3.1792 $$\int \frac{(a c+(b c+a d) x+b d x^2)^3}{(a+b x)^6} \, dx$$

Optimal. Leaf size=78 $\frac{3 d^2 (b c-a d) \log (a+b x)}{b^4}-\frac{3 d (b c-a d)^2}{b^4 (a+b x)}-\frac{(b c-a d)^3}{2 b^4 (a+b x)^2}+\frac{d^3 x}{b^3}$

[Out]

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

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Rubi [A]  time = 0.0606103, antiderivative size = 78, 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} $\frac{3 d^2 (b c-a d) \log (a+b x)}{b^4}-\frac{3 d (b c-a d)^2}{b^4 (a+b x)}-\frac{(b c-a d)^3}{2 b^4 (a+b x)^2}+\frac{d^3 x}{b^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^6,x]

[Out]

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

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

Mathematica [A]  time = 0.0403758, size = 114, normalized size = 1.46 $\frac{a^2 b d^2 (9 c-4 d x)-5 a^3 d^3+a b^2 d \left (-3 c^2+12 c d x+4 d^2 x^2\right )-6 d^2 (a+b x)^2 (a d-b c) \log (a+b x)+b^3 \left (-\left (6 c^2 d x+c^3-2 d^3 x^3\right )\right )}{2 b^4 (a+b x)^2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^6,x]

[Out]

(-5*a^3*d^3 + a^2*b*d^2*(9*c - 4*d*x) + a*b^2*d*(-3*c^2 + 12*c*d*x + 4*d^2*x^2) - b^3*(c^3 + 6*c^2*d*x - 2*d^3
*x^3) - 6*d^2*(-(b*c) + a*d)*(a + b*x)^2*Log[a + b*x])/(2*b^4*(a + b*x)^2)

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Maple [B]  time = 0.048, size = 160, normalized size = 2.1 \begin{align*}{\frac{{d}^{3}x}{{b}^{3}}}+{\frac{{a}^{3}{d}^{3}}{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}-{\frac{3\,{a}^{2}c{d}^{2}}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}}+{\frac{3\,a{c}^{2}d}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{{c}^{3}}{2\,b \left ( bx+a \right ) ^{2}}}-3\,{\frac{{d}^{3}\ln \left ( bx+a \right ) a}{{b}^{4}}}+3\,{\frac{{d}^{2}\ln \left ( bx+a \right ) c}{{b}^{3}}}-3\,{\frac{{a}^{2}{d}^{3}}{{b}^{4} \left ( bx+a \right ) }}+6\,{\frac{ac{d}^{2}}{{b}^{3} \left ( bx+a \right ) }}-3\,{\frac{{c}^{2}d}{{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)^3/(b*x+a)^6,x)

[Out]

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

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Maxima [A]  time = 1.04744, size = 169, normalized size = 2.17 \begin{align*} \frac{d^{3} x}{b^{3}} - \frac{b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 9 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3} + 6 \,{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{2 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} + \frac{3 \,{\left (b c d^{2} - a d^{3}\right )} \log \left (b x + a\right )}{b^{4}} \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)^3/(b*x+a)^6,x, algorithm="maxima")

[Out]

d^3*x/b^3 - 1/2*(b^3*c^3 + 3*a*b^2*c^2*d - 9*a^2*b*c*d^2 + 5*a^3*d^3 + 6*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^
3)*x)/(b^6*x^2 + 2*a*b^5*x + a^2*b^4) + 3*(b*c*d^2 - a*d^3)*log(b*x + a)/b^4

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Fricas [B]  time = 1.6434, size = 375, normalized size = 4.81 \begin{align*} \frac{2 \, b^{3} d^{3} x^{3} + 4 \, a b^{2} d^{3} x^{2} - b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3} - 2 \,{\left (3 \, b^{3} c^{2} d - 6 \, a b^{2} c d^{2} + 2 \, a^{2} b d^{3}\right )} x + 6 \,{\left (a^{2} b c d^{2} - a^{3} d^{3} +{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 2 \,{\left (a b^{2} c d^{2} - a^{2} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{2 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\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)^3/(b*x+a)^6,x, algorithm="fricas")

[Out]

1/2*(2*b^3*d^3*x^3 + 4*a*b^2*d^3*x^2 - b^3*c^3 - 3*a*b^2*c^2*d + 9*a^2*b*c*d^2 - 5*a^3*d^3 - 2*(3*b^3*c^2*d -
6*a*b^2*c*d^2 + 2*a^2*b*d^3)*x + 6*(a^2*b*c*d^2 - a^3*d^3 + (b^3*c*d^2 - a*b^2*d^3)*x^2 + 2*(a*b^2*c*d^2 - a^2
*b*d^3)*x)*log(b*x + a))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4)

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Sympy [A]  time = 1.57843, size = 128, normalized size = 1.64 \begin{align*} - \frac{5 a^{3} d^{3} - 9 a^{2} b c d^{2} + 3 a b^{2} c^{2} d + b^{3} c^{3} + x \left (6 a^{2} b d^{3} - 12 a b^{2} c d^{2} + 6 b^{3} c^{2} d\right )}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} + \frac{d^{3} x}{b^{3}} - \frac{3 d^{2} \left (a d - b c\right ) \log{\left (a + b x \right )}}{b^{4}} \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)**3/(b*x+a)**6,x)

[Out]

-(5*a**3*d**3 - 9*a**2*b*c*d**2 + 3*a*b**2*c**2*d + b**3*c**3 + x*(6*a**2*b*d**3 - 12*a*b**2*c*d**2 + 6*b**3*c
**2*d))/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + d**3*x/b**3 - 3*d**2*(a*d - b*c)*log(a + b*x)/b**4

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Giac [A]  time = 1.20084, size = 151, normalized size = 1.94 \begin{align*} \frac{d^{3} x}{b^{3}} + \frac{3 \,{\left (b c d^{2} - a d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4}} - \frac{b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 9 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3} + 6 \,{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{2 \,{\left (b x + a\right )}^{2} b^{4}} \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)^3/(b*x+a)^6,x, algorithm="giac")

[Out]

d^3*x/b^3 + 3*(b*c*d^2 - a*d^3)*log(abs(b*x + a))/b^4 - 1/2*(b^3*c^3 + 3*a*b^2*c^2*d - 9*a^2*b*c*d^2 + 5*a^3*d
^3 + 6*(b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x)/((b*x + a)^2*b^4)