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

Optimal. Leaf size=179 \[ \frac{x \left (3 a^2 d^2+4 a b c d+3 b^2 c^2\right )}{b^4 d^4}-\frac{a^6}{b^5 (a+b x) (b c-a d)^2}-\frac{2 a^5 (3 b c-2 a d) \log (a+b x)}{b^5 (b c-a d)^3}-\frac{x^2 (a d+b c)}{b^3 d^3}-\frac{c^6}{d^5 (c+d x) (b c-a d)^2}-\frac{2 c^5 (2 b c-3 a d) \log (c+d x)}{d^5 (b c-a d)^3}+\frac{x^3}{3 b^2 d^2} \]

[Out]

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

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Rubi [A]  time = 0.233144, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {88} \[ \frac{x \left (3 a^2 d^2+4 a b c d+3 b^2 c^2\right )}{b^4 d^4}-\frac{a^6}{b^5 (a+b x) (b c-a d)^2}-\frac{2 a^5 (3 b c-2 a d) \log (a+b x)}{b^5 (b c-a d)^3}-\frac{x^2 (a d+b c)}{b^3 d^3}-\frac{c^6}{d^5 (c+d x) (b c-a d)^2}-\frac{2 c^5 (2 b c-3 a d) \log (c+d x)}{d^5 (b c-a d)^3}+\frac{x^3}{3 b^2 d^2} \]

Antiderivative was successfully verified.

[In]

Int[x^6/((a + b*x)^2*(c + d*x)^2),x]

[Out]

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

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

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

Mathematica [A]  time = 0.21016, size = 179, normalized size = 1. \[ \frac{x \left (3 a^2 d^2+4 a b c d+3 b^2 c^2\right )}{b^4 d^4}-\frac{a^6}{b^5 (a+b x) (b c-a d)^2}+\frac{2 a^5 (2 a d-3 b c) \log (a+b x)}{b^5 (b c-a d)^3}-\frac{x^2 (a d+b c)}{b^3 d^3}-\frac{c^6}{d^5 (c+d x) (b c-a d)^2}+\frac{2 c^5 (2 b c-3 a d) \log (c+d x)}{d^5 (a d-b c)^3}+\frac{x^3}{3 b^2 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.013, size = 222, normalized size = 1.2 \begin{align*}{\frac{{x}^{3}}{3\,{b}^{2}{d}^{2}}}-{\frac{a{x}^{2}}{{d}^{2}{b}^{3}}}-{\frac{c{x}^{2}}{{d}^{3}{b}^{2}}}+3\,{\frac{{a}^{2}x}{{b}^{4}{d}^{2}}}+4\,{\frac{acx}{{b}^{3}{d}^{3}}}+3\,{\frac{{c}^{2}x}{{b}^{2}{d}^{4}}}-{\frac{{c}^{6}}{{d}^{5} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}-6\,{\frac{{c}^{5}\ln \left ( dx+c \right ) a}{{d}^{4} \left ( ad-bc \right ) ^{3}}}+4\,{\frac{{c}^{6}\ln \left ( dx+c \right ) b}{{d}^{5} \left ( ad-bc \right ) ^{3}}}-{\frac{{a}^{6}}{{b}^{5} \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) }}-4\,{\frac{{a}^{6}\ln \left ( bx+a \right ) d}{{b}^{5} \left ( ad-bc \right ) ^{3}}}+6\,{\frac{{a}^{5}\ln \left ( bx+a \right ) c}{{b}^{4} \left ( ad-bc \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(b*x+a)^2/(d*x+c)^2,x)

[Out]

