3.15.95 \(\int \frac {1}{(a+b x)^4 (a c+(b c+a d) x+b d x^2)} \, dx\)

Optimal. Leaf size=130 \[ \frac {d^4 \log (a+b x)}{(b c-a d)^5}-\frac {d^4 \log (c+d x)}{(b c-a d)^5}+\frac {d^3}{(a+b x) (b c-a d)^4}-\frac {d^2}{2 (a+b x)^2 (b c-a d)^3}+\frac {d}{3 (a+b x)^3 (b c-a d)^2}-\frac {1}{4 (a+b x)^4 (b c-a d)} \]

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Rubi [A]  time = 0.09, antiderivative size = 130, normalized size of antiderivative = 1.00, 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, 44} \begin {gather*} \frac {d^3}{(a+b x) (b c-a d)^4}-\frac {d^2}{2 (a+b x)^2 (b c-a d)^3}+\frac {d^4 \log (a+b x)}{(b c-a d)^5}-\frac {d^4 \log (c+d x)}{(b c-a d)^5}+\frac {d}{3 (a+b x)^3 (b c-a d)^2}-\frac {1}{4 (a+b x)^4 (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^4*(a*c + (b*c + a*d)*x + b*d*x^2)),x]

[Out]

-1/(4*(b*c - a*d)*(a + b*x)^4) + d/(3*(b*c - a*d)^2*(a + b*x)^3) - d^2/(2*(b*c - a*d)^3*(a + b*x)^2) + d^3/((b
*c - a*d)^4*(a + b*x)) + (d^4*Log[a + b*x])/(b*c - a*d)^5 - (d^4*Log[c + d*x])/(b*c - a*d)^5

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

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]

Rubi steps

\begin {align*} \int \frac {1}{(a+b x)^4 \left (a c+(b c+a d) x+b d x^2\right )} \, dx &=\int \frac {1}{(a+b x)^5 (c+d x)} \, dx\\ &=\int \left (\frac {b}{(b c-a d) (a+b x)^5}-\frac {b d}{(b c-a d)^2 (a+b x)^4}+\frac {b d^2}{(b c-a d)^3 (a+b x)^3}-\frac {b d^3}{(b c-a d)^4 (a+b x)^2}+\frac {b d^4}{(b c-a d)^5 (a+b x)}-\frac {d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx\\ &=-\frac {1}{4 (b c-a d) (a+b x)^4}+\frac {d}{3 (b c-a d)^2 (a+b x)^3}-\frac {d^2}{2 (b c-a d)^3 (a+b x)^2}+\frac {d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4 \log (a+b x)}{(b c-a d)^5}-\frac {d^4 \log (c+d x)}{(b c-a d)^5}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 130, normalized size = 1.00 \begin {gather*} \frac {d^4 \log (a+b x)}{(b c-a d)^5}-\frac {d^4 \log (c+d x)}{(b c-a d)^5}+\frac {d^3}{(a+b x) (b c-a d)^4}-\frac {d^2}{2 (a+b x)^2 (b c-a d)^3}+\frac {d}{3 (a+b x)^3 (b c-a d)^2}+\frac {1}{4 (a+b x)^4 (a d-b c)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^4*(a*c + (b*c + a*d)*x + b*d*x^2)),x]

[Out]

1/(4*(-(b*c) + a*d)*(a + b*x)^4) + d/(3*(b*c - a*d)^2*(a + b*x)^3) - d^2/(2*(b*c - a*d)^3*(a + b*x)^2) + d^3/(
(b*c - a*d)^4*(a + b*x)) + (d^4*Log[a + b*x])/(b*c - a*d)^5 - (d^4*Log[c + d*x])/(b*c - a*d)^5

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(a+b x)^4 \left (a c+(b c+a d) x+b d x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/((a + b*x)^4*(a*c + (b*c + a*d)*x + b*d*x^2)),x]

[Out]

IntegrateAlgebraic[1/((a + b*x)^4*(a*c + (b*c + a*d)*x + b*d*x^2)), x]

