*e^6)*x^4 - 10*(2*c^3*d^3*e^3 - 5*b*c^2*d^2*e^4 + 4*(b^2*c + a*c^2)*d*e^5
- (b^3 + 6*a*b*c)*e^6)*x^3 + 30*(2*c^3*d^4*e^2 - 5*b*c^2*d^3*e^3 + 4*(b^2*c + a*c^2)*d^2*e^4 - (b^3 + 6*a*b*c
)*d*e^5 + 2*(a*b^2 + a^2*c)*e^6)*x^2 + 20*(5*c^3*d^5*e - 12*b*c^2*d^4*e^2 + 9*(b^2*c + a*c^2)*d^3*e^3 - 2*(b^3
+ 6*a*b*c)*d^2*e^4 + 3*(a*b^2 + a^2*c)*d*e^5)*x - 60*(2*c^3*d^6 - 5*b*c^2*d^5*e - a^2*b*d*e^5 + 4*(b^2*c + a*
c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 2*(a*b^2 + a^2*c)*d^2*e^4 + (2*c^3*d^5*e - 5*b*c^2*d^4*e^2 - a^2*b*e^
6 + 4*(b^2*c + a*c^2)*d^3*e^3 - (b^3 + 6*a*b*c)*d^2*e^4 + 2*(a*b^2 + a^2*c)*d*e^5)*x)*log(e*x + d))/(e^8*x + d
*e^7)

