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

Optimal. Leaf size=245 \[ \frac{a^7}{2 b^5 (a+b x)^2 (b c-a d)^3}-\frac{a^6 (7 b c-4 a d)}{b^5 (a+b x) (b c-a d)^4}+\frac{3 c^5 \left (7 a^2 d^2-7 a b c d+2 b^2 c^2\right ) \log (c+d x)}{d^5 (b c-a d)^5}-\frac{3 a^5 \left (2 a^2 d^2-7 a b c d+7 b^2 c^2\right ) \log (a+b x)}{b^5 (b c-a d)^5}-\frac{3 x (a d+b c)}{b^4 d^4}-\frac{c^7}{2 d^5 (c+d x)^2 (b c-a d)^3}+\frac{c^6 (4 b c-7 a d)}{d^5 (c+d x) (b c-a d)^4}+\frac{x^2}{2 b^3 d^3} \]

[Out]

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

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Rubi [A]  time = 0.776104, antiderivative size = 245, 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 \[ \frac{a^7}{2 b^5 (a+b x)^2 (b c-a d)^3}-\frac{a^6 (7 b c-4 a d)}{b^5 (a+b x) (b c-a d)^4}+\frac{3 c^5 \left (7 a^2 d^2-7 a b c d+2 b^2 c^2\right ) \log (c+d x)}{d^5 (b c-a d)^5}-\frac{3 a^5 \left (2 a^2 d^2-7 a b c d+7 b^2 c^2\right ) \log (a+b x)}{b^5 (b c-a d)^5}-\frac{3 x (a d+b c)}{b^4 d^4}-\frac{c^7}{2 d^5 (c+d x)^2 (b c-a d)^3}+\frac{c^6 (4 b c-7 a d)}{d^5 (c+d x) (b c-a d)^4}+\frac{x^2}{2 b^3 d^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(b*x+a)**3/(d*x+c)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.814025, size = 241, normalized size = 0.98 \[ \frac{1}{2} \left (\frac{a^7}{b^5 (a+b x)^2 (b c-a d)^3}+\frac{2 a^6 (4 a d-7 b c)}{b^5 (a+b x) (b c-a d)^4}-\frac{6 c^5 \left (7 a^2 d^2-7 a b c d+2 b^2 c^2\right ) \log (c+d x)}{d^5 (a d-b c)^5}-\frac{6 a^5 \left (2 a^2 d^2-7 a b c d+7 b^2 c^2\right ) \log (a+b x)}{b^5 (b c-a d)^5}-\frac{6 x (a d+b c)}{b^4 d^4}+\frac{c^7}{d^5 (c+d x)^2 (a d-b c)^3}+\frac{2 c^6 (4 b c-7 a d)}{d^5 (c+d x) (b c-a d)^4}+\frac{x^2}{b^3 d^3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.031, size = 347, normalized size = 1.4 \[{\frac{{x}^{2}}{2\,{b}^{3}{d}^{3}}}-3\,{\frac{ax}{{d}^{3}{b}^{4}}}-3\,{\frac{cx}{{d}^{4}{b}^{3}}}-7\,{\frac{{c}^{6}a}{{d}^{4} \left ( ad-bc \right ) ^{4} \left ( dx+c \right ) }}+4\,{\frac{{c}^{7}b}{ \left ( ad-bc \right ) ^{4}{d}^{5} \left ( dx+c \right ) }}+{\frac{{c}^{7}}{2\,{d}^{5} \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) ^{2}}}-21\,{\frac{{c}^{5}\ln \left ( dx+c \right ){a}^{2}}{{d}^{3} \left ( ad-bc \right ) ^{5}}}+21\,{\frac{{c}^{6}\ln \left ( dx+c \right ) ab}{{d}^{4} \left ( ad-bc \right ) ^{5}}}-6\,{\frac{{c}^{7}\ln \left ( dx+c \right ){b}^{2}}{{d}^{5} \left ( ad-bc \right ) ^{5}}}-{\frac{{a}^{7}}{2\,{b}^{5} \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) ^{2}}}+6\,{\frac{{a}^{7}\ln \left ( bx+a \right ){d}^{2}}{{b}^{5} \left ( ad-bc \right ) ^{5}}}-21\,{\frac{{a}^{6}\ln \left ( bx+a \right ) cd}{{b}^{4} \left ( ad-bc \right ) ^{5}}}+21\,{\frac{{a}^{5}\ln \left ( bx+a \right ){c}^{2}}{{b}^{3} \left ( ad-bc \right ) ^{5}}}+4\,{\frac{{a}^{7}d}{ \left ( ad-bc \right ) ^{4}{b}^{5} \left ( bx+a \right ) }}-7\,{\frac{{a}^{6}c}{{b}^{4} \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(b*x+a)^3/(d*x+c)^3,x)

