3.19.9 \(\int \frac {(a+b x+c x^2)^3}{(d+e x)^5} \, dx\)

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

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Rubi [A]  time = 0.27, 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 = {698} \begin {gather*} \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 )}{e^7 (d+e x)}-\frac {3 \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 )}{2 e^7 (d+e x)^2}+\frac {3 c \log (d+e x) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 (d+e x)^3}-\frac {\left (a e^2-b d e+c d^2\right )^3}{4 e^7 (d+e x)^4}-\frac {c^2 x (5 c d-3 b e)}{e^6}+\frac {c^3 x^2}{2 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^3/(d + e*x)^5,x]

[Out]

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

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Mathematica [A]  time = 0.17, size = 402, normalized size = 1.60 \begin {gather*} \frac {-c e^2 \left (a^2 e^2 \left (d^2+4 d e x+6 e^2 x^2\right )+6 a b e \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )+b^2 (-d) \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )\right )-e^3 \left (a^3 e^3+a^2 b e^2 (d+4 e x)+a b^2 e \left (d^2+4 d e x+6 e^2 x^2\right )+b^3 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )+12 c (d+e x)^4 \log (d+e x) \left (c e (a e-5 b d)+b^2 e^2+5 c^2 d^2\right )+c^2 e \left (a d e \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )-b \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 )\right )+c^3 \left (57 d^6+168 d^5 e x+132 d^4 e^2 x^2-32 d^3 e^3 x^3-68 d^2 e^4 x^4-12 d e^5 x^5+2 e^6 x^6\right )}{4 e^7 (d+e x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^3/(d + e*x)^5,x]

[Out]

(c^3*(57*d^6 + 168*d^5*e*x + 132*d^4*e^2*x^2 - 32*d^3*e^3*x^3 - 68*d^2*e^4*x^4 - 12*d*e^5*x^5 + 2*e^6*x^6) - e
^3*(a^3*e^3 + a^2*b*e^2*(d + 4*e*x) + a*b^2*e*(d^2 + 4*d*e*x + 6*e^2*x^2) + b^3*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2
+ 4*e^3*x^3)) - c*e^2*(a^2*e^2*(d^2 + 4*d*e*x + 6*e^2*x^2) + 6*a*b*e*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x
^3) - b^2*d*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3)) + c^2*e*(a*d*e*(25*d^3 + 88*d^2*e*x + 108*d*e^
2*x^2 + 48*e^3*x^3) - b*(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))
+ 12*c*(5*c^2*d^2 + b^2*e^2 + c*e*(-5*b*d + a*e))*(d + e*x)^4*Log[d + e*x])/(4*e^7*(d + e*x)^4)

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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 )^3}{(d+e x)^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x + c*x^2)^3/(d + e*x)^5,x]

[Out]

IntegrateAlgebraic[(a + b*x + c*x^2)^3/(d + e*x)^5, x]

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fricas [B]  time = 0.40, size = 647, normalized size = 2.58 \begin {gather*} \frac {2 \, c^{3} e^{6} x^{6} + 57 \, c^{3} d^{6} - 77 \, b c^{2} d^{5} e - a^{2} b d e^{5} - a^{3} e^{6} + 25 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 12 \, {\left (c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} - 4 \, {\left (17 \, c^{3} d^{2} e^{4} - 12 \, b c^{2} d e^{5}\right )} x^{4} - 4 \, {\left (8 \, c^{3} d^{3} e^{3} + 12 \, b c^{2} d^{2} e^{4} - 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} + {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 6 \, {\left (22 \, c^{3} d^{4} e^{2} - 42 \, b c^{2} d^{3} e^{3} + 18 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - {\left (b^{3} + 6 \, a b c\right )} d e^{5} - {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 4 \, {\left (42 \, c^{3} d^{5} e - 62 \, b c^{2} d^{4} e^{2} - a^{2} b e^{6} + 22 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} - {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x + 12 \, {\left (5 \, c^{3} d^{6} - 5 \, b c^{2} d^{5} e + {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + {\left (5 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} + {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 4 \, {\left (5 \, c^{3} d^{3} e^{3} - 5 \, b c^{2} d^{2} e^{4} + {\left (b^{2} c + a c^{2}\right )} d e^{5}\right )} x^{3} + 6 \, {\left (5 \, c^{3} d^{4} e^{2} - 5 \, b c^{2} d^{3} e^{3} + {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4}\right )} x^{2} + 4 \, {\left (5 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} + {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{4 \, {\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} 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)^5,x, algorithm="fricas")

