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

Optimal. Leaf size=131 \[ \frac {5 e^4 (a+b x)^3 (b d-a e)}{3 b^6}+\frac {5 e^3 (a+b x)^2 (b d-a e)^2}{b^6}-\frac {(b d-a e)^5}{b^6 (a+b x)}+\frac {5 e (b d-a e)^4 \log (a+b x)}{b^6}+\frac {e^5 (a+b x)^4}{4 b^6}+\frac {10 e^2 x (b d-a e)^3}{b^5} \]

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Rubi [A]  time = 0.15, antiderivative size = 131, 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 (a+b x)^3 (b d-a e)}{3 b^6}+\frac {5 e^3 (a+b x)^2 (b d-a e)^2}{b^6}+\frac {10 e^2 x (b d-a e)^3}{b^5}-\frac {(b d-a e)^5}{b^6 (a+b x)}+\frac {5 e (b d-a e)^4 \log (a+b x)}{b^6}+\frac {e^5 (a+b x)^4}{4 b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.08, size = 230, normalized size = 1.76 \begin {gather*} \frac {12 a^5 e^5-12 a^4 b e^4 (5 d+4 e x)+30 a^3 b^2 e^3 \left (4 d^2+6 d e x-e^2 x^2\right )+10 a^2 b^3 e^2 \left (-12 d^3-24 d^2 e x+12 d e^2 x^2+e^3 x^3\right )-5 a b^4 e \left (-12 d^4-24 d^3 e x+36 d^2 e^2 x^2+8 d e^3 x^3+e^4 x^4\right )+60 e (a+b x) (b d-a e)^4 \log (a+b x)+b^5 \left (-12 d^5+120 d^3 e^2 x^2+60 d^2 e^3 x^3+20 d e^4 x^4+3 e^5 x^5\right )}{12 b^6 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(12*a^5*e^5 - 12*a^4*b*e^4*(5*d + 4*e*x) + 30*a^3*b^2*e^3*(4*d^2 + 6*d*e*x - e^2*x^2) + 10*a^2*b^3*e^2*(-12*d^
3 - 24*d^2*e*x + 12*d*e^2*x^2 + e^3*x^3) - 5*a*b^4*e*(-12*d^4 - 24*d^3*e*x + 36*d^2*e^2*x^2 + 8*d*e^3*x^3 + e^
4*x^4) + b^5*(-12*d^5 + 120*d^3*e^2*x^2 + 60*d^2*e^3*x^3 + 20*d*e^4*x^4 + 3*e^5*x^5) + 60*e*(b*d - a*e)^4*(a +
b*x)*Log[a + b*x])/(12*b^6*(a + b*x))

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

[Out]

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

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fricas [B]  time = 0.39, size = 375, normalized size = 2.86 \begin {gather*} \frac {3 \, b^{5} e^{5} x^{5} - 12 \, b^{5} d^{5} + 60 \, a b^{4} d^{4} e - 120 \, a^{2} b^{3} d^{3} e^{2} + 120 \, a^{3} b^{2} d^{2} e^{3} - 60 \, a^{4} b d e^{4} + 12 \, a^{5} e^{5} + 5 \, {\left (4 \, b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 10 \, {\left (6 \, b^{5} d^{2} e^{3} - 4 \, a b^{4} d e^{4} + a^{2} b^{3} e^{5}\right )} x^{3} + 30 \, {\left (4 \, b^{5} d^{3} e^{2} - 6 \, a b^{4} d^{2} e^{3} + 4 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 12 \, {\left (10 \, a b^{4} d^{3} e^{2} - 20 \, a^{2} b^{3} d^{2} e^{3} + 15 \, a^{3} b^{2} d e^{4} - 4 \, a^{4} b e^{5}\right )} x + 60 \, {\left (a b^{4} d^{4} e - 4 \, a^{2} b^{3} d^{3} e^{2} + 6 \, a^{3} b^{2} d^{2} e^{3} - 4 \, a^{4} b d e^{4} + a^{5} e^{5} + {\left (b^{5} d^{4} e - 4 \, a b^{4} d^{3} e^{2} + 6 \, a^{2} b^{3} d^{2} e^{3} - 4 \, a^{3} b^{2} d e^{4} + a^{4} b e^{5}\right )} x\right )} \log \left (b x + a\right )}{12 \, {\left (b^{7} x + a 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),x, algorithm="fricas")

