3.291 \(\int \frac {x^6}{(a+b x)^2 (c+d x)^3} \, dx\)

Optimal. Leaf size=198 \[ -\frac {a^6}{b^4 (a+b x) (b c-a d)^3}-\frac {3 a^5 (2 b c-a d) \log (a+b x)}{b^4 (b c-a d)^4}+\frac {3 c^4 \left (5 a^2 d^2-6 a b c d+2 b^2 c^2\right ) \log (c+d x)}{d^5 (b c-a d)^4}-\frac {x (2 a d+3 b c)}{b^3 d^4}-\frac {c^6}{2 d^5 (c+d x)^2 (b c-a d)^2}+\frac {2 c^5 (2 b c-3 a d)}{d^5 (c+d x) (b c-a d)^3}+\frac {x^2}{2 b^2 d^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.27, antiderivative size = 198, normalized size of antiderivative = 1.00, 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} \[ \frac {3 c^4 \left (5 a^2 d^2-6 a b c d+2 b^2 c^2\right ) \log (c+d x)}{d^5 (b c-a d)^4}-\frac {a^6}{b^4 (a+b x) (b c-a d)^3}-\frac {3 a^5 (2 b c-a d) \log (a+b x)}{b^4 (b c-a d)^4}-\frac {x (2 a d+3 b c)}{b^3 d^4}+\frac {2 c^5 (2 b c-3 a d)}{d^5 (c+d x) (b c-a d)^3}-\frac {c^6}{2 d^5 (c+d x)^2 (b c-a d)^2}+\frac {x^2}{2 b^2 d^3} \]

Antiderivative was successfully verified.

[In]

Int[x^6/((a + b*x)^2*(c + d*x)^3),x]

[Out]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.31, size = 198, normalized size = 1.00 \[ -\frac {a^6}{b^4 (a+b x) (b c-a d)^3}+\frac {3 a^5 (a d-2 b c) \log (a+b x)}{b^4 (b c-a d)^4}+\frac {3 c^4 \left (5 a^2 d^2-6 a b c d+2 b^2 c^2\right ) \log (c+d x)}{d^5 (b c-a d)^4}-\frac {x (2 a d+3 b c)}{b^3 d^4}-\frac {c^6}{2 d^5 (c+d x)^2 (b c-a d)^2}+\frac {6 a c^5 d-4 b c^6}{d^5 (c+d x) (a d-b c)^3}+\frac {x^2}{2 b^2 d^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/((a + b*x)^2*(c + d*x)^3),x]

