3.242 \(\int \frac{(b x+c x^2)^2}{(d+e x)^7} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0921652, antiderivative size = 137, normalized size of antiderivative = 1., 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} \[ -\frac{b^2 e^2-6 b c d e+6 c^2 d^2}{4 e^5 (d+e x)^4}-\frac{d^2 (c d-b e)^2}{6 e^5 (d+e x)^6}+\frac{2 c (2 c d-b e)}{3 e^5 (d+e x)^3}+\frac{2 d (c d-b e) (2 c d-b e)}{5 e^5 (d+e x)^5}-\frac{c^2}{2 e^5 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^2/(d + e*x)^7,x]

[Out]

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

Mathematica [A]  time = 0.040776, size = 116, normalized size = 0.85 \[ -\frac{b^2 e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+2 b c e \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )+2 c^2 \left (15 d^2 e^2 x^2+6 d^3 e x+d^4+20 d e^3 x^3+15 e^4 x^4\right )}{60 e^5 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.052, size = 143, normalized size = 1. \begin{align*} -{\frac{{c}^{2}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}+{\frac{2\,d \left ({b}^{2}{e}^{2}-3\,bcde+2\,{c}^{2}{d}^{2} \right ) }{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}-{\frac{2\,c \left ( be-2\,cd \right ) }{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{{d}^{2} \left ({b}^{2}{e}^{2}-2\,bcde+{c}^{2}{d}^{2} \right ) }{6\,{e}^{5} \left ( ex+d \right ) ^{6}}}-{\frac{{b}^{2}{e}^{2}-6\,bcde+6\,{c}^{2}{d}^{2}}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.15776, size = 258, normalized size = 1.88 \begin{align*} -\frac{30 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 40 \,{\left (c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 15 \,{\left (2 \, c^{2} d^{2} e^{2} + 2 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} + 6 \,{\left (2 \, c^{2} d^{3} e + 2 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x}{60 \,{\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.63062, size = 396, normalized size = 2.89 \begin{align*} -\frac{30 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 40 \,{\left (c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 15 \,{\left (2 \, c^{2} d^{2} e^{2} + 2 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} + 6 \,{\left (2 \, c^{2} d^{3} e + 2 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x}{60 \,{\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 13.8266, size = 204, normalized size = 1.49 \begin{align*} - \frac{b^{2} d^{2} e^{2} + 2 b c d^{3} e + 2 c^{2} d^{4} + 30 c^{2} e^{4} x^{4} + x^{3} \left (40 b c e^{4} + 40 c^{2} d e^{3}\right ) + x^{2} \left (15 b^{2} e^{4} + 30 b c d e^{3} + 30 c^{2} d^{2} e^{2}\right ) + x \left (6 b^{2} d e^{3} + 12 b c d^{2} e^{2} + 12 c^{2} d^{3} e\right )}{60 d^{6} e^{5} + 360 d^{5} e^{6} x + 900 d^{4} e^{7} x^{2} + 1200 d^{3} e^{8} x^{3} + 900 d^{2} e^{9} x^{4} + 360 d e^{10} x^{5} + 60 e^{11} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b**2*d**2*e**2 + 2*b*c*d**3*e + 2*c**2*d**4 + 30*c**2*e**4*x**4 + x**3*(40*b*c*e**4 + 40*c**2*d*e**3) + x**2
*(15*b**2*e**4 + 30*b*c*d*e**3 + 30*c**2*d**2*e**2) + x*(6*b**2*d*e**3 + 12*b*c*d**2*e**2 + 12*c**2*d**3*e))/(
60*d**6*e**5 + 360*d**5*e**6*x + 900*d**4*e**7*x**2 + 1200*d**3*e**8*x**3 + 900*d**2*e**9*x**4 + 360*d*e**10*x
**5 + 60*e**11*x**6)

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Giac [A]  time = 1.28522, size = 178, normalized size = 1.3 \begin{align*} -\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} + 6 \, b^{2} d x e^{3} + b^{2} d^{2} e^{2}\right )} e^{\left (-5\right )}}{60 \,{\left (x e + d\right )}^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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