[Out]

1/12*(3*b^5*e^5*x^5 - 12*b^5*d^5 + 60*a*b^4*d^4*e - 120*a^2*b^3*d^3*e^2 + 120*a^3*b^2*d^2*e^3 - 60*a^4*b*d*e^4
+ 12*a^5*e^5 + 5*(4*b^5*d*e^4 - a*b^4*e^5)*x^4 + 10*(6*b^5*d^2*e^3 - 4*a*b^4*d*e^4 + a^2*b^3*e^5)*x^3 + 30*(4
*b^5*d^3*e^2 - 6*a*b^4*d^2*e^3 + 4*a^2*b^3*d*e^4 - a^3*b^2*e^5)*x^2 + 12*(10*a*b^4*d^3*e^2 - 20*a^2*b^3*d^2*e^
3 + 15*a^3*b^2*d*e^4 - 4*a^4*b*e^5)*x + 60*(a*b^4*d^4*e - 4*a^2*b^3*d^3*e^2 + 6*a^3*b^2*d^2*e^3 - 4*a^4*b*d*e^
4 + a^5*e^5 + (b^5*d^4*e - 4*a*b^4*d^3*e^2 + 6*a^2*b^3*d^2*e^3 - 4*a^3*b^2*d*e^4 + a^4*b*e^5)*x)*log(b*x + a))
/(b^7*x + a*b^6)

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giac [B]  time = 0.19, size = 257, normalized size = 1.96 \begin {gather*} \frac {5 \, {\left (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}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} - \frac {b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}}{{\left (b x + a\right )} b^{6}} + \frac {3 \, b^{6} x^{4} e^{5} + 20 \, b^{6} d x^{3} e^{4} + 60 \, b^{6} d^{2} x^{2} e^{3} + 120 \, b^{6} d^{3} x e^{2} - 8 \, a b^{5} x^{3} e^{5} - 60 \, a b^{5} d x^{2} e^{4} - 240 \, a b^{5} d^{2} x e^{3} + 18 \, a^{2} b^{4} x^{2} e^{5} + 180 \, a^{2} b^{4} d x e^{4} - 48 \, a^{3} b^{3} x e^{5}}{12 \, b^{8}} \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),x, algorithm="giac")

[Out]

5*(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)*log(abs(b*x + a))/b^6 - (b^5*d^5
- 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)/((b*x + a)*b^6) + 1/12*(
3*b^6*x^4*e^5 + 20*b^6*d*x^3*e^4 + 60*b^6*d^2*x^2*e^3 + 120*b^6*d^3*x*e^2 - 8*a*b^5*x^3*e^5 - 60*a*b^5*d*x^2*e
^4 - 240*a*b^5*d^2*x*e^3 + 18*a^2*b^4*x^2*e^5 + 180*a^2*b^4*d*x*e^4 - 48*a^3*b^3*x*e^5)/b^8