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

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Maxima [A]  time = 1.11317, size = 474, normalized size = 2.65 \begin{align*} -\frac{2 \,{\left (3 \, a^{5} b c - 2 \, a^{6} d\right )} \log \left (b x + a\right )}{b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}} - \frac{2 \,{\left (2 \, b c^{6} - 3 \, a c^{5} d\right )} \log \left (d x + c\right )}{b^{3} c^{3} d^{5} - 3 \, a b^{2} c^{2} d^{6} + 3 \, a^{2} b c d^{7} - a^{3} d^{8}} - \frac{a b^{5} c^{6} + a^{6} c d^{5} +{\left (b^{6} c^{6} + a^{6} d^{6}\right )} x}{a b^{7} c^{3} d^{5} - 2 \, a^{2} b^{6} c^{2} d^{6} + a^{3} b^{5} c d^{7} +{\left (b^{8} c^{2} d^{6} - 2 \, a b^{7} c d^{7} + a^{2} b^{6} d^{8}\right )} x^{2} +{\left (b^{8} c^{3} d^{5} - a b^{7} c^{2} d^{6} - a^{2} b^{6} c d^{7} + a^{3} b^{5} d^{8}\right )} x} + \frac{b^{2} d^{2} x^{3} - 3 \,{\left (b^{2} c d + a b d^{2}\right )} x^{2} + 3 \,{\left (3 \, b^{2} c^{2} + 4 \, a b c d + 3 \, a^{2} d^{2}\right )} x}{3 \, b^{4} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x+a)^2/(d*x+c)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.80467, size = 1364, normalized size = 7.62 \begin{align*} -\frac{3 \, a b^{6} c^{7} - 3 \, a^{2} b^{5} c^{6} d + 3 \, a^{6} b c^{2} d^{5} - 3 \, a^{7} c d^{6} -{\left (b^{7} c^{3} d^{4} - 3 \, a b^{6} c^{2} d^{5} + 3 \, a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{5} + 2 \,{\left (b^{7} c^{4} d^{3} - 2 \, a b^{6} c^{3} d^{4} + 2 \, a^{3} b^{4} c d^{6} - a^{4} b^{3} d^{7}\right )} x^{4} -{\left (6 \, b^{7} c^{5} d^{2} - 11 \, a b^{6} c^{4} d^{3} + 3 \, a^{2} b^{5} c^{3} d^{4} - 3 \, a^{3} b^{4} c^{2} d^{5} + 11 \, a^{4} b^{3} c d^{6} - 6 \, a^{5} b^{2} d^{7}\right )} x^{3} - 9 \,{\left (b^{7} c^{6} d - a b^{6} c^{5} d^{2} - a^{2} b^{5} c^{4} d^{3} + a^{4} b^{3} c^{2} d^{5} + a^{5} b^{2} c d^{6} - a^{6} b d^{7}\right )} x^{2} + 3 \,{\left (b^{7} c^{7} - 4 \, a b^{6} c^{6} d + 5 \, a^{2} b^{5} c^{5} d^{2} - 5 \, a^{5} b^{2} c^{2} d^{5} + 4 \, a^{6} b c d^{6} - a^{7} d^{7}\right )} x + 6 \,{\left (3 \, a^{6} b c^{2} d^{5} - 2 \, a^{7} c d^{6} +{\left (3 \, a^{5} b^{2} c d^{6} - 2 \, a^{6} b d^{7}\right )} x^{2} +{\left (3 \, a^{5} b^{2} c^{2} d^{5} + a^{6} b c d^{6} - 2 \, a^{7} d^{7}\right )} x\right )} \log \left (b x + a\right ) + 6 \,{\left (2 \, a b^{6} c^{7} - 3 \, a^{2} b^{5} c^{6} d +{\left (2 \, b^{7} c^{6} d - 3 \, a b^{6} c^{5} d^{2}\right )} x^{2} +{\left (2 \, b^{7} c^{7} - a b^{6} c^{6} d - 3 \, a^{2} b^{5} c^{5} d^{2}\right )} x\right )} \log \left (d x + c\right )}{3 \,{\left (a b^{8} c^{4} d^{5} - 3 \, a^{2} b^{7} c^{3} d^{6} + 3 \, a^{3} b^{6} c^{2} d^{7} - a^{4} b^{5} c d^{8} +{\left (b^{9} c^{3} d^{6} - 3 \, a b^{8} c^{2} d^{7} + 3 \, a^{2} b^{7} c d^{8} - a^{3} b^{6} d^{9}\right )} x^{2} +{\left (b^{9} c^{4} d^{5} - 2 \, a b^{8} c^{3} d^{6} + 2 \, a^{3} b^{6} c d^{8} - a^{4} b^{5} d^{9}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x+a)^2/(d*x+c)^2,x, algorithm="fricas")

[Out]

