3.1819 \(\int \frac{(a+b x)^7}{(a c+(b c+a d) x+b d x^2)^3} \, dx\)

Optimal. Leaf size=103 \[ -\frac{b^3 x (3 b c-4 a d)}{d^4}+\frac{6 b^2 (b c-a d)^2 \log (c+d x)}{d^5}+\frac{4 b (b c-a d)^3}{d^5 (c+d x)}-\frac{(b c-a d)^4}{2 d^5 (c+d x)^2}+\frac{b^4 x^2}{2 d^3} \]

[Out]

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

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Rubi [A]  time = 0.0881793, antiderivative size = 103, 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} \[ -\frac{b^3 x (3 b c-4 a d)}{d^4}+\frac{6 b^2 (b c-a d)^2 \log (c+d x)}{d^5}+\frac{4 b (b c-a d)^3}{d^5 (c+d x)}-\frac{(b c-a d)^4}{2 d^5 (c+d x)^2}+\frac{b^4 x^2}{2 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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{(a+b x)^7}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx &=\int \frac{(a+b x)^4}{(c+d x)^3} \, dx\\ &=\int \left (-\frac{b^3 (3 b c-4 a d)}{d^4}+\frac{b^4 x}{d^3}+\frac{(-b c+a d)^4}{d^4 (c+d x)^3}-\frac{4 b (b c-a d)^3}{d^4 (c+d x)^2}+\frac{6 b^2 (b c-a d)^2}{d^4 (c+d x)}\right ) \, dx\\ &=-\frac{b^3 (3 b c-4 a d) x}{d^4}+\frac{b^4 x^2}{2 d^3}-\frac{(b c-a d)^4}{2 d^5 (c+d x)^2}+\frac{4 b (b c-a d)^3}{d^5 (c+d x)}+\frac{6 b^2 (b c-a d)^2 \log (c+d x)}{d^5}\\ \end{align*}

Mathematica [A]  time = 0.0540216, size = 167, normalized size = 1.62 \[ \frac{6 a^2 b^2 c d^2 (3 c+4 d x)-4 a^3 b d^3 (c+2 d x)-a^4 d^4+4 a b^3 d \left (-4 c^2 d x-5 c^3+4 c d^2 x^2+2 d^3 x^3\right )+12 b^2 (c+d x)^2 (b c-a d)^2 \log (c+d x)+b^4 \left (-11 c^2 d^2 x^2+2 c^3 d x+7 c^4-4 c d^3 x^3+d^4 x^4\right )}{2 d^5 (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(a^4*d^4) - 4*a^3*b*d^3*(c + 2*d*x) + 6*a^2*b^2*c*d^2*(3*c + 4*d*x) + 4*a*b^3*d*(-5*c^3 - 4*c^2*d*x + 4*c*d^
2*x^2 + 2*d^3*x^3) + b^4*(7*c^4 + 2*c^3*d*x - 11*c^2*d^2*x^2 - 4*c*d^3*x^3 + d^4*x^4) + 12*b^2*(b*c - a*d)^2*(
c + d*x)^2*Log[c + d*x])/(2*d^5*(c + d*x)^2)

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Maple [B]  time = 0.048, size = 245, normalized size = 2.4 \begin{align*}{\frac{{b}^{4}{x}^{2}}{2\,{d}^{3}}}+4\,{\frac{a{b}^{3}x}{{d}^{3}}}-3\,{\frac{{b}^{4}xc}{{d}^{4}}}+6\,{\frac{{b}^{2}\ln \left ( dx+c \right ){a}^{2}}{{d}^{3}}}-12\,{\frac{{b}^{3}\ln \left ( dx+c \right ) ca}{{d}^{4}}}+6\,{\frac{{b}^{4}\ln \left ( dx+c \right ){c}^{2}}{{d}^{5}}}-{\frac{{a}^{4}}{2\,d \left ( dx+c \right ) ^{2}}}+2\,{\frac{c{a}^{3}b}{{d}^{2} \left ( dx+c \right ) ^{2}}}-3\,{\frac{{a}^{2}{b}^{2}{c}^{2}}{{d}^{3} \left ( dx+c \right ) ^{2}}}+2\,{\frac{a{b}^{3}{c}^{3}}{{d}^{4} \left ( dx+c \right ) ^{2}}}-{\frac{{b}^{4}{c}^{4}}{2\,{d}^{5} \left ( dx+c \right ) ^{2}}}-4\,{\frac{{a}^{3}b}{{d}^{2} \left ( dx+c \right ) }}+12\,{\frac{{b}^{2}c{a}^{2}}{{d}^{3} \left ( dx+c \right ) }}-12\,{\frac{a{b}^{3}{c}^{2}}{{d}^{4} \left ( dx+c \right ) }}+4\,{\frac{{b}^{4}{c}^{3}}{{d}^{5} \left ( dx+c \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^7/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x)

[Out]

1/2*b^4*x^2/d^3+4*b^3/d^3*a*x-3*b^4/d^4*x*c+6*b^2/d^3*ln(d*x+c)*a^2-12*b^3/d^4*ln(d*x+c)*c*a+6*b^4/d^5*ln(d*x+
c)*c^2-1/2/d/(d*x+c)^2*a^4+2/d^2/(d*x+c)^2*c*a^3*b-3/d^3/(d*x+c)^2*a^2*b^2*c^2+2/d^4/(d*x+c)^2*a*b^3*c^3-1/2/d
^5/(d*x+c)^2*b^4*c^4-4*b/d^2/(d*x+c)*a^3+12*b^2/d^3/(d*x+c)*c*a^2-12*b^3/d^4/(d*x+c)*a*c^2+4*b^4/d^5/(d*x+c)*c
^3

