3.1793 \(\int \frac{(a c+(b c+a d) x+b d x^2)^3}{(a+b x)^7} \, dx\)

Optimal. Leaf size=86 \[ -\frac{3 d^2 (b c-a d)}{b^4 (a+b x)}-\frac{3 d (b c-a d)^2}{2 b^4 (a+b x)^2}-\frac{(b c-a d)^3}{3 b^4 (a+b x)^3}+\frac{d^3 \log (a+b x)}{b^4} \]

[Out]

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

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Rubi [A]  time = 0.0600761, antiderivative size = 86, 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} \[ -\frac{3 d^2 (b c-a d)}{b^4 (a+b x)}-\frac{3 d (b c-a d)^2}{2 b^4 (a+b x)^2}-\frac{(b c-a d)^3}{3 b^4 (a+b x)^3}+\frac{d^3 \log (a+b x)}{b^4} \]

Antiderivative was successfully verified.

[In]

Int[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^7,x]

[Out]

-(b*c - a*d)^3/(3*b^4*(a + b*x)^3) - (3*d*(b*c - a*d)^2)/(2*b^4*(a + b*x)^2) - (3*d^2*(b*c - a*d))/(b^4*(a + b
*x)) + (d^3*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)^7} \, dx &=\int \frac{(c+d x)^3}{(a+b x)^4} \, dx\\ &=\int \left (\frac{(b c-a d)^3}{b^3 (a+b x)^4}+\frac{3 d (b c-a d)^2}{b^3 (a+b x)^3}+\frac{3 d^2 (b c-a d)}{b^3 (a+b x)^2}+\frac{d^3}{b^3 (a+b x)}\right ) \, dx\\ &=-\frac{(b c-a d)^3}{3 b^4 (a+b x)^3}-\frac{3 d (b c-a d)^2}{2 b^4 (a+b x)^2}-\frac{3 d^2 (b c-a d)}{b^4 (a+b x)}+\frac{d^3 \log (a+b x)}{b^4}\\ \end{align*}

Mathematica [A]  time = 0.0410163, size = 80, normalized size = 0.93 \[ \frac{6 d^3 \log (a+b x)-\frac{(b c-a d) \left (11 a^2 d^2+a b d (5 c+27 d x)+b^2 \left (2 c^2+9 c d x+18 d^2 x^2\right )\right )}{(a+b x)^3}}{6 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)^3/(a + b*x)^7,x]

[Out]

(-(((b*c - a*d)*(11*a^2*d^2 + a*b*d*(5*c + 27*d*x) + b^2*(2*c^2 + 9*c*d*x + 18*d^2*x^2)))/(a + b*x)^3) + 6*d^3
*Log[a + b*x])/(6*b^4)

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Maple [B]  time = 0.045, size = 166, normalized size = 1.9 \begin{align*} -{\frac{3\,{a}^{2}{d}^{3}}{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}+3\,{\frac{ac{d}^{2}}{{b}^{3} \left ( bx+a \right ) ^{2}}}-{\frac{3\,{c}^{2}d}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}+{\frac{{a}^{3}{d}^{3}}{3\,{b}^{4} \left ( bx+a \right ) ^{3}}}-{\frac{{a}^{2}c{d}^{2}}{{b}^{3} \left ( bx+a \right ) ^{3}}}+{\frac{a{c}^{2}d}{{b}^{2} \left ( bx+a \right ) ^{3}}}-{\frac{{c}^{3}}{3\,b \left ( bx+a \right ) ^{3}}}+{\frac{{d}^{3}\ln \left ( bx+a \right ) }{{b}^{4}}}+3\,{\frac{a{d}^{3}}{{b}^{4} \left ( bx+a \right ) }}-3\,{\frac{c{d}^{2}}{{b}^{3} \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)^7,x)

[Out]

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

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Maxima [A]  time = 1.14136, size = 192, normalized size = 2.23 \begin{align*} -\frac{2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - 11 \, a^{3} d^{3} + 18 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 9 \,{\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3}\right )} x}{6 \,{\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} + \frac{d^{3} \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)^7,x, algorithm="maxima")

[Out]

-1/6*(2*b^3*c^3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 11*a^3*d^3 + 18*(b^3*c*d^2 - a*b^2*d^3)*x^2 + 9*(b^3*c^2*d +
2*a*b^2*c*d^2 - 3*a^2*b*d^3)*x)/(b^7*x^3 + 3*a*b^6*x^2 + 3*a^2*b^5*x + a^3*b^4) + d^3*log(b*x + a)/b^4

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Fricas [B]  time = 1.55612, size = 360, normalized size = 4.19 \begin{align*} -\frac{2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - 11 \, a^{3} d^{3} + 18 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 9 \,{\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3}\right )} x - 6 \,{\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \log \left (b x + a\right )}{6 \,{\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} 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)^7,x, algorithm="fricas")

[Out]

-1/6*(2*b^3*c^3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 11*a^3*d^3 + 18*(b^3*c*d^2 - a*b^2*d^3)*x^2 + 9*(b^3*c^2*d +
2*a*b^2*c*d^2 - 3*a^2*b*d^3)*x - 6*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a))/(b
^7*x^3 + 3*a*b^6*x^2 + 3*a^2*b^5*x + a^3*b^4)

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Sympy [A]  time = 2.22226, size = 148, normalized size = 1.72 \begin{align*} \frac{11 a^{3} d^{3} - 6 a^{2} b c d^{2} - 3 a b^{2} c^{2} d - 2 b^{3} c^{3} + x^{2} \left (18 a b^{2} d^{3} - 18 b^{3} c d^{2}\right ) + x \left (27 a^{2} b d^{3} - 18 a b^{2} c d^{2} - 9 b^{3} c^{2} d\right )}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} + \frac{d^{3} \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)**7,x)

[Out]

(11*a**3*d**3 - 6*a**2*b*c*d**2 - 3*a*b**2*c**2*d - 2*b**3*c**3 + x**2*(18*a*b**2*d**3 - 18*b**3*c*d**2) + x*(
27*a**2*b*d**3 - 18*a*b**2*c*d**2 - 9*b**3*c**2*d))/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x*
*3) + d**3*log(a + b*x)/b**4

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Giac [A]  time = 1.20789, size = 159, normalized size = 1.85 \begin{align*} \frac{d^{3} \log \left ({\left | b x + a \right |}\right )}{b^{4}} - \frac{18 \,{\left (b^{2} c d^{2} - a b d^{3}\right )} x^{2} + 9 \,{\left (b^{2} c^{2} d + 2 \, a b c d^{2} - 3 \, a^{2} d^{3}\right )} x + \frac{2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - 11 \, a^{3} d^{3}}{b}}{6 \,{\left (b x + a\right )}^{3} 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)^3/(b*x+a)^7,x, algorithm="giac")

[Out]

d^3*log(abs(b*x + a))/b^4 - 1/6*(18*(b^2*c*d^2 - a*b*d^3)*x^2 + 9*(b^2*c^2*d + 2*a*b*c*d^2 - 3*a^2*d^3)*x + (2
*b^3*c^3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 11*a^3*d^3)/b)/((b*x + a)^3*b^3)