3.12.75 \(\int \frac {(a^2+2 a b x+b^2 x^2)^2}{(d+e x)^4} \, dx\)

Optimal. Leaf size=103 \[ -\frac {4 b^3 (b d-a e) \log (d+e x)}{e^5}-\frac {6 b^2 (b d-a e)^2}{e^5 (d+e x)}+\frac {2 b (b d-a e)^3}{e^5 (d+e x)^2}-\frac {(b d-a e)^4}{3 e^5 (d+e x)^3}+\frac {b^4 x}{e^4} \]

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Rubi [A]  time = 0.08, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 43} \begin {gather*} -\frac {6 b^2 (b d-a e)^2}{e^5 (d+e x)}-\frac {4 b^3 (b d-a e) \log (d+e x)}{e^5}+\frac {2 b (b d-a e)^3}{e^5 (d+e x)^2}-\frac {(b d-a e)^4}{3 e^5 (d+e x)^3}+\frac {b^4 x}{e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^4,x]

[Out]

(b^4*x)/e^4 - (b*d - a*e)^4/(3*e^5*(d + e*x)^3) + (2*b*(b*d - a*e)^3)/(e^5*(d + e*x)^2) - (6*b^2*(b*d - a*e)^2
)/(e^5*(d + e*x)) - (4*b^3*(b*d - a*e)*Log[d + e*x])/e^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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

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Mathematica [A]  time = 0.07, size = 163, normalized size = 1.58 \begin {gather*} -\frac {a^4 e^4+2 a^3 b e^3 (d+3 e x)+6 a^2 b^2 e^2 \left (d^2+3 d e x+3 e^2 x^2\right )-2 a b^3 d e \left (11 d^2+27 d e x+18 e^2 x^2\right )+12 b^3 (d+e x)^3 (b d-a e) \log (d+e x)+b^4 \left (13 d^4+27 d^3 e x+9 d^2 e^2 x^2-9 d e^3 x^3-3 e^4 x^4\right )}{3 e^5 (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^4,x]

[Out]

-1/3*(a^4*e^4 + 2*a^3*b*e^3*(d + 3*e*x) + 6*a^2*b^2*e^2*(d^2 + 3*d*e*x + 3*e^2*x^2) - 2*a*b^3*d*e*(11*d^2 + 27
*d*e*x + 18*e^2*x^2) + b^4*(13*d^4 + 27*d^3*e*x + 9*d^2*e^2*x^2 - 9*d*e^3*x^3 - 3*e^4*x^4) + 12*b^3*(b*d - a*e
)*(d + e*x)^3*Log[d + e*x])/(e^5*(d + e*x)^3)

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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)^4} \, 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)^4,x]

[Out]

IntegrateAlgebraic[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^4, x]

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fricas [B]  time = 0.38, size = 292, normalized size = 2.83 \begin {gather*} \frac {3 \, b^{4} e^{4} x^{4} + 9 \, b^{4} d e^{3} x^{3} - 13 \, b^{4} d^{4} + 22 \, a b^{3} d^{3} e - 6 \, a^{2} b^{2} d^{2} e^{2} - 2 \, a^{3} b d e^{3} - a^{4} e^{4} - 9 \, {\left (b^{4} d^{2} e^{2} - 4 \, a b^{3} d e^{3} + 2 \, a^{2} b^{2} e^{4}\right )} x^{2} - 3 \, {\left (9 \, b^{4} d^{3} e - 18 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} + 2 \, a^{3} b e^{4}\right )} x - 12 \, {\left (b^{4} d^{4} - a b^{3} d^{3} e + {\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 3 \, {\left (b^{4} d^{2} e^{2} - a b^{3} d e^{3}\right )} x^{2} + 3 \, {\left (b^{4} d^{3} e - a b^{3} d^{2} e^{2}\right )} x\right )} \log \left (e x + d\right )}{3 \, {\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} 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)^4,x, algorithm="fricas")

[Out]