-1/3*(3*a*b^6*c^7 - 3*a^2*b^5*c^6*d + 3*a^6*b*c^2*d^5 - 3*a^7*c*d^6 - (b^7*c^3*d^4 - 3*a*b^6*c^2*d^5 + 3*a^2*b
^5*c*d^6 - a^3*b^4*d^7)*x^5 + 2*(b^7*c^4*d^3 - 2*a*b^6*c^3*d^4 + 2*a^3*b^4*c*d^6 - a^4*b^3*d^7)*x^4 - (6*b^7*c
^5*d^2 - 11*a*b^6*c^4*d^3 + 3*a^2*b^5*c^3*d^4 - 3*a^3*b^4*c^2*d^5 + 11*a^4*b^3*c*d^6 - 6*a^5*b^2*d^7)*x^3 - 9*
(b^7*c^6*d - a*b^6*c^5*d^2 - a^2*b^5*c^4*d^3 + a^4*b^3*c^2*d^5 + a^5*b^2*c*d^6 - a^6*b*d^7)*x^2 + 3*(b^7*c^7 -
4*a*b^6*c^6*d + 5*a^2*b^5*c^5*d^2 - 5*a^5*b^2*c^2*d^5 + 4*a^6*b*c*d^6 - a^7*d^7)*x + 6*(3*a^6*b*c^2*d^5 - 2*a
^7*c*d^6 + (3*a^5*b^2*c*d^6 - 2*a^6*b*d^7)*x^2 + (3*a^5*b^2*c^2*d^5 + a^6*b*c*d^6 - 2*a^7*d^7)*x)*log(b*x + a)
+ 6*(2*a*b^6*c^7 - 3*a^2*b^5*c^6*d + (2*b^7*c^6*d - 3*a*b^6*c^5*d^2)*x^2 + (2*b^7*c^7 - a*b^6*c^6*d - 3*a^2*b
^5*c^5*d^2)*x)*log(d*x + c))/(a*b^8*c^4*d^5 - 3*a^2*b^7*c^3*d^6 + 3*a^3*b^6*c^2*d^7 - a^4*b^5*c*d^8 + (b^9*c^3
*d^6 - 3*a*b^8*c^2*d^7 + 3*a^2*b^7*c*d^8 - a^3*b^6*d^9)*x^2 + (b^9*c^4*d^5 - 2*a*b^8*c^3*d^6 + 2*a^3*b^6*c*d^8
- a^4*b^5*d^9)*x)

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Sympy [B]  time = 7.91092, size = 768, normalized size = 4.29 \begin{align*} - \frac{2 a^{5} \left (2 a d - 3 b c\right ) \log{\left (x + \frac{\frac{2 a^{9} d^{8} \left (2 a d - 3 b c\right )}{b \left (a d - b c\right )^{3}} - \frac{8 a^{8} c d^{7} \left (2 a d - 3 b c\right )}{\left (a d - b c\right )^{3}} + \frac{12 a^{7} b c^{2} d^{6} \left (2 a d - 3 b c\right )}{\left (a d - b c\right )^{3}} - \frac{8 a^{6} b^{2} c^{3} d^{5} \left (2 a d - 3 b c\right )}{\left (a d - b c\right )^{3}} + 4 a^{6} c d^{5} + \frac{2 a^{5} b^{3} c^{4} d^{4} \left (2 a d - 3 b c\right )}{\left (a d - b c\right )^{3}} - 6 a^{5} b c^{2} d^{4} - 6 a^{2} b^{4} c^{5} d + 4 a b^{5} c^{6}}{4 a^{6} d^{6} - 6 a^{5} b c d^{5} - 6 a b^{5} c^{5} d + 4 b^{6} c^{6}} \right )}}{b^{5} \left (a d - b c\right )^{3}} - \frac{2 c^{5} \left (3 a d - 2 b c\right ) \log{\left (x + \frac{4 a^{6} c d^{5} - 6 a^{5} b c^{2} d^{4} + \frac{2 a^{4} b^{4} c^{5} d^{3} \left (3 a d - 2 b c\right )}{\left (a d - b c\right )^{3}} - \frac{8 a^{3} b^{5} c^{6} d^{2} \left (3 a d - 2 b c\right )}{\left (a d - b c\right )^{3}} + \frac{12 a^{2} b^{6} c^{7} d \left (3 a d - 2 b c\right )}{\left (a d - b c\right )^{3}} - 6 a^{2} b^{4} c^{5} d - \frac{8 a b^{7} c^{8} \left (3 a d - 2 b c\right )}{\left (a d - b c\right )^{3}} + 4 a b^{5} c^{6} + \frac{2 b^{8} c^{9} \left (3 a d - 2 b c\right )}{d \left (a d - b c\right )^{3}}}{4 a^{6} d^{6} - 6 a^{5} b c d^{5} - 6 a b^{5} c^{5} d + 4 b^{6} c^{6}} \right )}}{d^{5} \left (a d - b c\right )^{3}} - \frac{a^{6} c d^{5} + a b^{5} c^{6} + x \left (a^{6} d^{6} + b^{6} c^{6}\right )}{a^{3} b^{5} c d^{7} - 2 a^{2} b^{6} c^{2} d^{6} + a b^{7} c^{3} d^{5} + x^{2} \left (a^{2} b^{6} d^{8} - 2 a b^{7} c d^{7} + b^{8} c^{2} d^{6}\right ) + x \left (a^{3} b^{5} d^{8} - a^{2} b^{6} c d^{7} - a b^{7} c^{2} d^{6} + b^{8} c^{3} d^{5}\right )} + \frac{x^{3}}{3 b^{2} d^{2}} - \frac{x^{2} \left (a d + b c\right )}{b^{3} d^{3}} + \frac{x \left (3 a^{2} d^{2} + 4 a b c d + 3 b^{2} c^{2}\right )}{b^{4} d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(b*x+a)**2/(d*x+c)**2,x)

