3.3.78 \(\int \frac {x^5}{(a+b x)^2 (c+d x)^3} \, dx\)

Optimal. Leaf size=173 \[ \frac {a^5}{b^3 (a+b x) (b c-a d)^3}+\frac {a^4 (5 b c-2 a d) \log (a+b x)}{b^3 (b c-a d)^4}-\frac {c^3 \left (10 a^2 d^2-10 a b c d+3 b^2 c^2\right ) \log (c+d x)}{d^4 (b c-a d)^4}+\frac {c^5}{2 d^4 (c+d x)^2 (b c-a d)^2}-\frac {c^4 (3 b c-5 a d)}{d^4 (c+d x) (b c-a d)^3}+\frac {x}{b^2 d^3} \]

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Rubi [A]  time = 0.20, antiderivative size = 173, 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} \begin {gather*} -\frac {c^3 \left (10 a^2 d^2-10 a b c d+3 b^2 c^2\right ) \log (c+d x)}{d^4 (b c-a d)^4}+\frac {a^5}{b^3 (a+b x) (b c-a d)^3}+\frac {a^4 (5 b c-2 a d) \log (a+b x)}{b^3 (b c-a d)^4}-\frac {c^4 (3 b c-5 a d)}{d^4 (c+d x) (b c-a d)^3}+\frac {c^5}{2 d^4 (c+d x)^2 (b c-a d)^2}+\frac {x}{b^2 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5/((a + b*x)^2*(c + d*x)^3),x]

[Out]

x/(b^2*d^3) + a^5/(b^3*(b*c - a*d)^3*(a + b*x)) + c^5/(2*d^4*(b*c - a*d)^2*(c + d*x)^2) - (c^4*(3*b*c - 5*a*d)
)/(d^4*(b*c - a*d)^3*(c + d*x)) + (a^4*(5*b*c - 2*a*d)*Log[a + b*x])/(b^3*(b*c - a*d)^4) - (c^3*(3*b^2*c^2 - 1
0*a*b*c*d + 10*a^2*d^2)*Log[c + d*x])/(d^4*(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^5}{(a+b x)^2 (c+d x)^3} \, dx &=\int \left (\frac {1}{b^2 d^3}-\frac {a^5}{b^2 (b c-a d)^3 (a+b x)^2}-\frac {a^4 (-5 b c+2 a d)}{b^2 (b c-a d)^4 (a+b x)}-\frac {c^5}{d^3 (-b c+a d)^2 (c+d x)^3}-\frac {c^4 (3 b c-5 a d)}{d^3 (-b c+a d)^3 (c+d x)^2}-\frac {c^3 \left (3 b^2 c^2-10 a b c d+10 a^2 d^2\right )}{d^3 (-b c+a d)^4 (c+d x)}\right ) \, dx\\ &=\frac {x}{b^2 d^3}+\frac {a^5}{b^3 (b c-a d)^3 (a+b x)}+\frac {c^5}{2 d^4 (b c-a d)^2 (c+d x)^2}-\frac {c^4 (3 b c-5 a d)}{d^4 (b c-a d)^3 (c+d x)}+\frac {a^4 (5 b c-2 a d) \log (a+b x)}{b^3 (b c-a d)^4}-\frac {c^3 \left (3 b^2 c^2-10 a b c d+10 a^2 d^2\right ) \log (c+d x)}{d^4 (b c-a d)^4}\\ \end {align*}

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Mathematica [A]  time = 0.27, size = 172, normalized size = 0.99 \begin {gather*} \frac {a^5}{b^3 (a+b x) (b c-a d)^3}+\frac {a^4 (5 b c-2 a d) \log (a+b x)}{b^3 (b c-a d)^4}-\frac {c^3 \left (10 a^2 d^2-10 a b c d+3 b^2 c^2\right ) \log (c+d x)}{d^4 (b c-a d)^4}+\frac {c^5}{2 d^4 (c+d x)^2 (b c-a d)^2}+\frac {c^4 (3 b c-5 a d)}{d^4 (c+d x) (a d-b c)^3}+\frac {x}{b^2 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5/((a + b*x)^2*(c + d*x)^3),x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^5}{(a+b x)^2 (c+d x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^5/((a + b*x)^2*(c + d*x)^3),x]

[Out]

IntegrateAlgebraic[x^5/((a + b*x)^2*(c + d*x)^3), x]