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 1.10, size = 1081, normalized size = 5.46 \[ \frac {7 \, a b^{6} c^{8} - 18 \, a^{2} b^{5} c^{7} d + 11 \, a^{3} b^{4} c^{6} d^{2} - 2 \, a^{6} b c^{3} d^{5} + 2 \, a^{7} c^{2} d^{6} + {\left (b^{7} c^{4} d^{4} - 4 \, a b^{6} c^{3} d^{5} + 6 \, a^{2} b^{5} c^{2} d^{6} - 4 \, a^{3} b^{4} c d^{7} + a^{4} b^{3} d^{8}\right )} x^{5} - {\left (4 \, b^{7} c^{5} d^{3} - 13 \, a b^{6} c^{4} d^{4} + 12 \, a^{2} b^{5} c^{3} d^{5} + 2 \, a^{3} b^{4} c^{2} d^{6} - 8 \, a^{4} b^{3} c d^{7} + 3 \, a^{5} b^{2} d^{8}\right )} x^{4} - {\left (11 \, b^{7} c^{6} d^{2} - 32 \, a b^{6} c^{5} d^{3} + 22 \, a^{2} b^{5} c^{4} d^{4} + 12 \, a^{3} b^{4} c^{3} d^{5} - 13 \, a^{4} b^{3} c^{2} d^{6} - 4 \, a^{5} b^{2} c d^{7} + 4 \, a^{6} b d^{8}\right )} x^{3} + {\left (2 \, b^{7} c^{7} d - 11 \, a b^{6} c^{6} d^{2} + 28 \, a^{2} b^{5} c^{5} d^{3} - 34 \, a^{3} b^{4} c^{4} d^{4} + 6 \, a^{4} b^{3} c^{3} d^{5} + 17 \, a^{5} b^{2} c^{2} d^{6} - 10 \, a^{6} b c d^{7} + 2 \, a^{7} d^{8}\right )} x^{2} + {\left (7 \, b^{7} c^{8} - 16 \, a b^{6} c^{7} d + 11 \, a^{2} b^{5} c^{6} d^{2} - 8 \, a^{3} b^{4} c^{5} d^{3} + 10 \, a^{5} b^{2} c^{3} d^{5} - 8 \, a^{6} b c^{2} d^{6} + 4 \, a^{7} c d^{7}\right )} x - 6 \, {\left (2 \, a^{6} b c^{3} d^{5} - a^{7} c^{2} d^{6} + {\left (2 \, a^{5} b^{2} c d^{7} - a^{6} b d^{8}\right )} x^{3} + {\left (4 \, a^{5} b^{2} c^{2} d^{6} - a^{7} d^{8}\right )} x^{2} + {\left (2 \, a^{5} b^{2} c^{3} d^{5} + 3 \, a^{6} b c^{2} d^{6} - 2 \, a^{7} c d^{7}\right )} x\right )} \log \left (b x + a\right ) + 6 \, {\left (2 \, a b^{6} c^{8} - 6 \, a^{2} b^{5} c^{7} d + 5 \, a^{3} b^{4} c^{6} d^{2} + {\left (2 \, b^{7} c^{6} d^{2} - 6 \, a b^{6} c^{5} d^{3} + 5 \, a^{2} b^{5} c^{4} d^{4}\right )} x^{3} + {\left (4 \, b^{7} c^{7} d - 10 \, a b^{6} c^{6} d^{2} + 4 \, a^{2} b^{5} c^{5} d^{3} + 5 \, a^{3} b^{4} c^{4} d^{4}\right )} x^{2} + {\left (2 \, b^{7} c^{8} - 2 \, a b^{6} c^{7} d - 7 \, a^{2} b^{5} c^{6} d^{2} + 10 \, a^{3} b^{4} c^{5} d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a b^{8} c^{6} d^{5} - 4 \, a^{2} b^{7} c^{5} d^{6} + 6 \, a^{3} b^{6} c^{4} d^{7} - 4 \, a^{4} b^{5} c^{3} d^{8} + a^{5} b^{4} c^{2} d^{9} + {\left (b^{9} c^{4} d^{7} - 4 \, a b^{8} c^{3} d^{8} + 6 \, a^{2} b^{7} c^{2} d^{9} - 4 \, a^{3} b^{6} c d^{10} + a^{4} b^{5} d^{11}\right )} x^{3} + {\left (2 \, b^{9} c^{5} d^{6} - 7 \, a b^{8} c^{4} d^{7} + 8 \, a^{2} b^{7} c^{3} d^{8} - 2 \, a^{3} b^{6} c^{2} d^{9} - 2 \, a^{4} b^{5} c d^{10} + a^{5} b^{4} d^{11}\right )} x^{2} + {\left (b^{9} c^{6} d^{5} - 2 \, a b^{8} c^{5} d^{6} - 2 \, a^{2} b^{7} c^{4} d^{7} + 8 \, a^{3} b^{6} c^{3} d^{8} - 7 \, a^{4} b^{5} c^{2} d^{9} + 2 \, a^{5} b^{4} c d^{10}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x+a)^2/(d*x+c)^3,x, algorithm="fricas")

[Out]