[Out]

-2*a**5*(2*a*d - 3*b*c)*log(x + (2*a**9*d**8*(2*a*d - 3*b*c)/(b*(a*d - b*c)**3) - 8*a**8*c*d**7*(2*a*d - 3*b*c
)/(a*d - b*c)**3 + 12*a**7*b*c**2*d**6*(2*a*d - 3*b*c)/(a*d - b*c)**3 - 8*a**6*b**2*c**3*d**5*(2*a*d - 3*b*c)/
(a*d - b*c)**3 + 4*a**6*c*d**5 + 2*a**5*b**3*c**4*d**4*(2*a*d - 3*b*c)/(a*d - b*c)**3 - 6*a**5*b*c**2*d**4 - 6
*a**2*b**4*c**5*d + 4*a*b**5*c**6)/(4*a**6*d**6 - 6*a**5*b*c*d**5 - 6*a*b**5*c**5*d + 4*b**6*c**6))/(b**5*(a*d
- b*c)**3) - 2*c**5*(3*a*d - 2*b*c)*log(x + (4*a**6*c*d**5 - 6*a**5*b*c**2*d**4 + 2*a**4*b**4*c**5*d**3*(3*a*
d - 2*b*c)/(a*d - b*c)**3 - 8*a**3*b**5*c**6*d**2*(3*a*d - 2*b*c)/(a*d - b*c)**3 + 12*a**2*b**6*c**7*d*(3*a*d
- 2*b*c)/(a*d - b*c)**3 - 6*a**2*b**4*c**5*d - 8*a*b**7*c**8*(3*a*d - 2*b*c)/(a*d - b*c)**3 + 4*a*b**5*c**6 +
2*b**8*c**9*(3*a*d - 2*b*c)/(d*(a*d - b*c)**3))/(4*a**6*d**6 - 6*a**5*b*c*d**5 - 6*a*b**5*c**5*d + 4*b**6*c**6
))/(d**5*(a*d - b*c)**3) - (a**6*c*d**5 + a*b**5*c**6 + x*(a**6*d**6 + b**6*c**6))/(a**3*b**5*c*d**7 - 2*a**2*
b**6*c**2*d**6 + a*b**7*c**3*d**5 + x**2*(a**2*b**6*d**8 - 2*a*b**7*c*d**7 + b**8*c**2*d**6) + x*(a**3*b**5*d*
*8 - a**2*b**6*c*d**7 - a*b**7*c**2*d**6 + b**8*c**3*d**5)) + x**3/(3*b**2*d**2) - x**2*(a*d + b*c)/(b**3*d**3
) + x*(3*a**2*d**2 + 4*a*b*c*d + 3*b**2*c**2)/(b**4*d**4)

