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

Optimal. Leaf size=106 \[ \frac {b^3 \log (a+b x)}{(b d-a e)^4}-\frac {b^3 \log (d+e x)}{(b d-a e)^4}+\frac {b^2}{(d+e x) (b d-a e)^3}+\frac {b}{2 (d+e x)^2 (b d-a e)^2}+\frac {1}{3 (d+e x)^3 (b d-a e)} \]

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Rubi [A]  time = 0.07, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 44} \begin {gather*} \frac {b^2}{(d+e x) (b d-a e)^3}+\frac {b^3 \log (a+b x)}{(b d-a e)^4}-\frac {b^3 \log (d+e x)}{(b d-a e)^4}+\frac {b}{2 (d+e x)^2 (b d-a e)^2}+\frac {1}{3 (d+e x)^3 (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

1/(3*(b*d - a*e)*(d + e*x)^3) + b/(2*(b*d - a*e)^2*(d + e*x)^2) + b^2/((b*d - a*e)^3*(d + e*x)) + (b^3*Log[a +
b*x])/(b*d - a*e)^4 - (b^3*Log[d + e*x])/(b*d - a*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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {a+b x}{(d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac {1}{(a+b x) (d+e x)^4} \, dx\\ &=\int \left (\frac {b^4}{(b d-a e)^4 (a+b x)}-\frac {e}{(b d-a e) (d+e x)^4}-\frac {b e}{(b d-a e)^2 (d+e x)^3}-\frac {b^2 e}{(b d-a e)^3 (d+e x)^2}-\frac {b^3 e}{(b d-a e)^4 (d+e x)}\right ) \, dx\\ &=\frac {1}{3 (b d-a e) (d+e x)^3}+\frac {b}{2 (b d-a e)^2 (d+e x)^2}+\frac {b^2}{(b d-a e)^3 (d+e x)}+\frac {b^3 \log (a+b x)}{(b d-a e)^4}-\frac {b^3 \log (d+e x)}{(b d-a e)^4}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 106, normalized size = 1.00 \begin {gather*} \frac {b^3 \log (a+b x)}{(b d-a e)^4}-\frac {b^3 \log (d+e x)}{(b d-a e)^4}+\frac {b^2}{(d+e x) (b d-a e)^3}+\frac {b}{2 (d+e x)^2 (b d-a e)^2}-\frac {1}{3 (d+e x)^3 (a e-b d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*1/((-(b*d) + a*e)*(d + e*x)^3) + b/(2*(b*d - a*e)^2*(d + e*x)^2) + b^2/((b*d - a*e)^3*(d + e*x)) + (b^3*L
og[a + b*x])/(b*d - a*e)^4 - (b^3*Log[d + e*x])/(b*d - a*e)^4

