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

Optimal. Leaf size=231 \[ \frac{3 x \left (a^2 d^2+2 a b c d+2 b^2 c^2\right )}{b^4 d^5}-\frac{c^5 \left (21 a^2 d^2-28 a b c d+10 b^2 c^2\right ) \log (c+d x)}{d^6 (b c-a d)^4}+\frac{a^7}{b^5 (a+b x) (b c-a d)^3}+\frac{a^6 (7 b c-4 a d) \log (a+b x)}{b^5 (b c-a d)^4}-\frac{x^2 (2 a d+3 b c)}{2 b^3 d^4}-\frac{c^6 (5 b c-7 a d)}{d^6 (c+d x) (b c-a d)^3}+\frac{c^7}{2 d^6 (c+d x)^2 (b c-a d)^2}+\frac{x^3}{3 b^2 d^3} \]

[Out]

(3*(2*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*x)/(b^4*d^5) - ((3*b*c + 2*a*d)*x^2)/(2*b^3*d^4) + x^3/(3*b^2*d^3) + a^7/
(b^5*(b*c - a*d)^3*(a + b*x)) + c^7/(2*d^6*(b*c - a*d)^2*(c + d*x)^2) - (c^6*(5*b*c - 7*a*d))/(d^6*(b*c - a*d)
^3*(c + d*x)) + (a^6*(7*b*c - 4*a*d)*Log[a + b*x])/(b^5*(b*c - a*d)^4) - (c^5*(10*b^2*c^2 - 28*a*b*c*d + 21*a^
2*d^2)*Log[c + d*x])/(d^6*(b*c - a*d)^4)

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Rubi [A]  time = 0.367475, antiderivative size = 231, 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} \[ \frac{3 x \left (a^2 d^2+2 a b c d+2 b^2 c^2\right )}{b^4 d^5}-\frac{c^5 \left (21 a^2 d^2-28 a b c d+10 b^2 c^2\right ) \log (c+d x)}{d^6 (b c-a d)^4}+\frac{a^7}{b^5 (a+b x) (b c-a d)^3}+\frac{a^6 (7 b c-4 a d) \log (a+b x)}{b^5 (b c-a d)^4}-\frac{x^2 (2 a d+3 b c)}{2 b^3 d^4}-\frac{c^6 (5 b c-7 a d)}{d^6 (c+d x) (b c-a d)^3}+\frac{c^7}{2 d^6 (c+d x)^2 (b c-a d)^2}+\frac{x^3}{3 b^2 d^3} \]

Antiderivative was successfully verified.

[In]

Int[x^7/((a + b*x)^2*(c + d*x)^3),x]

[Out]

(3*(2*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*x)/(b^4*d^5) - ((3*b*c + 2*a*d)*x^2)/(2*b^3*d^4) + x^3/(3*b^2*d^3) + a^7/
(b^5*(b*c - a*d)^3*(a + b*x)) + c^7/(2*d^6*(b*c - a*d)^2*(c + d*x)^2) - (c^6*(5*b*c - 7*a*d))/(d^6*(b*c - a*d)
^3*(c + d*x)) + (a^6*(7*b*c - 4*a*d)*Log[a + b*x])/(b^5*(b*c - a*d)^4) - (c^5*(10*b^2*c^2 - 28*a*b*c*d + 21*a^
2*d^2)*Log[c + d*x])/(d^6*(b*c - a*d)^4)

