3.293 \(\int \frac {x^4}{(a+b x)^2 (c+d x)^3} \, dx\)

Optimal. Leaf size=164 \[ -\frac {a^4}{b^2 (a+b x) (b c-a d)^3}-\frac {a^3 (4 b c-a d) \log (a+b x)}{b^2 (b c-a d)^4}+\frac {c^2 \left (6 a^2 d^2-4 a b c d+b^2 c^2\right ) \log (c+d x)}{d^3 (b c-a d)^4}-\frac {c^4}{2 d^3 (c+d x)^2 (b c-a d)^2}+\frac {2 c^3 (b c-2 a d)}{d^3 (c+d x) (b c-a d)^3} \]

[Out]

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

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Rubi [A]  time = 0.18, antiderivative size = 164, 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} \[ \frac {c^2 \left (6 a^2 d^2-4 a b c d+b^2 c^2\right ) \log (c+d x)}{d^3 (b c-a d)^4}-\frac {a^4}{b^2 (a+b x) (b c-a d)^3}-\frac {a^3 (4 b c-a d) \log (a+b x)}{b^2 (b c-a d)^4}+\frac {2 c^3 (b c-2 a d)}{d^3 (c+d x) (b c-a d)^3}-\frac {c^4}{2 d^3 (c+d x)^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Int[x^4/((a + b*x)^2*(c + d*x)^3),x]

[Out]

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

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Mathematica [A]  time = 0.28, size = 162, normalized size = 0.99 \[ -\frac {a^4}{b^2 (a+b x) (b c-a d)^3}+\frac {a^3 (a d-4 b c) \log (a+b x)}{b^2 (b c-a d)^4}+\frac {c^2 \left (6 a^2 d^2-4 a b c d+b^2 c^2\right ) \log (c+d x)}{d^3 (b c-a d)^4}-\frac {c^4}{2 d^3 (c+d x)^2 (b c-a d)^2}-\frac {2 c^3 (b c-2 a d)}{d^3 (c+d x) (a d-b c)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/((a + b*x)^2*(c + d*x)^3),x]

[Out]