1/2*(7*a*b^6*c^8 - 18*a^2*b^5*c^7*d + 11*a^3*b^4*c^6*d^2 - 2*a^6*b*c^3*d^5 + 2*a^7*c^2*d^6 + (b^7*c^4*d^4 - 4*
a*b^6*c^3*d^5 + 6*a^2*b^5*c^2*d^6 - 4*a^3*b^4*c*d^7 + a^4*b^3*d^8)*x^5 - (4*b^7*c^5*d^3 - 13*a*b^6*c^4*d^4 + 1
2*a^2*b^5*c^3*d^5 + 2*a^3*b^4*c^2*d^6 - 8*a^4*b^3*c*d^7 + 3*a^5*b^2*d^8)*x^4 - (11*b^7*c^6*d^2 - 32*a*b^6*c^5*
d^3 + 22*a^2*b^5*c^4*d^4 + 12*a^3*b^4*c^3*d^5 - 13*a^4*b^3*c^2*d^6 - 4*a^5*b^2*c*d^7 + 4*a^6*b*d^8)*x^3 + (2*b
^7*c^7*d - 11*a*b^6*c^6*d^2 + 28*a^2*b^5*c^5*d^3 - 34*a^3*b^4*c^4*d^4 + 6*a^4*b^3*c^3*d^5 + 17*a^5*b^2*c^2*d^6
- 10*a^6*b*c*d^7 + 2*a^7*d^8)*x^2 + (7*b^7*c^8 - 16*a*b^6*c^7*d + 11*a^2*b^5*c^6*d^2 - 8*a^3*b^4*c^5*d^3 + 10
*a^5*b^2*c^3*d^5 - 8*a^6*b*c^2*d^6 + 4*a^7*c*d^7)*x - 6*(2*a^6*b*c^3*d^5 - a^7*c^2*d^6 + (2*a^5*b^2*c*d^7 - a^
6*b*d^8)*x^3 + (4*a^5*b^2*c^2*d^6 - a^7*d^8)*x^2 + (2*a^5*b^2*c^3*d^5 + 3*a^6*b*c^2*d^6 - 2*a^7*c*d^7)*x)*log(
b*x + a) + 6*(2*a*b^6*c^8 - 6*a^2*b^5*c^7*d + 5*a^3*b^4*c^6*d^2 + (2*b^7*c^6*d^2 - 6*a*b^6*c^5*d^3 + 5*a^2*b^5
*c^4*d^4)*x^3 + (4*b^7*c^7*d - 10*a*b^6*c^6*d^2 + 4*a^2*b^5*c^5*d^3 + 5*a^3*b^4*c^4*d^4)*x^2 + (2*b^7*c^8 - 2*
a*b^6*c^7*d - 7*a^2*b^5*c^6*d^2 + 10*a^3*b^4*c^5*d^3)*x)*log(d*x + c))/(a*b^8*c^6*d^5 - 4*a^2*b^7*c^5*d^6 + 6*
a^3*b^6*c^4*d^7 - 4*a^4*b^5*c^3*d^8 + a^5*b^4*c^2*d^9 + (b^9*c^4*d^7 - 4*a*b^8*c^3*d^8 + 6*a^2*b^7*c^2*d^9 - 4
*a^3*b^6*c*d^10 + a^4*b^5*d^11)*x^3 + (2*b^9*c^5*d^6 - 7*a*b^8*c^4*d^7 + 8*a^2*b^7*c^3*d^8 - 2*a^3*b^6*c^2*d^9
- 2*a^4*b^5*c*d^10 + a^5*b^4*d^11)*x^2 + (b^9*c^6*d^5 - 2*a*b^8*c^5*d^6 - 2*a^2*b^7*c^4*d^7 + 8*a^3*b^6*c^3*d
^8 - 7*a^4*b^5*c^2*d^9 + 2*a^5*b^4*c*d^10)*x)