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maple [B]  time = 0.05, size = 326, normalized size = 2.49 \begin {gather*} \frac {e^{5} x^{4}}{4 b^{2}}-\frac {2 a \,e^{5} x^{3}}{3 b^{3}}+\frac {5 d \,e^{4} x^{3}}{3 b^{2}}+\frac {3 a^{2} e^{5} x^{2}}{2 b^{4}}-\frac {5 a d \,e^{4} x^{2}}{b^{3}}+\frac {5 d^{2} e^{3} x^{2}}{b^{2}}+\frac {a^{5} e^{5}}{\left (b x +a \right ) b^{6}}-\frac {5 a^{4} d \,e^{4}}{\left (b x +a \right ) b^{5}}+\frac {5 a^{4} e^{5} \ln \left (b x +a \right )}{b^{6}}+\frac {10 a^{3} d^{2} e^{3}}{\left (b x +a \right ) b^{4}}-\frac {20 a^{3} d \,e^{4} \ln \left (b x +a \right )}{b^{5}}-\frac {4 a^{3} e^{5} x}{b^{5}}-\frac {10 a^{2} d^{3} e^{2}}{\left (b x +a \right ) b^{3}}+\frac {30 a^{2} d^{2} e^{3} \ln \left (b x +a \right )}{b^{4}}+\frac {15 a^{2} d \,e^{4} x}{b^{4}}+\frac {5 a \,d^{4} e}{\left (b x +a \right ) b^{2}}-\frac {20 a \,d^{3} e^{2} \ln \left (b x +a \right )}{b^{3}}-\frac {20 a \,d^{2} e^{3} x}{b^{3}}-\frac {d^{5}}{\left (b x +a \right ) b}+\frac {5 d^{4} e \ln \left (b x +a \right )}{b^{2}}+\frac {10 d^{3} e^{2} x}{b^{2}} \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),x)

[Out]

1/4*e^5/b^2*x^4-2/3*e^5/b^3*x^3*a+5/3*e^4/b^2*x^3*d+3/2*e^5/b^4*x^2*a^2-5*e^4/b^3*x^2*a*d+5*e^3/b^2*x^2*d^2-4*
e^5/b^5*a^3*x+15*e^4/b^4*a^2*d*x-20*e^3/b^3*a*d^2*x+10*e^2/b^2*d^3*x+5/b^6*e^5*ln(b*x+a)*a^4-20/b^5*e^4*ln(b*x
+a)*d*a^3+30/b^4*e^3*ln(b*x+a)*a^2*d^2-20/b^3*e^2*ln(b*x+a)*a*d^3+5/b^2*e*ln(b*x+a)*d^4+1/b^6/(b*x+a)*a^5*e^5-
5/b^5/(b*x+a)*a^4*d*e^4+10/b^4/(b*x+a)*a^3*d^2*e^3-10/b^3/(b*x+a)*a^2*d^3*e^2+5/b^2/(b*x+a)*a*d^4*e-1/b/(b*x+a
)*d^5

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maxima [B]  time = 1.39, size = 265, normalized size = 2.02 \begin {gather*} -\frac {b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}}{b^{7} x + a b^{6}} + \frac {3 \, b^{3} e^{5} x^{4} + 4 \, {\left (5 \, b^{3} d e^{4} - 2 \, a b^{2} e^{5}\right )} x^{3} + 6 \, {\left (10 \, b^{3} d^{2} e^{3} - 10 \, a b^{2} d e^{4} + 3 \, a^{2} b e^{5}\right )} x^{2} + 12 \, {\left (10 \, b^{3} d^{3} e^{2} - 20 \, a b^{2} d^{2} e^{3} + 15 \, a^{2} b d e^{4} - 4 \, a^{3} e^{5}\right )} x}{12 \, b^{5}} + \frac {5 \, {\left (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}\right )} \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),x, algorithm="maxima")

[Out]

-(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)/(b^7*x + a*b^6)
+ 1/12*(3*b^3*e^5*x^4 + 4*(5*b^3*d*e^4 - 2*a*b^2*e^5)*x^3 + 6*(10*b^3*d^2*e^3 - 10*a*b^2*d*e^4 + 3*a^2*b*e^5)
*x^2 + 12*(10*b^3*d^3*e^2 - 20*a*b^2*d^2*e^3 + 15*a^2*b*d*e^4 - 4*a^3*e^5)*x)/b^5 + 5*(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)*log(b*x + a)/b^6