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^7}{(a+b x)^2 (c+d x)^3} \, dx &=\int \left (\frac{3 \left (2 b^2 c^2+2 a b c d+a^2 d^2\right )}{b^4 d^5}-\frac{(3 b c+2 a d) x}{b^3 d^4}+\frac{x^2}{b^2 d^3}-\frac{a^7}{b^4 (b c-a d)^3 (a+b x)^2}-\frac{a^6 (-7 b c+4 a d)}{b^4 (b c-a d)^4 (a+b x)}-\frac{c^7}{d^5 (-b c+a d)^2 (c+d x)^3}-\frac{c^6 (5 b c-7 a d)}{d^5 (-b c+a d)^3 (c+d x)^2}-\frac{c^5 \left (10 b^2 c^2-28 a b c d+21 a^2 d^2\right )}{d^5 (-b c+a d)^4 (c+d x)}\right ) \, dx\\ &=\frac{3 \left (2 b^2 c^2+2 a b c d+a^2 d^2\right ) x}{b^4 d^5}-\frac{(3 b c+2 a d) x^2}{2 b^3 d^4}+\frac{x^3}{3 b^2 d^3}+\frac{a^7}{b^5 (b c-a d)^3 (a+b x)}+\frac{c^7}{2 d^6 (b c-a d)^2 (c+d x)^2}-\frac{c^6 (5 b c-7 a d)}{d^6 (b c-a d)^3 (c+d x)}+\frac{a^6 (7 b c-4 a d) \log (a+b x)}{b^5 (b c-a d)^4}-\frac{c^5 \left (10 b^2 c^2-28 a b c d+21 a^2 d^2\right ) \log (c+d x)}{d^6 (b c-a d)^4}\\ \end{align*}

Mathematica [A]  time = 0.310156, size = 230, normalized size = 1. \[ \frac{3 x \left (a^2 d^2+2 a b c d+2 b^2 c^2\right )}{b^4 d^5}-\frac{c^5 \left (21 a^2 d^2-28 a b c d+10 b^2 c^2\right ) \log (c+d x)}{d^6 (b c-a d)^4}+\frac{a^7}{b^5 (a+b x) (b c-a d)^3}+\frac{a^6 (7 b c-4 a d) \log (a+b x)}{b^5 (b c-a d)^4}-\frac{x^2 (2 a d+3 b c)}{2 b^3 d^4}+\frac{c^6 (5 b c-7 a d)}{d^6 (c+d x) (a d-b c)^3}+\frac{c^7}{2 d^6 (c+d x)^2 (b c-a d)^2}+\frac{x^3}{3 b^2 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(2*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*x)/(b^4*d^5) - ((3*b*c + 2*a*d)*x^2)/(2*b^3*d^4) + x^3/(3*b^2*d^3) + a^7/
(b^5*(b*c - a*d)^3*(a + b*x)) + c^7/(2*d^6*(b*c - a*d)^2*(c + d*x)^2) + (c^6*(5*b*c - 7*a*d))/(d^6*(-(b*c) + a