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

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fricas [B]  time = 1.03, size = 797, normalized size = 4.86 \[ \frac {3 \, a b^{4} c^{6} - 10 \, a^{2} b^{3} c^{5} d + 7 \, a^{3} b^{2} c^{4} d^{2} - 2 \, a^{4} b c^{3} d^{3} + 2 \, a^{5} c^{2} d^{4} + 2 \, {\left (2 \, b^{5} c^{5} d - 6 \, a b^{4} c^{4} d^{2} + 4 \, a^{2} b^{3} c^{3} d^{3} - a^{4} b c d^{5} + a^{5} d^{6}\right )} x^{2} + {\left (3 \, b^{5} c^{6} - 6 \, a b^{4} c^{5} d - 5 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 4 \, a^{4} b c^{2} d^{4} + 4 \, a^{5} c d^{5}\right )} x - 2 \, {\left (4 \, a^{4} b c^{3} d^{3} - a^{5} c^{2} d^{4} + {\left (4 \, a^{3} b^{2} c d^{5} - a^{4} b d^{6}\right )} x^{3} + {\left (8 \, a^{3} b^{2} c^{2} d^{4} + 2 \, a^{4} b c d^{5} - a^{5} d^{6}\right )} x^{2} + {\left (4 \, a^{3} b^{2} c^{3} d^{3} + 7 \, a^{4} b c^{2} d^{4} - 2 \, a^{5} c d^{5}\right )} x\right )} \log \left (b x + a\right ) + 2 \, {\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} + {\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4}\right )} x^{3} + {\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} + 8 \, a^{2} b^{3} c^{3} d^{3} + 6 \, a^{3} b^{2} c^{2} d^{4}\right )} x^{2} + {\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 2 \, a^{2} b^{3} c^{4} d^{2} + 12 \, a^{3} b^{2} c^{3} d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a b^{6} c^{6} d^{3} - 4 \, a^{2} b^{5} c^{5} d^{4} + 6 \, a^{3} b^{4} c^{4} d^{5} - 4 \, a^{4} b^{3} c^{3} d^{6} + a^{5} b^{2} c^{2} d^{7} + {\left (b^{7} c^{4} d^{5} - 4 \, a b^{6} c^{3} d^{6} + 6 \, a^{2} b^{5} c^{2} d^{7} - 4 \, a^{3} b^{4} c d^{8} + a^{4} b^{3} d^{9}\right )} x^{3} + {\left (2 \, b^{7} c^{5} d^{4} - 7 \, a b^{6} c^{4} d^{5} + 8 \, a^{2} b^{5} c^{3} d^{6} - 2 \, a^{3} b^{4} c^{2} d^{7} - 2 \, a^{4} b^{3} c d^{8} + a^{5} b^{2} d^{9}\right )} x^{2} + {\left (b^{7} c^{6} d^{3} - 2 \, a b^{6} c^{5} d^{4} - 2 \, a^{2} b^{5} c^{4} d^{5} + 8 \, a^{3} b^{4} c^{3} d^{6} - 7 \, a^{4} b^{3} c^{2} d^{7} + 2 \, a^{5} b^{2} c d^{8}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x+a)^2/(d*x+c)^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.95, size = 310, normalized size = 1.89 \[ -\frac {a^{4} b^{3}}{{\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} {\left (b x + a\right )}} + \frac {{\left (b^{3} c^{4} - 4 \, a b^{2} c^{3} d + 6 \, a^{2} b c^{2} d^{2}\right )} \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{5} c^{4} d^{3} - 4 \, a b^{4} c^{3} d^{4} + 6 \, a^{2} b^{3} c^{2} d^{5} - 4 \, a^{3} b^{2} c d^{6} + a^{4} b d^{7}} - \frac {\log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{2} d^{3}} - \frac {3 \, b^{2} c^{4} d^{2} - 8 \, a b c^{3} d^{3} + \frac {2 \, {\left (b^{4} c^{5} d - 5 \, a b^{3} c^{4} d^{2} + 4 \, a^{2} b^{2} c^{3} d^{3}\right )}}{{\left (b x + a\right )} b}}{2 \, {\left (b c - a d\right )}^{4} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}^{2} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x+a)^2/(d*x+c)^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.01, size = 231, normalized size = 1.41 \[ \frac {a^{4} d \ln \left (b x +a \right )}{\left (a d -b c \right )^{4} b^{2}}-\frac {4 a^{3} c \ln \left (b x +a \right )}{\left (a d -b c \right )^{4} b}+\frac {6 a^{2} c^{2} \ln \left (d x +c \right )}{\left (a d -b c \right )^{4} d}-\frac {4 a b \,c^{3} \ln \left (d x +c \right )}{\left (a d -b c \right )^{4} d^{2}}+\frac {b^{2} c^{4} \ln \left (d x +c \right )}{\left (a d -b c \right )^{4} d^{3}}+\frac {a^{4}}{\left (a d -b c \right )^{3} \left (b x +a \right ) b^{2}}+\frac {4 a \,c^{3}}{\left (a d -b c \right )^{3} \left (d x +c \right ) d^{2}}-\frac {2 b \,c^{4}}{\left (a d -b c \right )^{3} \left (d x +c \right ) d^{3}}-\frac {c^{4}}{2 \left (a d -b c \right )^{2} \left (d x +c \right )^{2} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(b*x+a)^2/(d*x+c)^3,x)

[Out]