________________________________________________________________________________________

giac [B]  time = 0.22, size = 541, normalized size = 2.11 \begin {gather*} \frac {1}{20} \, {\left (4 \, c^{3} - \frac {15 \, {\left (2 \, c^{3} d e - b c^{2} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac {20 \, {\left (5 \, c^{3} d^{2} e^{2} - 5 \, b c^{2} d e^{3} + b^{2} c e^{4} + a c^{2} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {10 \, {\left (20 \, c^{3} d^{3} e^{3} - 30 \, 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 )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} + \frac {60 \, {\left (5 \, c^{3} d^{4} e^{4} - 10 \, b c^{2} d^{3} e^{5} + 6 \, b^{2} c d^{2} e^{6} + 6 \, a c^{2} d^{2} e^{6} - b^{3} d e^{7} - 6 \, a b c d e^{7} + a b^{2} e^{8} + a^{2} c e^{8}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}}\right )} {\left (x e + d\right )}^{5} e^{\left (-7\right )} + 3 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} + 4 \, a c^{2} d^{3} e^{2} - b^{3} d^{2} e^{3} - 6 \, a b c d^{2} e^{3} + 2 \, a b^{2} d e^{4} + 2 \, a^{2} c d e^{4} - a^{2} b e^{5}\right )} e^{\left (-7\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - {\left (\frac {c^{3} d^{6} e^{5}}{x e + d} - \frac {3 \, b c^{2} d^{5} e^{6}}{x e + d} + \frac {3 \, b^{2} c d^{4} e^{7}}{x e + d} + \frac {3 \, a c^{2} d^{4} e^{7}}{x e + d} - \frac {b^{3} d^{3} e^{8}}{x e + d} - \frac {6 \, a b c d^{3} e^{8}}{x e + d} + \frac {3 \, a b^{2} d^{2} e^{9}}{x e + d} + \frac {3 \, a^{2} c d^{2} e^{9}}{x e + d} - \frac {3 \, a^{2} b d e^{10}}{x e + d} + \frac {a^{3} e^{11}}{x e + d}\right )} e^{\left (-12\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)^2,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [B]  time = 0.07, size = 585, normalized size = 2.29 \begin {gather*} \frac {c^{3} x^{5}}{5 e^{2}}+\frac {3 b \,c^{2} x^{4}}{4 e^{2}}-\frac {c^{3} d \,x^{4}}{2 e^{3}}+\frac {a \,c^{2} x^{3}}{e^{2}}+\frac {b^{2} c \,x^{3}}{e^{2}}-\frac {2 b \,c^{2} d \,x^{3}}{e^{3}}+\frac {c^{3} d^{2} x^{3}}{e^{4}}+\frac {3 a b c \,x^{2}}{e^{2}}-\frac {3 a \,c^{2} d \,x^{2}}{e^{3}}+\frac {b^{3} x^{2}}{2 e^{2}}-\frac {3 b^{2} c d \,x^{2}}{e^{3}}+\frac {9 b \,c^{2} d^{2} x^{2}}{2 e^{4}}-\frac {2 c^{3} d^{3} x^{2}}{e^{5}}-\frac {a^{3}}{\left (e x +d \right ) e}+\frac {3 a^{2} b d}{\left (e x +d \right ) e^{2}}+\frac {3 a^{2} b \ln \left (e x +d \right )}{e^{2}}-\frac {3 a^{2} c \,d^{2}}{\left (e x +d \right ) e^{3}}-\frac {6 a^{2} c d \ln \left (e x +d \right )}{e^{3}}+\frac {3 a^{2} c x}{e^{2}}-\frac {3 a \,b^{2} d^{2}}{\left (e x +d \right ) e^{3}}-\frac {6 a \,b^{2} d \ln \left (e x +d \right )}{e^{3}}+\frac {3 a \,b^{2} x}{e^{2}}+\frac {6 a b c \,d^{3}}{\left (e x +d \right ) e^{4}}+\frac {18 a b c \,d^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {12 a b c d x}{e^{3}}-\frac {3 a \,c^{2} d^{4}}{\left (e x +d \right ) e^{5}}-\frac {12 a \,c^{2} d^{3} \ln \left (e x +d \right )}{e^{5}}+\frac {9 a \,c^{2} d^{2} x}{e^{4}}+\frac {b^{3} d^{3}}{\left (e x +d \right ) e^{4}}+\frac {3 b^{3} d^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {2 b^{3} d x}{e^{3}}-\frac {3 b^{2} c \,d^{4}}{\left (e x +d \right ) e^{5}}-\frac {12 b^{2} c \,d^{3} \ln \left (e x +d \right )}{e^{5}}+\frac {9 b^{2} c \,d^{2} x}{e^{4}}+\frac {3 b \,c^{2} d^{5}}{\left (e x +d \right ) e^{6}}+\frac {15 b \,c^{2} d^{4} \ln \left (e x +d \right )}{e^{6}}-\frac {12 b \,c^{2} d^{3} x}{e^{5}}-\frac {c^{3} d^{6}}{\left (e x +d \right ) e^{7}}-\frac {6 c^{3} d^{5} \ln \left (e x +d \right )}{e^{7}}+\frac {5 c^{3} d^{4} x}{e^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6/e^4/(e*x+d)*a*b*c*d^3-12/e^3*a*b*c*d*x+18/e^4*ln(e*x+d)*a*b*c*d^2+1/e^2*x^3*a*c^2+3/e^2*ln(e*x+d)*a^2*b+3/e^
2*a^2*c*x+3/e^2*a*b^2*x-1/(e*x+d)*c^3*d^6/e^7+3*b^3*d^2/e^4*ln(e*x+d)-6*c^3*d^5/e^7*ln(e*x+d)+b^2*c/e^2*x^3+c^
3*d^2/e^4*x^3-2*c^3*d^3/e^5*x^2-2*b^3*d/e^3*x+5*c^3*d^4/e^6*x+1/(e*x+d)*b^3*d^3/e^4+3/4*b*c^2/e^2*x^4+1/2*b^3/
e^2*x^2-1/e/(e*x+d)*a^3+3/e^2*x^2*a*b*c-6/e^3*ln(e*x+d)*a*b^2*d-12/e^5*ln(e*x+d)*a*c^2*d^3-3/e^3*x^2*a*c^2*d-2
*b*c^2*d/e^3*x^3-3*b^2*c*d/e^3*x^2+9/2*b*c^2*d^2/e^4*x^2+9*b^2*c*d^2/e^4*x-12*b*c^2*d^3/e^5*x-3/(e*x+d)*b^2*c*
d^4/e^5+3/(e*x+d)*b*c^2*d^5/e^6-12*b^2*c*d^3/e^5*ln(e*x+d)+15*b*c^2*d^4/e^6*ln(e*x+d)+9/e^4*a*c^2*d^2*x-6/e^3*
ln(e*x+d)*a^2*c*d-3/e^5/(e*x+d)*a*c^2*d^4+3/e^2/(e*x+d)*d*a^2*b-3/e^3/(e*x+d)*a^2*c*d^2-3/e^3/(e*x+d)*a*b^2*d^
2+1/5*c^3/e^2*x^5-1/2*c^3*d/e^3*x^4

