3.13.30 \(\int \frac {(d+e x)^5}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=138 \[ -\frac {5 e^4 (b d-a e)}{b^6 (a+b x)}-\frac {5 e^3 (b d-a e)^2}{b^6 (a+b x)^2}-\frac {10 e^2 (b d-a e)^3}{3 b^6 (a+b x)^3}-\frac {5 e (b d-a e)^4}{4 b^6 (a+b x)^4}-\frac {(b d-a e)^5}{5 b^6 (a+b x)^5}+\frac {e^5 \log (a+b x)}{b^6} \]

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Rubi [A]  time = 0.12, antiderivative size = 138, 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 {5 e^4 (b d-a e)}{b^6 (a+b x)}-\frac {5 e^3 (b d-a e)^2}{b^6 (a+b x)^2}-\frac {10 e^2 (b d-a e)^3}{3 b^6 (a+b x)^3}-\frac {5 e (b d-a e)^4}{4 b^6 (a+b x)^4}-\frac {(b d-a e)^5}{5 b^6 (a+b x)^5}+\frac {e^5 \log (a+b x)}{b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^5/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

-(b*d - a*e)^5/(5*b^6*(a + b*x)^5) - (5*e*(b*d - a*e)^4)/(4*b^6*(a + b*x)^4) - (10*e^2*(b*d - a*e)^3)/(3*b^6*(
a + b*x)^3) - (5*e^3*(b*d - a*e)^2)/(b^6*(a + b*x)^2) - (5*e^4*(b*d - a*e))/(b^6*(a + b*x)) + (e^5*Log[a + b*x
])/b^6

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 {(d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {(d+e x)^5}{(a+b x)^6} \, dx\\ &=\int \left (\frac {(b d-a e)^5}{b^5 (a+b x)^6}+\frac {5 e (b d-a e)^4}{b^5 (a+b x)^5}+\frac {10 e^2 (b d-a e)^3}{b^5 (a+b x)^4}+\frac {10 e^3 (b d-a e)^2}{b^5 (a+b x)^3}+\frac {5 e^4 (b d-a e)}{b^5 (a+b x)^2}+\frac {e^5}{b^5 (a+b x)}\right ) \, dx\\ &=-\frac {(b d-a e)^5}{5 b^6 (a+b x)^5}-\frac {5 e (b d-a e)^4}{4 b^6 (a+b x)^4}-\frac {10 e^2 (b d-a e)^3}{3 b^6 (a+b x)^3}-\frac {5 e^3 (b d-a e)^2}{b^6 (a+b x)^2}-\frac {5 e^4 (b d-a e)}{b^6 (a+b x)}+\frac {e^5 \log (a+b x)}{b^6}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 171, normalized size = 1.24 \begin {gather*} \frac {e^5 \log (a+b x)}{b^6}-\frac {(b d-a e) \left (137 a^4 e^4+a^3 b e^3 (77 d+625 e x)+a^2 b^2 e^2 \left (47 d^2+325 d e x+1100 e^2 x^2\right )+a b^3 e \left (27 d^3+175 d^2 e x+500 d e^2 x^2+900 e^3 x^3\right )+b^4 \left (12 d^4+75 d^3 e x+200 d^2 e^2 x^2+300 d e^3 x^3+300 e^4 x^4\right )\right )}{60 b^6 (a+b x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^5/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

-1/60*((b*d - a*e)*(137*a^4*e^4 + a^3*b*e^3*(77*d + 625*e*x) + a^2*b^2*e^2*(47*d^2 + 325*d*e*x + 1100*e^2*x^2)
+ a*b^3*e*(27*d^3 + 175*d^2*e*x + 500*d*e^2*x^2 + 900*e^3*x^3) + b^4*(12*d^4 + 75*d^3*e*x + 200*d^2*e^2*x^2 +
300*d*e^3*x^3 + 300*e^4*x^4)))/(b^6*(a + b*x)^5) + (e^5*Log[a + b*x])/b^6

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^5}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)^5/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