[Out]

1/4*(2*c^3*e^6*x^6 + 57*c^3*d^6 - 77*b*c^2*d^5*e - a^2*b*d*e^5 - a^3*e^6 + 25*(b^2*c + a*c^2)*d^4*e^2 - (b^3 +
6*a*b*c)*d^3*e^3 - (a*b^2 + a^2*c)*d^2*e^4 - 12*(c^3*d*e^5 - b*c^2*e^6)*x^5 - 4*(17*c^3*d^2*e^4 - 12*b*c^2*d*
e^5)*x^4 - 4*(8*c^3*d^3*e^3 + 12*b*c^2*d^2*e^4 - 12*(b^2*c + a*c^2)*d*e^5 + (b^3 + 6*a*b*c)*e^6)*x^3 + 6*(22*c
^3*d^4*e^2 - 42*b*c^2*d^3*e^3 + 18*(b^2*c + a*c^2)*d^2*e^4 - (b^3 + 6*a*b*c)*d*e^5 - (a*b^2 + a^2*c)*e^6)*x^2
+ 4*(42*c^3*d^5*e - 62*b*c^2*d^4*e^2 - a^2*b*e^6 + 22*(b^2*c + a*c^2)*d^3*e^3 - (b^3 + 6*a*b*c)*d^2*e^4 - (a*b
^2 + a^2*c)*d*e^5)*x + 12*(5*c^3*d^6 - 5*b*c^2*d^5*e + (b^2*c + a*c^2)*d^4*e^2 + (5*c^3*d^2*e^4 - 5*b*c^2*d*e^
5 + (b^2*c + a*c^2)*e^6)*x^4 + 4*(5*c^3*d^3*e^3 - 5*b*c^2*d^2*e^4 + (b^2*c + a*c^2)*d*e^5)*x^3 + 6*(5*c^3*d^4*
e^2 - 5*b*c^2*d^3*e^3 + (b^2*c + a*c^2)*d^2*e^4)*x^2 + 4*(5*c^3*d^5*e - 5*b*c^2*d^4*e^2 + (b^2*c + a*c^2)*d^3*
e^3)*x)*log(e*x + d))/(e^11*x^4 + 4*d*e^10*x^3 + 6*d^2*e^9*x^2 + 4*d^3*e^8*x + d^4*e^7)