[Out]

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

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Maxima [A]  time = 1.42103, size = 1135, normalized size = 4.63 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(7*a^5*b^2*c^2 - 7*a^6*b*c*d + 2*a^7*d^2)*log(b*x + a)/(b^10*c^5 - 5*a*b^9*c^
4*d + 10*a^2*b^8*c^3*d^2 - 10*a^3*b^7*c^2*d^3 + 5*a^4*b^6*c*d^4 - a^5*b^5*d^5) +
3*(2*b^2*c^7 - 7*a*b*c^6*d + 7*a^2*c^5*d^2)*log(d*x + c)/(b^5*c^5*d^5 - 5*a*b^4
*c^4*d^6 + 10*a^2*b^3*c^3*d^7 - 10*a^3*b^2*c^2*d^8 + 5*a^4*b*c*d^9 - a^5*d^10) +
1/2*(7*a^2*b^6*c^8 - 13*a^3*b^5*c^7*d - 13*a^7*b*c^3*d^5 + 7*a^8*c^2*d^6 + 2*(4
*b^8*c^7*d - 7*a*b^7*c^6*d^2 - 7*a^6*b^2*c*d^7 + 4*a^7*b*d^8)*x^3 + (7*b^8*c^8 +
3*a*b^7*c^7*d - 28*a^2*b^6*c^6*d^2 - 28*a^6*b^2*c^2*d^6 + 3*a^7*b*c*d^7 + 7*a^8
*d^8)*x^2 + 2*(7*a*b^7*c^8 - 9*a^2*b^6*c^7*d - 7*a^3*b^5*c^6*d^2 - 7*a^6*b^2*c^3
*d^5 - 9*a^7*b*c^2*d^6 + 7*a^8*c*d^7)*x)/(a^2*b^9*c^6*d^5 - 4*a^3*b^8*c^5*d^6 +
6*a^4*b^7*c^4*d^7 - 4*a^5*b^6*c^3*d^8 + a^6*b^5*c^2*d^9 + (b^11*c^4*d^7 - 4*a*b^
10*c^3*d^8 + 6*a^2*b^9*c^2*d^9 - 4*a^3*b^8*c*d^10 + a^4*b^7*d^11)*x^4 + 2*(b^11*
c^5*d^6 - 3*a*b^10*c^4*d^7 + 2*a^2*b^9*c^3*d^8 + 2*a^3*b^8*c^2*d^9 - 3*a^4*b^7*c
*d^10 + a^5*b^6*d^11)*x^3 + (b^11*c^6*d^5 - 9*a^2*b^9*c^4*d^7 + 16*a^3*b^8*c^3*d
^8 - 9*a^4*b^7*c^2*d^9 + a^6*b^5*d^11)*x^2 + 2*(a*b^10*c^6*d^5 - 3*a^2*b^9*c^5*d
^6 + 2*a^3*b^8*c^4*d^7 + 2*a^4*b^7*c^3*d^8 - 3*a^5*b^6*c^2*d^9 + a^6*b^5*c*d^10)
*x) + 1/2*(b*d*x^2 - 6*(b*c + a*d)*x)/(b^4*d^4)