IntegrateAlgebraic[(d + e*x)^5/(a^2 + 2*a*b*x + b^2*x^2)^3, x]

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fricas [B]  time = 0.41, size = 372, normalized size = 2.70 \begin {gather*} -\frac {12 \, b^{5} d^{5} + 15 \, a b^{4} d^{4} e + 20 \, a^{2} b^{3} d^{3} e^{2} + 30 \, a^{3} b^{2} d^{2} e^{3} + 60 \, a^{4} b d e^{4} - 137 \, a^{5} e^{5} + 300 \, {\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 300 \, {\left (b^{5} d^{2} e^{3} + 2 \, a b^{4} d e^{4} - 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 100 \, {\left (2 \, b^{5} d^{3} e^{2} + 3 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} - 11 \, a^{3} b^{2} e^{5}\right )} x^{2} + 25 \, {\left (3 \, b^{5} d^{4} e + 4 \, a b^{4} d^{3} e^{2} + 6 \, a^{2} b^{3} d^{2} e^{3} + 12 \, a^{3} b^{2} d e^{4} - 25 \, a^{4} b e^{5}\right )} x - 60 \, {\left (b^{5} e^{5} x^{5} + 5 \, a b^{4} e^{5} x^{4} + 10 \, a^{2} b^{3} e^{5} x^{3} + 10 \, a^{3} b^{2} e^{5} x^{2} + 5 \, a^{4} b e^{5} x + a^{5} e^{5}\right )} \log \left (b x + a\right )}{60 \, {\left (b^{11} x^{5} + 5 \, a b^{10} x^{4} + 10 \, a^{2} b^{9} x^{3} + 10 \, a^{3} b^{8} x^{2} + 5 \, a^{4} b^{7} x + a^{5} b^{6}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

-1/60*(12*b^5*d^5 + 15*a*b^4*d^4*e + 20*a^2*b^3*d^3*e^2 + 30*a^3*b^2*d^2*e^3 + 60*a^4*b*d*e^4 - 137*a^5*e^5 +
300*(b^5*d*e^4 - a*b^4*e^5)*x^4 + 300*(b^5*d^2*e^3 + 2*a*b^4*d*e^4 - 3*a^2*b^3*e^5)*x^3 + 100*(2*b^5*d^3*e^2 +
3*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 - 11*a^3*b^2*e^5)*x^2 + 25*(3*b^5*d^4*e + 4*a*b^4*d^3*e^2 + 6*a^2*b^3*d^2*e
^3 + 12*a^3*b^2*d*e^4 - 25*a^4*b*e^5)*x - 60*(b^5*e^5*x^5 + 5*a*b^4*e^5*x^4 + 10*a^2*b^3*e^5*x^3 + 10*a^3*b^2*
e^5*x^2 + 5*a^4*b*e^5*x + a^5*e^5)*log(b*x + a))/(b^11*x^5 + 5*a*b^10*x^4 + 10*a^2*b^9*x^3 + 10*a^3*b^8*x^2 +
5*a^4*b^7*x + a^5*b^6)

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giac [A]  time = 0.18, size = 248, normalized size = 1.80 \begin {gather*} \frac {e^{5} \log \left ({\left | b x + a \right |}\right )}{b^{6}} - \frac {300 \, {\left (b^{4} d e^{4} - a b^{3} e^{5}\right )} x^{4} + 300 \, {\left (b^{4} d^{2} e^{3} + 2 \, a b^{3} d e^{4} - 3 \, a^{2} b^{2} e^{5}\right )} x^{3} + 100 \, {\left (2 \, b^{4} d^{3} e^{2} + 3 \, a b^{3} d^{2} e^{3} + 6 \, a^{2} b^{2} d e^{4} - 11 \, a^{3} b e^{5}\right )} x^{2} + 25 \, {\left (3 \, b^{4} d^{4} e + 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} + 12 \, a^{3} b d e^{4} - 25 \, a^{4} e^{5}\right )} x + \frac {12 \, b^{5} d^{5} + 15 \, a b^{4} d^{4} e + 20 \, a^{2} b^{3} d^{3} e^{2} + 30 \, a^{3} b^{2} d^{2} e^{3} + 60 \, a^{4} b d e^{4} - 137 \, a^{5} e^{5}}{b}}{60 \, {\left (b x + a\right )}^{5} b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