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fricas [B]  time = 0.41, size = 657, normalized size = 5.05 \begin {gather*} -\frac {3 \, b^{4} c^{4} - 16 \, a b^{3} c^{3} d + 36 \, a^{2} b^{2} c^{2} d^{2} - 48 \, a^{3} b c d^{3} + 25 \, a^{4} d^{4} - 12 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{3} + 6 \, {\left (b^{4} c^{2} d^{2} - 8 \, a b^{3} c d^{3} + 7 \, a^{2} b^{2} d^{4}\right )} x^{2} - 4 \, {\left (b^{4} c^{3} d - 6 \, a b^{3} c^{2} d^{2} + 18 \, a^{2} b^{2} c d^{3} - 13 \, a^{3} b d^{4}\right )} x - 12 \, {\left (b^{4} d^{4} x^{4} + 4 \, a b^{3} d^{4} x^{3} + 6 \, a^{2} b^{2} d^{4} x^{2} + 4 \, a^{3} b d^{4} x + a^{4} d^{4}\right )} \log \left (b x + a\right ) + 12 \, {\left (b^{4} d^{4} x^{4} + 4 \, a b^{3} d^{4} x^{3} + 6 \, a^{2} b^{2} d^{4} x^{2} + 4 \, a^{3} b d^{4} x + a^{4} d^{4}\right )} \log \left (d x + c\right )}{12 \, {\left (a^{4} b^{5} c^{5} - 5 \, a^{5} b^{4} c^{4} d + 10 \, a^{6} b^{3} c^{3} d^{2} - 10 \, a^{7} b^{2} c^{2} d^{3} + 5 \, a^{8} b c d^{4} - a^{9} d^{5} + {\left (b^{9} c^{5} - 5 \, a b^{8} c^{4} d + 10 \, a^{2} b^{7} c^{3} d^{2} - 10 \, a^{3} b^{6} c^{2} d^{3} + 5 \, a^{4} b^{5} c d^{4} - a^{5} b^{4} d^{5}\right )} x^{4} + 4 \, {\left (a b^{8} c^{5} - 5 \, a^{2} b^{7} c^{4} d + 10 \, a^{3} b^{6} c^{3} d^{2} - 10 \, a^{4} b^{5} c^{2} d^{3} + 5 \, a^{5} b^{4} c d^{4} - a^{6} b^{3} d^{5}\right )} x^{3} + 6 \, {\left (a^{2} b^{7} c^{5} - 5 \, a^{3} b^{6} c^{4} d + 10 \, a^{4} b^{5} c^{3} d^{2} - 10 \, a^{5} b^{4} c^{2} d^{3} + 5 \, a^{6} b^{3} c d^{4} - a^{7} b^{2} d^{5}\right )} x^{2} + 4 \, {\left (a^{3} b^{6} c^{5} - 5 \, a^{4} b^{5} c^{4} d + 10 \, a^{5} b^{4} c^{3} d^{2} - 10 \, a^{6} b^{3} c^{2} d^{3} + 5 \, a^{7} b^{2} c d^{4} - a^{8} b d^{5}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^4/(a*c+(a*d+b*c)*x+b*d*x^2),x, algorithm="fricas")

[Out]

-1/12*(3*b^4*c^4 - 16*a*b^3*c^3*d + 36*a^2*b^2*c^2*d^2 - 48*a^3*b*c*d^3 + 25*a^4*d^4 - 12*(b^4*c*d^3 - a*b^3*d
^4)*x^3 + 6*(b^4*c^2*d^2 - 8*a*b^3*c*d^3 + 7*a^2*b^2*d^4)*x^2 - 4*(b^4*c^3*d - 6*a*b^3*c^2*d^2 + 18*a^2*b^2*c*
d^3 - 13*a^3*b*d^4)*x - 12*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*log(b
*x + a) + 12*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*log(d*x + c))/(a^4*
b^5*c^5 - 5*a^5*b^4*c^4*d + 10*a^6*b^3*c^3*d^2 - 10*a^7*b^2*c^2*d^3 + 5*a^8*b*c*d^4 - a^9*d^5 + (b^9*c^5 - 5*a
*b^8*c^4*d + 10*a^2*b^7*c^3*d^2 - 10*a^3*b^6*c^2*d^3 + 5*a^4*b^5*c*d^4 - a^5*b^4*d^5)*x^4 + 4*(a*b^8*c^5 - 5*a
^2*b^7*c^4*d + 10*a^3*b^6*c^3*d^2 - 10*a^4*b^5*c^2*d^3 + 5*a^5*b^4*c*d^4 - a^6*b^3*d^5)*x^3 + 6*(a^2*b^7*c^5 -
5*a^3*b^6*c^4*d + 10*a^4*b^5*c^3*d^2 - 10*a^5*b^4*c^2*d^3 + 5*a^6*b^3*c*d^4 - a^7*b^2*d^5)*x^2 + 4*(a^3*b^6*c
^5 - 5*a^4*b^5*c^4*d + 10*a^5*b^4*c^3*d^2 - 10*a^6*b^3*c^2*d^3 + 5*a^7*b^2*c*d^4 - a^8*b*d^5)*x)