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Giac [B]  time = 1.19122, size = 698, normalized size = 3.9 \begin{align*} -\frac{a^{6} b^{5}}{{\left (b^{12} c^{2} - 2 \, a b^{11} c d + a^{2} b^{10} d^{2}\right )}{\left (b x + a\right )}} - \frac{2 \,{\left (2 \, b^{2} c^{6} - 3 \, a b c^{5} d\right )} \log \left ({\left | \frac{b c}{b x + a} - \frac{a d}{b x + a} + d \right |}\right )}{b^{4} c^{3} d^{5} - 3 \, a b^{3} c^{2} d^{6} + 3 \, a^{2} b^{2} c d^{7} - a^{3} b d^{8}} + \frac{2 \,{\left (2 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3}\right )} \log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{5} d^{5}} + \frac{{\left (b^{3} c^{3} d^{4} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7} - \frac{2 \, b^{5} c^{4} d^{3} + a b^{4} c^{3} d^{4} - 15 \, a^{2} b^{3} c^{2} d^{5} + 19 \, a^{3} b^{2} c d^{6} - 7 \, a^{4} b d^{7}}{{\left (b x + a\right )} b} + \frac{3 \,{\left (2 \, b^{7} c^{5} d^{2} - a b^{6} c^{4} d^{3} - a^{2} b^{5} c^{3} d^{4} - 11 \, a^{3} b^{4} c^{2} d^{5} + 19 \, a^{4} b^{3} c d^{6} - 8 \, a^{5} b^{2} d^{7}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac{3 \,{\left (4 \, b^{9} c^{6} d - 6 \, a b^{8} c^{5} d^{2} + 15 \, a^{4} b^{5} c^{2} d^{5} - 18 \, a^{5} b^{4} c d^{6} + 6 \, a^{6} b^{3} d^{7}\right )}}{{\left (b x + a\right )}^{3} b^{3}}\right )}{\left (b x + a\right )}^{3}}{3 \,{\left (b c - a d\right )}^{3} b^{5}{\left (\frac{b c}{b x + a} - \frac{a d}{b x + a} + d\right )} d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x+a)^2/(d*x+c)^2,x, algorithm="giac")

[Out]

-a^6*b^5/((b^12*c^2 - 2*a*b^11*c*d + a^2*b^10*d^2)*(b*x + a)) - 2*(2*b^2*c^6 - 3*a*b*c^5*d)*log(abs(b*c/(b*x +
a) - a*d/(b*x + a) + d))/(b^4*c^3*d^5 - 3*a*b^3*c^2*d^6 + 3*a^2*b^2*c*d^7 - a^3*b*d^8) + 2*(2*b^3*c^3 + 3*a*b
^2*c^2*d + 3*a^2*b*c*d^2 + 2*a^3*d^3)*log(abs(b*x + a)/((b*x + a)^2*abs(b)))/(b^5*d^5) + 1/3*(b^3*c^3*d^4 - 3*
a*b^2*c^2*d^5 + 3*a^2*b*c*d^6 - a^3*d^7 - (2*b^5*c^4*d^3 + a*b^4*c^3*d^4 - 15*a^2*b^3*c^2*d^5 + 19*a^3*b^2*c*d
^6 - 7*a^4*b*d^7)/((b*x + a)*b) + 3*(2*b^7*c^5*d^2 - a*b^6*c^4*d^3 - a^2*b^5*c^3*d^4 - 11*a^3*b^4*c^2*d^5 + 19
*a^4*b^3*c*d^6 - 8*a^5*b^2*d^7)/((b*x + a)^2*b^2) + 3*(4*b^9*c^6*d - 6*a*b^8*c^5*d^2 + 15*a^4*b^5*c^2*d^5 - 18
*a^5*b^4*c*d^6 + 6*a^6*b^3*d^7)/((b*x + a)^3*b^3))*(b*x + a)^3/((b*c - a*d)^3*b^5*(b*c/(b*x + a) - a*d/(b*x +
a) + d)*d^5)