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b x}{(d+e x)^4 \left (a^2+2 a b x+b^2 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.40, size = 425, normalized size = 4.01 \begin {gather*} \frac {11 \, b^{3} d^{3} - 18 \, a b^{2} d^{2} e + 9 \, a^{2} b d e^{2} - 2 \, a^{3} e^{3} + 6 \, {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 3 \, {\left (5 \, b^{3} d^{2} e - 6 \, 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 (b x + a\right ) - 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 (b^{4} d^{7} - 4 \, a b^{3} d^{6} e + 6 \, a^{2} b^{2} d^{5} e^{2} - 4 \, a^{3} b d^{4} e^{3} + a^{4} d^{3} e^{4} + {\left (b^{4} d^{4} e^{3} - 4 \, a b^{3} d^{3} e^{4} + 6 \, a^{2} b^{2} d^{2} e^{5} - 4 \, a^{3} b d e^{6} + a^{4} e^{7}\right )} x^{3} + 3 \, {\left (b^{4} d^{5} e^{2} - 4 \, a b^{3} d^{4} e^{3} + 6 \, a^{2} b^{2} d^{3} e^{4} - 4 \, a^{3} b d^{2} e^{5} + a^{4} d e^{6}\right )} x^{2} + 3 \, {\left (b^{4} d^{6} e - 4 \, a b^{3} d^{5} e^{2} + 6 \, a^{2} b^{2} d^{4} e^{3} - 4 \, a^{3} b d^{3} e^{4} + a^{4} d^{2} e^{5}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.17, size = 238, normalized size = 2.25 \begin {gather*} \frac {b^{4} \log \left ({\left | b x + a \right |}\right )}{b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}} - \frac {b^{3} e \log \left ({\left | x e + d \right |}\right )}{b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}} + \frac {11 \, b^{3} d^{3} - 18 \, a b^{2} d^{2} e + 9 \, a^{2} b d e^{2} - 2 \, a^{3} e^{3} + 6 \, {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 3 \, {\left (5 \, b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{6 \, {\left (b d - a e\right )}^{4} {\left (x e + d\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.10, size = 104, normalized size = 0.98 \begin {gather*} \frac {b^{3} \ln \left (b x +a \right )}{\left (a e -b d \right )^{4}}-\frac {b^{3} \ln \left (e x +d \right )}{\left (a e -b d \right )^{4}}-\frac {b^{2}}{\left (a e -b d \right )^{3} \left (e x +d \right )}+\frac {b}{2 \left (a e -b d \right )^{2} \left (e x +d \right )^{2}}-\frac {1}{3 \left (a e -b d \right ) \left (e x +d \right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.62, size = 362, normalized size = 3.42 \begin {gather*} \frac {b^{3} \log \left (b x + a\right )}{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}} - \frac {b^{3} \log \left (e x + d\right )}{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}} + \frac {6 \, b^{2} e^{2} x^{2} + 11 \, b^{2} d^{2} - 7 \, a b d e + 2 \, a^{2} e^{2} + 3 \, {\left (5 \, b^{2} d e - a b e^{2}\right )} x}{6 \, {\left (b^{3} d^{6} - 3 \, a b^{2} d^{5} e + 3 \, a^{2} b d^{4} e^{2} - a^{3} d^{3} e^{3} + {\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}\right )} x^{3} + 3 \, {\left (b^{3} d^{4} e^{2} - 3 \, a b^{2} d^{3} e^{3} + 3 \, a^{2} b d^{2} e^{4} - a^{3} d e^{5}\right )} x^{2} + 3 \, {\left (b^{3} d^{5} e - 3 \, a b^{2} d^{4} e^{2} + 3 \, a^{2} b d^{3} e^{3} - a^{3} d^{2} e^{4}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.22, size = 313, normalized size = 2.95 \begin {gather*} \frac {2\,b^3\,\mathrm {atanh}\left (\frac {a^4\,e^4-2\,a^3\,b\,d\,e^3+2\,a\,b^3\,d^3\,e-b^4\,d^4}{{\left (a\,e-b\,d\right )}^4}+\frac {2\,b\,e\,x\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{{\left (a\,e-b\,d\right )}^4}\right )}{{\left (a\,e-b\,d\right )}^4}-\frac {\frac {2\,a^2\,e^2-7\,a\,b\,d\,e+11\,b^2\,d^2}{6\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}-\frac {b\,x\,\left (a\,e^2-5\,b\,d\,e\right )}{2\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}+\frac {b^2\,e^2\,x^2}{a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*b^3*atanh((a^4*e^4 - b^4*d^4 + 2*a*b^3*d^3*e - 2*a^3*b*d*e^3)/(a*e - b*d)^4 + (2*b*e*x*(a^3*e^3 - b^3*d^3 +