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giac [B]  time = 0.17, size = 338, normalized size = 2.60 \begin {gather*} \frac {b d^{4} \log \left ({\left | b x + a \right |}\right )}{b^{6} c^{5} - 5 \, a b^{5} c^{4} d + 10 \, a^{2} b^{4} c^{3} d^{2} - 10 \, a^{3} b^{3} c^{2} d^{3} + 5 \, a^{4} b^{2} c d^{4} - a^{5} b d^{5}} - \frac {d^{5} \log \left ({\left | d x + c \right |}\right )}{b^{5} c^{5} d - 5 \, a b^{4} c^{4} d^{2} + 10 \, a^{2} b^{3} c^{3} d^{3} - 10 \, a^{3} b^{2} c^{2} d^{4} + 5 \, a^{4} b c d^{5} - a^{5} d^{6}} - \frac {3 \, b^{4} c^{4} - 16 \, a b^{3} c^{3} d + 36 \, a^{2} b^{2} c^{2} d^{2} - 48 \, a^{3} b c d^{3} + 25 \, a^{4} d^{4} - 12 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{3} + 6 \, {\left (b^{4} c^{2} d^{2} - 8 \, a b^{3} c d^{3} + 7 \, a^{2} b^{2} d^{4}\right )} x^{2} - 4 \, {\left (b^{4} c^{3} d - 6 \, a b^{3} c^{2} d^{2} + 18 \, a^{2} b^{2} c d^{3} - 13 \, a^{3} b d^{4}\right )} x}{12 \, {\left (b c - a d\right )}^{5} {\left (b x + a\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^4/(a*c+(a*d+b*c)*x+b*d*x^2),x, algorithm="giac")

[Out]

b*d^4*log(abs(b*x + a))/(b^6*c^5 - 5*a*b^5*c^4*d + 10*a^2*b^4*c^3*d^2 - 10*a^3*b^3*c^2*d^3 + 5*a^4*b^2*c*d^4 -
a^5*b*d^5) - d^5*log(abs(d*x + c))/(b^5*c^5*d - 5*a*b^4*c^4*d^2 + 10*a^2*b^3*c^3*d^3 - 10*a^3*b^2*c^2*d^4 + 5
*a^4*b*c*d^5 - a^5*d^6) - 1/12*(3*b^4*c^4 - 16*a*b^3*c^3*d + 36*a^2*b^2*c^2*d^2 - 48*a^3*b*c*d^3 + 25*a^4*d^4
- 12*(b^4*c*d^3 - a*b^3*d^4)*x^3 + 6*(b^4*c^2*d^2 - 8*a*b^3*c*d^3 + 7*a^2*b^2*d^4)*x^2 - 4*(b^4*c^3*d - 6*a*b^
3*c^2*d^2 + 18*a^2*b^2*c*d^3 - 13*a^3*b*d^4)*x)/((b*c - a*d)^5*(b*x + a)^4)