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giac [B]  time = 0.19, size = 687, normalized size = 2.74 \begin {gather*} \frac {1}{2} \, {\left (c^{3} - \frac {6 \, {\left (2 \, c^{3} d e - b c^{2} e^{2}\right )} e^{\left (-1\right )}}{x e + d}\right )} {\left (x e + d\right )}^{2} e^{\left (-7\right )} - 3 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2} + a c^{2} e^{2}\right )} e^{\left (-7\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + \frac {1}{4} \, {\left (\frac {80 \, c^{3} d^{3} e^{29}}{x e + d} - \frac {30 \, c^{3} d^{4} e^{29}}{{\left (x e + d\right )}^{2}} + \frac {8 \, c^{3} d^{5} e^{29}}{{\left (x e + d\right )}^{3}} - \frac {c^{3} d^{6} e^{29}}{{\left (x e + d\right )}^{4}} - \frac {120 \, b c^{2} d^{2} e^{30}}{x e + d} + \frac {60 \, b c^{2} d^{3} e^{30}}{{\left (x e + d\right )}^{2}} - \frac {20 \, b c^{2} d^{4} e^{30}}{{\left (x e + d\right )}^{3}} + \frac {3 \, b c^{2} d^{5} e^{30}}{{\left (x e + d\right )}^{4}} + \frac {48 \, b^{2} c d e^{31}}{x e + d} + \frac {48 \, a c^{2} d e^{31}}{x e + d} - \frac {36 \, b^{2} c d^{2} e^{31}}{{\left (x e + d\right )}^{2}} - \frac {36 \, a c^{2} d^{2} e^{31}}{{\left (x e + d\right )}^{2}} + \frac {16 \, b^{2} c d^{3} e^{31}}{{\left (x e + d\right )}^{3}} + \frac {16 \, a c^{2} d^{3} e^{31}}{{\left (x e + d\right )}^{3}} - \frac {3 \, b^{2} c d^{4} e^{31}}{{\left (x e + d\right )}^{4}} - \frac {3 \, a c^{2} d^{4} e^{31}}{{\left (x e + d\right )}^{4}} - \frac {4 \, b^{3} e^{32}}{x e + d} - \frac {24 \, a b c e^{32}}{x e + d} + \frac {6 \, b^{3} d e^{32}}{{\left (x e + d\right )}^{2}} + \frac {36 \, a b c d e^{32}}{{\left (x e + d\right )}^{2}} - \frac {4 \, b^{3} d^{2} e^{32}}{{\left (x e + d\right )}^{3}} - \frac {24 \, a b c d^{2} e^{32}}{{\left (x e + d\right )}^{3}} + \frac {b^{3} d^{3} e^{32}}{{\left (x e + d\right )}^{4}} + \frac {6 \, a b c d^{3} e^{32}}{{\left (x e + d\right )}^{4}} - \frac {6 \, a b^{2} e^{33}}{{\left (x e + d\right )}^{2}} - \frac {6 \, a^{2} c e^{33}}{{\left (x e + d\right )}^{2}} + \frac {8 \, a b^{2} d e^{33}}{{\left (x e + d\right )}^{3}} + \frac {8 \, a^{2} c d e^{33}}{{\left (x e + d\right )}^{3}} - \frac {3 \, a b^{2} d^{2} e^{33}}{{\left (x e + d\right )}^{4}} - \frac {3 \, a^{2} c d^{2} e^{33}}{{\left (x e + d\right )}^{4}} - \frac {4 \, a^{2} b e^{34}}{{\left (x e + d\right )}^{3}} + \frac {3 \, a^{2} b d e^{34}}{{\left (x e + d\right )}^{4}} - \frac {a^{3} e^{35}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-36\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)^5,x, algorithm="giac")

[Out]

1/2*(c^3 - 6*(2*c^3*d*e - b*c^2*e^2)*e^(-1)/(x*e + d))*(x*e + d)^2*e^(-7) - 3*(5*c^3*d^2 - 5*b*c^2*d*e + b^2*c
*e^2 + a*c^2*e^2)*e^(-7)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) + 1/4*(80*c^3*d^3*e^29/(x*e + d) - 30*c^3*d^4*e^
29/(x*e + d)^2 + 8*c^3*d^5*e^29/(x*e + d)^3 - c^3*d^6*e^29/(x*e + d)^4 - 120*b*c^2*d^2*e^30/(x*e + d) + 60*b*c
^2*d^3*e^30/(x*e + d)^2 - 20*b*c^2*d^4*e^30/(x*e + d)^3 + 3*b*c^2*d^5*e^30/(x*e + d)^4 + 48*b^2*c*d*e^31/(x*e
+ d) + 48*a*c^2*d*e^31/(x*e + d) - 36*b^2*c*d^2*e^31/(x*e + d)^2 - 36*a*c^2*d^2*e^31/(x*e + d)^2 + 16*b^2*c*d^
3*e^31/(x*e + d)^3 + 16*a*c^2*d^3*e^31/(x*e + d)^3 - 3*b^2*c*d^4*e^31/(x*e + d)^4 - 3*a*c^2*d^4*e^31/(x*e + d)
^4 - 4*b^3*e^32/(x*e + d) - 24*a*b*c*e^32/(x*e + d) + 6*b^3*d*e^32/(x*e + d)^2 + 36*a*b*c*d*e^32/(x*e + d)^2 -
4*b^3*d^2*e^32/(x*e + d)^3 - 24*a*b*c*d^2*e^32/(x*e + d)^3 + b^3*d^3*e^32/(x*e + d)^4 + 6*a*b*c*d^3*e^32/(x*e
+ d)^4 - 6*a*b^2*e^33/(x*e + d)^2 - 6*a^2*c*e^33/(x*e + d)^2 + 8*a*b^2*d*e^33/(x*e + d)^3 + 8*a^2*c*d*e^33/(x
*e + d)^3 - 3*a*b^2*d^2*e^33/(x*e + d)^4 - 3*a^2*c*d^2*e^33/(x*e + d)^4 - 4*a^2*b*e^34/(x*e + d)^3 + 3*a^2*b*d
*e^34/(x*e + d)^4 - a^3*e^35/(x*e + d)^4)*e^(-36)

