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

Optimal. Leaf size=58 \[ \frac {b (a+b x)^5}{30 (d+e x)^5 (b d-a e)^2}+\frac {(a+b x)^5}{6 (d+e x)^6 (b d-a e)} \]

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Rubi [A]  time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {27, 45, 37} \begin {gather*} \frac {b (a+b x)^5}{30 (d+e x)^5 (b d-a e)^2}+\frac {(a+b x)^5}{6 (d+e x)^6 (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rubi steps

\begin {align*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^7} \, dx &=\int \frac {(a+b x)^4}{(d+e x)^7} \, dx\\ &=\frac {(a+b x)^5}{6 (b d-a e) (d+e x)^6}+\frac {b \int \frac {(a+b x)^4}{(d+e x)^6} \, dx}{6 (b d-a e)}\\ &=\frac {(a+b x)^5}{6 (b d-a e) (d+e x)^6}+\frac {b (a+b x)^5}{30 (b d-a e)^2 (d+e x)^5}\\ \end {align*}

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Mathematica [B]  time = 0.05, size = 144, normalized size = 2.48 \begin {gather*} -\frac {5 a^4 e^4+4 a^3 b e^3 (d+6 e x)+3 a^2 b^2 e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+2 a b^3 e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+b^4 \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 )}{30 e^5 (d+e x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.40, size = 236, normalized size = 4.07 \begin {gather*} -\frac {15 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 2 \, a b^{3} d^{3} e + 3 \, a^{2} b^{2} d^{2} e^{2} + 4 \, a^{3} b d e^{3} + 5 \, a^{4} e^{4} + 20 \, {\left (b^{4} d e^{3} + 2 \, a b^{3} e^{4}\right )} x^{3} + 15 \, {\left (b^{4} d^{2} e^{2} + 2 \, a b^{3} d e^{3} + 3 \, a^{2} b^{2} e^{4}\right )} x^{2} + 6 \, {\left (b^{4} d^{3} e + 2 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} + 4 \, a^{3} b e^{4}\right )} x}{30 \, {\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(15*b^4*e^4*x^4 + b^4*d^4 + 2*a*b^3*d^3*e + 3*a^2*b^2*d^2*e^2 + 4*a^3*b*d*e^3 + 5*a^4*e^4 + 20*(b^4*d*e^
3 + 2*a*b^3*e^4)*x^3 + 15*(b^4*d^2*e^2 + 2*a*b^3*d*e^3 + 3*a^2*b^2*e^4)*x^2 + 6*(b^4*d^3*e + 2*a*b^3*d^2*e^2 +
3*a^2*b^2*d*e^3 + 4*a^3*b*e^4)*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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giac [B]  time = 0.18, size = 174, normalized size = 3.00 \begin {gather*} -\frac {{\left (15 \, b^{4} x^{4} e^{4} + 20 \, b^{4} d x^{3} e^{3} + 15 \, b^{4} d^{2} x^{2} e^{2} + 6 \, b^{4} d^{3} x e + b^{4} d^{4} + 40 \, a b^{3} x^{3} e^{4} + 30 \, a b^{3} d x^{2} e^{3} + 12 \, a b^{3} d^{2} x e^{2} + 2 \, a b^{3} d^{3} e + 45 \, a^{2} b^{2} x^{2} e^{4} + 18 \, a^{2} b^{2} d x e^{3} + 3 \, a^{2} b^{2} d^{2} e^{2} + 24 \, a^{3} b x e^{4} + 4 \, a^{3} b d e^{3} + 5 \, a^{4} e^{4}\right )} e^{\left (-5\right )}}{30 \, {\left (x e + d\right )}^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(15*b^4*x^4*e^4 + 20*b^4*d*x^3*e^3 + 15*b^4*d^2*x^2*e^2 + 6*b^4*d^3*x*e + b^4*d^4 + 40*a*b^3*x^3*e^4 + 3
0*a*b^3*d*x^2*e^3 + 12*a*b^3*d^2*x*e^2 + 2*a*b^3*d^3*e + 45*a^2*b^2*x^2*e^4 + 18*a^2*b^2*d*x*e^3 + 3*a^2*b^2*d
^2*e^2 + 24*a^3*b*x*e^4 + 4*a^3*b*d*e^3 + 5*a^4*e^4)*e^(-5)/(x*e + d)^6