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

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maxima [B]  time = 1.18, size = 518, normalized size = 3.16 \[ -\frac {{\left (4 \, a^{3} b c - a^{4} d\right )} \log \left (b x + a\right )}{b^{6} c^{4} - 4 \, a b^{5} c^{3} d + 6 \, a^{2} b^{4} c^{2} d^{2} - 4 \, a^{3} b^{3} c d^{3} + a^{4} b^{2} d^{4}} + \frac {{\left (b^{2} c^{4} - 4 \, a b c^{3} d + 6 \, a^{2} c^{2} d^{2}\right )} \log \left (d x + c\right )}{b^{4} c^{4} d^{3} - 4 \, a b^{3} c^{3} d^{4} + 6 \, a^{2} b^{2} c^{2} d^{5} - 4 \, a^{3} b c d^{6} + a^{4} d^{7}} + \frac {3 \, a b^{3} c^{5} - 7 \, a^{2} b^{2} c^{4} d - 2 \, a^{4} c^{2} d^{3} + 2 \, {\left (2 \, b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} - a^{4} d^{5}\right )} x^{2} + {\left (3 \, b^{4} c^{5} - 3 \, a b^{3} c^{4} d - 8 \, a^{2} b^{2} c^{3} d^{2} - 4 \, a^{4} c d^{4}\right )} x}{2 \, {\left (a b^{5} c^{5} d^{3} - 3 \, a^{2} b^{4} c^{4} d^{4} + 3 \, a^{3} b^{3} c^{3} d^{5} - a^{4} b^{2} c^{2} d^{6} + {\left (b^{6} c^{3} d^{5} - 3 \, a b^{5} c^{2} d^{6} + 3 \, a^{2} b^{4} c d^{7} - a^{3} b^{3} d^{8}\right )} x^{3} + {\left (2 \, b^{6} c^{4} d^{4} - 5 \, a b^{5} c^{3} d^{5} + 3 \, a^{2} b^{4} c^{2} d^{6} + a^{3} b^{3} c d^{7} - a^{4} b^{2} d^{8}\right )} x^{2} + {\left (b^{6} c^{5} d^{3} - a b^{5} c^{4} d^{4} - 3 \, a^{2} b^{4} c^{3} d^{5} + 5 \, a^{3} b^{3} c^{2} d^{6} - 2 \, a^{4} b^{2} c d^{7}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x+a)^2/(d*x+c)^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.79, size = 468, normalized size = 2.85 \[ \frac {\frac {x^2\,\left (a^4\,d^4+4\,a\,b^3\,c^3\,d-2\,b^4\,c^4\right )}{b^2\,d^2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {a\,c^2\,\left (2\,a^3\,d^3+7\,a\,b^2\,c^2\,d-3\,b^3\,c^3\right )}{2\,b^2\,d^3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {c\,x\,\left (4\,a^4\,d^4+8\,a^2\,b^2\,c^2\,d^2+3\,a\,b^3\,c^3\,d-3\,b^4\,c^4\right )}{2\,b^2\,d^3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}}{x\,\left (b\,c^2+2\,a\,d\,c\right )+a\,c^2+x^2\,\left (a\,d^2+2\,b\,c\,d\right )+b\,d^2\,x^3}+\frac {a^4\,d\,\ln \left (a+b\,x\right )}{a^4\,b^2\,d^4-4\,a^3\,b^3\,c\,d^3+6\,a^2\,b^4\,c^2\,d^2-4\,a\,b^5\,c^3\,d+b^6\,c^4}+\frac {c^2\,\ln \left (c+d\,x\right )\,\left (6\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2\right )}{d^3\,{\left (a\,d-b\,c\right )}^4}-\frac {4\,a^3\,b\,c\,\ln \left (a+b\,x\right )}{a^4\,b^2\,d^4-4\,a^3\,b^3\,c\,d^3+6\,a^2\,b^4\,c^2\,d^2-4\,a\,b^5\,c^3\,d+b^6\,c^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/((a + b*x)^2*(c + d*x)^3),x)

[Out]

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

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sympy [B]  time = 33.49, size = 1083, normalized size = 6.60 \[ \frac {a^{3} \left (a d - 4 b c\right ) \log {\left (x + \frac {\frac {a^{8} d^{7} \left (a d - 4 b c\right )}{b \left (a d - b c\right )^{4}} - \frac {5 a^{7} c d^{6} \left (a d - 4 b c\right )}{\left (a d - b c\right )^{4}} + \frac {10 a^{6} b c^{2} d^{5} \left (a d - 4 b c\right )}{\left (a d - b c\right )^{4}} - \frac {10 a^{5} b^{2} c^{3} d^{4} \left (a d - 4 b c\right )}{\left (a d - b c\right )^{4}} + \frac {5 a^{4} b^{3} c^{4} d^{3} \left (a d - 4 b c\right )}{\left (a d - b c\right )^{4}} + a^{4} c d^{3} - \frac {a^{3} b^{4} c^{5} d^{2} \left (a d - 4 b c\right )}{\left (a d - b c\right )^{4}} - 10 a^{3} b c^{2} d^{2} + 4 a^{2} b^{2} c^{3} d - a b^{3} c^{4}}{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 )}}{b^{2} \left (a d - b c\right )^{4}} + \frac {c^{2} \left (6 a^{2} d^{2} - 4 a b c d + b^{2} c^{2}\right ) \log {\left (x + \frac {\frac {a^{5} b c^{2} d^{4} \left (6 a^{2} d^{2} - 4 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{4}} - \frac {5 a^{4} b^{2} c^{3} d^{3} \left (6 a^{2} d^{2} - 4 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{4}} + a^{4} c d^{3} + \frac {10 a^{3} b^{3} c^{4} d^{2} \left (6 a^{2} d^{2} - 4 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{4}} - 10 a^{3} b c^{2} d^{2} - \frac {10 a^{2} b^{4} c^{5} d \left (6 a^{2} d^{2} - 4 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{4}} + 4 a^{2} b^{2} c^{3} d + \frac {5 a b^{5} c^{6} \left (6 a^{2} d^{2} - 4 a b c d + b^{2} c^{2}\right )}{\left (a d - b c\right )^{4}} - a b^{3} c^{4} - \frac {b^{6} c^{7} \left (6 a^{2} d^{2} - 4 a b c d + b^{2} c^{2}\right )}{d \left (a d - b c\right )^{4}}}{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 )}}{d^{3} \left (a d - b c\right )^{4}} + \frac {2 a^{4} c^{2} d^{3} + 7 a^{2} b^{2} c^{4} d - 3 a b^{3} c^{5} + x^{2} \left (2 a^{4} d^{5} + 8 a b^{3} c^{3} d^{2} - 4 b^{4} c^{4} d\right ) + x \left (4 a^{4} c d^{4} + 8 a^{2} b^{2} c^{3} d^{2} + 3 a b^{3} c^{4} d - 3 b^{4} c^{5}\right )}{2 a^{4} b^{2} c^{2} d^{6} - 6 a^{3} b^{3} c^{3} d^{5} + 6 a^{2} b^{4} c^{4} d^{4} - 2 a b^{5} c^{5} d^{3} + x^{3} \left (2 a^{3} b^{3} d^{8} - 6 a^{2} b^{4} c d^{7} + 6 a b^{5} c^{2} d^{6} - 2 b^{6} c^{3} d^{5}\right ) + x^{2} \left (2 a^{4} b^{2} d^{8} - 2 a^{3} b^{3} c d^{7} - 6 a^{2} b^{4} c^{2} d^{6} + 10 a b^{5} c^{3} d^{5} - 4 b^{6} c^{4} d^{4}\right ) + x \left (4 a^{4} b^{2} c d^{7} - 10 a^{3} b^{3} c^{2} d^{6} + 6 a^{2} b^{4} c^{3} d^{5} + 2 a b^{5} c^{4} d^{4} - 2 b^{6} c^{5} d^{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(b*x+a)**2/(d*x+c)**3,x)