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maple [B]  time = 0.05, size = 678, normalized size = 2.70 \begin {gather*} -\frac {a^{3}}{4 \left (e x +d \right )^{4} e}+\frac {3 a^{2} b d}{4 \left (e x +d \right )^{4} e^{2}}-\frac {3 a^{2} c \,d^{2}}{4 \left (e x +d \right )^{4} e^{3}}-\frac {3 a \,b^{2} d^{2}}{4 \left (e x +d \right )^{4} e^{3}}+\frac {3 a b c \,d^{3}}{2 \left (e x +d \right )^{4} e^{4}}-\frac {3 a \,c^{2} d^{4}}{4 \left (e x +d \right )^{4} e^{5}}+\frac {b^{3} d^{3}}{4 \left (e x +d \right )^{4} e^{4}}-\frac {3 b^{2} c \,d^{4}}{4 \left (e x +d \right )^{4} e^{5}}+\frac {3 b \,c^{2} d^{5}}{4 \left (e x +d \right )^{4} e^{6}}-\frac {c^{3} d^{6}}{4 \left (e x +d \right )^{4} e^{7}}-\frac {a^{2} b}{\left (e x +d \right )^{3} e^{2}}+\frac {2 a^{2} c d}{\left (e x +d \right )^{3} e^{3}}+\frac {2 a \,b^{2} d}{\left (e x +d \right )^{3} e^{3}}-\frac {6 a b c \,d^{2}}{\left (e x +d \right )^{3} e^{4}}+\frac {4 a \,c^{2} d^{3}}{\left (e x +d \right )^{3} e^{5}}-\frac {b^{3} d^{2}}{\left (e x +d \right )^{3} e^{4}}+\frac {4 b^{2} c \,d^{3}}{\left (e x +d \right )^{3} e^{5}}-\frac {5 b \,c^{2} d^{4}}{\left (e x +d \right )^{3} e^{6}}+\frac {2 c^{3} d^{5}}{\left (e x +d \right )^{3} e^{7}}-\frac {3 a^{2} c}{2 \left (e x +d \right )^{2} e^{3}}-\frac {3 a \,b^{2}}{2 \left (e x +d \right )^{2} e^{3}}+\frac {9 a b c d}{\left (e x +d \right )^{2} e^{4}}-\frac {9 a \,c^{2} d^{2}}{\left (e x +d \right )^{2} e^{5}}+\frac {3 b^{3} d}{2 \left (e x +d \right )^{2} e^{4}}-\frac {9 b^{2} c \,d^{2}}{\left (e x +d \right )^{2} e^{5}}+\frac {15 b \,c^{2} d^{3}}{\left (e x +d \right )^{2} e^{6}}-\frac {15 c^{3} d^{4}}{2 \left (e x +d \right )^{2} e^{7}}+\frac {c^{3} x^{2}}{2 e^{5}}-\frac {6 a b c}{\left (e x +d \right ) e^{4}}+\frac {12 a \,c^{2} d}{\left (e x +d \right ) e^{5}}+\frac {3 a \,c^{2} \ln \left (e x +d \right )}{e^{5}}-\frac {b^{3}}{\left (e x +d \right ) e^{4}}+\frac {12 b^{2} c d}{\left (e x +d \right ) e^{5}}+\frac {3 b^{2} c \ln \left (e x +d \right )}{e^{5}}-\frac {30 b \,c^{2} d^{2}}{\left (e x +d \right ) e^{6}}-\frac {15 b \,c^{2} d \ln \left (e x +d \right )}{e^{6}}+\frac {3 b \,c^{2} x}{e^{5}}+\frac {20 c^{3} d^{3}}{\left (e x +d \right ) e^{7}}+\frac {15 c^{3} d^{2} \ln \left (e x +d \right )}{e^{7}}-\frac {5 c^{3} d 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)^5,x)