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fricas [B]  time = 1.19, size = 981, normalized size = 5.67 \begin {gather*} -\frac {5 \, a b^{5} c^{7} - 14 \, a^{2} b^{4} c^{6} d + 9 \, a^{3} b^{3} c^{5} d^{2} - 2 \, a^{5} b c^{3} d^{4} + 2 \, a^{6} c^{2} d^{5} - 2 \, {\left (b^{6} c^{4} d^{3} - 4 \, a b^{5} c^{3} d^{4} + 6 \, a^{2} b^{4} c^{2} d^{5} - 4 \, a^{3} b^{3} c d^{6} + a^{4} b^{2} d^{7}\right )} x^{4} - 2 \, {\left (2 \, b^{6} c^{5} d^{2} - 7 \, a b^{5} c^{4} d^{3} + 8 \, a^{2} b^{4} c^{3} d^{4} - 2 \, a^{3} b^{3} c^{2} d^{5} - 2 \, a^{4} b^{2} c d^{6} + a^{5} b d^{7}\right )} x^{3} + 2 \, {\left (2 \, b^{6} c^{6} d - 6 \, a b^{5} c^{5} d^{2} + 7 \, a^{2} b^{4} c^{4} d^{3} - 8 \, a^{3} b^{3} c^{3} d^{4} + 7 \, a^{4} b^{2} c^{2} d^{5} - 3 \, a^{5} b c d^{6} + a^{6} d^{7}\right )} x^{2} + {\left (5 \, b^{6} c^{7} - 10 \, a b^{5} c^{6} d + a^{2} b^{4} c^{5} d^{2} - 2 \, a^{3} b^{3} c^{4} d^{3} + 8 \, a^{4} b^{2} c^{3} d^{4} - 6 \, a^{5} b c^{2} d^{5} + 4 \, a^{6} c d^{6}\right )} x - 2 \, {\left (5 \, a^{5} b c^{3} d^{4} - 2 \, a^{6} c^{2} d^{5} + {\left (5 \, a^{4} b^{2} c d^{6} - 2 \, a^{5} b d^{7}\right )} x^{3} + {\left (10 \, a^{4} b^{2} c^{2} d^{5} + a^{5} b c d^{6} - 2 \, a^{6} d^{7}\right )} x^{2} + {\left (5 \, a^{4} b^{2} c^{3} d^{4} + 8 \, a^{5} b c^{2} d^{5} - 4 \, a^{6} c d^{6}\right )} x\right )} \log \left (b x + a\right ) + 2 \, {\left (3 \, a b^{5} c^{7} - 10 \, a^{2} b^{4} c^{6} d + 10 \, a^{3} b^{3} c^{5} d^{2} + {\left (3 \, b^{6} c^{5} d^{2} - 10 \, a b^{5} c^{4} d^{3} + 10 \, a^{2} b^{4} c^{3} d^{4}\right )} x^{3} + {\left (6 \, b^{6} c^{6} d - 17 \, a b^{5} c^{5} d^{2} + 10 \, a^{2} b^{4} c^{4} d^{3} + 10 \, a^{3} b^{3} c^{3} d^{4}\right )} x^{2} + {\left (3 \, b^{6} c^{7} - 4 \, a b^{5} c^{6} d - 10 \, a^{2} b^{4} c^{5} d^{2} + 20 \, a^{3} b^{3} c^{4} d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (a b^{7} c^{6} d^{4} - 4 \, a^{2} b^{6} c^{5} d^{5} + 6 \, a^{3} b^{5} c^{4} d^{6} - 4 \, a^{4} b^{4} c^{3} d^{7} + a^{5} b^{3} c^{2} d^{8} + {\left (b^{8} c^{4} d^{6} - 4 \, a b^{7} c^{3} d^{7} + 6 \, a^{2} b^{6} c^{2} d^{8} - 4 \, a^{3} b^{5} c d^{9} + a^{4} b^{4} d^{10}\right )} x^{3} + {\left (2 \, b^{8} c^{5} d^{5} - 7 \, a b^{7} c^{4} d^{6} + 8 \, a^{2} b^{6} c^{3} d^{7} - 2 \, a^{3} b^{5} c^{2} d^{8} - 2 \, a^{4} b^{4} c d^{9} + a^{5} b^{3} d^{10}\right )} x^{2} + {\left (b^{8} c^{6} d^{4} - 2 \, a b^{7} c^{5} d^{5} - 2 \, a^{2} b^{6} c^{4} d^{6} + 8 \, a^{3} b^{5} c^{3} d^{7} - 7 \, a^{4} b^{4} c^{2} d^{8} + 2 \, a^{5} b^{3} c d^{9}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)^2/(d*x+c)^3,x, algorithm="fricas")

[Out]