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Fricas [A]  time = 0.367481, size = 2115, normalized size = 8.63 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(7*a^2*b^7*c^9 - 20*a^3*b^6*c^8*d + 13*a^4*b^5*c^7*d^2 - 13*a^7*b^2*c^4*d^5
+ 20*a^8*b*c^3*d^6 - 7*a^9*c^2*d^7 + (b^9*c^5*d^4 - 5*a*b^8*c^4*d^5 + 10*a^2*b^7
*c^3*d^6 - 10*a^3*b^6*c^2*d^7 + 5*a^4*b^5*c*d^8 - a^5*b^4*d^9)*x^6 - 4*(b^9*c^6*
d^3 - 4*a*b^8*c^5*d^4 + 5*a^2*b^7*c^4*d^5 - 5*a^4*b^5*c^2*d^7 + 4*a^5*b^4*c*d^8
- a^6*b^3*d^9)*x^5 - (11*b^9*c^7*d^2 - 35*a*b^8*c^6*d^3 + 21*a^2*b^7*c^5*d^4 + 3
5*a^3*b^6*c^4*d^5 - 35*a^4*b^5*c^3*d^6 - 21*a^5*b^4*c^2*d^7 + 35*a^6*b^3*c*d^8 -
11*a^7*b^2*d^9)*x^4 + 2*(b^9*c^8*d - 10*a*b^8*c^7*d^2 + 33*a^2*b^7*c^6*d^3 - 43
*a^3*b^6*c^5*d^4 + 43*a^5*b^4*c^3*d^6 - 33*a^6*b^3*c^2*d^7 + 10*a^7*b^2*c*d^8 -
a^8*b*d^9)*x^3 + (7*b^9*c^9 - 16*a*b^8*c^8*d + 6*a^2*b^7*c^7*d^2 + 11*a^3*b^6*c^
6*d^3 - 50*a^4*b^5*c^5*d^4 + 50*a^5*b^4*c^4*d^5 - 11*a^6*b^3*c^3*d^6 - 6*a^7*b^2
*c^2*d^7 + 16*a^8*b*c*d^8 - 7*a^9*d^9)*x^2 + 2*(7*a*b^8*c^9 - 19*a^2*b^7*c^8*d +
14*a^3*b^6*c^7*d^2 - 8*a^4*b^5*c^6*d^3 + 8*a^6*b^3*c^4*d^5 - 14*a^7*b^2*c^3*d^6
+ 19*a^8*b*c^2*d^7 - 7*a^9*c*d^8)*x - 6*(7*a^7*b^2*c^4*d^5 - 7*a^8*b*c^3*d^6 +
2*a^9*c^2*d^7 + (7*a^5*b^4*c^2*d^7 - 7*a^6*b^3*c*d^8 + 2*a^7*b^2*d^9)*x^4 + 2*(7
*a^5*b^4*c^3*d^6 - 5*a^7*b^2*c*d^8 + 2*a^8*b*d^9)*x^3 + (7*a^5*b^4*c^4*d^5 + 21*
a^6*b^3*c^3*d^6 - 19*a^7*b^2*c^2*d^7 + a^8*b*c*d^8 + 2*a^9*d^9)*x^2 + 2*(7*a^6*b
^3*c^4*d^5 - 5*a^8*b*c^2*d^7 + 2*a^9*c*d^8)*x)*log(b*x + a) + 6*(2*a^2*b^7*c^9 -
7*a^3*b^6*c^8*d + 7*a^4*b^5*c^7*d^2 + (2*b^9*c^7*d^2 - 7*a*b^8*c^6*d^3 + 7*a^2*
b^7*c^5*d^4)*x^4 + 2*(2*b^9*c^8*d - 5*a*b^8*c^7*d^2 + 7*a^3*b^6*c^5*d^4)*x^3 + (
2*b^9*c^9 + a*b^8*c^8*d - 19*a^2*b^7*c^7*d^2 + 21*a^3*b^6*c^6*d^3 + 7*a^4*b^5*c^
5*d^4)*x^2 + 2*(2*a*b^8*c^9 - 5*a^2*b^7*c^8*d + 7*a^4*b^5*c^6*d^3)*x)*log(d*x +
c))/(a^2*b^10*c^7*d^5 - 5*a^3*b^9*c^6*d^6 + 10*a^4*b^8*c^5*d^7 - 10*a^5*b^7*c^4*
d^8 + 5*a^6*b^6*c^3*d^9 - a^7*b^5*c^2*d^10 + (b^12*c^5*d^7 - 5*a*b^11*c^4*d^8 +
10*a^2*b^10*c^3*d^9 - 10*a^3*b^9*c^2*d^10 + 5*a^4*b^8*c*d^11 - a^5*b^7*d^12)*x^4
+ 2*(b^12*c^6*d^6 - 4*a*b^11*c^5*d^7 + 5*a^2*b^10*c^4*d^8 - 5*a^4*b^8*c^2*d^10
+ 4*a^5*b^7*c*d^11 - a^6*b^6*d^12)*x^3 + (b^12*c^7*d^5 - a*b^11*c^6*d^6 - 9*a^2*
b^10*c^5*d^7 + 25*a^3*b^9*c^4*d^8 - 25*a^4*b^8*c^3*d^9 + 9*a^5*b^7*c^2*d^10 + a^
6*b^6*c*d^11 - a^7*b^5*d^12)*x^2 + 2*(a*b^11*c^7*d^5 - 4*a^2*b^10*c^6*d^6 + 5*a^
3*b^9*c^5*d^7 - 5*a^5*b^7*c^3*d^9 + 4*a^6*b^6*c^2*d^10 - a^7*b^5*c*d^11)*x)