[Out]

-3/4/e^5/(e*x+d)^4*d^4*b^2*c+3/4/e^6/(e*x+d)^4*b*c^2*d^5+12/e^5/(e*x+d)*b^2*c*d-30/e^6/(e*x+d)*b*c^2*d^2-6/e^4
/(e*x+d)^3*a*b*c*d^2+3/2/e^4/(e*x+d)^4*d^3*a*b*c+9/e^4/(e*x+d)^2*a*b*c*d-6/e^4/(e*x+d)*a*b*c+12/e^5/(e*x+d)*c^
2*a*d-15*c^2/e^6*ln(e*x+d)*b*d+4/e^5/(e*x+d)^3*b^2*c*d^3+3*c/e^5*ln(e*x+d)*b^2+3*c^2/e^5*ln(e*x+d)*a-3/2/e^3/(
e*x+d)^2*a*b^2+3/2/e^4/(e*x+d)^2*b^3*d-15/2/e^7/(e*x+d)^2*c^3*d^4+3*c^2/e^5*x*b+20/e^7/(e*x+d)*c^3*d^3+15*c^3/
e^7*ln(e*x+d)*d^2+1/4/e^4/(e*x+d)^4*d^3*b^3-1/4/e^7/(e*x+d)^4*c^3*d^6-1/e^2/(e*x+d)^3*a^2*b-1/e^4/(e*x+d)^3*b^
3*d^2+2/e^7/(e*x+d)^3*c^3*d^5-3/2/e^3/(e*x+d)^2*a^2*c-5/e^6/(e*x+d)^3*b*c^2*d^4+3/4/e^2/(e*x+d)^4*d*a^2*b-3/4/
e^3/(e*x+d)^4*a^2*c*d^2-1/4/e/(e*x+d)^4*a^3-1/e^4/(e*x+d)*b^3+1/2*c^3/e^5*x^2+2/e^3/(e*x+d)^3*a^2*c*d+2/e^3/(e
*x+d)^3*a*b^2*d+4/e^5/(e*x+d)^3*a*c^2*d^3-3/4/e^3/(e*x+d)^4*d^2*a*b^2-3/4/e^5/(e*x+d)^4*a*c^2*d^4-9/e^5/(e*x+d
)^2*a*c^2*d^2-9/e^5/(e*x+d)^2*b^2*c*d^2+15/e^6/(e*x+d)^2*b*c^2*d^3-5*c^3*d/e^6*x

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maxima [A]  time = 1.20, size = 441, normalized size = 1.76 \begin {gather*} \frac {57 \, c^{3} d^{6} - 77 \, b c^{2} d^{5} e - a^{2} b d e^{5} - a^{3} e^{6} + 25 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 4 \, {\left (20 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} - {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 6 \, {\left (35 \, c^{3} d^{4} e^{2} - 50 \, b c^{2} d^{3} e^{3} + 18 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - {\left (b^{3} + 6 \, a b c\right )} d e^{5} - {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 4 \, {\left (47 \, c^{3} d^{5} e - 65 \, b c^{2} d^{4} e^{2} - a^{2} b e^{6} + 22 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} - {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{4 \, {\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} + \frac {c^{3} e x^{2} - 2 \, {\left (5 \, c^{3} d - 3 \, b c^{2} e\right )} x}{2 \, e^{6}} + \frac {3 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + {\left (b^{2} c + a c^{2}\right )} e^{2}\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)^5,x, algorithm="maxima")

[Out]

