3.22.28 \(\int \frac {(a+b x+c x^2)^2}{(d+e x)^6} \, dx\) [2128]

Optimal. Leaf size=151 \[ -\frac {\left (c d^2-b d e+a e^2\right )^2}{5 e^5 (d+e x)^5}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )}{2 e^5 (d+e x)^4}-\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{3 e^5 (d+e x)^3}+\frac {c (2 c d-b e)}{e^5 (d+e x)^2}-\frac {c^2}{e^5 (d+e x)} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 151, 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 {-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{3 e^5 (d+e x)^3}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{2 e^5 (d+e x)^4}-\frac {\left (a e^2-b d e+c d^2\right )^2}{5 e^5 (d+e x)^5}+\frac {c (2 c d-b e)}{e^5 (d+e x)^2}-\frac {c^2}{e^5 (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.05, size = 160, normalized size = 1.06 \begin {gather*} -\frac {6 c^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 )+e^2 \left (6 a^2 e^2+3 a b e (d+5 e x)+b^2 \left (d^2+5 d e x+10 e^2 x^2\right )\right )+c e \left (2 a e \left (d^2+5 d e x+10 e^2 x^2\right )+3 b \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )}{30 e^5 (d+e x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/30*(6*c^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) + e^2*(6*a^2*e^2 + 3*a*b*e*(d + 5*e
*x) + b^2*(d^2 + 5*d*e*x + 10*e^2*x^2)) + c*e*(2*a*e*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*b*(d^3 + 5*d^2*e*x + 10*
d*e^2*x^2 + 10*e^3*x^3)))/(e^5*(d + e*x)^5)

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Maple [A]
time = 0.78, size = 195, normalized size = 1.29