1/3*(3*b^4*e^4*x^4 + 9*b^4*d*e^3*x^3 - 13*b^4*d^4 + 22*a*b^3*d^3*e - 6*a^2*b^2*d^2*e^2 - 2*a^3*b*d*e^3 - a^4*e
^4 - 9*(b^4*d^2*e^2 - 4*a*b^3*d*e^3 + 2*a^2*b^2*e^4)*x^2 - 3*(9*b^4*d^3*e - 18*a*b^3*d^2*e^2 + 6*a^2*b^2*d*e^3
+ 2*a^3*b*e^4)*x - 12*(b^4*d^4 - a*b^3*d^3*e + (b^4*d*e^3 - a*b^3*e^4)*x^3 + 3*(b^4*d^2*e^2 - a*b^3*d*e^3)*x^
2 + 3*(b^4*d^3*e - a*b^3*d^2*e^2)*x)*log(e*x + d))/(e^8*x^3 + 3*d*e^7*x^2 + 3*d^2*e^6*x + d^3*e^5)

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giac [A]  time = 0.18, size = 170, normalized size = 1.65 \begin {gather*} b^{4} x e^{\left (-4\right )} - 4 \, {\left (b^{4} d - a b^{3} e\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) - \frac {{\left (13 \, b^{4} d^{4} - 22 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 2 \, a^{3} b d e^{3} + a^{4} e^{4} + 18 \, {\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 6 \, {\left (5 \, b^{4} d^{3} e - 9 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} e^{\left (-5\right )}}{3 \, {\left (x e + d\right )}^{3}} \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)^4,x, algorithm="giac")

[Out]

b^4*x*e^(-4) - 4*(b^4*d - a*b^3*e)*e^(-5)*log(abs(x*e + d)) - 1/3*(13*b^4*d^4 - 22*a*b^3*d^3*e + 6*a^2*b^2*d^2
*e^2 + 2*a^3*b*d*e^3 + a^4*e^4 + 18*(b^4*d^2*e^2 - 2*a*b^3*d*e^3 + a^2*b^2*e^4)*x^2 + 6*(5*b^4*d^3*e - 9*a*b^3
*d^2*e^2 + 3*a^2*b^2*d*e^3 + a^3*b*e^4)*x)*e^(-5)/(x*e + d)^3

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maple [B]  time = 0.05, size = 255, normalized size = 2.48 \begin {gather*} -\frac {a^{4}}{3 \left (e x +d \right )^{3} e}+\frac {4 a^{3} b d}{3 \left (e x +d \right )^{3} e^{2}}-\frac {2 a^{2} b^{2} d^{2}}{\left (e x +d \right )^{3} e^{3}}+\frac {4 a \,b^{3} d^{3}}{3 \left (e x +d \right )^{3} e^{4}}-\frac {b^{4} d^{4}}{3 \left (e x +d \right )^{3} e^{5}}-\frac {2 a^{3} b}{\left (e x +d \right )^{2} e^{2}}+\frac {6 a^{2} b^{2} d}{\left (e x +d \right )^{2} e^{3}}-\frac {6 a \,b^{3} d^{2}}{\left (e x +d \right )^{2} e^{4}}+\frac {2 b^{4} d^{3}}{\left (e x +d \right )^{2} e^{5}}-\frac {6 a^{2} b^{2}}{\left (e x +d \right ) e^{3}}+\frac {12 a \,b^{3} d}{\left (e x +d \right ) e^{4}}+\frac {4 a \,b^{3} \ln \left (e x +d \right )}{e^{4}}-\frac {6 b^{4} d^{2}}{\left (e x +d \right ) e^{5}}-\frac {4 b^{4} d \ln \left (e x +d \right )}{e^{5}}+\frac {b^{4} x}{e^{4}} \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)^4,x)

[Out]

b^4*x/e^4-6*b^2/e^3/(e*x+d)*a^2+12*b^3/e^4/(e*x+d)*a*d-6*b^4/e^5/(e*x+d)*d^2-2*b/e^2/(e*x+d)^2*a^3+6*b^2/e^3/(
e*x+d)^2*a^2*d-6*b^3/e^4/(e*x+d)^2*a*d^2+2*b^4/e^5/(e*x+d)^2*d^3-1/3/e/(e*x+d)^3*a^4+4/3/e^2/(e*x+d)^3*d*a^3*b
-2/e^3/(e*x+d)^3*d^2*b^2*a^2+4/3/e^4/(e*x+d)^3*d^3*a*b^3-1/3/e^5/(e*x+d)^3*b^4*d^4+4*b^3/e^4*ln(e*x+d)*a-4*b^4
/e^5*ln(e*x+d)*d

