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

Optimal. Leaf size=75 \[ -\frac {(b d-a e)^3}{b^4 (a+b x)}+\frac {3 e (b d-a e)^2 \log (a+b x)}{b^4}+\frac {e^2 x (3 b d-2 a e)}{b^3}+\frac {e^3 x^2}{2 b^2} \]

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Rubi [A]  time = 0.06, antiderivative size = 75, 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 {e^2 x (3 b d-2 a e)}{b^3}-\frac {(b d-a e)^3}{b^4 (a+b x)}+\frac {3 e (b d-a e)^2 \log (a+b x)}{b^4}+\frac {e^3 x^2}{2 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.05, size = 72, normalized size = 0.96 \begin {gather*} \frac {2 b e^2 x (3 b d-2 a e)-\frac {2 (b d-a e)^3}{a+b x}+6 e (b d-a e)^2 \log (a+b x)+b^2 e^3 x^2}{2 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

[Out]

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

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fricas [B]  time = 0.40, size = 173, normalized size = 2.31 \begin {gather*} \frac {b^{3} e^{3} x^{3} - 2 \, b^{3} d^{3} + 6 \, a b^{2} d^{2} e - 6 \, a^{2} b d e^{2} + 2 \, a^{3} e^{3} + 3 \, {\left (2 \, b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 2 \, {\left (3 \, a b^{2} d e^{2} - 2 \, a^{2} b e^{3}\right )} x + 6 \, {\left (a b^{2} d^{2} e - 2 \, a^{2} b d e^{2} + a^{3} e^{3} + {\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \log \left (b x + a\right )}{2 \, {\left (b^{5} x + a b^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.15, size = 114, normalized size = 1.52 \begin {gather*} \frac {3 \, {\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4}} + \frac {b^{2} x^{2} e^{3} + 6 \, b^{2} d x e^{2} - 4 \, a b x e^{3}}{2 \, b^{4}} - \frac {b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}}{{\left (b x + a\right )} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.05, size = 149, normalized size = 1.99 \begin {gather*} \frac {e^{3} x^{2}}{2 b^{2}}+\frac {a^{3} e^{3}}{\left (b x +a \right ) b^{4}}-\frac {3 a^{2} d \,e^{2}}{\left (b x +a \right ) b^{3}}+\frac {3 a^{2} e^{3} \ln \left (b x +a \right )}{b^{4}}+\frac {3 a \,d^{2} e}{\left (b x +a \right ) b^{2}}-\frac {6 a d \,e^{2} \ln \left (b x +a \right )}{b^{3}}-\frac {2 a \,e^{3} x}{b^{3}}-\frac {d^{3}}{\left (b x +a \right ) b}+\frac {3 d^{2} e \ln \left (b x +a \right )}{b^{2}}+\frac {3 d \,e^{2} x}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.40, size = 118, normalized size = 1.57 \begin {gather*} -\frac {b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}}{b^{5} x + a b^{4}} + \frac {b e^{3} x^{2} + 2 \, {\left (3 \, b d e^{2} - 2 \, a e^{3}\right )} x}{2 \, b^{3}} + \frac {3 \, {\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )} \log \left (b x + a\right )}{b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.07, size = 123, normalized size = 1.64 \begin {gather*} \frac {\ln \left (a+b\,x\right )\,\left (3\,a^2\,e^3-6\,a\,b\,d\,e^2+3\,b^2\,d^2\,e\right )}{b^4}-x\,\left (\frac {2\,a\,e^3}{b^3}-\frac {3\,d\,e^2}{b^2}\right )+\frac {a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3}{b\,\left (x\,b^4+a\,b^3\right )}+\frac {e^3\,x^2}{2\,b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.51, size = 102, normalized size = 1.36 \begin {gather*} x \left (- \frac {2 a e^{3}}{b^{3}} + \frac {3 d e^{2}}{b^{2}}\right ) + \frac {a^{3} e^{3} - 3 a^{2} b d e^{2} + 3 a b^{2} d^{2} e - b^{3} d^{3}}{a b^{4} + b^{5} x} + \frac {e^{3} x^{2}}{2 b^{2}} + \frac {3 e \left (a e - b d\right )^{2} \log {\left (a + b x \right )}}{b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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