________________________________________________________________________________________

giac [B]  time = 1.10, size = 621, normalized size = 3.14 \[ -\frac {a^{6} b^{5}}{{\left (b^{12} c^{3} - 3 \, a b^{11} c^{2} d + 3 \, a^{2} b^{10} c d^{2} - a^{3} b^{9} d^{3}\right )} {\left (b x + a\right )}} + \frac {3 \, {\left (2 \, b^{3} c^{6} - 6 \, a b^{2} c^{5} d + 5 \, a^{2} b c^{4} d^{2}\right )} \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{5} c^{4} d^{5} - 4 \, a b^{4} c^{3} d^{6} + 6 \, a^{2} b^{3} c^{2} d^{7} - 4 \, a^{3} b^{2} c d^{8} + a^{4} b d^{9}} - \frac {3 \, {\left (2 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{4} d^{5}} + \frac {{\left (b^{4} c^{4} d^{3} - 4 \, a b^{3} c^{3} d^{4} + 6 \, a^{2} b^{2} c^{2} d^{5} - 4 \, a^{3} b c d^{6} + a^{4} d^{7} - \frac {4 \, {\left (b^{6} c^{5} d^{2} - 2 \, a b^{5} c^{4} d^{3} - 2 \, a^{2} b^{4} c^{3} d^{4} + 8 \, a^{3} b^{3} c^{2} d^{5} - 7 \, a^{4} b^{2} c d^{6} + 2 \, a^{5} b d^{7}\right )}}{{\left (b x + a\right )} b} - \frac {18 \, b^{8} c^{6} d - 54 \, a b^{7} c^{5} d^{2} + 45 \, a^{2} b^{6} c^{4} d^{3} + 20 \, a^{3} b^{5} c^{3} d^{4} - 75 \, a^{4} b^{4} c^{2} d^{5} + 54 \, a^{5} b^{3} c d^{6} - 13 \, a^{6} b^{2} d^{7}}{{\left (b x + a\right )}^{2} b^{2}} - \frac {6 \, {\left (2 \, b^{10} c^{7} - 8 \, a b^{9} c^{6} d + 11 \, a^{2} b^{8} c^{5} d^{2} - 5 \, a^{3} b^{7} c^{4} d^{3} - 5 \, a^{4} b^{6} c^{3} d^{4} + 9 \, a^{5} b^{5} c^{2} d^{5} - 5 \, a^{6} b^{4} c d^{6} + a^{7} b^{3} d^{7}\right )}}{{\left (b x + a\right )}^{3} b^{3}}\right )} {\left (b x + a\right )}^{2}}{2 \, {\left (b c - a d\right )}^{4} b^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}^{2} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x+a)^2/(d*x+c)^3,x, algorithm="giac")

[Out]

-a^6*b^5/((b^12*c^3 - 3*a*b^11*c^2*d + 3*a^2*b^10*c*d^2 - a^3*b^9*d^3)*(b*x + a)) + 3*(2*b^3*c^6 - 6*a*b^2*c^5
*d + 5*a^2*b*c^4*d^2)*log(abs(b*c/(b*x + a) - a*d/(b*x + a) + d))/(b^5*c^4*d^5 - 4*a*b^4*c^3*d^6 + 6*a^2*b^3*c
^2*d^7 - 4*a^3*b^2*c*d^8 + a^4*b*d^9) - 3*(2*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*log(abs(b*x + a)/((b*x + a)^2*abs(
b)))/(b^4*d^5) + 1/2*(b^4*c^4*d^3 - 4*a*b^3*c^3*d^4 + 6*a^2*b^2*c^2*d^5 - 4*a^3*b*c*d^6 + a^4*d^7 - 4*(b^6*c^5
*d^2 - 2*a*b^5*c^4*d^3 - 2*a^2*b^4*c^3*d^4 + 8*a^3*b^3*c^2*d^5 - 7*a^4*b^2*c*d^6 + 2*a^5*b*d^7)/((b*x + a)*b)
- (18*b^8*c^6*d - 54*a*b^7*c^5*d^2 + 45*a^2*b^6*c^4*d^3 + 20*a^3*b^5*c^3*d^4 - 75*a^4*b^4*c^2*d^5 + 54*a^5*b^3
*c*d^6 - 13*a^6*b^2*d^7)/((b*x + a)^2*b^2) - 6*(2*b^10*c^7 - 8*a*b^9*c^6*d + 11*a^2*b^8*c^5*d^2 - 5*a^3*b^7*c^
4*d^3 - 5*a^4*b^6*c^3*d^4 + 9*a^5*b^5*c^2*d^5 - 5*a^6*b^4*c*d^6 + a^7*b^3*d^7)/((b*x + a)^3*b^3))*(b*x + a)^2/
((b*c - a*d)^4*b^4*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*d^4)