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Maxima [A]  time = 1.08637, size = 258, normalized size = 2.5 \begin{align*} \frac{7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4} + 8 \,{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x}{2 \,{\left (d^{7} x^{2} + 2 \, c d^{6} x + c^{2} d^{5}\right )}} + \frac{b^{4} d x^{2} - 2 \,{\left (3 \, b^{4} c - 4 \, a b^{3} d\right )} x}{2 \, d^{4}} + \frac{6 \,{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \log \left (d x + c\right )}{d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="maxima")

[Out]

1/2*(7*b^4*c^4 - 20*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 - a^4*d^4 + 8*(b^4*c^3*d - 3*a*b^3*c^2*d^
2 + 3*a^2*b^2*c*d^3 - a^3*b*d^4)*x)/(d^7*x^2 + 2*c*d^6*x + c^2*d^5) + 1/2*(b^4*d*x^2 - 2*(3*b^4*c - 4*a*b^3*d)
*x)/d^4 + 6*(b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*log(d*x + c)/d^5

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Fricas [B]  time = 1.49897, size = 586, normalized size = 5.69 \begin{align*} \frac{b^{4} d^{4} x^{4} + 7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4} - 4 \,{\left (b^{4} c d^{3} - 2 \, a b^{3} d^{4}\right )} x^{3} -{\left (11 \, b^{4} c^{2} d^{2} - 16 \, a b^{3} c d^{3}\right )} x^{2} + 2 \,{\left (b^{4} c^{3} d - 8 \, a b^{3} c^{2} d^{2} + 12 \, a^{2} b^{2} c d^{3} - 4 \, a^{3} b d^{4}\right )} x + 12 \,{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + a^{2} b^{2} c^{2} d^{2} +{\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} + 2 \,{\left (b^{4} c^{3} d - 2 \, a b^{3} c^{2} d^{2} + a^{2} b^{2} c d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \,{\left (d^{7} x^{2} + 2 \, c d^{6} x + c^{2} d^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="fricas")

[Out]

1/2*(b^4*d^4*x^4 + 7*b^4*c^4 - 20*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 - a^4*d^4 - 4*(b^4*c*d^3 -
2*a*b^3*d^4)*x^3 - (11*b^4*c^2*d^2 - 16*a*b^3*c*d^3)*x^2 + 2*(b^4*c^3*d - 8*a*b^3*c^2*d^2 + 12*a^2*b^2*c*d^3 -
4*a^3*b*d^4)*x + 12*(b^4*c^4 - 2*a*b^3*c^3*d + a^2*b^2*c^2*d^2 + (b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*
x^2 + 2*(b^4*c^3*d - 2*a*b^3*c^2*d^2 + a^2*b^2*c*d^3)*x)*log(d*x + c))/(d^7*x^2 + 2*c*d^6*x + c^2*d^5)

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Sympy [A]  time = 1.80211, size = 184, normalized size = 1.79 \begin{align*} \frac{b^{4} x^{2}}{2 d^{3}} + \frac{6 b^{2} \left (a d - b c\right )^{2} \log{\left (c + d x \right )}}{d^{5}} - \frac{a^{4} d^{4} + 4 a^{3} b c d^{3} - 18 a^{2} b^{2} c^{2} d^{2} + 20 a b^{3} c^{3} d - 7 b^{4} c^{4} + x \left (8 a^{3} b d^{4} - 24 a^{2} b^{2} c d^{3} + 24 a b^{3} c^{2} d^{2} - 8 b^{4} c^{3} d\right )}{2 c^{2} d^{5} + 4 c d^{6} x + 2 d^{7} x^{2}} + \frac{x \left (4 a b^{3} d - 3 b^{4} c\right )}{d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**7/(a*c+(a*d+b*c)*x+b*d*x**2)**3,x)

[Out]

b**4*x**2/(2*d**3) + 6*b**2*(a*d - b*c)**2*log(c + d*x)/d**5 - (a**4*d**4 + 4*a**3*b*c*d**3 - 18*a**2*b**2*c**
2*d**2 + 20*a*b**3*c**3*d - 7*b**4*c**4 + x*(8*a**3*b*d**4 - 24*a**2*b**2*c*d**3 + 24*a*b**3*c**2*d**2 - 8*b**
4*c**3*d))/(2*c**2*d**5 + 4*c*d**6*x + 2*d**7*x**2) + x*(4*a*b**3*d - 3*b**4*c)/d**4

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Giac [A]  time = 1.26366, size = 247, normalized size = 2.4 \begin{align*} \frac{6 \,{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{5}} + \frac{b^{4} d^{3} x^{2} - 6 \, b^{4} c d^{2} x + 8 \, a b^{3} d^{3} x}{2 \, d^{6}} + \frac{7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4} + 8 \,{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x}{2 \,{\left (d x + c\right )}^{2} d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="giac")

[Out]

6*(b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*log(abs(d*x + c))/d^5 + 1/2*(b^4*d^3*x^2 - 6*b^4*c*d^2*x + 8*a*b^3*d^3
*x)/d^6 + 1/2*(7*b^4*c^4 - 20*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 - a^4*d^4 + 8*(b^4*c^3*d - 3*a*
b^3*c^2*d^2 + 3*a^2*b^2*c*d^3 - a^3*b*d^4)*x)/((d*x + c)^2*d^5)