3.243 \(\int \frac {(b x+c x^2)^2}{(d+e x)^8} \, dx\)

Optimal. Leaf size=137 \[ -\frac {b^2 e^2-6 b c d e+6 c^2 d^2}{5 e^5 (d+e x)^5}-\frac {d^2 (c d-b e)^2}{7 e^5 (d+e x)^7}+\frac {c (2 c d-b e)}{2 e^5 (d+e x)^4}+\frac {d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^6}-\frac {c^2}{3 e^5 (d+e x)^3} \]

[Out]

-1/7*d^2*(-b*e+c*d)^2/e^5/(e*x+d)^7+1/3*d*(-b*e+c*d)*(-b*e+2*c*d)/e^5/(e*x+d)^6+1/5*(-b^2*e^2+6*b*c*d*e-6*c^2*
d^2)/e^5/(e*x+d)^5+1/2*c*(-b*e+2*c*d)/e^5/(e*x+d)^4-1/3*c^2/e^5/(e*x+d)^3

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Rubi [A]  time = 0.09, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {698} \[ -\frac {b^2 e^2-6 b c d e+6 c^2 d^2}{5 e^5 (d+e x)^5}-\frac {d^2 (c d-b e)^2}{7 e^5 (d+e x)^7}+\frac {c (2 c d-b e)}{2 e^5 (d+e x)^4}+\frac {d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^6}-\frac {c^2}{3 e^5 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^2/(d + e*x)^8,x]

[Out]

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

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 (b x+c x^2\right )^2}{(d+e x)^8} \, dx &=\int \left (\frac {d^2 (c d-b e)^2}{e^4 (d+e x)^8}+\frac {2 d (c d-b e) (-2 c d+b e)}{e^4 (d+e x)^7}+\frac {6 c^2 d^2-6 b c d e+b^2 e^2}{e^4 (d+e x)^6}-\frac {2 c (2 c d-b e)}{e^4 (d+e x)^5}+\frac {c^2}{e^4 (d+e x)^4}\right ) \, dx\\ &=-\frac {d^2 (c d-b e)^2}{7 e^5 (d+e x)^7}+\frac {d (c d-b e) (2 c d-b e)}{3 e^5 (d+e x)^6}-\frac {6 c^2 d^2-6 b c d e+b^2 e^2}{5 e^5 (d+e x)^5}+\frac {c (2 c d-b e)}{2 e^5 (d+e x)^4}-\frac {c^2}{3 e^5 (d+e x)^3}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 117, normalized size = 0.85 \[ -\frac {2 b^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+3 b c e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+2 c^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )}{210 e^5 (d+e x)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x + c*x^2)^2/(d + e*x)^8,x]

[Out]

-1/210*(2*b^2*e^2*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*b*c*e*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 2*c^2
*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4))/(e^5*(d + e*x)^7)

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fricas [A]  time = 0.75, size = 207, normalized size = 1.51 \[ -\frac {70 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 3 \, b c d^{3} e + 2 \, b^{2} d^{2} e^{2} + 35 \, {\left (2 \, c^{2} d e^{3} + 3 \, b c e^{4}\right )} x^{3} + 21 \, {\left (2 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + 2 \, b^{2} e^{4}\right )} x^{2} + 7 \, {\left (2 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + 2 \, b^{2} d e^{3}\right )} x}{210 \, {\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^2/(e*x+d)^8,x, algorithm="fricas")

[Out]

-1/210*(70*c^2*e^4*x^4 + 2*c^2*d^4 + 3*b*c*d^3*e + 2*b^2*d^2*e^2 + 35*(2*c^2*d*e^3 + 3*b*c*e^4)*x^3 + 21*(2*c^
2*d^2*e^2 + 3*b*c*d*e^3 + 2*b^2*e^4)*x^2 + 7*(2*c^2*d^3*e + 3*b*c*d^2*e^2 + 2*b^2*d*e^3)*x)/(e^12*x^7 + 7*d*e^
11*x^6 + 21*d^2*e^10*x^5 + 35*d^3*e^9*x^4 + 35*d^4*e^8*x^3 + 21*d^5*e^7*x^2 + 7*d^6*e^6*x + d^7*e^5)

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giac [A]  time = 0.15, size = 133, normalized size = 0.97 \[ -\frac {{\left (70 \, c^{2} x^{4} e^{4} + 70 \, c^{2} d x^{3} e^{3} + 42 \, c^{2} d^{2} x^{2} e^{2} + 14 \, c^{2} d^{3} x e + 2 \, c^{2} d^{4} + 105 \, b c x^{3} e^{4} + 63 \, b c d x^{2} e^{3} + 21 \, b c d^{2} x e^{2} + 3 \, b c d^{3} e + 42 \, b^{2} x^{2} e^{4} + 14 \, b^{2} d x e^{3} + 2 \, b^{2} d^{2} e^{2}\right )} e^{\left (-5\right )}}{210 \, {\left (x e + d\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^2/(e*x+d)^8,x, algorithm="giac")

[Out]

-1/210*(70*c^2*x^4*e^4 + 70*c^2*d*x^3*e^3 + 42*c^2*d^2*x^2*e^2 + 14*c^2*d^3*x*e + 2*c^2*d^4 + 105*b*c*x^3*e^4
+ 63*b*c*d*x^2*e^3 + 21*b*c*d^2*x*e^2 + 3*b*c*d^3*e + 42*b^2*x^2*e^4 + 14*b^2*d*x*e^3 + 2*b^2*d^2*e^2)*e^(-5)/
(x*e + d)^7

