3.15.71 \(\int \frac {(a^2+2 a b x+b^2 x^2)^2}{(d+e x)^2} \, dx\) [1471]

Optimal. Leaf size=104 \[ \frac {6 b^2 (b d-a e)^2 x}{e^4}-\frac {(b d-a e)^4}{e^5 (d+e x)}-\frac {2 b^3 (b d-a e) (d+e x)^2}{e^5}+\frac {b^4 (d+e x)^3}{3 e^5}-\frac {4 b (b d-a e)^3 \log (d+e x)}{e^5} \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 104, 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, 45} \begin {gather*} -\frac {2 b^3 (d+e x)^2 (b d-a e)}{e^5}+\frac {6 b^2 x (b d-a e)^2}{e^4}-\frac {(b d-a e)^4}{e^5 (d+e x)}-\frac {4 b (b d-a e)^3 \log (d+e x)}{e^5}+\frac {b^4 (d+e x)^3}{3 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(6*b^2*(b*d - a*e)^2*x)/e^4 - (b*d - a*e)^4/(e^5*(d + e*x)) - (2*b^3*(b*d - a*e)*(d + e*x)^2)/e^5 + (b^4*(d +
e*x)^3)/(3*e^5) - (4*b*(b*d - a*e)^3*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 45

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

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Mathematica [A]
time = 0.04, size = 165, normalized size = 1.59 \begin {gather*} \frac {12 a^3 b d e^3-3 a^4 e^4+18 a^2 b^2 e^2 \left (-d^2+d e x+e^2 x^2\right )+6 a b^3 e \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+b^4 \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )-12 b (b d-a e)^3 (d+e x) \log (d+e x)}{3 e^5 (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.63, size = 175, normalized size = 1.68

method	result	size
default	\(\frac {b^{2} \left (\frac {1}{3} b^{2} e^{2} x^{3}+2 a b \,e^{2} x^{2}-b^{2} d e \,x^{2}+6 a^{2} e^{2} x -8 a b d e x +3 x \,b^{2} d^{2}\right )}{e^{4}}-\frac {e^{4} a^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}{e^{5} \left (e x +d \right )}+\frac {4 b \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \ln \left (e x +d \right )}{e^{5}}\)	\(175\)
norman	\(\frac {\frac {\left (e^{4} a^{4}-4 a^{3} b d \,e^{3}+12 a^{2} b^{2} d^{2} e^{2}-12 a \,b^{3} d^{3} e +4 b^{4} d^{4}\right ) x}{d \,e^{4}}+\frac {b^{4} x^{4}}{3 e}+\frac {2 b^{2} \left (3 a^{2} e^{2}-3 a b d e +b^{2} d^{2}\right ) x^{2}}{e^{3}}+\frac {2 b^{3} \left (3 a e -b d \right ) x^{3}}{3 e^{2}}}{e x +d}+\frac {4 b \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \ln \left (e x +d \right )}{e^{5}}\)	\(181\)
risch	\(\frac {b^{4} x^{3}}{3 e^{2}}+\frac {2 b^{3} a \,x^{2}}{e^{2}}-\frac {b^{4} d \,x^{2}}{e^{3}}+\frac {6 b^{2} a^{2} x}{e^{2}}-\frac {8 b^{3} a d x}{e^{3}}+\frac {3 b^{4} x \,d^{2}}{e^{4}}-\frac {a^{4}}{e \left (e x +d \right )}+\frac {4 a^{3} b d}{e^{2} \left (e x +d \right )}-\frac {6 a^{2} b^{2} d^{2}}{e^{3} \left (e x +d \right )}+\frac {4 a \,b^{3} d^{3}}{e^{4} \left (e x +d \right )}-\frac {b^{4} d^{4}}{e^{5} \left (e x +d \right )}+\frac {4 b \ln \left (e x +d \right ) a^{3}}{e^{2}}-\frac {12 b^{2} \ln \left (e x +d \right ) a^{2} d}{e^{3}}+\frac {12 b^{3} \ln \left (e x +d \right ) a \,d^{2}}{e^{4}}-\frac {4 b^{4} \ln \left (e x +d \right ) d^{3}}{e^{5}}\)	\(230\)

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)^2,x,method=_RETURNVERBOSE)

[Out]

b^2/e^4*(1/3*b^2*e^2*x^3+2*a*b*e^2*x^2-b^2*d*e*x^2+6*a^2*e^2*x-8*a*b*d*e*x+3*x*b^2*d^2)-(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)+4*b/e^5*(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)*ln
(e*x+d)