e^5*log(abs(b*x + a))/b^6 - 1/60*(300*(b^4*d*e^4 - a*b^3*e^5)*x^4 + 300*(b^4*d^2*e^3 + 2*a*b^3*d*e^4 - 3*a^2*b
^2*e^5)*x^3 + 100*(2*b^4*d^3*e^2 + 3*a*b^3*d^2*e^3 + 6*a^2*b^2*d*e^4 - 11*a^3*b*e^5)*x^2 + 25*(3*b^4*d^4*e + 4
*a*b^3*d^3*e^2 + 6*a^2*b^2*d^2*e^3 + 12*a^3*b*d*e^4 - 25*a^4*e^5)*x + (12*b^5*d^5 + 15*a*b^4*d^4*e + 20*a^2*b^
3*d^3*e^2 + 30*a^3*b^2*d^2*e^3 + 60*a^4*b*d*e^4 - 137*a^5*e^5)/b)/((b*x + a)^5*b^5)

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maple [B]  time = 0.05, size = 377, normalized size = 2.73 \begin {gather*} \frac {a^{5} e^{5}}{5 \left (b x +a \right )^{5} b^{6}}-\frac {a^{4} d \,e^{4}}{\left (b x +a \right )^{5} b^{5}}+\frac {2 a^{3} d^{2} e^{3}}{\left (b x +a \right )^{5} b^{4}}-\frac {2 a^{2} d^{3} e^{2}}{\left (b x +a \right )^{5} b^{3}}+\frac {a \,d^{4} e}{\left (b x +a \right )^{5} b^{2}}-\frac {d^{5}}{5 \left (b x +a \right )^{5} b}-\frac {5 a^{4} e^{5}}{4 \left (b x +a \right )^{4} b^{6}}+\frac {5 a^{3} d \,e^{4}}{\left (b x +a \right )^{4} b^{5}}-\frac {15 a^{2} d^{2} e^{3}}{2 \left (b x +a \right )^{4} b^{4}}+\frac {5 a \,d^{3} e^{2}}{\left (b x +a \right )^{4} b^{3}}-\frac {5 d^{4} e}{4 \left (b x +a \right )^{4} b^{2}}+\frac {10 a^{3} e^{5}}{3 \left (b x +a \right )^{3} b^{6}}-\frac {10 a^{2} d \,e^{4}}{\left (b x +a \right )^{3} b^{5}}+\frac {10 a \,d^{2} e^{3}}{\left (b x +a \right )^{3} b^{4}}-\frac {10 d^{3} e^{2}}{3 \left (b x +a \right )^{3} b^{3}}-\frac {5 a^{2} e^{5}}{\left (b x +a \right )^{2} b^{6}}+\frac {10 a d \,e^{4}}{\left (b x +a \right )^{2} b^{5}}-\frac {5 d^{2} e^{3}}{\left (b x +a \right )^{2} b^{4}}+\frac {5 a \,e^{5}}{\left (b x +a \right ) b^{6}}-\frac {5 d \,e^{4}}{\left (b x +a \right ) b^{5}}+\frac {e^{5} \ln \left (b x +a \right )}{b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

1/5/b^6/(b*x+a)^5*a^5*e^5-1/b^5/(b*x+a)^5*a^4*d*e^4+2/b^4/(b*x+a)^5*a^3*d^2*e^3-2/b^3/(b*x+a)^5*a^2*d^3*e^2+1/
b^2/(b*x+a)^5*a*d^4*e-1/5/b/(b*x+a)^5*d^5+10/3*e^5/b^6/(b*x+a)^3*a^3-10*e^4/b^5/(b*x+a)^3*a^2*d+10*e^3/b^4/(b*
x+a)^3*a*d^2-10/3*e^2/b^3/(b*x+a)^3*d^3+e^5*ln(b*x+a)/b^6-5/4*e^5/b^6/(b*x+a)^4*a^4+5*e^4/b^5/(b*x+a)^4*d*a^3-
15/2*e^3/b^4/(b*x+a)^4*a^2*d^2+5*e^2/b^3/(b*x+a)^4*a*d^3-5/4*e/b^2/(b*x+a)^4*d^4+5/b^6*e^5/(b*x+a)*a-5/b^5*e^4
/(b*x+a)*d-5*e^5/b^6/(b*x+a)^2*a^2+10*e^4/b^5/(b*x+a)^2*a*d-5*e^3/b^4/(b*x+a)^2*d^2