________________________________________________________________________________________

maple [A]  time = 0.02, size = 264, normalized size = 1.33 \[ \frac {3 a^{6} d \ln \left (b x +a \right )}{\left (a d -b c \right )^{4} b^{4}}-\frac {6 a^{5} c \ln \left (b x +a \right )}{\left (a d -b c \right )^{4} b^{3}}+\frac {15 a^{2} c^{4} \ln \left (d x +c \right )}{\left (a d -b c \right )^{4} d^{3}}-\frac {18 a b \,c^{5} \ln \left (d x +c \right )}{\left (a d -b c \right )^{4} d^{4}}+\frac {6 b^{2} c^{6} \ln \left (d x +c \right )}{\left (a d -b c \right )^{4} d^{5}}+\frac {a^{6}}{\left (a d -b c \right )^{3} \left (b x +a \right ) b^{4}}+\frac {6 a \,c^{5}}{\left (a d -b c \right )^{3} \left (d x +c \right ) d^{4}}-\frac {4 b \,c^{6}}{\left (a d -b c \right )^{3} \left (d x +c \right ) d^{5}}-\frac {c^{6}}{2 \left (a d -b c \right )^{2} \left (d x +c \right )^{2} d^{5}}+\frac {x^{2}}{2 b^{2} d^{3}}-\frac {2 a x}{b^{3} d^{3}}-\frac {3 c x}{b^{2} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(b*x+a)^2/(d*x+c)^3,x)

[Out]

1/2*x^2/b^2/d^3-2/b^3/d^3*x*a-3/b^2/d^4*x*c-1/2/d^5*c^6/(a*d-b*c)^2/(d*x+c)^2+15/d^3*c^4/(a*d-b*c)^4*ln(d*x+c)
*a^2-18/d^4*c^5/(a*d-b*c)^4*ln(d*x+c)*a*b+6/d^5*c^6/(a*d-b*c)^4*ln(d*x+c)*b^2+6/d^4*c^5/(a*d-b*c)^3/(d*x+c)*a-
4/d^5*c^6/(a*d-b*c)^3/(d*x+c)*b+1/b^4*a^6/(a*d-b*c)^3/(b*x+a)+3/b^4*a^6/(a*d-b*c)^4*ln(b*x+a)*d-6/b^3*a^5/(a*d
-b*c)^4*ln(b*x+a)*c