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mupad [B]  time = 0.55, size = 326, normalized size = 2.49 \begin {gather*} x\,\left (\frac {10\,d^3\,e^2}{b^2}-\frac {2\,a\,\left (\frac {2\,a\,\left (\frac {2\,a\,e^5}{b^3}-\frac {5\,d\,e^4}{b^2}\right )}{b}-\frac {a^2\,e^5}{b^4}+\frac {10\,d^2\,e^3}{b^2}\right )}{b}+\frac {a^2\,\left (\frac {2\,a\,e^5}{b^3}-\frac {5\,d\,e^4}{b^2}\right )}{b^2}\right )-x^3\,\left (\frac {2\,a\,e^5}{3\,b^3}-\frac {5\,d\,e^4}{3\,b^2}\right )+x^2\,\left (\frac {a\,\left (\frac {2\,a\,e^5}{b^3}-\frac {5\,d\,e^4}{b^2}\right )}{b}-\frac {a^2\,e^5}{2\,b^4}+\frac {5\,d^2\,e^3}{b^2}\right )+\frac {\ln \left (a+b\,x\right )\,\left (5\,a^4\,e^5-20\,a^3\,b\,d\,e^4+30\,a^2\,b^2\,d^2\,e^3-20\,a\,b^3\,d^3\,e^2+5\,b^4\,d^4\,e\right )}{b^6}+\frac {a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5}{b\,\left (x\,b^6+a\,b^5\right )}+\frac {e^5\,x^4}{4\,b^2} \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),x)

[Out]

x*((10*d^3*e^2)/b^2 - (2*a*((2*a*((2*a*e^5)/b^3 - (5*d*e^4)/b^2))/b - (a^2*e^5)/b^4 + (10*d^2*e^3)/b^2))/b + (
a^2*((2*a*e^5)/b^3 - (5*d*e^4)/b^2))/b^2) - x^3*((2*a*e^5)/(3*b^3) - (5*d*e^4)/(3*b^2)) + x^2*((a*((2*a*e^5)/b
^3 - (5*d*e^4)/b^2))/b - (a^2*e^5)/(2*b^4) + (5*d^2*e^3)/b^2) + (log(a + b*x)*(5*a^4*e^5 + 5*b^4*d^4*e - 20*a*
b^3*d^3*e^2 + 30*a^2*b^2*d^2*e^3 - 20*a^3*b*d*e^4))/b^6 + (a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2
*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4)/(b*(a*b^5 + b^6*x)) + (e^5*x^4)/(4*b^2)

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sympy [A]  time = 0.94, size = 231, normalized size = 1.76 \begin {gather*} x^{3} \left (- \frac {2 a e^{5}}{3 b^{3}} + \frac {5 d e^{4}}{3 b^{2}}\right ) + x^{2} \left (\frac {3 a^{2} e^{5}}{2 b^{4}} - \frac {5 a d e^{4}}{b^{3}} + \frac {5 d^{2} e^{3}}{b^{2}}\right ) + x \left (- \frac {4 a^{3} e^{5}}{b^{5}} + \frac {15 a^{2} d e^{4}}{b^{4}} - \frac {20 a d^{2} e^{3}}{b^{3}} + \frac {10 d^{3} e^{2}}{b^{2}}\right ) + \frac {a^{5} e^{5} - 5 a^{4} b d e^{4} + 10 a^{3} b^{2} d^{2} e^{3} - 10 a^{2} b^{3} d^{3} e^{2} + 5 a b^{4} d^{4} e - b^{5} d^{5}}{a b^{6} + b^{7} x} + \frac {e^{5} x^{4}}{4 b^{2}} + \frac {5 e \left (a e - b d\right )^{4} \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),x)

[Out]

x**3*(-2*a*e**5/(3*b**3) + 5*d*e**4/(3*b**2)) + x**2*(3*a**2*e**5/(2*b**4) - 5*a*d*e**4/b**3 + 5*d**2*e**3/b**
2) + x*(-4*a**3*e**5/b**5 + 15*a**2*d*e**4/b**4 - 20*a*d**2*e**3/b**3 + 10*d**3*e**2/b**2) + (a**5*e**5 - 5*a*
*4*b*d*e**4 + 10*a**3*b**2*d**2*e**3 - 10*a**2*b**3*d**3*e**2 + 5*a*b**4*d**4*e - b**5*d**5)/(a*b**6 + b**7*x)
+ e**5*x**4/(4*b**2) + 5*e*(a*e - b*d)**4*log(a + b*x)/b**6

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