________________________________________________________________________________________

maxima [A]  time = 1.09, size = 410, normalized size = 1.60 \begin {gather*} -\frac {c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}}{e^{8} x + d e^{7}} + \frac {4 \, c^{3} e^{4} x^{5} - 5 \, {\left (2 \, c^{3} d e^{3} - 3 \, b c^{2} e^{4}\right )} x^{4} + 20 \, {\left (c^{3} d^{2} e^{2} - 2 \, b c^{2} d e^{3} + {\left (b^{2} c + a c^{2}\right )} e^{4}\right )} x^{3} - 10 \, {\left (4 \, c^{3} d^{3} e - 9 \, b c^{2} d^{2} e^{2} + 6 \, {\left (b^{2} c + a c^{2}\right )} d e^{3} - {\left (b^{3} + 6 \, a b c\right )} e^{4}\right )} x^{2} + 20 \, {\left (5 \, c^{3} d^{4} - 12 \, b c^{2} d^{3} e + 9 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} - 2 \, {\left (b^{3} + 6 \, a b c\right )} d e^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} e^{4}\right )} x}{20 \, e^{6}} - \frac {3 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e - a^{2} b e^{5} + 4 \, {\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}\right )} \log \left (e x + d\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)^2,x, algorithm="maxima")

[Out]

-(c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*
(a*b^2 + a^2*c)*d^2*e^4)/(e^8*x + d*e^7) + 1/20*(4*c^3*e^4*x^5 - 5*(2*c^3*d*e^3 - 3*b*c^2*e^4)*x^4 + 20*(c^3*d
^2*e^2 - 2*b*c^2*d*e^3 + (b^2*c + a*c^2)*e^4)*x^3 - 10*(4*c^3*d^3*e - 9*b*c^2*d^2*e^2 + 6*(b^2*c + a*c^2)*d*e^
3 - (b^3 + 6*a*b*c)*e^4)*x^2 + 20*(5*c^3*d^4 - 12*b*c^2*d^3*e + 9*(b^2*c + a*c^2)*d^2*e^2 - 2*(b^3 + 6*a*b*c)*
d*e^3 + 3*(a*b^2 + a^2*c)*e^4)*x)/e^6 - 3*(2*c^3*d^5 - 5*b*c^2*d^4*e - a^2*b*e^5 + 4*(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)*log(e*x + d)/e^7

________________________________________________________________________________________

mupad [B]  time = 0.70, size = 592, normalized size = 2.31 \begin {gather*} x^2\,\left (\frac {b^3+6\,a\,c\,b}{2\,e^2}+\frac {d\,\left (\frac {2\,d\,\left (\frac {3\,b\,c^2}{e^2}-\frac {2\,c^3\,d}{e^3}\right )}{e}+\frac {c^3\,d^2}{e^4}-\frac {3\,c\,\left (b^2+a\,c\right )}{e^2}\right )}{e}-\frac {d^2\,\left (\frac {3\,b\,c^2}{e^2}-\frac {2\,c^3\,d}{e^3}\right )}{2\,e^2}\right )+x^4\,\left (\frac {3\,b\,c^2}{4\,e^2}-\frac {c^3\,d}{2\,e^3}\right )-x^3\,\left (\frac {2\,d\,\left (\frac {3\,b\,c^2}{e^2}-\frac {2\,c^3\,d}{e^3}\right )}{3\,e}+\frac {c^3\,d^2}{3\,e^4}-\frac {c\,\left (b^2+a\,c\right )}{e^2}\right )+x\,\left (\frac {d^2\,\left (\frac {2\,d\,\left (\frac {3\,b\,c^2}{e^2}-\frac {2\,c^3\,d}{e^3}\right )}{e}+\frac {c^3\,d^2}{e^4}-\frac {3\,c\,\left (b^2+a\,c\right )}{e^2}\right )}{e^2}-\frac {2\,d\,\left (\frac {b^3+6\,a\,c\,b}{e^2}+\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {3\,b\,c^2}{e^2}-\frac {2\,c^3\,d}{e^3}\right )}{e}+\frac {c^3\,d^2}{e^4}-\frac {3\,c\,\left (b^2+a\,c\right )}{e^2}\right )}{e}-\frac {d^2\,\left (\frac {3\,b\,c^2}{e^2}-\frac {2\,c^3\,d}{e^3}\right )}{e^2}\right )}{e}+\frac {3\,a\,\left (b^2+a\,c\right )}{e^2}\right )-\frac {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\,\left (x\,e^7+d\,e^6\right )}+\frac {c^3\,x^5}{5\,e^2}-\frac {\ln \left (d+e\,x\right )\,\left (-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\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)^2,x)