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Maxima [A]
time = 0.28, size = 176, normalized size = 1.69 \begin {gather*} -4 \, {\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )} e^{\left (-5\right )} \log \left (x e + d\right ) + \frac {1}{3} \, {\left (b^{4} x^{3} e^{2} - 3 \, {\left (b^{4} d e - 2 \, a b^{3} e^{2}\right )} x^{2} + 3 \, {\left (3 \, b^{4} d^{2} - 8 \, a b^{3} d e + 6 \, a^{2} b^{2} e^{2}\right )} x\right )} e^{\left (-4\right )} - \frac {b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}}{x e^{6} + d 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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 251 vs. \(2 (105) = 210\).
time = 2.67, size = 251, normalized size = 2.41 \begin {gather*} -\frac {3 \, b^{4} d^{4} - {\left (b^{4} x^{4} + 6 \, a b^{3} x^{3} + 18 \, a^{2} b^{2} x^{2} - 3 \, a^{4}\right )} e^{4} + 2 \, {\left (b^{4} d x^{3} + 9 \, a b^{3} d x^{2} - 9 \, a^{2} b^{2} d x - 6 \, a^{3} b d\right )} e^{3} - 6 \, {\left (b^{4} d^{2} x^{2} - 4 \, a b^{3} d^{2} x - 3 \, a^{2} b^{2} d^{2}\right )} e^{2} - 3 \, {\left (3 \, b^{4} d^{3} x + 4 \, a b^{3} d^{3}\right )} e + 12 \, {\left (b^{4} d^{4} - a^{3} b x e^{4} + {\left (3 \, a^{2} b^{2} d x - a^{3} b d\right )} e^{3} - 3 \, {\left (a b^{3} d^{2} x - a^{2} b^{2} d^{2}\right )} e^{2} + {\left (b^{4} d^{3} x - 3 \, a b^{3} d^{3}\right )} e\right )} \log \left (x e + d\right )}{3 \, {\left (x e^{6} + d 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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.44, size = 155, normalized size = 1.49 \begin {gather*} \frac {b^{4} x^{3}}{3 e^{2}} + \frac {4 b \left (a e - b d\right )^{3} \log {\left (d + e x \right )}}{e^{5}} + x^{2} \cdot \left (\frac {2 a b^{3}}{e^{2}} - \frac {b^{4} d}{e^{3}}\right ) + x \left (\frac {6 a^{2} b^{2}}{e^{2}} - \frac {8 a b^{3} d}{e^{3}} + \frac {3 b^{4} d^{2}}{e^{4}}\right ) + \frac {- 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}}{d e^{5} + e^{6} x} \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)**2,x)

[Out]

b**4*x**3/(3*e**2) + 4*b*(a*e - b*d)**3*log(d + e*x)/e**5 + x**2*(2*a*b**3/e**2 - b**4*d/e**3) + x*(6*a**2*b**
2/e**2 - 8*a*b**3*d/e**3 + 3*b**4*d**2/e**4) + (-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)/(d*e**5 + e**6*x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 239 vs. \(2 (105) = 210\).
time = 1.45, size = 239, normalized size = 2.30 \begin {gather*} \frac {1}{3} \, {\left (b^{4} - \frac {6 \, {\left (b^{4} d e - a b^{3} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac {18 \, {\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}}\right )} {\left (x e + d\right )}^{3} e^{\left (-5\right )} + 4 \, {\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )} e^{\left (-5\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - {\left (\frac {b^{4} d^{4} e^{3}}{x e + d} - \frac {4 \, a b^{3} d^{3} e^{4}}{x e + d} + \frac {6 \, a^{2} b^{2} d^{2} e^{5}}{x e + d} - \frac {4 \, a^{3} b d e^{6}}{x e + d} + \frac {a^{4} e^{7}}{x e + d}\right )} e^{\left (-8\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)^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.55, size = 203, normalized size = 1.95 \begin {gather*} x^2\,\left (\frac {2\,a\,b^3}{e^2}-\frac {b^4\,d}{e^3}\right )-x\,\left (\frac {2\,d\,\left (\frac {4\,a\,b^3}{e^2}-\frac {2\,b^4\,d}{e^3}\right )}{e}-\frac {6\,a^2\,b^2}{e^2}+\frac {b^4\,d^2}{e^4}\right )+\frac {b^4\,x^3}{3\,e^2}-\frac {\ln \left (d+e\,x\right )\,\left (-4\,a^3\,b\,e^3+12\,a^2\,b^2\,d\,e^2-12\,a\,b^3\,d^2\,e+4\,b^4\,d^3\right )}{e^5}-\frac {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\,\left (x\,e^5+d\,e^4\right )} \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)^2,x)

[Out]

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

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