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Sympy [A]  time = 109.558, size = 1737, normalized size = 7.09 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(b*x+a)**3/(d*x+c)**3,x)

[Out]

3*a**5*(2*a**2*d**2 - 7*a*b*c*d + 7*b**2*c**2)*log(x + (3*a**11*d**10*(2*a**2*d*
*2 - 7*a*b*c*d + 7*b**2*c**2)/(b*(a*d - b*c)**5) - 18*a**10*c*d**9*(2*a**2*d**2
- 7*a*b*c*d + 7*b**2*c**2)/(a*d - b*c)**5 + 45*a**9*b*c**2*d**8*(2*a**2*d**2 - 7
*a*b*c*d + 7*b**2*c**2)/(a*d - b*c)**5 - 60*a**8*b**2*c**3*d**7*(2*a**2*d**2 - 7
*a*b*c*d + 7*b**2*c**2)/(a*d - b*c)**5 + 45*a**7*b**3*c**4*d**6*(2*a**2*d**2 - 7
*a*b*c*d + 7*b**2*c**2)/(a*d - b*c)**5 + 6*a**7*c*d**6 - 18*a**6*b**4*c**5*d**5*
(2*a**2*d**2 - 7*a*b*c*d + 7*b**2*c**2)/(a*d - b*c)**5 - 21*a**6*b*c**2*d**5 + 3
*a**5*b**5*c**6*d**4*(2*a**2*d**2 - 7*a*b*c*d + 7*b**2*c**2)/(a*d - b*c)**5 + 21
*a**5*b**2*c**3*d**4 + 21*a**3*b**4*c**5*d**2 - 21*a**2*b**5*c**6*d + 6*a*b**6*c
**7)/(6*a**7*d**7 - 21*a**6*b*c*d**6 + 21*a**5*b**2*c**2*d**5 + 21*a**2*b**5*c**
5*d**2 - 21*a*b**6*c**6*d + 6*b**7*c**7))/(b**5*(a*d - b*c)**5) - 3*c**5*(7*a**2
*d**2 - 7*a*b*c*d + 2*b**2*c**2)*log(x + (6*a**7*c*d**6 - 3*a**6*b**4*c**5*d**5*
(7*a**2*d**2 - 7*a*b*c*d + 2*b**2*c**2)/(a*d - b*c)**5 - 21*a**6*b*c**2*d**5 + 1
8*a**5*b**5*c**6*d**4*(7*a**2*d**2 - 7*a*b*c*d + 2*b**2*c**2)/(a*d - b*c)**5 + 2
1*a**5*b**2*c**3*d**4 - 45*a**4*b**6*c**7*d**3*(7*a**2*d**2 - 7*a*b*c*d + 2*b**2
*c**2)/(a*d - b*c)**5 + 60*a**3*b**7*c**8*d**2*(7*a**2*d**2 - 7*a*b*c*d + 2*b**2
*c**2)/(a*d - b*c)**5 + 21*a**3*b**4*c**5*d**2 - 45*a**2*b**8*c**9*d*(7*a**2*d**
