3.16.7 \(\int \frac {(d+e x)^3}{a^2+2 a b x+b^2 x^2} \, dx\) [1507]

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

[Out]

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

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Rubi [A]
time = 0.05, 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, 45} \begin {gather*} -\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} \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 45

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 = 0.03, size = 72, normalized size = 0.96 \begin {gather*} \frac {2 b e^2 (3 b d-2 a e) x+b^2 e^3 x^2-\frac {2 (b d-a e)^3}{a+b x}+6 e (b d-a e)^2 \log (a+b x)}{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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Maple [A]
time = 0.66, size = 109, normalized size = 1.45

method	result	size
default	\(-\frac {e^{2} \left (-\frac {1}{2} b e \,x^{2}+2 a e x -3 x b d \right )}{b^{3}}-\frac {-e^{3} a^{3}+3 a^{2} b d \,e^{2}-3 a \,b^{2} d^{2} e +b^{3} d^{3}}{b^{4} \left (b x +a \right )}+\frac {3 e \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \ln \left (b x +a \right )}{b^{4}}\)	\(109\)
norman	\(\frac {\frac {3 e^{3} a^{3}-6 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}}{b^{4}}+\frac {e^{3} x^{3}}{2 b}-\frac {3 e^{2} \left (a e -2 b d \right ) x^{2}}{2 b^{2}}}{b x +a}+\frac {3 e \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \ln \left (b x +a \right )}{b^{4}}\)	\(115\)
risch	\(\frac {e^{3} x^{2}}{2 b^{2}}-\frac {2 e^{3} a x}{b^{3}}+\frac {3 e^{2} x d}{b^{2}}+\frac {e^{3} a^{3}}{b^{4} \left (b x +a \right )}-\frac {3 a^{2} d \,e^{2}}{b^{3} \left (b x +a \right )}+\frac {3 a \,d^{2} e}{b^{2} \left (b x +a \right )}-\frac {d^{3}}{b \left (b x +a \right )}+\frac {3 e^{3} \ln \left (b x +a \right ) a^{2}}{b^{4}}-\frac {6 e^{2} \ln \left (b x +a \right ) a d}{b^{3}}+\frac {3 e \ln \left (b x +a \right ) d^{2}}{b^{2}}\)	\(149\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 113, normalized size = 1.51 \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 x^{2} e^{3} + 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*x^2*e^3 + 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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (75) = 150\).
time = 2.36, size = 155, normalized size = 2.07 \begin {gather*} -\frac {2 \, b^{3} d^{3} - 6 \, a b^{2} d^{2} e - {\left (b^{3} x^{3} - 3 \, a b^{2} x^{2} - 4 \, a^{2} b x + 2 \, a^{3}\right )} e^{3} - 6 \, {\left (b^{3} d x^{2} + a b^{2} d x - a^{2} b d\right )} e^{2} - 6 \, {\left ({\left (a^{2} b x + a^{3}\right )} e^{3} - 2 \, {\left (a b^{2} d x + a^{2} b d\right )} e^{2} + {\left (b^{3} d^{2} x + a b^{2} d^{2}\right )} e\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*(2*b^3*d^3 - 6*a*b^2*d^2*e - (b^3*x^3 - 3*a*b^2*x^2 - 4*a^2*b*x + 2*a^3)*e^3 - 6*(b^3*d*x^2 + a*b^2*d*x -
a^2*b*d)*e^2 - 6*((a^2*b*x + a^3)*e^3 - 2*(a*b^2*d*x + a^2*b*d)*e^2 + (b^3*d^2*x + a*b^2*d^2)*e)*log(b*x + a)
)/(b^5*x + a*b^4)

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Sympy [A]
time = 0.29, 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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Giac [A]
time = 1.21, 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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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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