3*a*b^2*d^2*e - 3*a^2*b*d*e^2))/(a*e - b*d)^4))/(a*e - b*d)^4 - ((2*a^2*e^2 + 11*b^2*d^2 - 7*a*b*d*e)/(6*(a^3
*e^3 - b^3*d^3 + 3*a*b^2*d^2*e - 3*a^2*b*d*e^2)) - (b*x*(a*e^2 - 5*b*d*e))/(2*(a^3*e^3 - b^3*d^3 + 3*a*b^2*d^2
*e - 3*a^2*b*d*e^2)) + (b^2*e^2*x^2)/(a^3*e^3 - b^3*d^3 + 3*a*b^2*d^2*e - 3*a^2*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 [B]  time = 1.53, size = 570, normalized size = 5.38 \begin {gather*} - \frac {b^{3} \log {\left (x + \frac {- \frac {a^{5} b^{3} e^{5}}{\left (a e - b d\right )^{4}} + \frac {5 a^{4} b^{4} d e^{4}}{\left (a e - b d\right )^{4}} - \frac {10 a^{3} b^{5} d^{2} e^{3}}{\left (a e - b d\right )^{4}} + \frac {10 a^{2} b^{6} d^{3} e^{2}}{\left (a e - b d\right )^{4}} - \frac {5 a b^{7} d^{4} e}{\left (a e - b d\right )^{4}} + a b^{3} e + \frac {b^{8} d^{5}}{\left (a e - b d\right )^{4}} + b^{4} d}{2 b^{4} e} \right )}}{\left (a e - b d\right )^{4}} + \frac {b^{3} \log {\left (x + \frac {\frac {a^{5} b^{3} e^{5}}{\left (a e - b d\right )^{4}} - \frac {5 a^{4} b^{4} d e^{4}}{\left (a e - b d\right )^{4}} + \frac {10 a^{3} b^{5} d^{2} e^{3}}{\left (a e - b d\right )^{4}} - \frac {10 a^{2} b^{6} d^{3} e^{2}}{\left (a e - b d\right )^{4}} + \frac {5 a b^{7} d^{4} e}{\left (a e - b d\right )^{4}} + a b^{3} e - \frac {b^{8} d^{5}}{\left (a e - b d\right )^{4}} + b^{4} d}{2 b^{4} e} \right )}}{\left (a e - b d\right )^{4}} + \frac {- 2 a^{2} e^{2} + 7 a b d e - 11 b^{2} d^{2} - 6 b^{2} e^{2} x^{2} + x \left (3 a b e^{2} - 15 b^{2} d e\right )}{6 a^{3} d^{3} e^{3} - 18 a^{2} b d^{4} e^{2} + 18 a b^{2} d^{5} e - 6 b^{3} d^{6} + x^{3} \left (6 a^{3} e^{6} - 18 a^{2} b d e^{5} + 18 a b^{2} d^{2} e^{4} - 6 b^{3} d^{3} e^{3}\right ) + x^{2} \left (18 a^{3} d e^{5} - 54 a^{2} b d^{2} e^{4} + 54 a b^{2} d^{3} e^{3} - 18 b^{3} d^{4} e^{2}\right ) + x \left (18 a^{3} d^{2} e^{4} - 54 a^{2} b d^{3} e^{3} + 54 a b^{2} d^{4} e^{2} - 18 b^{3} d^{5} e\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b**3*log(x + (-a**5*b**3*e**5/(a*e - b*d)**4 + 5*a**4*b**4*d*e**4/(a*e - b*d)**4 - 10*a**3*b**5*d**2*e**3/(a*
e - b*d)**4 + 10*a**2*b**6*d**3*e**2/(a*e - b*d)**4 - 5*a*b**7*d**4*e/(a*e - b*d)**4 + a*b**3*e + b**8*d**5/(a
*e - b*d)**4 + b**4*d)/(2*b**4*e))/(a*e - b*d)**4 + b**3*log(x + (a**5*b**3*e**5/(a*e - b*d)**4 - 5*a**4*b**4*
d*e**4/(a*e - b*d)**4 + 10*a**3*b**5*d**2*e**3/(a*e - b*d)**4 - 10*a**2*b**6*d**3*e**2/(a*e - b*d)**4 + 5*a*b*
*7*d**4*e/(a*e - b*d)**4 + a*b**3*e - b**8*d**5/(a*e - b*d)**4 + b**4*d)/(2*b**4*e))/(a*e - b*d)**4 + (-2*a**2
*e**2 + 7*a*b*d*e - 11*b**2*d**2 - 6*b**2*e**2*x**2 + x*(3*a*b*e**2 - 15*b**2*d*e))/(6*a**3*d**3*e**3 - 18*a**
2*b*d**4*e**2 + 18*a*b**2*d**5*e - 6*b**3*d**6 + x**3*(6*a**3*e**6 - 18*a**2*b*d*e**5 + 18*a*b**2*d**2*e**4 -
6*b**3*d**3*e**3) + x**2*(18*a**3*d*e**5 - 54*a**2*b*d**2*e**4 + 54*a*b**2*d**3*e**3 - 18*b**3*d**4*e**2) + x*
(18*a**3*d**2*e**4 - 54*a**2*b*d**3*e**3 + 54*a*b**2*d**4*e**2 - 18*b**3*d**5*e))

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