________________________________________________________________________________________

maxima [B]  time = 1.42, size = 547, normalized size = 2.76 \[ -\frac {3 \, {\left (2 \, a^{5} b c - a^{6} d\right )} \log \left (b x + a\right )}{b^{8} c^{4} - 4 \, a b^{7} c^{3} d + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{3} b^{5} c d^{3} + a^{4} b^{4} d^{4}} + \frac {3 \, {\left (2 \, b^{2} c^{6} - 6 \, a b c^{5} d + 5 \, a^{2} c^{4} d^{2}\right )} \log \left (d x + c\right )}{b^{4} c^{4} d^{5} - 4 \, a b^{3} c^{3} d^{6} + 6 \, a^{2} b^{2} c^{2} d^{7} - 4 \, a^{3} b c d^{8} + a^{4} d^{9}} + \frac {7 \, a b^{5} c^{7} - 11 \, a^{2} b^{4} c^{6} d - 2 \, a^{6} c^{2} d^{5} + 2 \, {\left (4 \, b^{6} c^{6} d - 6 \, a b^{5} c^{5} d^{2} - a^{6} d^{7}\right )} x^{2} + {\left (7 \, b^{6} c^{7} - 3 \, a b^{5} c^{6} d - 12 \, a^{2} b^{4} c^{5} d^{2} - 4 \, a^{6} c d^{6}\right )} x}{2 \, {\left (a b^{7} c^{5} d^{5} - 3 \, a^{2} b^{6} c^{4} d^{6} + 3 \, a^{3} b^{5} c^{3} d^{7} - a^{4} b^{4} c^{2} d^{8} + {\left (b^{8} c^{3} d^{7} - 3 \, a b^{7} c^{2} d^{8} + 3 \, a^{2} b^{6} c d^{9} - a^{3} b^{5} d^{10}\right )} x^{3} + {\left (2 \, b^{8} c^{4} d^{6} - 5 \, a b^{7} c^{3} d^{7} + 3 \, a^{2} b^{6} c^{2} d^{8} + a^{3} b^{5} c d^{9} - a^{4} b^{4} d^{10}\right )} x^{2} + {\left (b^{8} c^{5} d^{5} - a b^{7} c^{4} d^{6} - 3 \, a^{2} b^{6} c^{3} d^{7} + 5 \, a^{3} b^{5} c^{2} d^{8} - 2 \, a^{4} b^{4} c d^{9}\right )} x\right )}} + \frac {b d x^{2} - 2 \, {\left (3 \, b c + 2 \, a d\right )} x}{2 \, b^{3} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b*x+a)^2/(d*x+c)^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.83, size = 508, normalized size = 2.57 \[ \frac {\frac {2\,a^6\,c^2\,d^5+11\,a^2\,b^4\,c^6\,d-7\,a\,b^5\,c^7}{2\,b\,d\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {x^2\,\left (a^6\,d^6+6\,a\,b^5\,c^5\,d-4\,b^6\,c^6\right )}{b\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {x\,\left (4\,a^6\,c\,d^6+12\,a^2\,b^4\,c^5\,d^2+3\,a\,b^5\,c^6\,d-7\,b^6\,c^7\right )}{2\,b\,d\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}}{x^2\,\left (2\,c\,b^4\,d^5+a\,b^3\,d^6\right )+x\,\left (b^4\,c^2\,d^4+2\,a\,b^3\,c\,d^5\right )+b^4\,d^6\,x^3+a\,b^3\,c^2\,d^4}+\frac {\ln \left (c+d\,x\right )\,\left (15\,a^2\,c^4\,d^2-18\,a\,b\,c^5\,d+6\,b^2\,c^6\right )}{a^4\,d^9-4\,a^3\,b\,c\,d^8+6\,a^2\,b^2\,c^2\,d^7-4\,a\,b^3\,c^3\,d^6+b^4\,c^4\,d^5}+\frac {x^2}{2\,b^2\,d^3}+\frac {\ln \left (a+b\,x\right )\,\left (3\,a^6\,d-6\,a^5\,b\,c\right )}{a^4\,b^4\,d^4-4\,a^3\,b^5\,c\,d^3+6\,a^2\,b^6\,c^2\,d^2-4\,a\,b^7\,c^3\,d+b^8\,c^4}-\frac {x\,\left (2\,a\,d+3\,b\,c\right )}{b^3\,d^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/((a + b*x)^2*(c + d*x)^3),x)