1/4*(57*c^3*d^6 - 77*b*c^2*d^5*e - a^2*b*d*e^5 - a^3*e^6 + 25*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^
3 - (a*b^2 + a^2*c)*d^2*e^4 + 4*(20*c^3*d^3*e^3 - 30*b*c^2*d^2*e^4 + 12*(b^2*c + a*c^2)*d*e^5 - (b^3 + 6*a*b*c
)*e^6)*x^3 + 6*(35*c^3*d^4*e^2 - 50*b*c^2*d^3*e^3 + 18*(b^2*c + a*c^2)*d^2*e^4 - (b^3 + 6*a*b*c)*d*e^5 - (a*b^
2 + a^2*c)*e^6)*x^2 + 4*(47*c^3*d^5*e - 65*b*c^2*d^4*e^2 - a^2*b*e^6 + 22*(b^2*c + a*c^2)*d^3*e^3 - (b^3 + 6*a
*b*c)*d^2*e^4 - (a*b^2 + a^2*c)*d*e^5)*x)/(e^11*x^4 + 4*d*e^10*x^3 + 6*d^2*e^9*x^2 + 4*d^3*e^8*x + d^4*e^7) +
1/2*(c^3*e*x^2 - 2*(5*c^3*d - 3*b*c^2*e)*x)/e^6 + 3*(5*c^3*d^2 - 5*b*c^2*d*e + (b^2*c + a*c^2)*e^2)*log(e*x +
d)/e^7

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mupad [B]  time = 0.14, size = 475, normalized size = 1.89 \begin {gather*} x\,\left (\frac {3\,b\,c^2}{e^5}-\frac {5\,c^3\,d}{e^6}\right )-\frac {x\,\left (a^2\,b\,e^5+a^2\,c\,d\,e^4+a\,b^2\,d\,e^4+6\,a\,b\,c\,d^2\,e^3-22\,a\,c^2\,d^3\,e^2+b^3\,d^2\,e^3-22\,b^2\,c\,d^3\,e^2+65\,b\,c^2\,d^4\,e-47\,c^3\,d^5\right )+\frac {a^3\,e^6+a^2\,b\,d\,e^5+a^2\,c\,d^2\,e^4+a\,b^2\,d^2\,e^4+6\,a\,b\,c\,d^3\,e^3-25\,a\,c^2\,d^4\,e^2+b^3\,d^3\,e^3-25\,b^2\,c\,d^4\,e^2+77\,b\,c^2\,d^5\,e-57\,c^3\,d^6}{4\,e}+x^2\,\left (\frac {3\,a^2\,c\,e^5}{2}+\frac {3\,a\,b^2\,e^5}{2}+9\,a\,b\,c\,d\,e^4-27\,a\,c^2\,d^2\,e^3+\frac {3\,b^3\,d\,e^4}{2}-27\,b^2\,c\,d^2\,e^3+75\,b\,c^2\,d^3\,e^2-\frac {105\,c^3\,d^4\,e}{2}\right )+x^3\,\left (b^3\,e^5-12\,b^2\,c\,d\,e^4+30\,b\,c^2\,d^2\,e^3+6\,a\,b\,c\,e^5-20\,c^3\,d^3\,e^2-12\,a\,c^2\,d\,e^4\right )}{d^4\,e^6+4\,d^3\,e^7\,x+6\,d^2\,e^8\,x^2+4\,d\,e^9\,x^3+e^{10}\,x^4}+\frac {\ln \left (d+e\,x\right )\,\left (3\,b^2\,c\,e^2-15\,b\,c^2\,d\,e+15\,c^3\,d^2+3\,a\,c^2\,e^2\right )}{e^7}+\frac {c^3\,x^2}{2\,e^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^3/(d + e*x)^5,x)

[Out]

x*((3*b*c^2)/e^5 - (5*c^3*d)/e^6) - (x*(a^2*b*e^5 - 47*c^3*d^5 + b^3*d^2*e^3 - 22*a*c^2*d^3*e^2 - 22*b^2*c*d^3
*e^2 + a*b^2*d*e^4 + a^2*c*d*e^4 + 65*b*c^2*d^4*e + 6*a*b*c*d^2*e^3) + (a^3*e^6 - 57*c^3*d^6 + b^3*d^3*e^3 + a
*b^2*d^2*e^4 - 25*a*c^2*d^4*e^2 + a^2*c*d^2*e^4 - 25*b^2*c*d^4*e^2 + a^2*b*d*e^5 + 77*b*c^2*d^5*e + 6*a*b*c*d^
3*e^3)/(4*e) + x^2*((3*a*b^2*e^5)/2 + (3*a^2*c*e^5)/2 + (3*b^3*d*e^4)/2 - (105*c^3*d^4*e)/2 - 27*a*c^2*d^2*e^3
+ 75*b*c^2*d^3*e^2 - 27*b^2*c*d^2*e^3 + 9*a*b*c*d*e^4) + x^3*(b^3*e^5 - 20*c^3*d^3*e^2 + 30*b*c^2*d^2*e^3 + 6
*a*b*c*e^5 - 12*a*c^2*d*e^4 - 12*b^2*c*d*e^4))/(d^4*e^6 + e^10*x^4 + 4*d^3*e^7*x + 4*d*e^9*x^3 + 6*d^2*e^8*x^2
) + (log(d + e*x)*(15*c^3*d^2 + 3*a*c^2*e^2 + 3*b^2*c*e^2 - 15*b*c^2*d*e))/e^7 + (c^3*x^2)/(2*e^5)