[Out]

x^2*((b^3 + 6*a*b*c)/(2*e^2) + (d*((2*d*((3*b*c^2)/e^2 - (2*c^3*d)/e^3))/e + (c^3*d^2)/e^4 - (3*c*(a*c + b^2))
/e^2))/e - (d^2*((3*b*c^2)/e^2 - (2*c^3*d)/e^3))/(2*e^2)) + x^4*((3*b*c^2)/(4*e^2) - (c^3*d)/(2*e^3)) - x^3*((
2*d*((3*b*c^2)/e^2 - (2*c^3*d)/e^3))/(3*e) + (c^3*d^2)/(3*e^4) - (c*(a*c + b^2))/e^2) + x*((d^2*((2*d*((3*b*c^
2)/e^2 - (2*c^3*d)/e^3))/e + (c^3*d^2)/e^4 - (3*c*(a*c + b^2))/e^2))/e^2 - (2*d*((b^3 + 6*a*b*c)/e^2 + (2*d*((
2*d*((3*b*c^2)/e^2 - (2*c^3*d)/e^3))/e + (c^3*d^2)/e^4 - (3*c*(a*c + b^2))/e^2))/e - (d^2*((3*b*c^2)/e^2 - (2*
c^3*d)/e^3))/e^2))/e + (3*a*(a*c + b^2))/e^2) - (a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d
^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3)/(e*(d*e^6 + e^7*
x)) + (c^3*x^5)/(5*e^2) - (log(d + e*x)*(6*c^3*d^5 - 3*a^2*b*e^5 - 3*b^3*d^2*e^3 + 12*a*c^2*d^3*e^2 + 12*b^2*c
*d^3*e^2 + 6*a*b^2*d*e^4 + 6*a^2*c*d*e^4 - 15*b*c^2*d^4*e - 18*a*b*c*d^2*e^3))/e^7

________________________________________________________________________________________

sympy [A]  time = 2.18, size = 411, normalized size = 1.61 \begin {gather*} \frac {c^{3} x^{5}}{5 e^{2}} + x^{4} \left (\frac {3 b c^{2}}{4 e^{2}} - \frac {c^{3} d}{2 e^{3}}\right ) + x^{3} \left (\frac {a c^{2}}{e^{2}} + \frac {b^{2} c}{e^{2}} - \frac {2 b c^{2} d}{e^{3}} + \frac {c^{3} d^{2}}{e^{4}}\right ) + x^{2} \left (\frac {3 a b c}{e^{2}} - \frac {3 a c^{2} d}{e^{3}} + \frac {b^{3}}{2 e^{2}} - \frac {3 b^{2} c d}{e^{3}} + \frac {9 b c^{2} d^{2}}{2 e^{4}} - \frac {2 c^{3} d^{3}}{e^{5}}\right ) + x \left (\frac {3 a^{2} c}{e^{2}} + \frac {3 a b^{2}}{e^{2}} - \frac {12 a b c d}{e^{3}} + \frac {9 a c^{2} d^{2}}{e^{4}} - \frac {2 b^{3} d}{e^{3}} + \frac {9 b^{2} c d^{2}}{e^{4}} - \frac {12 b c^{2} d^{3}}{e^{5}} + \frac {5 c^{3} d^{4}}{e^{6}}\right ) + \frac {- 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}}{d e^{7} + e^{8} x} + \frac {3 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2} \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)**2,x)

[Out]

c**3*x**5/(5*e**2) + x**4*(3*b*c**2/(4*e**2) - c**3*d/(2*e**3)) + x**3*(a*c**2/e**2 + b**2*c/e**2 - 2*b*c**2*d
/e**3 + c**3*d**2/e**4) + x**2*(3*a*b*c/e**2 - 3*a*c**2*d/e**3 + b**3/(2*e**2) - 3*b**2*c*d/e**3 + 9*b*c**2*d*
*2/(2*e**4) - 2*c**3*d**3/e**5) + x*(3*a**2*c/e**2 + 3*a*b**2/e**2 - 12*a*b*c*d/e**3 + 9*a*c**2*d**2/e**4 - 2*
b**3*d/e**3 + 9*b**2*c*d**2/e**4 - 12*b*c**2*d**3/e**5 + 5*c**3*d**4/e**6) + (-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)/(d*e**7 + e**8*x) + 3*(b*e - 2*c*d)*(a*e**2 - b*d*e + c*d**2)**2*log(d
+ e*x)/e**7

________________________________________________________________________________________