3.2127 \(\int \frac {(a+b x+c x^2)^2}{(d+e x)^5} \, dx\)

Optimal. Leaf size=150 \[ -\frac {-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{2 e^5 (d+e x)^2}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5 (d+e x)^3}-\frac {\left (a e^2-b d e+c d^2\right )^2}{4 e^5 (d+e x)^4}+\frac {2 c (2 c d-b e)}{e^5 (d+e x)}+\frac {c^2 \log (d+e x)}{e^5} \]

[Out]

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

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Rubi [A]  time = 0.12, antiderivative size = 150, 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} \[ -\frac {-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{2 e^5 (d+e x)^2}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5 (d+e x)^3}-\frac {\left (a e^2-b d e+c d^2\right )^2}{4 e^5 (d+e x)^4}+\frac {2 c (2 c d-b e)}{e^5 (d+e x)}+\frac {c^2 \log (d+e x)}{e^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.07, size = 170, normalized size = 1.13 \[ \frac {-e^2 \left (3 a^2 e^2+2 a b e (d+4 e x)+b^2 \left (d^2+4 d e x+6 e^2 x^2\right )\right )-2 c e \left (a e \left (d^2+4 d e x+6 e^2 x^2\right )+3 b \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )+c^2 d \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )+12 c^2 (d+e x)^4 \log (d+e x)}{12 e^5 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.80, size = 263, normalized size = 1.75 \[ \frac {25 \, c^{2} d^{4} - 6 \, b c d^{3} e - 2 \, a b d e^{3} - 3 \, a^{2} e^{4} - {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 24 \, {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} x^{3} + 6 \, {\left (18 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} - {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 4 \, {\left (22 \, c^{2} d^{3} e - 6 \, b c d^{2} e^{2} - 2 \, a b e^{4} - {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x + 12 \, {\left (c^{2} e^{4} x^{4} + 4 \, c^{2} d e^{3} x^{3} + 6 \, c^{2} d^{2} e^{2} x^{2} + 4 \, c^{2} d^{3} e x + c^{2} d^{4}\right )} \log \left (e x + d\right )}{12 \, {\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(25*c^2*d^4 - 6*b*c*d^3*e - 2*a*b*d*e^3 - 3*a^2*e^4 - (b^2 + 2*a*c)*d^2*e^2 + 24*(2*c^2*d*e^3 - b*c*e^4)*
x^3 + 6*(18*c^2*d^2*e^2 - 6*b*c*d*e^3 - (b^2 + 2*a*c)*e^4)*x^2 + 4*(22*c^2*d^3*e - 6*b*c*d^2*e^2 - 2*a*b*e^4 -
(b^2 + 2*a*c)*d*e^3)*x + 12*(c^2*e^4*x^4 + 4*c^2*d*e^3*x^3 + 6*c^2*d^2*e^2*x^2 + 4*c^2*d^3*e*x + c^2*d^4)*log
(e*x + d))/(e^9*x^4 + 4*d*e^8*x^3 + 6*d^2*e^7*x^2 + 4*d^3*e^6*x + d^4*e^5)