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 104.17, size = 1187, normalized size = 5.99 \[ \frac {3 a^{5} \left (a d - 2 b c\right ) \log {\left (x + \frac {\frac {3 a^{10} d^{9} \left (a d - 2 b c\right )}{b \left (a d - b c\right )^{4}} - \frac {15 a^{9} c d^{8} \left (a d - 2 b c\right )}{\left (a d - b c\right )^{4}} + \frac {30 a^{8} b c^{2} d^{7} \left (a d - 2 b c\right )}{\left (a d - b c\right )^{4}} - \frac {30 a^{7} b^{2} c^{3} d^{6} \left (a d - 2 b c\right )}{\left (a d - b c\right )^{4}} + \frac {15 a^{6} b^{3} c^{4} d^{5} \left (a d - 2 b c\right )}{\left (a d - b c\right )^{4}} + 3 a^{6} c d^{5} - \frac {3 a^{5} b^{4} c^{5} d^{4} \left (a d - 2 b c\right )}{\left (a d - b c\right )^{4}} - 6 a^{5} b c^{2} d^{4} - 15 a^{3} b^{3} c^{4} d^{2} + 18 a^{2} b^{4} c^{5} d - 6 a b^{5} c^{6}}{3 a^{6} d^{6} - 6 a^{5} b c d^{5} - 15 a^{2} b^{4} c^{4} d^{2} + 18 a b^{5} c^{5} d - 6 b^{6} c^{6}} \right )}}{b^{4} \left (a d - b c\right )^{4}} + \frac {3 c^{4} \left (5 a^{2} d^{2} - 6 a b c d + 2 b^{2} c^{2}\right ) \log {\left (x + \frac {3 a^{6} c d^{5} + \frac {3 a^{5} b^{3} c^{4} d^{4} \left (5 a^{2} d^{2} - 6 a b c d + 2 b^{2} c^{2}\right )}{\left (a d - b c\right )^{4}} - 6 a^{5} b c^{2} d^{4} - \frac {15 a^{4} b^{4} c^{5} d^{3} \left (5 a^{2} d^{2} - 6 a b c d + 2 b^{2} c^{2}\right )}{\left (a d - b c\right )^{4}} + \frac {30 a^{3} b^{5} c^{6} d^{2} \left (5 a^{2} d^{2} - 6 a b c d + 2 b^{2} c^{2}\right )}{\left (a d - b c\right )^{4}} - 15 a^{3} b^{3} c^{4} d^{2} - \frac {30 a^{2} b^{6} c^{7} d \left (5 a^{2} d^{2} - 6 a b c d + 2 b^{2} c^{2}\right )}{\left (a d - b c\right )^{4}} + 18 a^{2} b^{4} c^{5} d + \frac {15 a b^{7} c^{8} \left (5 a^{2} d^{2} - 6 a b c d + 2 b^{2} c^{2}\right )}{\left (a d - b c\right )^{4}} - 6 a b^{5} c^{6} - \frac {3 b^{8} c^{9} \left (5 a^{2} d^{2} - 6 a b c d + 2 b^{2} c^{2}\right )}{d \left (a d - b c\right )^{4}}}{3 a^{6} d^{6} - 6 a^{5} b c d^{5} - 15 a^{2} b^{4} c^{4} d^{2} + 18 a b^{5} c^{5} d - 6 b^{6} c^{6}} \right )}}{d^{5} \left (a d - b c\right )^{4}} + x \left (- \frac {2 a}{b^{3} d^{3}} - \frac {3 c}{b^{2} d^{4}}\right ) + \frac {2 a^{6} c^{2} d^{5} + 11 a^{2} b^{4} c^{6} d - 7 a b^{5} c^{7} + x^{2} \left (2 a^{6} d^{7} + 12 a b^{5} c^{5} d^{2} - 8 b^{6} c^{6} d\right ) + x \left (4 a^{6} c d^{6} + 12 a^{2} b^{4} c^{5} d^{2} + 3 a b^{5} c^{6} d - 7 b^{6} c^{7}\right )}{2 a^{4} b^{4} c^{2} d^{8} - 6 a^{3} b^{5} c^{3} d^{7} + 6 a^{2} b^{6} c^{4} d^{6} - 2 a b^{7} c^{5} d^{5} + x^{3} \left (2 a^{3} b^{5} d^{10} - 6 a^{2} b^{6} c d^{9} + 6 a b^{7} c^{2} d^{8} - 2 b^{8} c^{3} d^{7}\right ) + x^{2} \left (2 a^{4} b^{4} d^{10} - 2 a^{3} b^{5} c d^{9} - 6 a^{2} b^{6} c^{2} d^{8} + 10 a b^{7} c^{3} d^{7} - 4 b^{8} c^{4} d^{6}\right ) + x \left (4 a^{4} b^{4} c d^{9} - 10 a^{3} b^{5} c^{2} d^{8} + 6 a^{2} b^{6} c^{3} d^{7} + 2 a b^{7} c^{4} d^{6} - 2 b^{8} c^{5} d^{5}\right )} + \frac {x^{2}}{2 b^{2} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(b*x+a)**2/(d*x+c)**3,x)

[Out]

