### 3.1777 $$\int \frac{(a c+(b c+a d) x+b d x^2)^2}{(a+b x)^5} \, dx$$

Optimal. Leaf size=59 $-\frac{2 d (b c-a d)}{b^3 (a+b x)}-\frac{(b c-a d)^2}{2 b^3 (a+b x)^2}+\frac{d^2 \log (a+b x)}{b^3}$

[Out]

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

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Rubi [A]  time = 0.0431223, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.069, Rules used = {626, 43} $-\frac{2 d (b c-a d)}{b^3 (a+b x)}-\frac{(b c-a d)^2}{2 b^3 (a+b x)^2}+\frac{d^2 \log (a+b x)}{b^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 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{\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^5} \, dx &=\int \frac{(c+d x)^2}{(a+b x)^3} \, dx\\ &=\int \left (\frac{(b c-a d)^2}{b^2 (a+b x)^3}+\frac{2 d (b c-a d)}{b^2 (a+b x)^2}+\frac{d^2}{b^2 (a+b x)}\right ) \, dx\\ &=-\frac{(b c-a d)^2}{2 b^3 (a+b x)^2}-\frac{2 d (b c-a d)}{b^3 (a+b x)}+\frac{d^2 \log (a+b x)}{b^3}\\ \end{align*}

Mathematica [A]  time = 0.0241648, size = 49, normalized size = 0.83 $\frac{2 d^2 \log (a+b x)-\frac{(b c-a d) (3 a d+b (c+4 d x))}{(a+b x)^2}}{2 b^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.045, size = 92, normalized size = 1.6 \begin{align*} -{\frac{{a}^{2}{d}^{2}}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}}+{\frac{acd}{{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{{c}^{2}}{2\,b \left ( bx+a \right ) ^{2}}}+{\frac{{d}^{2}\ln \left ( bx+a \right ) }{{b}^{3}}}+2\,{\frac{a{d}^{2}}{{b}^{3} \left ( bx+a \right ) }}-2\,{\frac{cd}{{b}^{2} \left ( bx+a \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.994966, size = 107, normalized size = 1.81 \begin{align*} -\frac{b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2} + 4 \,{\left (b^{2} c d - a b d^{2}\right )} x}{2 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} + \frac{d^{2} \log \left (b x + a\right )}{b^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.60843, size = 207, normalized size = 3.51 \begin{align*} -\frac{b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2} + 4 \,{\left (b^{2} c d - a b d^{2}\right )} x - 2 \,{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right )}{2 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.853165, size = 80, normalized size = 1.36 \begin{align*} \frac{3 a^{2} d^{2} - 2 a b c d - b^{2} c^{2} + x \left (4 a b d^{2} - 4 b^{2} c d\right )}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac{d^{2} \log{\left (a + b x \right )}}{b^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19446, size = 149, normalized size = 2.53 \begin{align*} -\frac{d^{2} \log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{3}} - \frac{\frac{b^{5} c^{2}}{{\left (b x + a\right )}^{2}} + \frac{4 \, b^{4} c d}{b x + a} - \frac{2 \, a b^{4} c d}{{\left (b x + a\right )}^{2}} - \frac{4 \, a b^{3} d^{2}}{b x + a} + \frac{a^{2} b^{3} d^{2}}{{\left (b x + a\right )}^{2}}}{2 \, b^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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