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giac [B]  time = 0.18, size = 304, normalized size = 2.03 \[ -c^{2} e^{\left (-5\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + \frac {1}{12} \, {\left (\frac {48 \, c^{2} d e^{15}}{x e + d} - \frac {36 \, c^{2} d^{2} e^{15}}{{\left (x e + d\right )}^{2}} + \frac {16 \, c^{2} d^{3} e^{15}}{{\left (x e + d\right )}^{3}} - \frac {3 \, c^{2} d^{4} e^{15}}{{\left (x e + d\right )}^{4}} - \frac {24 \, b c e^{16}}{x e + d} + \frac {36 \, b c d e^{16}}{{\left (x e + d\right )}^{2}} - \frac {24 \, b c d^{2} e^{16}}{{\left (x e + d\right )}^{3}} + \frac {6 \, b c d^{3} e^{16}}{{\left (x e + d\right )}^{4}} - \frac {6 \, b^{2} e^{17}}{{\left (x e + d\right )}^{2}} - \frac {12 \, a c e^{17}}{{\left (x e + d\right )}^{2}} + \frac {8 \, b^{2} d e^{17}}{{\left (x e + d\right )}^{3}} + \frac {16 \, a c d e^{17}}{{\left (x e + d\right )}^{3}} - \frac {3 \, b^{2} d^{2} e^{17}}{{\left (x e + d\right )}^{4}} - \frac {6 \, a c d^{2} e^{17}}{{\left (x e + d\right )}^{4}} - \frac {8 \, a b e^{18}}{{\left (x e + d\right )}^{3}} + \frac {6 \, a b d e^{18}}{{\left (x e + d\right )}^{4}} - \frac {3 \, a^{2} e^{19}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-20\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c^2*e^(-5)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) + 1/12*(48*c^2*d*e^15/(x*e + d) - 36*c^2*d^2*e^15/(x*e + d)^2
+ 16*c^2*d^3*e^15/(x*e + d)^3 - 3*c^2*d^4*e^15/(x*e + d)^4 - 24*b*c*e^16/(x*e + d) + 36*b*c*d*e^16/(x*e + d)^
2 - 24*b*c*d^2*e^16/(x*e + d)^3 + 6*b*c*d^3*e^16/(x*e + d)^4 - 6*b^2*e^17/(x*e + d)^2 - 12*a*c*e^17/(x*e + d)^
2 + 8*b^2*d*e^17/(x*e + d)^3 + 16*a*c*d*e^17/(x*e + d)^3 - 3*b^2*d^2*e^17/(x*e + d)^4 - 6*a*c*d^2*e^17/(x*e +
d)^4 - 8*a*b*e^18/(x*e + d)^3 + 6*a*b*d*e^18/(x*e + d)^4 - 3*a^2*e^19/(x*e + d)^4)*e^(-20)

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maple [A]  time = 0.08, size = 287, normalized size = 1.91 \[ -\frac {a^{2}}{4 \left (e x +d \right )^{4} e}+\frac {a b d}{2 \left (e x +d \right )^{4} e^{2}}-\frac {a c \,d^{2}}{2 \left (e x +d \right )^{4} e^{3}}-\frac {b^{2} d^{2}}{4 \left (e x +d \right )^{4} e^{3}}+\frac {b c \,d^{3}}{2 \left (e x +d \right )^{4} e^{4}}-\frac {c^{2} d^{4}}{4 \left (e x +d \right )^{4} e^{5}}-\frac {2 a b}{3 \left (e x +d \right )^{3} e^{2}}+\frac {4 a c d}{3 \left (e x +d \right )^{3} e^{3}}+\frac {2 b^{2} d}{3 \left (e x +d \right )^{3} e^{3}}-\frac {2 b c \,d^{2}}{\left (e x +d \right )^{3} e^{4}}+\frac {4 c^{2} d^{3}}{3 \left (e x +d \right )^{3} e^{5}}-\frac {a c}{\left (e x +d \right )^{2} e^{3}}-\frac {b^{2}}{2 \left (e x +d \right )^{2} e^{3}}+\frac {3 b c d}{\left (e x +d \right )^{2} e^{4}}-\frac {3 c^{2} d^{2}}{\left (e x +d \right )^{2} e^{5}}-\frac {2 b c}{\left (e x +d \right ) e^{4}}+\frac {4 c^{2} d}{\left (e x +d \right ) e^{5}}+\frac {c^{2} \ln \left (e x +d \right )}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.06, size = 215, normalized size = 1.43 \[ \frac {25 \, c^{2} d^{4} - 6 \, b c d^{3} e - 2 \, a b d e^{3} - 3 \, a^{2} e^{4} - {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 24 \, {\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} x^{3} + 6 \, {\left (18 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} - {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 4 \, {\left (22 \, c^{2} d^{3} e - 6 \, b c d^{2} e^{2} - 2 \, a b e^{4} - {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{12 \, {\left (e^{9} x^{4} + 4 \, d e^{8} x^{3} + 6 \, d^{2} e^{7} x^{2} + 4 \, d^{3} e^{6} x + d^{4} e^{5}\right )}} + \frac {c^{2} \log \left (e x + d\right )}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(25*c^2*d^4 - 6*b*c*d^3*e - 2*a*b*d*e^3 - 3*a^2*e^4 - (b^2 + 2*a*c)*d^2*e^2 + 24*(2*c^2*d*e^3 - b*c*e^4)*
x^3 + 6*(18*c^2*d^2*e^2 - 6*b*c*d*e^3 - (b^2 + 2*a*c)*e^4)*x^2 + 4*(22*c^2*d^3*e - 6*b*c*d^2*e^2 - 2*a*b*e^4 -
(b^2 + 2*a*c)*d*e^3)*x)/(e^9*x^4 + 4*d*e^8*x^3 + 6*d^2*e^7*x^2 + 4*d^3*e^6*x + d^4*e^5) + c^2*log(e*x + d)/e^
5