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maple [A]  time = 0.07, size = 125, normalized size = 0.96 \begin {gather*} -\frac {d^{4} \ln \left (b x +a \right )}{\left (a d -b c \right )^{5}}+\frac {d^{4} \ln \left (d x +c \right )}{\left (a d -b c \right )^{5}}+\frac {d^{3}}{\left (a d -b c \right )^{4} \left (b x +a \right )}+\frac {d^{2}}{2 \left (a d -b c \right )^{3} \left (b x +a \right )^{2}}+\frac {d}{3 \left (a d -b c \right )^{2} \left (b x +a \right )^{3}}+\frac {1}{4 \left (a d -b c \right ) \left (b x +a \right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^4/(a*c+(a*d+b*c)*x+b*d*x^2),x)

[Out]

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

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maxima [B]  time = 1.30, size = 558, normalized size = 4.29 \begin {gather*} \frac {d^{4} \log \left (b x + a\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} - \frac {d^{4} \log \left (d x + c\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} + \frac {12 \, b^{3} d^{3} x^{3} - 3 \, b^{3} c^{3} + 13 \, a b^{2} c^{2} d - 23 \, a^{2} b c d^{2} + 25 \, a^{3} d^{3} - 6 \, {\left (b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} + 4 \, {\left (b^{3} c^{2} d - 5 \, a b^{2} c d^{2} + 13 \, a^{2} b d^{3}\right )} x}{12 \, {\left (a^{4} b^{4} c^{4} - 4 \, a^{5} b^{3} c^{3} d + 6 \, a^{6} b^{2} c^{2} d^{2} - 4 \, a^{7} b c d^{3} + a^{8} d^{4} + {\left (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}\right )} x^{4} + 4 \, {\left (a b^{7} c^{4} - 4 \, a^{2} b^{6} c^{3} d + 6 \, a^{3} b^{5} c^{2} d^{2} - 4 \, a^{4} b^{4} c d^{3} + a^{5} b^{3} d^{4}\right )} x^{3} + 6 \, {\left (a^{2} b^{6} c^{4} - 4 \, a^{3} b^{5} c^{3} d + 6 \, a^{4} b^{4} c^{2} d^{2} - 4 \, a^{5} b^{3} c d^{3} + a^{6} b^{2} d^{4}\right )} x^{2} + 4 \, {\left (a^{3} b^{5} c^{4} - 4 \, a^{4} b^{4} c^{3} d + 6 \, a^{5} b^{3} c^{2} d^{2} - 4 \, a^{6} b^{2} c d^{3} + a^{7} b d^{4}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^4/(a*c+(a*d+b*c)*x+b*d*x^2),x, algorithm="maxima")

[Out]

d^4*log(b*x + a)/(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)
- d^4*log(d*x + c)/(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d
^5) + 1/12*(12*b^3*d^3*x^3 - 3*b^3*c^3 + 13*a*b^2*c^2*d - 23*a^2*b*c*d^2 + 25*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*b^2
*d^3)*x^2 + 4*(b^3*c^2*d - 5*a*b^2*c*d^2 + 13*a^2*b*d^3)*x)/(a^4*b^4*c^4 - 4*a^5*b^3*c^3*d + 6*a^6*b^2*c^2*d^2
- 4*a^7*b*c*d^3 + a^8*d^4 + (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)*x^4
+ 4*(a*b^7*c^4 - 4*a^2*b^6*c^3*d + 6*a^3*b^5*c^2*d^2 - 4*a^4*b^4*c*d^3 + a^5*b^3*d^4)*x^3 + 6*(a^2*b^6*c^4 -
4*a^3*b^5*c^3*d + 6*a^4*b^4*c^2*d^2 - 4*a^5*b^3*c*d^3 + a^6*b^2*d^4)*x^2 + 4*(a^3*b^5*c^4 - 4*a^4*b^4*c^3*d +
6*a^5*b^3*c^2*d^2 - 4*a^6*b^2*c*d^3 + a^7*b*d^4)*x)