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maxima [B]  time = 1.58, size = 310, normalized size = 2.25 \begin {gather*} -\frac {12 \, b^{5} d^{5} + 15 \, a b^{4} d^{4} e + 20 \, a^{2} b^{3} d^{3} e^{2} + 30 \, a^{3} b^{2} d^{2} e^{3} + 60 \, a^{4} b d e^{4} - 137 \, a^{5} e^{5} + 300 \, {\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 300 \, {\left (b^{5} d^{2} e^{3} + 2 \, a b^{4} d e^{4} - 3 \, a^{2} b^{3} e^{5}\right )} x^{3} + 100 \, {\left (2 \, b^{5} d^{3} e^{2} + 3 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} - 11 \, a^{3} b^{2} e^{5}\right )} x^{2} + 25 \, {\left (3 \, b^{5} d^{4} e + 4 \, a b^{4} d^{3} e^{2} + 6 \, a^{2} b^{3} d^{2} e^{3} + 12 \, a^{3} b^{2} d e^{4} - 25 \, a^{4} b e^{5}\right )} x}{60 \, {\left (b^{11} x^{5} + 5 \, a b^{10} x^{4} + 10 \, a^{2} b^{9} x^{3} + 10 \, a^{3} b^{8} x^{2} + 5 \, a^{4} b^{7} x + a^{5} b^{6}\right )}} + \frac {e^{5} \log \left (b x + a\right )}{b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

-1/60*(12*b^5*d^5 + 15*a*b^4*d^4*e + 20*a^2*b^3*d^3*e^2 + 30*a^3*b^2*d^2*e^3 + 60*a^4*b*d*e^4 - 137*a^5*e^5 +
300*(b^5*d*e^4 - a*b^4*e^5)*x^4 + 300*(b^5*d^2*e^3 + 2*a*b^4*d*e^4 - 3*a^2*b^3*e^5)*x^3 + 100*(2*b^5*d^3*e^2 +
3*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 - 11*a^3*b^2*e^5)*x^2 + 25*(3*b^5*d^4*e + 4*a*b^4*d^3*e^2 + 6*a^2*b^3*d^2*e
^3 + 12*a^3*b^2*d*e^4 - 25*a^4*b*e^5)*x)/(b^11*x^5 + 5*a*b^10*x^4 + 10*a^2*b^9*x^3 + 10*a^3*b^8*x^2 + 5*a^4*b^
7*x + a^5*b^6) + e^5*log(b*x + a)/b^6

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mupad [B]  time = 0.60, size = 261, normalized size = 1.89 \begin {gather*} \frac {e^5\,\ln \left (a+b\,x\right )}{b^6}-\frac {x\,\left (-\frac {125\,a^4\,b\,e^5}{12}+5\,a^3\,b^2\,d\,e^4+\frac {5\,a^2\,b^3\,d^2\,e^3}{2}+\frac {5\,a\,b^4\,d^3\,e^2}{3}+\frac {5\,b^5\,d^4\,e}{4}\right )-x^4\,\left (5\,a\,b^4\,e^5-5\,b^5\,d\,e^4\right )+x^3\,\left (-15\,a^2\,b^3\,e^5+10\,a\,b^4\,d\,e^4+5\,b^5\,d^2\,e^3\right )-\frac {137\,a^5\,e^5}{60}+\frac {b^5\,d^5}{5}+x^2\,\left (-\frac {55\,a^3\,b^2\,e^5}{3}+10\,a^2\,b^3\,d\,e^4+5\,a\,b^4\,d^2\,e^3+\frac {10\,b^5\,d^3\,e^2}{3}\right )+\frac {a^2\,b^3\,d^3\,e^2}{3}+\frac {a^3\,b^2\,d^2\,e^3}{2}+\frac {a\,b^4\,d^4\,e}{4}+a^4\,b\,d\,e^4}{b^6\,{\left (a+b\,x\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^5/(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