3*a**5*(a*d - 2*b*c)*log(x + (3*a**10*d**9*(a*d - 2*b*c)/(b*(a*d - b*c)**4) - 15*a**9*c*d**8*(a*d - 2*b*c)/(a*
d - b*c)**4 + 30*a**8*b*c**2*d**7*(a*d - 2*b*c)/(a*d - b*c)**4 - 30*a**7*b**2*c**3*d**6*(a*d - 2*b*c)/(a*d - b
*c)**4 + 15*a**6*b**3*c**4*d**5*(a*d - 2*b*c)/(a*d - b*c)**4 + 3*a**6*c*d**5 - 3*a**5*b**4*c**5*d**4*(a*d - 2*
b*c)/(a*d - b*c)**4 - 6*a**5*b*c**2*d**4 - 15*a**3*b**3*c**4*d**2 + 18*a**2*b**4*c**5*d - 6*a*b**5*c**6)/(3*a*
*6*d**6 - 6*a**5*b*c*d**5 - 15*a**2*b**4*c**4*d**2 + 18*a*b**5*c**5*d - 6*b**6*c**6))/(b**4*(a*d - b*c)**4) +
3*c**4*(5*a**2*d**2 - 6*a*b*c*d + 2*b**2*c**2)*log(x + (3*a**6*c*d**5 + 3*a**5*b**3*c**4*d**4*(5*a**2*d**2 - 6
*a*b*c*d + 2*b**2*c**2)/(a*d - b*c)**4 - 6*a**5*b*c**2*d**4 - 15*a**4*b**4*c**5*d**3*(5*a**2*d**2 - 6*a*b*c*d
+ 2*b**2*c**2)/(a*d - b*c)**4 + 30*a**3*b**5*c**6*d**2*(5*a**2*d**2 - 6*a*b*c*d + 2*b**2*c**2)/(a*d - b*c)**4
- 15*a**3*b**3*c**4*d**2 - 30*a**2*b**6*c**7*d*(5*a**2*d**2 - 6*a*b*c*d + 2*b**2*c**2)/(a*d - b*c)**4 + 18*a**
2*b**4*c**5*d + 15*a*b**7*c**8*(5*a**2*d**2 - 6*a*b*c*d + 2*b**2*c**2)/(a*d - b*c)**4 - 6*a*b**5*c**6 - 3*b**8
*c**9*(5*a**2*d**2 - 6*a*b*c*d + 2*b**2*c**2)/(d*(a*d - b*c)**4))/(3*a**6*d**6 - 6*a**5*b*c*d**5 - 15*a**2*b**
4*c**4*d**2 + 18*a*b**5*c**5*d - 6*b**6*c**6))/(d**5*(a*d - b*c)**4) + x*(-2*a/(b**3*d**3) - 3*c/(b**2*d**4))
+ (2*a**6*c**2*d**5 + 11*a**2*b**4*c**6*d - 7*a*b**5*c**7 + x**2*(2*a**6*d**7 + 12*a*b**5*c**5*d**2 - 8*b**6*c
**6*d) + x*(4*a**6*c*d**6 + 12*a**2*b**4*c**5*d**2 + 3*a*b**5*c**6*d - 7*b**6*c**7))/(2*a**4*b**4*c**2*d**8 -
6*a**3*b**5*c**3*d**7 + 6*a**2*b**6*c**4*d**6 - 2*a*b**7*c**5*d**5 + x**3*(2*a**3*b**5*d**10 - 6*a**2*b**6*c*d
**9 + 6*a*b**7*c**2*d**8 - 2*b**8*c**3*d**7) + x**2*(2*a**4*b**4*d**10 - 2*a**3*b**5*c*d**9 - 6*a**2*b**6*c**2
*d**8 + 10*a*b**7*c**3*d**7 - 4*b**8*c**4*d**6) + x*(4*a**4*b**4*c*d**9 - 10*a**3*b**5*c**2*d**8 + 6*a**2*b**6
*c**3*d**7 + 2*a*b**7*c**4*d**6 - 2*b**8*c**5*d**5)) + x**2/(2*b**2*d**3)

________________________________________________________________________________________