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maxima [A]  time = 1.40, size = 201, normalized size = 1.95 \begin {gather*} \frac {b^{4} x}{e^{4}} - \frac {13 \, b^{4} d^{4} - 22 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 2 \, a^{3} b d e^{3} + a^{4} e^{4} + 18 \, {\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 6 \, {\left (5 \, b^{4} d^{3} e - 9 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x}{3 \, {\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} - \frac {4 \, {\left (b^{4} d - a b^{3} e\right )} \log \left (e x + d\right )}{e^{5}} \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)^4,x, algorithm="maxima")

[Out]

b^4*x/e^4 - 1/3*(13*b^4*d^4 - 22*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 2*a^3*b*d*e^3 + a^4*e^4 + 18*(b^4*d^2*e^2 -
2*a*b^3*d*e^3 + a^2*b^2*e^4)*x^2 + 6*(5*b^4*d^3*e - 9*a*b^3*d^2*e^2 + 3*a^2*b^2*d*e^3 + a^3*b*e^4)*x)/(e^8*x^
3 + 3*d*e^7*x^2 + 3*d^2*e^6*x + d^3*e^5) - 4*(b^4*d - a*b^3*e)*log(e*x + d)/e^5

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mupad [B]  time = 0.60, size = 204, normalized size = 1.98 \begin {gather*} \frac {b^4\,x}{e^4}-\frac {\ln \left (d+e\,x\right )\,\left (4\,b^4\,d-4\,a\,b^3\,e\right )}{e^5}-\frac {\frac {a^4\,e^4+2\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-22\,a\,b^3\,d^3\,e+13\,b^4\,d^4}{3\,e}+x\,\left (2\,a^3\,b\,e^3+6\,a^2\,b^2\,d\,e^2-18\,a\,b^3\,d^2\,e+10\,b^4\,d^3\right )+x^2\,\left (6\,a^2\,b^2\,e^3-12\,a\,b^3\,d\,e^2+6\,b^4\,d^2\,e\right )}{d^3\,e^4+3\,d^2\,e^5\,x+3\,d\,e^6\,x^2+e^7\,x^3} \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)^4,x)

[Out]

(b^4*x)/e^4 - (log(d + e*x)*(4*b^4*d - 4*a*b^3*e))/e^5 - ((a^4*e^4 + 13*b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 22*a*b^3
*d^3*e + 2*a^3*b*d*e^3)/(3*e) + x*(10*b^4*d^3 + 2*a^3*b*e^3 + 6*a^2*b^2*d*e^2 - 18*a*b^3*d^2*e) + x^2*(6*b^4*d
^2*e + 6*a^2*b^2*e^3 - 12*a*b^3*d*e^2))/(d^3*e^4 + e^7*x^3 + 3*d^2*e^5*x + 3*d*e^6*x^2)

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sympy [B]  time = 1.98, size = 209, normalized size = 2.03 \begin {gather*} \frac {b^{4} x}{e^{4}} + \frac {4 b^{3} \left (a e - b d\right ) \log {\left (d + e x \right )}}{e^{5}} + \frac {- a^{4} e^{4} - 2 a^{3} b d e^{3} - 6 a^{2} b^{2} d^{2} e^{2} + 22 a b^{3} d^{3} e - 13 b^{4} d^{4} + x^{2} \left (- 18 a^{2} b^{2} e^{4} + 36 a b^{3} d e^{3} - 18 b^{4} d^{2} e^{2}\right ) + x \left (- 6 a^{3} b e^{4} - 18 a^{2} b^{2} d e^{3} + 54 a b^{3} d^{2} e^{2} - 30 b^{4} d^{3} e\right )}{3 d^{3} e^{5} + 9 d^{2} e^{6} x + 9 d e^{7} x^{2} + 3 e^{8} x^{3}} \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)**4,x)

[Out]

b**4*x/e**4 + 4*b**3*(a*e - b*d)*log(d + e*x)/e**5 + (-a**4*e**4 - 2*a**3*b*d*e**3 - 6*a**2*b**2*d**2*e**2 + 2
2*a*b**3*d**3*e - 13*b**4*d**4 + x**2*(-18*a**2*b**2*e**4 + 36*a*b**3*d*e**3 - 18*b**4*d**2*e**2) + x*(-6*a**3
*b*e**4 - 18*a**2*b**2*d*e**3 + 54*a*b**3*d**2*e**2 - 30*b**4*d**3*e))/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7
*x**2 + 3*e**8*x**3)

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