[Out]

a**3*(a*d - 4*b*c)*log(x + (a**8*d**7*(a*d - 4*b*c)/(b*(a*d - b*c)**4) - 5*a**7*c*d**6*(a*d - 4*b*c)/(a*d - b*
c)**4 + 10*a**6*b*c**2*d**5*(a*d - 4*b*c)/(a*d - b*c)**4 - 10*a**5*b**2*c**3*d**4*(a*d - 4*b*c)/(a*d - b*c)**4
+ 5*a**4*b**3*c**4*d**3*(a*d - 4*b*c)/(a*d - b*c)**4 + a**4*c*d**3 - a**3*b**4*c**5*d**2*(a*d - 4*b*c)/(a*d -
b*c)**4 - 10*a**3*b*c**2*d**2 + 4*a**2*b**2*c**3*d - a*b**3*c**4)/(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))/(b**2*(a*d - b*c)**4) + c**2*(6*a**2*d**2 - 4*a*b*c*d + b**2*c**2)*l
og(x + (a**5*b*c**2*d**4*(6*a**2*d**2 - 4*a*b*c*d + b**2*c**2)/(a*d - b*c)**4 - 5*a**4*b**2*c**3*d**3*(6*a**2*
d**2 - 4*a*b*c*d + b**2*c**2)/(a*d - b*c)**4 + a**4*c*d**3 + 10*a**3*b**3*c**4*d**2*(6*a**2*d**2 - 4*a*b*c*d +
b**2*c**2)/(a*d - b*c)**4 - 10*a**3*b*c**2*d**2 - 10*a**2*b**4*c**5*d*(6*a**2*d**2 - 4*a*b*c*d + b**2*c**2)/(
a*d - b*c)**4 + 4*a**2*b**2*c**3*d + 5*a*b**5*c**6*(6*a**2*d**2 - 4*a*b*c*d + b**2*c**2)/(a*d - b*c)**4 - a*b*
*3*c**4 - b**6*c**7*(6*a**2*d**2 - 4*a*b*c*d + b**2*c**2)/(d*(a*d - b*c)**4))/(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))/(d**3*(a*d - b*c)**4) + (2*a**4*c**2*d**3 + 7*a**2*b**2*c
**4*d - 3*a*b**3*c**5 + x**2*(2*a**4*d**5 + 8*a*b**3*c**3*d**2 - 4*b**4*c**4*d) + x*(4*a**4*c*d**4 + 8*a**2*b*
*2*c**3*d**2 + 3*a*b**3*c**4*d - 3*b**4*c**5))/(2*a**4*b**2*c**2*d**6 - 6*a**3*b**3*c**3*d**5 + 6*a**2*b**4*c*
*4*d**4 - 2*a*b**5*c**5*d**3 + x**3*(2*a**3*b**3*d**8 - 6*a**2*b**4*c*d**7 + 6*a*b**5*c**2*d**6 - 2*b**6*c**3*
d**5) + x**2*(2*a**4*b**2*d**8 - 2*a**3*b**3*c*d**7 - 6*a**2*b**4*c**2*d**6 + 10*a*b**5*c**3*d**5 - 4*b**6*c**
4*d**4) + x*(4*a**4*b**2*c*d**7 - 10*a**3*b**3*c**2*d**6 + 6*a**2*b**4*c**3*d**5 + 2*a*b**5*c**4*d**4 - 2*b**6
*c**5*d**3))

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