3.3.19 \(\int \frac {(b x+c x^2)^2}{(d+e x)^6} \, dx\)

Optimal. Leaf size=132 \[ -\frac {b^2 e^2-6 b c d e+6 c^2 d^2}{3 e^5 (d+e x)^3}-\frac {d^2 (c d-b e)^2}{5 e^5 (d+e x)^5}+\frac {c (2 c d-b e)}{e^5 (d+e x)^2}+\frac {d (c d-b e) (2 c d-b e)}{2 e^5 (d+e x)^4}-\frac {c^2}{e^5 (d+e x)} \]

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Rubi [A]  time = 0.09, antiderivative size = 132, 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} \begin {gather*} -\frac {b^2 e^2-6 b c d e+6 c^2 d^2}{3 e^5 (d+e x)^3}-\frac {d^2 (c d-b e)^2}{5 e^5 (d+e x)^5}+\frac {c (2 c d-b e)}{e^5 (d+e x)^2}+\frac {d (c d-b e) (2 c d-b e)}{2 e^5 (d+e x)^4}-\frac {c^2}{e^5 (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^2/(d + e*x)^6,x]

[Out]

-(d^2*(c*d - b*e)^2)/(5*e^5*(d + e*x)^5) + (d*(c*d - b*e)*(2*c*d - b*e))/(2*e^5*(d + e*x)^4) - (6*c^2*d^2 - 6*
b*c*d*e + b^2*e^2)/(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 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)^6} \, dx &=\int \left (\frac {d^2 (c d-b e)^2}{e^4 (d+e x)^6}+\frac {2 d (c d-b e) (-2 c d+b e)}{e^4 (d+e x)^5}+\frac {6 c^2 d^2-6 b c d e+b^2 e^2}{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 {d^2 (c d-b e)^2}{5 e^5 (d+e x)^5}+\frac {d (c d-b e) (2 c d-b e)}{2 e^5 (d+e x)^4}-\frac {6 c^2 d^2-6 b c d e+b^2 e^2}{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 = 116, normalized size = 0.88 \begin {gather*} -\frac {b^2 e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+3 b c e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+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 )}{30 e^5 (d+e x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b x+c x^2\right )^2}{(d+e x)^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 181, normalized size = 1.37 \begin {gather*} -\frac {30 \, c^{2} e^{4} x^{4} + 6 \, c^{2} d^{4} + 3 \, b c d^{3} e + b^{2} d^{2} e^{2} + 30 \, {\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 10 \, {\left (6 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} + 5 \, {\left (6 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x}{30 \, {\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.18, size = 132, normalized size = 1.00 \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} + 5 \, b^{2} d x e^{3} + b^{2} d^{2} e^{2}\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)^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 + 5*b^2*d*x*e^3 + b^2*d^2*e^2)*e^(-5)/(x*e
+ d)^5

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maple [A]  time = 0.07, size = 143, normalized size = 1.08 \begin {gather*} -\frac {c^{2}}{\left (e x +d \right ) e^{5}}-\frac {\left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) d^{2}}{5 \left (e x +d \right )^{5} e^{5}}-\frac {\left (b e -2 c d \right ) c}{\left (e x +d \right )^{2} e^{5}}+\frac {\left (b^{2} e^{2}-3 b c d e +2 c^{2} d^{2}\right ) d}{2 \left (e x +d \right )^{4} e^{5}}-\frac {b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}}{3 \left (e x +d \right )^{3} e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.46, size = 181, normalized size = 1.37 \begin {gather*} -\frac {30 \, c^{2} e^{4} x^{4} + 6 \, c^{2} d^{4} + 3 \, b c d^{3} e + b^{2} d^{2} e^{2} + 30 \, {\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 10 \, {\left (6 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} + 5 \, {\left (6 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x}{30 \, {\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.20, size = 169, normalized size = 1.28 \begin {gather*} -\frac {\frac {x^2\,\left (b^2\,e^2+3\,b\,c\,d\,e+6\,c^2\,d^2\right )}{3\,e^3}+\frac {c^2\,x^4}{e}+\frac {d^2\,\left (b^2\,e^2+3\,b\,c\,d\,e+6\,c^2\,d^2\right )}{30\,e^5}+\frac {c\,x^3\,\left (b\,e+2\,c\,d\right )}{e^2}+\frac {d\,x\,\left (b^2\,e^2+3\,b\,c\,d\,e+6\,c^2\,d^2\right )}{6\,e^4}}{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((b*x + c*x^2)^2/(d + e*x)^6,x)

[Out]

-((x^2*(b^2*e^2 + 6*c^2*d^2 + 3*b*c*d*e))/(3*e^3) + (c^2*x^4)/e + (d^2*(b^2*e^2 + 6*c^2*d^2 + 3*b*c*d*e))/(30*
e^5) + (c*x^3*(b*e + 2*c*d))/e^2 + (d*x*(b^2*e^2 + 6*c^2*d^2 + 3*b*c*d*e))/(6*e^4))/(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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sympy [A]  time = 2.85, size = 196, normalized size = 1.48 \begin {gather*} \frac {- 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 (- 10 b^{2} e^{4} - 30 b c d e^{3} - 60 c^{2} d^{2} e^{2}\right ) + x \left (- 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)**2/(e*x+d)**6,x)

[Out]

(-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*(-10*b**2*e**4 - 30*b*c*d*e**3 - 60*c**2*d**2*e**2) + x*(-5*b**2*d*e**3 - 15*b*c*d**2*e**2 - 30*c**2*d**3*e)
)/(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)

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