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maple [A]  time = 0.04, size = 143, normalized size = 1.04 \[ -\frac {c^{2}}{3 \left (e x +d \right )^{3} e^{5}}-\frac {\left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) d^{2}}{7 \left (e x +d \right )^{7} e^{5}}-\frac {\left (b e -2 c d \right ) c}{2 \left (e x +d \right )^{4} e^{5}}+\frac {\left (b^{2} e^{2}-3 b c d e +2 c^{2} d^{2}\right ) d}{3 \left (e x +d \right )^{6} e^{5}}-\frac {b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}}{5 \left (e x +d \right )^{5} e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^2/(e*x+d)^8,x)

[Out]

-1/3*c^2/e^5/(e*x+d)^3-1/5*(b^2*e^2-6*b*c*d*e+6*c^2*d^2)/e^5/(e*x+d)^5-1/2*c*(b*e-2*c*d)/e^5/(e*x+d)^4+1/3*d*(
b^2*e^2-3*b*c*d*e+2*c^2*d^2)/e^5/(e*x+d)^6-1/7*d^2*(b^2*e^2-2*b*c*d*e+c^2*d^2)/e^5/(e*x+d)^7

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maxima [A]  time = 1.49, size = 207, normalized size = 1.51 \[ -\frac {70 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 3 \, b c d^{3} e + 2 \, b^{2} d^{2} e^{2} + 35 \, {\left (2 \, c^{2} d e^{3} + 3 \, b c e^{4}\right )} x^{3} + 21 \, {\left (2 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + 2 \, b^{2} e^{4}\right )} x^{2} + 7 \, {\left (2 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + 2 \, b^{2} d e^{3}\right )} x}{210 \, {\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^2/(e*x+d)^8,x, algorithm="maxima")

[Out]

-1/210*(70*c^2*e^4*x^4 + 2*c^2*d^4 + 3*b*c*d^3*e + 2*b^2*d^2*e^2 + 35*(2*c^2*d*e^3 + 3*b*c*e^4)*x^3 + 21*(2*c^
2*d^2*e^2 + 3*b*c*d*e^3 + 2*b^2*e^4)*x^2 + 7*(2*c^2*d^3*e + 3*b*c*d^2*e^2 + 2*b^2*d*e^3)*x)/(e^12*x^7 + 7*d*e^
11*x^6 + 21*d^2*e^10*x^5 + 35*d^3*e^9*x^4 + 35*d^4*e^8*x^3 + 21*d^5*e^7*x^2 + 7*d^6*e^6*x + d^7*e^5)

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mupad [B]  time = 0.20, size = 197, normalized size = 1.44 \[ -\frac {\frac {x^2\,\left (2\,b^2\,e^2+3\,b\,c\,d\,e+2\,c^2\,d^2\right )}{10\,e^3}+\frac {c^2\,x^4}{3\,e}+\frac {d^2\,\left (2\,b^2\,e^2+3\,b\,c\,d\,e+2\,c^2\,d^2\right )}{210\,e^5}+\frac {c\,x^3\,\left (3\,b\,e+2\,c\,d\right )}{6\,e^2}+\frac {d\,x\,\left (2\,b^2\,e^2+3\,b\,c\,d\,e+2\,c^2\,d^2\right )}{30\,e^4}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^2/(d + e*x)^8,x)

[Out]

-((x^2*(2*b^2*e^2 + 2*c^2*d^2 + 3*b*c*d*e))/(10*e^3) + (c^2*x^4)/(3*e) + (d^2*(2*b^2*e^2 + 2*c^2*d^2 + 3*b*c*d
*e))/(210*e^5) + (c*x^3*(3*b*e + 2*c*d))/(6*e^2) + (d*x*(2*b^2*e^2 + 2*c^2*d^2 + 3*b*c*d*e))/(30*e^4))/(d^7 +
e^7*x^7 + 7*d*e^6*x^6 + 21*d^5*e^2*x^2 + 35*d^4*e^3*x^3 + 35*d^3*e^4*x^4 + 21*d^2*e^5*x^5 + 7*d^6*e*x)

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sympy [A]  time = 7.20, size = 221, normalized size = 1.61 \[ \frac {- 2 b^{2} d^{2} e^{2} - 3 b c d^{3} e - 2 c^{2} d^{4} - 70 c^{2} e^{4} x^{4} + x^{3} \left (- 105 b c e^{4} - 70 c^{2} d e^{3}\right ) + x^{2} \left (- 42 b^{2} e^{4} - 63 b c d e^{3} - 42 c^{2} d^{2} e^{2}\right ) + x \left (- 14 b^{2} d e^{3} - 21 b c d^{2} e^{2} - 14 c^{2} d^{3} e\right )}{210 d^{7} e^{5} + 1470 d^{6} e^{6} x + 4410 d^{5} e^{7} x^{2} + 7350 d^{4} e^{8} x^{3} + 7350 d^{3} e^{9} x^{4} + 4410 d^{2} e^{10} x^{5} + 1470 d e^{11} x^{6} + 210 e^{12} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x)**2/(e*x+d)**8,x)

[Out]

(-2*b**2*d**2*e**2 - 3*b*c*d**3*e - 2*c**2*d**4 - 70*c**2*e**4*x**4 + x**3*(-105*b*c*e**4 - 70*c**2*d*e**3) +
x**2*(-42*b**2*e**4 - 63*b*c*d*e**3 - 42*c**2*d**2*e**2) + x*(-14*b**2*d*e**3 - 21*b*c*d**2*e**2 - 14*c**2*d**
3*e))/(210*d**7*e**5 + 1470*d**6*e**6*x + 4410*d**5*e**7*x**2 + 7350*d**4*e**8*x**3 + 7350*d**3*e**9*x**4 + 44
10*d**2*e**10*x**5 + 1470*d*e**11*x**6 + 210*e**12*x**7)

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