*d)^3*(c + d*x)) + (a^6*(7*b*c - 4*a*d)*Log[a + b*x])/(b^5*(b*c - a*d)^4) - (c^5*(10*b^2*c^2 - 28*a*b*c*d + 21
*a^2*d^2)*Log[c + d*x])/(d^6*(b*c - a*d)^4)

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Maple [A]  time = 0.016, size = 304, normalized size = 1.3 \begin{align*}{\frac{{x}^{3}}{3\,{b}^{2}{d}^{3}}}-{\frac{a{x}^{2}}{{b}^{3}{d}^{3}}}-{\frac{3\,c{x}^{2}}{2\,{b}^{2}{d}^{4}}}+3\,{\frac{{a}^{2}x}{{b}^{4}{d}^{3}}}+6\,{\frac{acx}{{b}^{3}{d}^{4}}}+6\,{\frac{{c}^{2}x}{{b}^{2}{d}^{5}}}-7\,{\frac{{c}^{6}a}{{d}^{5} \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) }}+5\,{\frac{{c}^{7}b}{ \left ( ad-bc \right ) ^{3}{d}^{6} \left ( dx+c \right ) }}+{\frac{{c}^{7}}{2\,{d}^{6} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) ^{2}}}-21\,{\frac{{c}^{5}\ln \left ( dx+c \right ){a}^{2}}{{d}^{4} \left ( ad-bc \right ) ^{4}}}+28\,{\frac{{c}^{6}\ln \left ( dx+c \right ) ab}{{d}^{5} \left ( ad-bc \right ) ^{4}}}-10\,{\frac{{c}^{7}\ln \left ( dx+c \right ){b}^{2}}{{d}^{6} \left ( ad-bc \right ) ^{4}}}-{\frac{{a}^{7}}{{b}^{5} \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) }}-4\,{\frac{{a}^{7}\ln \left ( bx+a \right ) d}{{b}^{5} \left ( ad-bc \right ) ^{4}}}+7\,{\frac{{a}^{6}\ln \left ( bx+a \right ) c}{{b}^{4} \left ( ad-bc \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3/b^2/d^3-1/b^3/d^3*x^2*a-3/2/b^2/d^4*x^2*c+3/b^4/d^3*a^2*x+6/b^3/d^4*a*c*x+6/b^2/d^5*c^2*x-7/d^5*c^6/(a
*d-b*c)^3/(d*x+c)*a+5/d^6*c^7/(a*d-b*c)^3/(d*x+c)*b+1/2/d^6*c^7/(a*d-b*c)^2/(d*x+c)^2-21/d^4*c^5/(a*d-b*c)^4*l
n(d*x+c)*a^2+28/d^5*c^6/(a*d-b*c)^4*ln(d*x+c)*a*b-10/d^6*c^7/(a*d-b*c)^4*ln(d*x+c)*b^2-1/b^5*a^7/(a*d-b*c)^3/(
b*x+a)-4/b^5*a^7/(a*d-b*c)^4*ln(b*x+a)*d+7/b^4*a^6/(a*d-b*c)^4*ln(b*x+a)*c