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maple [B]  time = 0.05, size = 186, normalized size = 3.21 \begin {gather*} -\frac {b^{4}}{2 \left (e x +d \right )^{2} e^{5}}-\frac {4 \left (a e -b d \right ) b^{3}}{3 \left (e x +d \right )^{3} e^{5}}-\frac {3 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}}{2 \left (e x +d \right )^{4} e^{5}}-\frac {4 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) b}{5 \left (e x +d \right )^{5} e^{5}}-\frac {e^{4} a^{4}-4 d \,e^{3} a^{3} b +6 d^{2} e^{2} b^{2} a^{2}-4 d^{3} a \,b^{3} e +b^{4} d^{4}}{6 \left (e x +d \right )^{6} e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.48, size = 236, normalized size = 4.07 \begin {gather*} -\frac {15 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 2 \, a b^{3} d^{3} e + 3 \, a^{2} b^{2} d^{2} e^{2} + 4 \, a^{3} b d e^{3} + 5 \, a^{4} e^{4} + 20 \, {\left (b^{4} d e^{3} + 2 \, a b^{3} e^{4}\right )} x^{3} + 15 \, {\left (b^{4} d^{2} e^{2} + 2 \, a b^{3} d e^{3} + 3 \, a^{2} b^{2} e^{4}\right )} x^{2} + 6 \, {\left (b^{4} d^{3} e + 2 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} + 4 \, a^{3} b e^{4}\right )} x}{30 \, {\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(15*b^4*e^4*x^4 + b^4*d^4 + 2*a*b^3*d^3*e + 3*a^2*b^2*d^2*e^2 + 4*a^3*b*d*e^3 + 5*a^4*e^4 + 20*(b^4*d*e^
3 + 2*a*b^3*e^4)*x^3 + 15*(b^4*d^2*e^2 + 2*a*b^3*d*e^3 + 3*a^2*b^2*e^4)*x^2 + 6*(b^4*d^3*e + 2*a*b^3*d^2*e^2 +
3*a^2*b^2*d*e^3 + 4*a^3*b*e^4)*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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mupad [B]  time = 0.11, size = 226, normalized size = 3.90 \begin {gather*} -\frac {\frac {5\,a^4\,e^4+4\,a^3\,b\,d\,e^3+3\,a^2\,b^2\,d^2\,e^2+2\,a\,b^3\,d^3\,e+b^4\,d^4}{30\,e^5}+\frac {b^4\,x^4}{2\,e}+\frac {2\,b^3\,x^3\,\left (2\,a\,e+b\,d\right )}{3\,e^2}+\frac {b\,x\,\left (4\,a^3\,e^3+3\,a^2\,b\,d\,e^2+2\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{5\,e^4}+\frac {b^2\,x^2\,\left (3\,a^2\,e^2+2\,a\,b\,d\,e+b^2\,d^2\right )}{2\,e^3}}{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^2 + b^2*x^2 + 2*a*b*x)^2/(d + e*x)^7,x)

[Out]

-((5*a^4*e^4 + b^4*d^4 + 3*a^2*b^2*d^2*e^2 + 2*a*b^3*d^3*e + 4*a^3*b*d*e^3)/(30*e^5) + (b^4*x^4)/(2*e) + (2*b^
3*x^3*(2*a*e + b*d))/(3*e^2) + (b*x*(4*a^3*e^3 + b^3*d^3 + 2*a*b^2*d^2*e + 3*a^2*b*d*e^2))/(5*e^4) + (b^2*x^2*
(3*a^2*e^2 + b^2*d^2 + 2*a*b*d*e))/(2*e^3))/(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 + 1
5*d^2*e^4*x^4 + 6*d^5*e*x)

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sympy [B]  time = 5.92, size = 255, normalized size = 4.40 \begin {gather*} \frac {- 5 a^{4} e^{4} - 4 a^{3} b d e^{3} - 3 a^{2} b^{2} d^{2} e^{2} - 2 a b^{3} d^{3} e - b^{4} d^{4} - 15 b^{4} e^{4} x^{4} + x^{3} \left (- 40 a b^{3} e^{4} - 20 b^{4} d e^{3}\right ) + x^{2} \left (- 45 a^{2} b^{2} e^{4} - 30 a b^{3} d e^{3} - 15 b^{4} d^{2} e^{2}\right ) + x \left (- 24 a^{3} b e^{4} - 18 a^{2} b^{2} d e^{3} - 12 a b^{3} d^{2} e^{2} - 6 b^{4} d^{3} e\right )}{30 d^{6} e^{5} + 180 d^{5} e^{6} x + 450 d^{4} e^{7} x^{2} + 600 d^{3} e^{8} x^{3} + 450 d^{2} e^{9} x^{4} + 180 d e^{10} x^{5} + 30 e^{11} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-5*a**4*e**4 - 4*a**3*b*d*e**3 - 3*a**2*b**2*d**2*e**2 - 2*a*b**3*d**3*e - b**4*d**4 - 15*b**4*e**4*x**4 + x*
*3*(-40*a*b**3*e**4 - 20*b**4*d*e**3) + x**2*(-45*a**2*b**2*e**4 - 30*a*b**3*d*e**3 - 15*b**4*d**2*e**2) + x*(
-24*a**3*b*e**4 - 18*a**2*b**2*d*e**3 - 12*a*b**3*d**2*e**2 - 6*b**4*d**3*e))/(30*d**6*e**5 + 180*d**5*e**6*x
+ 450*d**4*e**7*x**2 + 600*d**3*e**8*x**3 + 450*d**2*e**9*x**4 + 180*d*e**10*x**5 + 30*e**11*x**6)

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