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sympy [B]  time = 108.83, size = 520, normalized size = 2.07 \begin {gather*} \frac {c^{3} x^{2}}{2 e^{5}} + \frac {3 c \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) \log {\left (d + e x \right )}}{e^{7}} + x \left (\frac {3 b c^{2}}{e^{5}} - \frac {5 c^{3} d}{e^{6}}\right ) + \frac {- a^{3} e^{6} - a^{2} b d e^{5} - a^{2} c d^{2} e^{4} - a b^{2} d^{2} e^{4} - 6 a b c d^{3} e^{3} + 25 a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} + 25 b^{2} c d^{4} e^{2} - 77 b c^{2} d^{5} e + 57 c^{3} d^{6} + x^{3} \left (- 24 a b c e^{6} + 48 a c^{2} d e^{5} - 4 b^{3} e^{6} + 48 b^{2} c d e^{5} - 120 b c^{2} d^{2} e^{4} + 80 c^{3} d^{3} e^{3}\right ) + x^{2} \left (- 6 a^{2} c e^{6} - 6 a b^{2} e^{6} - 36 a b c d e^{5} + 108 a c^{2} d^{2} e^{4} - 6 b^{3} d e^{5} + 108 b^{2} c d^{2} e^{4} - 300 b c^{2} d^{3} e^{3} + 210 c^{3} d^{4} e^{2}\right ) + x \left (- 4 a^{2} b e^{6} - 4 a^{2} c d e^{5} - 4 a b^{2} d e^{5} - 24 a b c d^{2} e^{4} + 88 a c^{2} d^{3} e^{3} - 4 b^{3} d^{2} e^{4} + 88 b^{2} c d^{3} e^{3} - 260 b c^{2} d^{4} e^{2} + 188 c^{3} d^{5} e\right )}{4 d^{4} e^{7} + 16 d^{3} e^{8} x + 24 d^{2} e^{9} x^{2} + 16 d e^{10} x^{3} + 4 e^{11} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**3/(e*x+d)**5,x)

[Out]

c**3*x**2/(2*e**5) + 3*c*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)*log(d + e*x)/e**7 + x*(3*b*c**2/e**5
- 5*c**3*d/e**6) + (-a**3*e**6 - a**2*b*d*e**5 - a**2*c*d**2*e**4 - a*b**2*d**2*e**4 - 6*a*b*c*d**3*e**3 + 25
*a*c**2*d**4*e**2 - b**3*d**3*e**3 + 25*b**2*c*d**4*e**2 - 77*b*c**2*d**5*e + 57*c**3*d**6 + x**3*(-24*a*b*c*e
**6 + 48*a*c**2*d*e**5 - 4*b**3*e**6 + 48*b**2*c*d*e**5 - 120*b*c**2*d**2*e**4 + 80*c**3*d**3*e**3) + x**2*(-6
*a**2*c*e**6 - 6*a*b**2*e**6 - 36*a*b*c*d*e**5 + 108*a*c**2*d**2*e**4 - 6*b**3*d*e**5 + 108*b**2*c*d**2*e**4 -
300*b*c**2*d**3*e**3 + 210*c**3*d**4*e**2) + x*(-4*a**2*b*e**6 - 4*a**2*c*d*e**5 - 4*a*b**2*d*e**5 - 24*a*b*c
*d**2*e**4 + 88*a*c**2*d**3*e**3 - 4*b**3*d**2*e**4 + 88*b**2*c*d**3*e**3 - 260*b*c**2*d**4*e**2 + 188*c**3*d*
*5*e))/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4)

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