method	result	size
risch	\(\frac {-\frac {c^{2} x^{4}}{e}-\frac {c \left (b e +2 c d \right ) x^{3}}{e^{2}}-\frac {\left (2 a c \,e^{2}+b^{2} e^{2}+3 b c d e +6 c^{2} d^{2}\right ) x^{2}}{3 e^{3}}-\frac {\left (3 a b \,e^{3}+2 a d \,e^{2} c +b^{2} d \,e^{2}+3 d^{2} e b c +6 c^{2} d^{3}\right ) x}{6 e^{4}}-\frac {6 a^{2} e^{4}+3 d \,e^{3} a b +2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}+3 d^{3} e b c +6 c^{2} d^{4}}{30 e^{5}}}{\left (e x +d \right )^{5}}\)	\(179\)
norman	\(\frac {-\frac {c^{2} x^{4}}{e}-\frac {\left (b c e +2 c^{2} d \right ) x^{3}}{e^{2}}-\frac {\left (2 a c \,e^{2}+b^{2} e^{2}+3 b c d e +6 c^{2} d^{2}\right ) x^{2}}{3 e^{3}}-\frac {\left (3 a b \,e^{3}+2 a d \,e^{2} c +b^{2} d \,e^{2}+3 d^{2} e b c +6 c^{2} d^{3}\right ) x}{6 e^{4}}-\frac {6 a^{2} e^{4}+3 d \,e^{3} a b +2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}+3 d^{3} e b c +6 c^{2} d^{4}}{30 e^{5}}}{\left (e x +d \right )^{5}}\)	\(181\)
gosper	\(-\frac {30 c^{2} x^{4} e^{4}+30 b c \,e^{4} x^{3}+60 c^{2} d \,e^{3} x^{3}+20 a c \,e^{4} x^{2}+10 b^{2} e^{4} x^{2}+30 b c d \,e^{3} x^{2}+60 c^{2} d^{2} x^{2} e^{2}+15 a b \,e^{4} x +10 a c d \,e^{3} x +5 b^{2} d \,e^{3} x +15 b c \,d^{2} e^{2} x +30 c^{2} d^{3} e x +6 a^{2} e^{4}+3 d \,e^{3} a b +2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}+3 d^{3} e b c +6 c^{2} d^{4}}{30 \left (e x +d \right )^{5} e^{5}}\)	\(193\)
default	\(-\frac {2 a c \,e^{2}+b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}}{3 e^{5} \left (e x +d \right )^{3}}-\frac {a^{2} e^{4}-2 d \,e^{3} a b +2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 d^{3} e b c +c^{2} d^{4}}{5 e^{5} \left (e x +d \right )^{5}}-\frac {c^{2}}{e^{5} \left (e x +d \right )}-\frac {c \left (b e -2 c d \right )}{e^{5} \left (e x +d \right )^{2}}-\frac {2 a b \,e^{3}-4 a d \,e^{2} c -2 b^{2} d \,e^{2}+6 d^{2} e b c -4 c^{2} d^{3}}{4 e^{5} \left (e x +d \right )^{4}}\)	\(195\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 210, normalized size = 1.39 \begin {gather*} -\frac {30 \, c^{2} x^{4} e^{4} + 6 \, c^{2} d^{4} + 3 \, b c d^{3} e + 30 \, {\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 3 \, a b d e^{3} + {\left (b^{2} e^{2} + 2 \, a c e^{2}\right )} d^{2} + 10 \, {\left (6 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + b^{2} e^{4} + 2 \, a c e^{4}\right )} x^{2} + 6 \, a^{2} e^{4} + 5 \, {\left (6 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + 3 \, a b e^{4} + {\left (b^{2} e^{3} + 2 \, a c e^{3}\right )} d\right )} x}{30 \, {\left (x^{5} e^{10} + 5 \, d x^{4} e^{9} + 10 \, d^{2} x^{3} e^{8} + 10 \, d^{3} x^{2} e^{7} + 5 \, d^{4} x e^{6} + d^{5} e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 2.28, size = 197, normalized size = 1.30 \begin {gather*} -\frac {6 \, c^{2} d^{4} + {\left (30 \, c^{2} x^{4} + 30 \, b c x^{3} + 15 \, a b x + 10 \, {\left (b^{2} + 2 \, a c\right )} x^{2} + 6 \, a^{2}\right )} e^{4} + {\left (60 \, c^{2} d x^{3} + 30 \, b c d x^{2} + 3 \, a b d + 5 \, {\left (b^{2} + 2 \, a c\right )} d x\right )} e^{3} + {\left (60 \, c^{2} d^{2} x^{2} + 15 \, b c d^{2} x + {\left (b^{2} + 2 \, a c\right )} d^{2}\right )} e^{2} + 3 \, {\left (10 \, c^{2} d^{3} x + b c d^{3}\right )} e}{30 \, {\left (x^{5} e^{10} + 5 \, d x^{4} e^{9} + 10 \, d^{2} x^{3} e^{8} + 10 \, d^{3} x^{2} e^{7} + 5 \, d^{4} x e^{6} + d^{5} e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 207.79, size = 253, normalized size = 1.68 \begin {gather*} \frac {- 6 a^{2} e^{4} - 3 a b d e^{3} - 2 a c d^{2} e^{2} - b^{2} d^{2} e^{2} - 3 b c d^{3} e - 6 c^{2} d^{4} - 30 c^{2} e^{4} x^{4} + x^{3} \left (- 30 b c e^{4} - 60 c^{2} d e^{3}\right ) + x^{2} \left (- 20 a c e^{4} - 10 b^{2} e^{4} - 30 b c d e^{3} - 60 c^{2} d^{2} e^{2}\right ) + x \left (- 15 a b e^{4} - 10 a c d e^{3} - 5 b^{2} d e^{3} - 15 b c d^{2} e^{2} - 30 c^{2} d^{3} e\right )}{30 d^{5} e^{5} + 150 d^{4} e^{6} x + 300 d^{3} e^{7} x^{2} + 300 d^{2} e^{8} x^{3} + 150 d e^{9} x^{4} + 30 e^{10} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**2/(e*x+d)**6,x)

[Out]

(-6*a**2*e**4 - 3*a*b*d*e**3 - 2*a*c*d**2*e**2 - b**2*d**2*e**2 - 3*b*c*d**3*e - 6*c**2*d**4 - 30*c**2*e**4*x*
*4 + x**3*(-30*b*c*e**4 - 60*c**2*d*e**3) + x**2*(-20*a*c*e**4 - 10*b**2*e**4 - 30*b*c*d*e**3 - 60*c**2*d**2*e
**2) + x*(-15*a*b*e**4 - 10*a*c*d*e**3 - 5*b**2*d*e**3 - 15*b*c*d**2*e**2 - 30*c**2*d**3*e))/(30*d**5*e**5 + 1
50*d**4*e**6*x + 300*d**3*e**7*x**2 + 300*d**2*e**8*x**3 + 150*d*e**9*x**4 + 30*e**10*x**5)

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Giac [A]
time = 1.54, size = 179, normalized size = 1.19 \begin {gather*} -\frac {{\left (30 \, c^{2} x^{4} e^{4} + 60 \, c^{2} d x^{3} e^{3} + 60 \, c^{2} d^{2} x^{2} e^{2} + 30 \, c^{2} d^{3} x e + 6 \, c^{2} d^{4} + 30 \, b c x^{3} e^{4} + 30 \, b c d x^{2} e^{3} + 15 \, b c d^{2} x e^{2} + 3 \, b c d^{3} e + 10 \, b^{2} x^{2} e^{4} + 20 \, a c x^{2} e^{4} + 5 \, b^{2} d x e^{3} + 10 \, a c d x e^{3} + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} + 15 \, a b x e^{4} + 3 \, a b d e^{3} + 6 \, a^{2} e^{4}\right )} e^{\left (-5\right )}}{30 \, {\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)^2/(e*x+d)^6,x, algorithm="giac")

[Out]

-1/30*(30*c^2*x^4*e^4 + 60*c^2*d*x^3*e^3 + 60*c^2*d^2*x^2*e^2 + 30*c^2*d^3*x*e + 6*c^2*d^4 + 30*b*c*x^3*e^4 +
30*b*c*d*x^2*e^3 + 15*b*c*d^2*x*e^2 + 3*b*c*d^3*e + 10*b^2*x^2*e^4 + 20*a*c*x^2*e^4 + 5*b^2*d*x*e^3 + 10*a*c*d
*x*e^3 + b^2*d^2*e^2 + 2*a*c*d^2*e^2 + 15*a*b*x*e^4 + 3*a*b*d*e^3 + 6*a^2*e^4)*e^(-5)/(x*e + d)^5

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Mupad [B]
time = 0.09, size = 221, normalized size = 1.46 \begin {gather*} -\frac {\frac {6\,a^2\,e^4+3\,a\,b\,d\,e^3+2\,a\,c\,d^2\,e^2+b^2\,d^2\,e^2+3\,b\,c\,d^3\,e+6\,c^2\,d^4}{30\,e^5}+\frac {x\,\left (b^2\,d\,e^2+3\,b\,c\,d^2\,e+3\,a\,b\,e^3+6\,c^2\,d^3+2\,a\,c\,d\,e^2\right )}{6\,e^4}+\frac {c^2\,x^4}{e}+\frac {x^2\,\left (b^2\,e^2+3\,b\,c\,d\,e+6\,c^2\,d^2+2\,a\,c\,e^2\right )}{3\,e^3}+\frac {c\,x^3\,\left (b\,e+2\,c\,d\right )}{e^2}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((6*a^2*e^4 + 6*c^2*d^4 + b^2*d^2*e^2 + 3*a*b*d*e^3 + 3*b*c*d^3*e + 2*a*c*d^2*e^2)/(30*e^5) + (x*(6*c^2*d^3 +
b^2*d*e^2 + 3*a*b*e^3 + 2*a*c*d*e^2 + 3*b*c*d^2*e))/(6*e^4) + (c^2*x^4)/e + (x^2*(b^2*e^2 + 6*c^2*d^2 + 2*a*c
*e^2 + 3*b*c*d*e))/(3*e^3) + (c*x^3*(b*e + 2*c*d))/e^2)/(d^5 + e^5*x^5 + 5*d*e^4*x^4 + 10*d^3*e^2*x^2 + 10*d^2
*e^3*x^3 + 5*d^4*e*x)

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