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Maxima [B]  time = 1.3117, size = 788, normalized size = 3.41 \begin{align*} \frac{{\left (7 \, a^{6} b c - 4 \, a^{7} d\right )} \log \left (b x + a\right )}{b^{9} c^{4} - 4 \, a b^{8} c^{3} d + 6 \, a^{2} b^{7} c^{2} d^{2} - 4 \, a^{3} b^{6} c d^{3} + a^{4} b^{5} d^{4}} - \frac{{\left (10 \, b^{2} c^{7} - 28 \, a b c^{6} d + 21 \, a^{2} c^{5} d^{2}\right )} \log \left (d x + c\right )}{b^{4} c^{4} d^{6} - 4 \, a b^{3} c^{3} d^{7} + 6 \, a^{2} b^{2} c^{2} d^{8} - 4 \, a^{3} b c d^{9} + a^{4} d^{10}} - \frac{9 \, a b^{6} c^{8} - 13 \, a^{2} b^{5} c^{7} d - 2 \, a^{7} c^{2} d^{6} + 2 \,{\left (5 \, b^{7} c^{7} d - 7 \, a b^{6} c^{6} d^{2} - a^{7} d^{8}\right )} x^{2} +{\left (9 \, b^{7} c^{8} - 3 \, a b^{6} c^{7} d - 14 \, a^{2} b^{5} c^{6} d^{2} - 4 \, a^{7} c d^{7}\right )} x}{2 \,{\left (a b^{8} c^{5} d^{6} - 3 \, a^{2} b^{7} c^{4} d^{7} + 3 \, a^{3} b^{6} c^{3} d^{8} - a^{4} b^{5} c^{2} d^{9} +{\left (b^{9} c^{3} d^{8} - 3 \, a b^{8} c^{2} d^{9} + 3 \, a^{2} b^{7} c d^{10} - a^{3} b^{6} d^{11}\right )} x^{3} +{\left (2 \, b^{9} c^{4} d^{7} - 5 \, a b^{8} c^{3} d^{8} + 3 \, a^{2} b^{7} c^{2} d^{9} + a^{3} b^{6} c d^{10} - a^{4} b^{5} d^{11}\right )} x^{2} +{\left (b^{9} c^{5} d^{6} - a b^{8} c^{4} d^{7} - 3 \, a^{2} b^{7} c^{3} d^{8} + 5 \, a^{3} b^{6} c^{2} d^{9} - 2 \, a^{4} b^{5} c d^{10}\right )} x\right )}} + \frac{2 \, b^{2} d^{2} x^{3} - 3 \,{\left (3 \, b^{2} c d + 2 \, a b d^{2}\right )} x^{2} + 18 \,{\left (2 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} x}{6 \, b^{4} d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(7*a^6*b*c - 4*a^7*d)*log(b*x + a)/(b^9*c^4 - 4*a*b^8*c^3*d + 6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*d^3 + a^4*b^5*d^
4) - (10*b^2*c^7 - 28*a*b*c^6*d + 21*a^2*c^5*d^2)*log(d*x + c)/(b^4*c^4*d^6 - 4*a*b^3*c^3*d^7 + 6*a^2*b^2*c^2*
d^8 - 4*a^3*b*c*d^9 + a^4*d^10) - 1/2*(9*a*b^6*c^8 - 13*a^2*b^5*c^7*d - 2*a^7*c^2*d^6 + 2*(5*b^7*c^7*d - 7*a*b
^6*c^6*d^2 - a^7*d^8)*x^2 + (9*b^7*c^8 - 3*a*b^6*c^7*d - 14*a^2*b^5*c^6*d^2 - 4*a^7*c*d^7)*x)/(a*b^8*c^5*d^6 -
3*a^2*b^7*c^4*d^7 + 3*a^3*b^6*c^3*d^8 - a^4*b^5*c^2*d^9 + (b^9*c^3*d^8 - 3*a*b^8*c^2*d^9 + 3*a^2*b^7*c*d^10 -
a^3*b^6*d^11)*x^3 + (2*b^9*c^4*d^7 - 5*a*b^8*c^3*d^8 + 3*a^2*b^7*c^2*d^9 + a^3*b^6*c*d^10 - a^4*b^5*d^11)*x^2
+ (b^9*c^5*d^6 - a*b^8*c^4*d^7 - 3*a^2*b^7*c^3*d^8 + 5*a^3*b^6*c^2*d^9 - 2*a^4*b^5*c*d^10)*x) + 1/6*(2*b^2*d^
2*x^3 - 3*(3*b^2*c*d + 2*a*b*d^2)*x^2 + 18*(2*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*x)/(b^4*d^5)