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mupad [B]  time = 0.83, size = 505, normalized size = 3.88 \begin {gather*} \frac {\frac {25\,a^3\,d^3-23\,a^2\,b\,c\,d^2+13\,a\,b^2\,c^2\,d-3\,b^3\,c^3}{12\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}-\frac {d^2\,x^2\,\left (b^3\,c-7\,a\,b^2\,d\right )}{2\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {d\,x\,\left (13\,a^2\,b\,d^2-5\,a\,b^2\,c\,d+b^3\,c^2\right )}{3\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {b^3\,d^3\,x^3}{a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4}}{a^4+4\,a^3\,b\,x+6\,a^2\,b^2\,x^2+4\,a\,b^3\,x^3+b^4\,x^4}-\frac {2\,d^4\,\mathrm {atanh}\left (\frac {a^5\,d^5-3\,a^4\,b\,c\,d^4+2\,a^3\,b^2\,c^2\,d^3+2\,a^2\,b^3\,c^3\,d^2-3\,a\,b^4\,c^4\,d+b^5\,c^5}{{\left (a\,d-b\,c\right )}^5}+\frac {2\,b\,d\,x\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{{\left (a\,d-b\,c\right )}^5}\right )}{{\left (a\,d-b\,c\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x)^4*(a*c + x*(a*d + b*c) + b*d*x^2)),x)

[Out]

((25*a^3*d^3 - 3*b^3*c^3 + 13*a*b^2*c^2*d - 23*a^2*b*c*d^2)/(12*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b
^3*c^3*d - 4*a^3*b*c*d^3)) - (d^2*x^2*(b^3*c - 7*a*b^2*d))/(2*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3
*c^3*d - 4*a^3*b*c*d^3)) + (d*x*(b^3*c^2 + 13*a^2*b*d^2 - 5*a*b^2*c*d))/(3*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*
d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) + (b^3*d^3*x^3)/(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d -
4*a^3*b*c*d^3))/(a^4 + b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x) - (2*d^4*atanh((a^5*d^5 + b^5*c^5 +
2*a^2*b^3*c^3*d^2 + 2*a^3*b^2*c^2*d^3 - 3*a*b^4*c^4*d - 3*a^4*b*c*d^4)/(a*d - b*c)^5 + (2*b*d*x*(a^4*d^4 + b^
4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3))/(a*d - b*c)^5))/(a*d - b*c)^5

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sympy [B]  time = 2.16, size = 802, normalized size = 6.17 \begin {gather*} \frac {d^{4} \log {\left (x + \frac {- \frac {a^{6} d^{10}}{\left (a d - b c\right )^{5}} + \frac {6 a^{5} b c d^{9}}{\left (a d - b c\right )^{5}} - \frac {15 a^{4} b^{2} c^{2} d^{8}}{\left (a d - b c\right )^{5}} + \frac {20 a^{3} b^{3} c^{3} d^{7}}{\left (a d - b c\right )^{5}} - \frac {15 a^{2} b^{4} c^{4} d^{6}}{\left (a d - b c\right )^{5}} + \frac {6 a b^{5} c^{5} d^{5}}{\left (a d - b c\right )^{5}} + a d^{5} - \frac {b^{6} c^{6} d^{4}}{\left (a d - b c\right )^{5}} + b c d^{4}}{2 b d^{5}} \right )}}{\left (a d - b c\right )^{5}} - \frac {d^{4} \log {\left (x + \frac {\frac {a^{6} d^{10}}{\left (a d - b c\right )^{5}} - \frac {6 a^{5} b c d^{9}}{\left (a d - b c\right )^{5}} + \frac {15 a^{4} b^{2} c^{2} d^{8}}{\left (a d - b c\right )^{5}} - \frac {20 a^{3} b^{3} c^{3} d^{7}}{\left (a d - b c\right )^{5}} + \frac {15 a^{2} b^{4} c^{4} d^{6}}{\left (a d - b c\right )^{5}} - \frac {6 a b^{5} c^{5} d^{5}}{\left (a d - b c\right )^{5}} + a d^{5} + \frac {b^{6} c^{6} d^{4}}{\left (a d - b c\right )^{5}} + b c d^{4}}{2 b d^{5}} \right )}}{\left (a d - b c\right )^{5}} + \frac {25 a^{3} d^{3} - 23 a^{2} b c d^{2} + 13 a b^{2} c^{2} d - 3 b^{3} c^{3} + 12 b^{3} d^{3} x^{3} + x^{2} \left (42 a b^{2} d^{3} - 6 b^{3} c d^{2}\right ) + x \left (52 a^{2} b d^{3} - 20 a b^{2} c d^{2} + 4 b^{3} c^{2} d\right )}{12 a^{8} d^{4} - 48 a^{7} b c d^{3} + 72 a^{6} b^{2} c^{2} d^{2} - 48 a^{5} b^{3} c^{3} d + 12 a^{4} b^{4} c^{4} + x^{4} \left (12 a^{4} b^{4} d^{4} - 48 a^{3} b^{5} c d^{3} + 72 a^{2} b^{6} c^{2} d^{2} - 48 a b^{7} c^{3} d + 12 b^{8} c^{4}\right ) + x^{3} \left (48 a^{5} b^{3} d^{4} - 192 a^{4} b^{4} c d^{3} + 288 a^{3} b^{5} c^{2} d^{2} - 192 a^{2} b^{6} c^{3} d + 48 a b^{7} c^{4}\right ) + x^{2} \left (72 a^{6} b^{2} d^{4} - 288 a^{5} b^{3} c d^{3} + 432 a^{4} b^{4} c^{2} d^{2} - 288 a^{3} b^{5} c^{3} d + 72 a^{2} b^{6} c^{4}\right ) + x \left (48 a^{7} b d^{4} - 192 a^{6} b^{2} c d^{3} + 288 a^{5} b^{3} c^{2} d^{2} - 192 a^{4} b^{4} c^{3} d + 48 a^{3} b^{5} c^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**4/(a*c+(a*d+b*c)*x+b*d*x**2),x)