(e^5*log(a + b*x))/b^6 - (x*((5*b^5*d^4*e)/4 - (125*a^4*b*e^5)/12 + (5*a*b^4*d^3*e^2)/3 + 5*a^3*b^2*d*e^4 + (5
*a^2*b^3*d^2*e^3)/2) - x^4*(5*a*b^4*e^5 - 5*b^5*d*e^4) + x^3*(5*b^5*d^2*e^3 - 15*a^2*b^3*e^5 + 10*a*b^4*d*e^4)
- (137*a^5*e^5)/60 + (b^5*d^5)/5 + x^2*((10*b^5*d^3*e^2)/3 - (55*a^3*b^2*e^5)/3 + 5*a*b^4*d^2*e^3 + 10*a^2*b^
3*d*e^4) + (a^2*b^3*d^3*e^2)/3 + (a^3*b^2*d^2*e^3)/2 + (a*b^4*d^4*e)/4 + a^4*b*d*e^4)/(b^6*(a + b*x)^5)

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sympy [B]  time = 9.86, size = 326, normalized size = 2.36 \begin {gather*} \frac {137 a^{5} e^{5} - 60 a^{4} b d e^{4} - 30 a^{3} b^{2} d^{2} e^{3} - 20 a^{2} b^{3} d^{3} e^{2} - 15 a b^{4} d^{4} e - 12 b^{5} d^{5} + x^{4} \left (300 a b^{4} e^{5} - 300 b^{5} d e^{4}\right ) + x^{3} \left (900 a^{2} b^{3} e^{5} - 600 a b^{4} d e^{4} - 300 b^{5} d^{2} e^{3}\right ) + x^{2} \left (1100 a^{3} b^{2} e^{5} - 600 a^{2} b^{3} d e^{4} - 300 a b^{4} d^{2} e^{3} - 200 b^{5} d^{3} e^{2}\right ) + x \left (625 a^{4} b e^{5} - 300 a^{3} b^{2} d e^{4} - 150 a^{2} b^{3} d^{2} e^{3} - 100 a b^{4} d^{3} e^{2} - 75 b^{5} d^{4} e\right )}{60 a^{5} b^{6} + 300 a^{4} b^{7} x + 600 a^{3} b^{8} x^{2} + 600 a^{2} b^{9} x^{3} + 300 a b^{10} x^{4} + 60 b^{11} x^{5}} + \frac {e^{5} \log {\left (a + b x \right )}}{b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**5/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

(137*a**5*e**5 - 60*a**4*b*d*e**4 - 30*a**3*b**2*d**2*e**3 - 20*a**2*b**3*d**3*e**2 - 15*a*b**4*d**4*e - 12*b*
*5*d**5 + x**4*(300*a*b**4*e**5 - 300*b**5*d*e**4) + x**3*(900*a**2*b**3*e**5 - 600*a*b**4*d*e**4 - 300*b**5*d
**2*e**3) + x**2*(1100*a**3*b**2*e**5 - 600*a**2*b**3*d*e**4 - 300*a*b**4*d**2*e**3 - 200*b**5*d**3*e**2) + x*
(625*a**4*b*e**5 - 300*a**3*b**2*d*e**4 - 150*a**2*b**3*d**2*e**3 - 100*a*b**4*d**3*e**2 - 75*b**5*d**4*e))/(6
0*a**5*b**6 + 300*a**4*b**7*x + 600*a**3*b**8*x**2 + 600*a**2*b**9*x**3 + 300*a*b**10*x**4 + 60*b**11*x**5) +
e**5*log(a + b*x)/b**6

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