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Fricas [B]  time = 4.55461, size = 2431, normalized size = 10.52 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(27*a*b^7*c^9 - 66*a^2*b^6*c^8*d + 39*a^3*b^5*c^7*d^2 - 6*a^7*b*c^3*d^6 + 6*a^8*c^2*d^7 - 2*(b^8*c^4*d^5
- 4*a*b^7*c^3*d^6 + 6*a^2*b^6*c^2*d^7 - 4*a^3*b^5*c*d^8 + a^4*b^4*d^9)*x^6 + (5*b^8*c^5*d^4 - 16*a*b^7*c^4*d^5
+ 14*a^2*b^6*c^3*d^6 + 4*a^3*b^5*c^2*d^7 - 11*a^4*b^4*c*d^8 + 4*a^5*b^3*d^9)*x^5 - (20*b^8*c^6*d^3 - 61*a*b^7
*c^5*d^4 + 56*a^2*b^6*c^4*d^5 - 14*a^3*b^5*c^3*d^6 + 16*a^4*b^4*c^2*d^7 - 29*a^5*b^3*c*d^8 + 12*a^6*b^2*d^9)*x
^4 - (63*b^8*c^7*d^2 - 166*a*b^7*c^6*d^3 + 94*a^2*b^6*c^5*d^4 + 42*a^3*b^5*c^4*d^5 + 7*a^4*b^4*c^3*d^6 - 46*a^
5*b^3*c^2*d^7 - 12*a^6*b^2*c*d^8 + 18*a^7*b*d^9)*x^3 - 3*(2*b^8*c^8*d + 9*a*b^7*c^7*d^2 - 46*a^2*b^6*c^6*d^3 +
50*a^3*b^5*c^5*d^4 - 7*a^5*b^3*c^3*d^6 - 20*a^6*b^2*c^2*d^7 + 14*a^7*b*c*d^8 - 2*a^8*d^9)*x^2 + 3*(9*b^8*c^9
- 24*a*b^7*c^8*d + 25*a^2*b^6*c^7*d^2 - 16*a^3*b^5*c^6*d^3 + 12*a^6*b^2*c^3*d^6 - 10*a^7*b*c^2*d^7 + 4*a^8*c*d
^8)*x - 6*(7*a^7*b*c^3*d^6 - 4*a^8*c^2*d^7 + (7*a^6*b^2*c*d^8 - 4*a^7*b*d^9)*x^3 + (14*a^6*b^2*c^2*d^7 - a^7*b
*c*d^8 - 4*a^8*d^9)*x^2 + (7*a^6*b^2*c^3*d^6 + 10*a^7*b*c^2*d^7 - 8*a^8*c*d^8)*x)*log(b*x + a) + 6*(10*a*b^7*c
^9 - 28*a^2*b^6*c^8*d + 21*a^3*b^5*c^7*d^2 + (10*b^8*c^7*d^2 - 28*a*b^7*c^6*d^3 + 21*a^2*b^6*c^5*d^4)*x^3 + (2
0*b^8*c^8*d - 46*a*b^7*c^7*d^2 + 14*a^2*b^6*c^6*d^3 + 21*a^3*b^5*c^5*d^4)*x^2 + (10*b^8*c^9 - 8*a*b^7*c^8*d -
35*a^2*b^6*c^7*d^2 + 42*a^3*b^5*c^6*d^3)*x)*log(d*x + c))/(a*b^9*c^6*d^6 - 4*a^2*b^8*c^5*d^7 + 6*a^3*b^7*c^4*d
^8 - 4*a^4*b^6*c^3*d^9 + a^5*b^5*c^2*d^10 + (b^10*c^4*d^8 - 4*a*b^9*c^3*d^9 + 6*a^2*b^8*c^2*d^10 - 4*a^3*b^7*c
*d^11 + a^4*b^6*d^12)*x^3 + (2*b^10*c^5*d^7 - 7*a*b^9*c^4*d^8 + 8*a^2*b^8*c^3*d^9 - 2*a^3*b^7*c^2*d^10 - 2*a^4
*b^6*c*d^11 + a^5*b^5*d^12)*x^2 + (b^10*c^6*d^6 - 2*a*b^9*c^5*d^7 - 2*a^2*b^8*c^4*d^8 + 8*a^3*b^7*c^3*d^9 - 7*
a^4*b^6*c^2*d^10 + 2*a^5*b^5*c*d^11)*x)