2 - 7*a*b*c*d + 2*b**2*c**2)/(a*d - b*c)**5 - 21*a**2*b**5*c**6*d + 18*a*b**9*c*
*10*(7*a**2*d**2 - 7*a*b*c*d + 2*b**2*c**2)/(a*d - b*c)**5 + 6*a*b**6*c**7 - 3*b
**10*c**11*(7*a**2*d**2 - 7*a*b*c*d + 2*b**2*c**2)/(d*(a*d - b*c)**5))/(6*a**7*d
**7 - 21*a**6*b*c*d**6 + 21*a**5*b**2*c**2*d**5 + 21*a**2*b**5*c**5*d**2 - 21*a*
b**6*c**6*d + 6*b**7*c**7))/(d**5*(a*d - b*c)**5) + (7*a**8*c**2*d**6 - 13*a**7*
b*c**3*d**5 - 13*a**3*b**5*c**7*d + 7*a**2*b**6*c**8 + x**3*(8*a**7*b*d**8 - 14*
a**6*b**2*c*d**7 - 14*a*b**7*c**6*d**2 + 8*b**8*c**7*d) + x**2*(7*a**8*d**8 + 3*
a**7*b*c*d**7 - 28*a**6*b**2*c**2*d**6 - 28*a**2*b**6*c**6*d**2 + 3*a*b**7*c**7*
d + 7*b**8*c**8) + x*(14*a**8*c*d**7 - 18*a**7*b*c**2*d**6 - 14*a**6*b**2*c**3*d
**5 - 14*a**3*b**5*c**6*d**2 - 18*a**2*b**6*c**7*d + 14*a*b**7*c**8))/(2*a**6*b*
*5*c**2*d**9 - 8*a**5*b**6*c**3*d**8 + 12*a**4*b**7*c**4*d**7 - 8*a**3*b**8*c**5
*d**6 + 2*a**2*b**9*c**6*d**5 + x**4*(2*a**4*b**7*d**11 - 8*a**3*b**8*c*d**10 +
12*a**2*b**9*c**2*d**9 - 8*a*b**10*c**3*d**8 + 2*b**11*c**4*d**7) + x**3*(4*a**5
*b**6*d**11 - 12*a**4*b**7*c*d**10 + 8*a**3*b**8*c**2*d**9 + 8*a**2*b**9*c**3*d*
*8 - 12*a*b**10*c**4*d**7 + 4*b**11*c**5*d**6) + x**2*(2*a**6*b**5*d**11 - 18*a*
*4*b**7*c**2*d**9 + 32*a**3*b**8*c**3*d**8 - 18*a**2*b**9*c**4*d**7 + 2*b**11*c*
*6*d**5) + x*(4*a**6*b**5*c*d**10 - 12*a**5*b**6*c**2*d**9 + 8*a**4*b**7*c**3*d*
*8 + 8*a**3*b**8*c**4*d**7 - 12*a**2*b**9*c**5*d**6 + 4*a*b**10*c**6*d**5)) + x*
*2/(2*b**3*d**3) - x*(3*a*d + 3*b*c)/(b**4*d**4)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError