3.22.29 \(\int \frac {(a+b x+c x^2)^2}{(d+e x)^7} \, dx\) [2129]

Optimal. Leaf size=156 \[ -\frac {\left (c d^2-b d e+a e^2\right )^2}{6 e^5 (d+e x)^6}+\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )}{5 e^5 (d+e x)^5}-\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{4 e^5 (d+e x)^4}+\frac {2 c (2 c d-b e)}{3 e^5 (d+e x)^3}-\frac {c^2}{2 e^5 (d+e x)^2} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 156, 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}{4 e^5 (d+e x)^4}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{5 e^5 (d+e x)^5}-\frac {\left (a e^2-b d e+c d^2\right )^2}{6 e^5 (d+e x)^6}+\frac {2 c (2 c d-b e)}{3 e^5 (d+e x)^3}-\frac {c^2}{2 e^5 (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^2/(d + e*x)^7,x]

[Out]

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

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

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Mathematica [A]
time = 0.04, size = 159, normalized size = 1.02 \begin {gather*} -\frac {2 c^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+e^2 \left (10 a^2 e^2+4 a b e (d+6 e x)+b^2 \left (d^2+6 d e x+15 e^2 x^2\right )\right )+2 c e \left (a e \left (d^2+6 d e x+15 e^2 x^2\right )+b \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )}{60 e^5 (d+e x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^2/(d + e*x)^7,x]

[Out]

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

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Maple [A]
time = 0.72, size = 195, normalized size = 1.25

method	result	size
risch	\(\frac {-\frac {c^{2} x^{4}}{2 e}-\frac {2 c \left (b e +c d \right ) x^{3}}{3 e^{2}}-\frac {\left (2 a c \,e^{2}+b^{2} e^{2}+2 b c d e +2 c^{2} d^{2}\right ) x^{2}}{4 e^{3}}-\frac {\left (4 a b \,e^{3}+2 a d \,e^{2} c +b^{2} d \,e^{2}+2 d^{2} e b c +2 c^{2} d^{3}\right ) x}{10 e^{4}}-\frac {10 a^{2} e^{4}+4 d \,e^{3} a b +2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}+2 d^{3} e b c +2 c^{2} d^{4}}{60 e^{5}}}{\left (e x +d \right )^{6}}\)	\(178\)
norman	\(\frac {-\frac {c^{2} x^{4}}{2 e}-\frac {2 \left (b c \,e^{2}+c^{2} d e \right ) x^{3}}{3 e^{3}}-\frac {\left (2 a c \,e^{3}+b^{2} e^{3}+2 d \,e^{2} b c +2 d^{2} e \,c^{2}\right ) x^{2}}{4 e^{4}}-\frac {\left (4 e^{4} a b +2 a c d \,e^{3}+b^{2} d \,e^{3}+2 d^{2} e^{2} b c +2 c^{2} d^{3} e \right ) x}{10 e^{5}}-\frac {10 a^{2} e^{5}+4 a b d \,e^{4}+2 a c \,e^{3} d^{2}+b^{2} d^{2} e^{3}+2 b c \,d^{3} e^{2}+2 c^{2} d^{4} e}{60 e^{6}}}{\left (e x +d \right )^{6}}\)	\(192\)
gosper	\(-\frac {30 c^{2} x^{4} e^{4}+40 b c \,e^{4} x^{3}+40 c^{2} d \,e^{3} x^{3}+30 a c \,e^{4} x^{2}+15 b^{2} e^{4} x^{2}+30 b c d \,e^{3} x^{2}+30 c^{2} d^{2} x^{2} e^{2}+24 a b \,e^{4} x +12 a c d \,e^{3} x +6 b^{2} d \,e^{3} x +12 b c \,d^{2} e^{2} x +12 c^{2} d^{3} e x +10 a^{2} e^{4}+4 d \,e^{3} a b +2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}+2 d^{3} e b c +2 c^{2} d^{4}}{60 e^{5} \left (e x +d \right )^{6}}\)	\(193\)
default	\(-\frac {a^{2} e^{4}-2 d \,e^{3} a b +2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 d^{3} e b c +c^{2} d^{4}}{6 e^{5} \left (e x +d \right )^{6}}-\frac {2 c \left (b e -2 c d \right )}{3 e^{5} \left (e x +d \right )^{3}}-\frac {2 a b \,e^{3}-4 a d \,e^{2} c -2 b^{2} d \,e^{2}+6 d^{2} e b c -4 c^{2} d^{3}}{5 e^{5} \left (e x +d \right )^{5}}-\frac {c^{2}}{2 e^{5} \left (e x +d \right )^{2}}-\frac {2 a c \,e^{2}+b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}}{4 e^{5} \left (e x +d \right )^{4}}\)	\(195\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^2/(e*x+d)^7,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, size = 219, normalized size = 1.40 \begin {gather*} -\frac {30 \, c^{2} x^{4} e^{4} + 2 \, c^{2} d^{4} + 2 \, b c d^{3} e + 40 \, {\left (c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 4 \, a b d e^{3} + {\left (b^{2} e^{2} + 2 \, a c e^{2}\right )} d^{2} + 15 \, {\left (2 \, c^{2} d^{2} e^{2} + 2 \, b c d e^{3} + b^{2} e^{4} + 2 \, a c e^{4}\right )} x^{2} + 10 \, a^{2} e^{4} + 6 \, {\left (2 \, c^{2} d^{3} e + 2 \, b c d^{2} e^{2} + 4 \, a b e^{4} + {\left (b^{2} e^{3} + 2 \, a c e^{3}\right )} d\right )} x}{60 \, {\left (x^{6} e^{11} + 6 \, d x^{5} e^{10} + 15 \, d^{2} x^{4} e^{9} + 20 \, d^{3} x^{3} e^{8} + 15 \, d^{4} x^{2} e^{7} + 6 \, d^{5} x e^{6} + d^{6} 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)^7,x, algorithm="maxima")

[Out]

-1/60*(30*c^2*x^4*e^4 + 2*c^2*d^4 + 2*b*c*d^3*e + 40*(c^2*d*e^3 + b*c*e^4)*x^3 + 4*a*b*d*e^3 + (b^2*e^2 + 2*a*
c*e^2)*d^2 + 15*(2*c^2*d^2*e^2 + 2*b*c*d*e^3 + b^2*e^4 + 2*a*c*e^4)*x^2 + 10*a^2*e^4 + 6*(2*c^2*d^3*e + 2*b*c*
d^2*e^2 + 4*a*b*e^4 + (b^2*e^3 + 2*a*c*e^3)*d)*x)/(x^6*e^11 + 6*d*x^5*e^10 + 15*d^2*x^4*e^9 + 20*d^3*x^3*e^8 +
15*d^4*x^2*e^7 + 6*d^5*x*e^6 + d^6*e^5)

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Fricas [A]
time = 2.18, size = 208, normalized size = 1.33 \begin {gather*} -\frac {2 \, c^{2} d^{4} + {\left (30 \, c^{2} x^{4} + 40 \, b c x^{3} + 24 \, a b x + 15 \, {\left (b^{2} + 2 \, a c\right )} x^{2} + 10 \, a^{2}\right )} e^{4} + 2 \, {\left (20 \, c^{2} d x^{3} + 15 \, b c d x^{2} + 2 \, a b d + 3 \, {\left (b^{2} + 2 \, a c\right )} d x\right )} e^{3} + {\left (30 \, c^{2} d^{2} x^{2} + 12 \, b c d^{2} x + {\left (b^{2} + 2 \, a c\right )} d^{2}\right )} e^{2} + 2 \, {\left (6 \, c^{2} d^{3} x + b c d^{3}\right )} e}{60 \, {\left (x^{6} e^{11} + 6 \, d x^{5} e^{10} + 15 \, d^{2} x^{4} e^{9} + 20 \, d^{3} x^{3} e^{8} + 15 \, d^{4} x^{2} e^{7} + 6 \, d^{5} x e^{6} + d^{6} 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)^7,x, algorithm="fricas")

[Out]

-1/60*(2*c^2*d^4 + (30*c^2*x^4 + 40*b*c*x^3 + 24*a*b*x + 15*(b^2 + 2*a*c)*x^2 + 10*a^2)*e^4 + 2*(20*c^2*d*x^3
+ 15*b*c*d*x^2 + 2*a*b*d + 3*(b^2 + 2*a*c)*d*x)*e^3 + (30*c^2*d^2*x^2 + 12*b*c*d^2*x + (b^2 + 2*a*c)*d^2)*e^2
+ 2*(6*c^2*d^3*x + b*c*d^3)*e)/(x^6*e^11 + 6*d*x^5*e^10 + 15*d^2*x^4*e^9 + 20*d^3*x^3*e^8 + 15*d^4*x^2*e^7 + 6
*d^5*x*e^6 + d^6*e^5)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**2/(e*x+d)**7,x)

[Out]

Timed out

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Giac [A]
time = 1.19, size = 179, normalized size = 1.15 \begin {gather*} -\frac {{\left (30 \, c^{2} x^{4} e^{4} + 40 \, c^{2} d x^{3} e^{3} + 30 \, c^{2} d^{2} x^{2} e^{2} + 12 \, c^{2} d^{3} x e + 2 \, c^{2} d^{4} + 40 \, b c x^{3} e^{4} + 30 \, b c d x^{2} e^{3} + 12 \, b c d^{2} x e^{2} + 2 \, b c d^{3} e + 15 \, b^{2} x^{2} e^{4} + 30 \, a c x^{2} e^{4} + 6 \, b^{2} d x e^{3} + 12 \, a c d x e^{3} + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} + 24 \, a b x e^{4} + 4 \, a b d e^{3} + 10 \, a^{2} e^{4}\right )} e^{\left (-5\right )}}{60 \, {\left (x e + d\right )}^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^2/(e*x+d)^7,x, algorithm="giac")

[Out]

-1/60*(30*c^2*x^4*e^4 + 40*c^2*d*x^3*e^3 + 30*c^2*d^2*x^2*e^2 + 12*c^2*d^3*x*e + 2*c^2*d^4 + 40*b*c*x^3*e^4 +
30*b*c*d*x^2*e^3 + 12*b*c*d^2*x*e^2 + 2*b*c*d^3*e + 15*b^2*x^2*e^4 + 30*a*c*x^2*e^4 + 6*b^2*d*x*e^3 + 12*a*c*d
*x*e^3 + b^2*d^2*e^2 + 2*a*c*d^2*e^2 + 24*a*b*x*e^4 + 4*a*b*d*e^3 + 10*a^2*e^4)*e^(-5)/(x*e + d)^6

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Mupad [B]
time = 0.71, size = 233, normalized size = 1.49 \begin {gather*} -\frac {\frac {10\,a^2\,e^4+4\,a\,b\,d\,e^3+2\,a\,c\,d^2\,e^2+b^2\,d^2\,e^2+2\,b\,c\,d^3\,e+2\,c^2\,d^4}{60\,e^5}+\frac {x\,\left (b^2\,d\,e^2+2\,b\,c\,d^2\,e+4\,a\,b\,e^3+2\,c^2\,d^3+2\,a\,c\,d\,e^2\right )}{10\,e^4}+\frac {c^2\,x^4}{2\,e}+\frac {x^2\,\left (b^2\,e^2+2\,b\,c\,d\,e+2\,c^2\,d^2+2\,a\,c\,e^2\right )}{4\,e^3}+\frac {2\,c\,x^3\,\left (b\,e+c\,d\right )}{3\,e^2}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^2/(d + e*x)^7,x)

[Out]

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

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