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Sympy [B]  time = 16.9454, size = 1221, normalized size = 5.29 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**6*(4*a*d - 7*b*c)*log(x + (a**11*d**10*(4*a*d - 7*b*c)/(b*(a*d - b*c)**4) - 5*a**10*c*d**9*(4*a*d - 7*b*c)
/(a*d - b*c)**4 + 10*a**9*b*c**2*d**8*(4*a*d - 7*b*c)/(a*d - b*c)**4 - 10*a**8*b**2*c**3*d**7*(4*a*d - 7*b*c)/
(a*d - b*c)**4 + 5*a**7*b**3*c**4*d**6*(4*a*d - 7*b*c)/(a*d - b*c)**4 + 4*a**7*c*d**6 - a**6*b**4*c**5*d**5*(4
*a*d - 7*b*c)/(a*d - b*c)**4 - 7*a**6*b*c**2*d**5 - 21*a**3*b**4*c**5*d**2 + 28*a**2*b**5*c**6*d - 10*a*b**6*c
**7)/(4*a**7*d**7 - 7*a**6*b*c*d**6 - 21*a**2*b**5*c**5*d**2 + 28*a*b**6*c**6*d - 10*b**7*c**7))/(b**5*(a*d -
b*c)**4) - c**5*(21*a**2*d**2 - 28*a*b*c*d + 10*b**2*c**2)*log(x + (4*a**7*c*d**6 - 7*a**6*b*c**2*d**5 + a**5*
b**4*c**5*d**4*(21*a**2*d**2 - 28*a*b*c*d + 10*b**2*c**2)/(a*d - b*c)**4 - 5*a**4*b**5*c**6*d**3*(21*a**2*d**2
- 28*a*b*c*d + 10*b**2*c**2)/(a*d - b*c)**4 + 10*a**3*b**6*c**7*d**2*(21*a**2*d**2 - 28*a*b*c*d + 10*b**2*c**
2)/(a*d - b*c)**4 - 21*a**3*b**4*c**5*d**2 - 10*a**2*b**7*c**8*d*(21*a**2*d**2 - 28*a*b*c*d + 10*b**2*c**2)/(a
*d - b*c)**4 + 28*a**2*b**5*c**6*d + 5*a*b**8*c**9*(21*a**2*d**2 - 28*a*b*c*d + 10*b**2*c**2)/(a*d - b*c)**4 -
10*a*b**6*c**7 - b**9*c**10*(21*a**2*d**2 - 28*a*b*c*d + 10*b**2*c**2)/(d*(a*d - b*c)**4))/(4*a**7*d**7 - 7*a
**6*b*c*d**6 - 21*a**2*b**5*c**5*d**2 + 28*a*b**6*c**6*d - 10*b**7*c**7))/(d**6*(a*d - b*c)**4) - (2*a**7*c**2
*d**6 + 13*a**2*b**5*c**7*d - 9*a*b**6*c**8 + x**2*(2*a**7*d**8 + 14*a*b**6*c**6*d**2 - 10*b**7*c**7*d) + x*(4
*a**7*c*d**7 + 14*a**2*b**5*c**6*d**2 + 3*a*b**6*c**7*d - 9*b**7*c**8))/(2*a**4*b**5*c**2*d**9 - 6*a**3*b**6*c
**3*d**8 + 6*a**2*b**7*c**4*d**7 - 2*a*b**8*c**5*d**6 + x**3*(2*a**3*b**6*d**11 - 6*a**2*b**7*c*d**10 + 6*a*b*
*8*c**2*d**9 - 2*b**9*c**3*d**8) + x**2*(2*a**4*b**5*d**11 - 2*a**3*b**6*c*d**10 - 6*a**2*b**7*c**2*d**9 + 10*
a*b**8*c**3*d**8 - 4*b**9*c**4*d**7) + x*(4*a**4*b**5*c*d**10 - 10*a**3*b**6*c**2*d**9 + 6*a**2*b**7*c**3*d**8
+ 2*a*b**8*c**4*d**7 - 2*b**9*c**5*d**6)) + x**3/(3*b**2*d**3) - x**2*(2*a*d + 3*b*c)/(2*b**3*d**4) + x*(3*a*
*2*d**2 + 6*a*b*c*d + 6*b**2*c**2)/(b**4*d**5)

