3.1904 \(\int \frac {(a+b x) (a^2+2 a b x+b^2 x^2)}{(d+e x)^4} \, dx\)

Optimal. Leaf size=86 \[ \frac {3 b^2 (b d-a e)}{e^4 (d+e x)}-\frac {3 b (b d-a e)^2}{2 e^4 (d+e x)^2}+\frac {(b d-a e)^3}{3 e^4 (d+e x)^3}+\frac {b^3 \log (d+e x)}{e^4} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {27, 43} \[ \frac {3 b^2 (b d-a e)}{e^4 (d+e x)}-\frac {3 b (b d-a e)^2}{2 e^4 (d+e x)^2}+\frac {(b d-a e)^3}{3 e^4 (d+e x)^3}+\frac {b^3 \log (d+e x)}{e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.04, size = 79, normalized size = 0.92 \[ \frac {\frac {(b d-a e) \left (2 a^2 e^2+a b e (5 d+9 e x)+b^2 \left (11 d^2+27 d e x+18 e^2 x^2\right )\right )}{(d+e x)^3}+6 b^3 \log (d+e x)}{6 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((b*d - a*e)*(2*a^2*e^2 + a*b*e*(5*d + 9*e*x) + b^2*(11*d^2 + 27*d*e*x + 18*e^2*x^2)))/(d + e*x)^3 + 6*b^3*Lo
g[d + e*x])/(6*e^4)

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fricas [B]  time = 0.98, size = 177, normalized size = 2.06 \[ \frac {11 \, b^{3} d^{3} - 6 \, a b^{2} d^{2} e - 3 \, a^{2} b d e^{2} - 2 \, a^{3} e^{3} + 18 \, {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 9 \, {\left (3 \, b^{3} d^{2} e - 2 \, a b^{2} d e^{2} - a^{2} b e^{3}\right )} x + 6 \, {\left (b^{3} e^{3} x^{3} + 3 \, b^{3} d e^{2} x^{2} + 3 \, b^{3} d^{2} e x + b^{3} d^{3}\right )} \log \left (e x + d\right )}{6 \, {\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

1/6*(11*b^3*d^3 - 6*a*b^2*d^2*e - 3*a^2*b*d*e^2 - 2*a^3*e^3 + 18*(b^3*d*e^2 - a*b^2*e^3)*x^2 + 9*(3*b^3*d^2*e
- 2*a*b^2*d*e^2 - a^2*b*e^3)*x + 6*(b^3*e^3*x^3 + 3*b^3*d*e^2*x^2 + 3*b^3*d^2*e*x + b^3*d^3)*log(e*x + d))/(e^
7*x^3 + 3*d*e^6*x^2 + 3*d^2*e^5*x + d^3*e^4)

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giac [A]  time = 0.16, size = 117, normalized size = 1.36 \[ b^{3} e^{\left (-4\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {{\left (18 \, {\left (b^{3} d e - a b^{2} e^{2}\right )} x^{2} + 9 \, {\left (3 \, b^{3} d^{2} - 2 \, a b^{2} d e - a^{2} b e^{2}\right )} x + {\left (11 \, b^{3} d^{3} - 6 \, a b^{2} d^{2} e - 3 \, a^{2} b d e^{2} - 2 \, a^{3} e^{3}\right )} e^{\left (-1\right )}\right )} e^{\left (-3\right )}}{6 \, {\left (x e + d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d)^4,x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.05, size = 166, normalized size = 1.93 \[ -\frac {a^{3}}{3 \left (e x +d \right )^{3} e}+\frac {a^{2} b d}{\left (e x +d \right )^{3} e^{2}}-\frac {a \,b^{2} d^{2}}{\left (e x +d \right )^{3} e^{3}}+\frac {b^{3} d^{3}}{3 \left (e x +d \right )^{3} e^{4}}-\frac {3 a^{2} b}{2 \left (e x +d \right )^{2} e^{2}}+\frac {3 a \,b^{2} d}{\left (e x +d \right )^{2} e^{3}}-\frac {3 b^{3} d^{2}}{2 \left (e x +d \right )^{2} e^{4}}-\frac {3 a \,b^{2}}{\left (e x +d \right ) e^{3}}+\frac {3 b^{3} d}{\left (e x +d \right ) e^{4}}+\frac {b^{3} \ln \left (e x +d \right )}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d)^4,x)