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mupad [B]  time = 0.71, size = 186, normalized size = 1.24 \[ \frac {c^2\,\ln \left (d+e\,x\right )}{e^5}-\frac {x^2\,\left (\frac {b^2\,e^4}{2}+3\,b\,c\,d\,e^3-9\,c^2\,d^2\,e^2+a\,c\,e^4\right )+x\,\left (\frac {b^2\,d\,e^3}{3}+2\,b\,c\,d^2\,e^2+\frac {2\,a\,b\,e^4}{3}-\frac {22\,c^2\,d^3\,e}{3}+\frac {2\,a\,c\,d\,e^3}{3}\right )-x^3\,\left (4\,c^2\,d\,e^3-2\,b\,c\,e^4\right )+\frac {a^2\,e^4}{4}-\frac {25\,c^2\,d^4}{12}+\frac {b^2\,d^2\,e^2}{12}+\frac {a\,b\,d\,e^3}{6}+\frac {b\,c\,d^3\,e}{2}+\frac {a\,c\,d^2\,e^2}{6}}{e^5\,{\left (d+e\,x\right )}^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^2*log(d + e*x))/e^5 - (x^2*((b^2*e^4)/2 - 9*c^2*d^2*e^2 + a*c*e^4 + 3*b*c*d*e^3) + x*((b^2*d*e^3)/3 - (22*c
^2*d^3*e)/3 + (2*a*b*e^4)/3 + (2*a*c*d*e^3)/3 + 2*b*c*d^2*e^2) - x^3*(4*c^2*d*e^3 - 2*b*c*e^4) + (a^2*e^4)/4 -
(25*c^2*d^4)/12 + (b^2*d^2*e^2)/12 + (a*b*d*e^3)/6 + (b*c*d^3*e)/2 + (a*c*d^2*e^2)/6)/(e^5*(d + e*x)^4)

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sympy [A]  time = 17.42, size = 238, normalized size = 1.59 \[ \frac {c^{2} \log {\left (d + e x \right )}}{e^{5}} + \frac {- 3 a^{2} e^{4} - 2 a b d e^{3} - 2 a c d^{2} e^{2} - b^{2} d^{2} e^{2} - 6 b c d^{3} e + 25 c^{2} d^{4} + x^{3} \left (- 24 b c e^{4} + 48 c^{2} d e^{3}\right ) + x^{2} \left (- 12 a c e^{4} - 6 b^{2} e^{4} - 36 b c d e^{3} + 108 c^{2} d^{2} e^{2}\right ) + x \left (- 8 a b e^{4} - 8 a c d e^{3} - 4 b^{2} d e^{3} - 24 b c d^{2} e^{2} + 88 c^{2} d^{3} e\right )}{12 d^{4} e^{5} + 48 d^{3} e^{6} x + 72 d^{2} e^{7} x^{2} + 48 d e^{8} x^{3} + 12 e^{9} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**2*log(d + e*x)/e**5 + (-3*a**2*e**4 - 2*a*b*d*e**3 - 2*a*c*d**2*e**2 - b**2*d**2*e**2 - 6*b*c*d**3*e + 25*c
**2*d**4 + x**3*(-24*b*c*e**4 + 48*c**2*d*e**3) + x**2*(-12*a*c*e**4 - 6*b**2*e**4 - 36*b*c*d*e**3 + 108*c**2*
d**2*e**2) + x*(-8*a*b*e**4 - 8*a*c*d*e**3 - 4*b**2*d*e**3 - 24*b*c*d**2*e**2 + 88*c**2*d**3*e))/(12*d**4*e**5
+ 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4)

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