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Giac [B]  time = 1.26496, size = 1004, normalized size = 4.35 \begin{align*} \frac{a^{7} b^{6}}{{\left (b^{14} c^{3} - 3 \, a b^{13} c^{2} d + 3 \, a^{2} b^{12} c d^{2} - a^{3} b^{11} d^{3}\right )}{\left (b x + a\right )}} - \frac{{\left (10 \, b^{3} c^{7} - 28 \, a b^{2} c^{6} d + 21 \, a^{2} b c^{5} d^{2}\right )} \log \left ({\left | \frac{b c}{b x + a} - \frac{a d}{b x + a} + d \right |}\right )}{b^{5} c^{4} d^{6} - 4 \, a b^{4} c^{3} d^{7} + 6 \, a^{2} b^{3} c^{2} d^{8} - 4 \, a^{3} b^{2} c d^{9} + a^{4} b d^{10}} + \frac{{\left (10 \, b^{3} c^{3} + 12 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + 4 \, 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^{6}} + \frac{{\left (2 \, b^{4} c^{4} d^{5} - 8 \, a b^{3} c^{3} d^{6} + 12 \, a^{2} b^{2} c^{2} d^{7} - 8 \, a^{3} b c d^{8} + 2 \, a^{4} d^{9} - \frac{5 \, b^{6} c^{5} d^{4} - 4 \, a b^{5} c^{4} d^{5} - 34 \, a^{2} b^{4} c^{3} d^{6} + 76 \, a^{3} b^{3} c^{2} d^{7} - 59 \, a^{4} b^{2} c d^{8} + 16 \, a^{5} b d^{9}}{{\left (b x + a\right )} b} + \frac{2 \,{\left (10 \, b^{8} c^{6} d^{3} - 18 \, a b^{7} c^{5} d^{4} + 3 \, a^{2} b^{6} c^{4} d^{5} - 32 \, a^{3} b^{5} c^{3} d^{6} + 108 \, a^{4} b^{4} c^{2} d^{7} - 102 \, a^{5} b^{3} c d^{8} + 31 \, a^{6} b^{2} d^{9}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac{3 \,{\left (30 \, b^{10} c^{7} d^{2} - 84 \, a b^{9} c^{6} d^{3} + 63 \, a^{2} b^{8} c^{5} d^{4} + 35 \, a^{4} b^{6} c^{3} d^{6} - 126 \, a^{5} b^{5} c^{2} d^{7} + 105 \, a^{6} b^{4} c d^{8} - 28 \, a^{7} b^{3} d^{9}\right )}}{{\left (b x + a\right )}^{3} b^{3}} + \frac{6 \,{\left (10 \, b^{12} c^{8} d - 38 \, a b^{11} c^{7} d^{2} + 49 \, a^{2} b^{10} c^{6} d^{3} - 21 \, a^{3} b^{9} c^{5} d^{4} - 21 \, a^{5} b^{7} c^{3} d^{6} + 42 \, a^{6} b^{6} c^{2} d^{7} - 27 \, a^{7} b^{5} c d^{8} + 6 \, a^{8} b^{4} d^{9}\right )}}{{\left (b x + a\right )}^{4} b^{4}}\right )}{\left (b x + a\right )}^{3}}{6 \,{\left (b c - a d\right )}^{4} b^{5}{\left (\frac{b c}{b x + a} - \frac{a d}{b x + a} + d\right )}^{2} d^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^7*b^6/((b^14*c^3 - 3*a*b^13*c^2*d + 3*a^2*b^12*c*d^2 - a^3*b^11*d^3)*(b*x + a)) - (10*b^3*c^7 - 28*a*b^2*c^6
*d + 21*a^2*b*c^5*d^2)*log(abs(b*c/(b*x + a) - a*d/(b*x + a) + d))/(b^5*c^4*d^6 - 4*a*b^4*c^3*d^7 + 6*a^2*b^3*
c^2*d^8 - 4*a^3*b^2*c*d^9 + a^4*b*d^10) + (10*b^3*c^3 + 12*a*b^2*c^2*d + 9*a^2*b*c*d^2 + 4*a^3*d^3)*log(abs(b*
x + a)/((b*x + a)^2*abs(b)))/(b^5*d^6) + 1/6*(2*b^4*c^4*d^5 - 8*a*b^3*c^3*d^6 + 12*a^2*b^2*c^2*d^7 - 8*a^3*b*c
*d^8 + 2*a^4*d^9 - (5*b^6*c^5*d^4 - 4*a*b^5*c^4*d^5 - 34*a^2*b^4*c^3*d^6 + 76*a^3*b^3*c^2*d^7 - 59*a^4*b^2*c*d
^8 + 16*a^5*b*d^9)/((b*x + a)*b) + 2*(10*b^8*c^6*d^3 - 18*a*b^7*c^5*d^4 + 3*a^2*b^6*c^4*d^5 - 32*a^3*b^5*c^3*d
^6 + 108*a^4*b^4*c^2*d^7 - 102*a^5*b^3*c*d^8 + 31*a^6*b^2*d^9)/((b*x + a)^2*b^2) + 3*(30*b^10*c^7*d^2 - 84*a*b
^9*c^6*d^3 + 63*a^2*b^8*c^5*d^4 + 35*a^4*b^6*c^3*d^6 - 126*a^5*b^5*c^2*d^7 + 105*a^6*b^4*c*d^8 - 28*a^7*b^3*d^
9)/((b*x + a)^3*b^3) + 6*(10*b^12*c^8*d - 38*a*b^11*c^7*d^2 + 49*a^2*b^10*c^6*d^3 - 21*a^3*b^9*c^5*d^4 - 21*a^
5*b^7*c^3*d^6 + 42*a^6*b^6*c^2*d^7 - 27*a^7*b^5*c*d^8 + 6*a^8*b^4*d^9)/((b*x + a)^4*b^4))*(b*x + a)^3/((b*c -
a*d)^4*b^5*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*d^6)