[Out]

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

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maxima [A]  time = 0.46, size = 143, normalized size = 1.66 \[ \frac {11 \, b^{3} d^{3} - 6 \, a b^{2} d^{2} e - 3 \, a^{2} b d e^{2} - 2 \, a^{3} e^{3} + 18 \, {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 9 \, {\left (3 \, b^{3} d^{2} e - 2 \, a b^{2} d e^{2} - a^{2} b e^{3}\right )} x}{6 \, {\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} + \frac {b^{3} \log \left (e x + d\right )}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)/(e*x+d)^4,x, algorithm="maxima")

[Out]

1/6*(11*b^3*d^3 - 6*a*b^2*d^2*e - 3*a^2*b*d*e^2 - 2*a^3*e^3 + 18*(b^3*d*e^2 - a*b^2*e^3)*x^2 + 9*(3*b^3*d^2*e
- 2*a*b^2*d*e^2 - a^2*b*e^3)*x)/(e^7*x^3 + 3*d*e^6*x^2 + 3*d^2*e^5*x + d^3*e^4) + b^3*log(e*x + d)/e^4

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mupad [B]  time = 0.09, size = 138, normalized size = 1.60 \[ \frac {b^3\,\ln \left (d+e\,x\right )}{e^4}-\frac {\frac {2\,a^3\,e^3+3\,a^2\,b\,d\,e^2+6\,a\,b^2\,d^2\,e-11\,b^3\,d^3}{6\,e^4}+\frac {3\,x\,\left (a^2\,b\,e^2+2\,a\,b^2\,d\,e-3\,b^3\,d^2\right )}{2\,e^3}+\frac {3\,b^2\,x^2\,\left (a\,e-b\,d\right )}{e^2}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x))/(d + e*x)^4,x)

[Out]

(b^3*log(d + e*x))/e^4 - ((2*a^3*e^3 - 11*b^3*d^3 + 6*a*b^2*d^2*e + 3*a^2*b*d*e^2)/(6*e^4) + (3*x*(a^2*b*e^2 -
3*b^3*d^2 + 2*a*b^2*d*e))/(2*e^3) + (3*b^2*x^2*(a*e - b*d))/e^2)/(d^3 + e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x)

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sympy [A]  time = 1.18, size = 148, normalized size = 1.72 \[ \frac {b^{3} \log {\left (d + e x \right )}}{e^{4}} + \frac {- 2 a^{3} e^{3} - 3 a^{2} b d e^{2} - 6 a b^{2} d^{2} e + 11 b^{3} d^{3} + x^{2} \left (- 18 a b^{2} e^{3} + 18 b^{3} d e^{2}\right ) + x \left (- 9 a^{2} b e^{3} - 18 a b^{2} d e^{2} + 27 b^{3} d^{2} e\right )}{6 d^{3} e^{4} + 18 d^{2} e^{5} x + 18 d e^{6} x^{2} + 6 e^{7} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)/(e*x+d)**4,x)

[Out]

b**3*log(d + e*x)/e**4 + (-2*a**3*e**3 - 3*a**2*b*d*e**2 - 6*a*b**2*d**2*e + 11*b**3*d**3 + x**2*(-18*a*b**2*e
**3 + 18*b**3*d*e**2) + x*(-9*a**2*b*e**3 - 18*a*b**2*d*e**2 + 27*b**3*d**2*e))/(6*d**3*e**4 + 18*d**2*e**5*x
+ 18*d*e**6*x**2 + 6*e**7*x**3)

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