[Out]

d**4*log(x + (-a**6*d**10/(a*d - b*c)**5 + 6*a**5*b*c*d**9/(a*d - b*c)**5 - 15*a**4*b**2*c**2*d**8/(a*d - b*c)
**5 + 20*a**3*b**3*c**3*d**7/(a*d - b*c)**5 - 15*a**2*b**4*c**4*d**6/(a*d - b*c)**5 + 6*a*b**5*c**5*d**5/(a*d
- b*c)**5 + a*d**5 - b**6*c**6*d**4/(a*d - b*c)**5 + b*c*d**4)/(2*b*d**5))/(a*d - b*c)**5 - d**4*log(x + (a**6
*d**10/(a*d - b*c)**5 - 6*a**5*b*c*d**9/(a*d - b*c)**5 + 15*a**4*b**2*c**2*d**8/(a*d - b*c)**5 - 20*a**3*b**3*
c**3*d**7/(a*d - b*c)**5 + 15*a**2*b**4*c**4*d**6/(a*d - b*c)**5 - 6*a*b**5*c**5*d**5/(a*d - b*c)**5 + a*d**5
+ b**6*c**6*d**4/(a*d - b*c)**5 + b*c*d**4)/(2*b*d**5))/(a*d - b*c)**5 + (25*a**3*d**3 - 23*a**2*b*c*d**2 + 13
*a*b**2*c**2*d - 3*b**3*c**3 + 12*b**3*d**3*x**3 + x**2*(42*a*b**2*d**3 - 6*b**3*c*d**2) + x*(52*a**2*b*d**3 -
20*a*b**2*c*d**2 + 4*b**3*c**2*d))/(12*a**8*d**4 - 48*a**7*b*c*d**3 + 72*a**6*b**2*c**2*d**2 - 48*a**5*b**3*c
**3*d + 12*a**4*b**4*c**4 + x**4*(12*a**4*b**4*d**4 - 48*a**3*b**5*c*d**3 + 72*a**2*b**6*c**2*d**2 - 48*a*b**7
*c**3*d + 12*b**8*c**4) + x**3*(48*a**5*b**3*d**4 - 192*a**4*b**4*c*d**3 + 288*a**3*b**5*c**2*d**2 - 192*a**2*
b**6*c**3*d + 48*a*b**7*c**4) + x**2*(72*a**6*b**2*d**4 - 288*a**5*b**3*c*d**3 + 432*a**4*b**4*c**2*d**2 - 288
*a**3*b**5*c**3*d + 72*a**2*b**6*c**4) + x*(48*a**7*b*d**4 - 192*a**6*b**2*c*d**3 + 288*a**5*b**3*c**2*d**2 -
192*a**4*b**4*c**3*d + 48*a**3*b**5*c**4))

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