-1/2*(5*a*b^5*c^7 - 14*a^2*b^4*c^6*d + 9*a^3*b^3*c^5*d^2 - 2*a^5*b*c^3*d^4 + 2*a^6*c^2*d^5 - 2*(b^6*c^4*d^3 -
4*a*b^5*c^3*d^4 + 6*a^2*b^4*c^2*d^5 - 4*a^3*b^3*c*d^6 + a^4*b^2*d^7)*x^4 - 2*(2*b^6*c^5*d^2 - 7*a*b^5*c^4*d^3
+ 8*a^2*b^4*c^3*d^4 - 2*a^3*b^3*c^2*d^5 - 2*a^4*b^2*c*d^6 + a^5*b*d^7)*x^3 + 2*(2*b^6*c^6*d - 6*a*b^5*c^5*d^2
+ 7*a^2*b^4*c^4*d^3 - 8*a^3*b^3*c^3*d^4 + 7*a^4*b^2*c^2*d^5 - 3*a^5*b*c*d^6 + a^6*d^7)*x^2 + (5*b^6*c^7 - 10*a
*b^5*c^6*d + a^2*b^4*c^5*d^2 - 2*a^3*b^3*c^4*d^3 + 8*a^4*b^2*c^3*d^4 - 6*a^5*b*c^2*d^5 + 4*a^6*c*d^6)*x - 2*(5
*a^5*b*c^3*d^4 - 2*a^6*c^2*d^5 + (5*a^4*b^2*c*d^6 - 2*a^5*b*d^7)*x^3 + (10*a^4*b^2*c^2*d^5 + a^5*b*c*d^6 - 2*a
^6*d^7)*x^2 + (5*a^4*b^2*c^3*d^4 + 8*a^5*b*c^2*d^5 - 4*a^6*c*d^6)*x)*log(b*x + a) + 2*(3*a*b^5*c^7 - 10*a^2*b^
4*c^6*d + 10*a^3*b^3*c^5*d^2 + (3*b^6*c^5*d^2 - 10*a*b^5*c^4*d^3 + 10*a^2*b^4*c^3*d^4)*x^3 + (6*b^6*c^6*d - 17
*a*b^5*c^5*d^2 + 10*a^2*b^4*c^4*d^3 + 10*a^3*b^3*c^3*d^4)*x^2 + (3*b^6*c^7 - 4*a*b^5*c^6*d - 10*a^2*b^4*c^5*d^
2 + 20*a^3*b^3*c^4*d^3)*x)*log(d*x + c))/(a*b^7*c^6*d^4 - 4*a^2*b^6*c^5*d^5 + 6*a^3*b^5*c^4*d^6 - 4*a^4*b^4*c^
3*d^7 + a^5*b^3*c^2*d^8 + (b^8*c^4*d^6 - 4*a*b^7*c^3*d^7 + 6*a^2*b^6*c^2*d^8 - 4*a^3*b^5*c*d^9 + a^4*b^4*d^10)
*x^3 + (2*b^8*c^5*d^5 - 7*a*b^7*c^4*d^6 + 8*a^2*b^6*c^3*d^7 - 2*a^3*b^5*c^2*d^8 - 2*a^4*b^4*c*d^9 + a^5*b^3*d^
10)*x^2 + (b^8*c^6*d^4 - 2*a*b^7*c^5*d^5 - 2*a^2*b^6*c^4*d^6 + 8*a^3*b^5*c^3*d^7 - 7*a^4*b^4*c^2*d^8 + 2*a^5*b
^3*c*d^9)*x)

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giac [B]  time = 1.10, size = 496, normalized size = 2.87 \begin {gather*} \frac {a^{5} b^{4}}{{\left (b^{10} c^{3} - 3 \, a b^{9} c^{2} d + 3 \, a^{2} b^{8} c d^{2} - a^{3} b^{7} d^{3}\right )} {\left (b x + a\right )}} - \frac {{\left (3 \, b^{3} c^{5} - 10 \, a b^{2} c^{4} d + 10 \, a^{2} b c^{3} 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^{4} - 4 \, a b^{4} c^{3} d^{5} + 6 \, a^{2} b^{3} c^{2} d^{6} - 4 \, a^{3} b^{2} c d^{7} + a^{4} b d^{8}} + \frac {{\left (3 \, b c + 2 \, a d\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{3} d^{4}} + \frac {{\left (2 \, b^{4} c^{4} d^{3} - 8 \, a b^{3} c^{3} d^{4} + 12 \, a^{2} b^{2} c^{2} d^{5} - 8 \, a^{3} b c d^{6} + 2 \, a^{4} d^{7} + \frac {9 \, b^{6} c^{5} d^{2} - 30 \, a b^{5} c^{4} d^{3} + 40 \, a^{2} b^{4} c^{3} d^{4} - 40 \, a^{3} b^{3} c^{2} d^{5} + 20 \, a^{4} b^{2} c d^{6} - 4 \, a^{5} b d^{7}}{{\left (b x + a\right )} b} + \frac {2 \, {\left (3 \, b^{8} c^{6} d - 13 \, a b^{7} c^{5} d^{2} + 20 \, a^{2} b^{6} c^{4} d^{3} - 20 \, a^{3} b^{5} c^{3} d^{4} + 15 \, a^{4} b^{4} c^{2} d^{5} - 6 \, a^{5} b^{3} c d^{6} + a^{6} b^{2} d^{7}\right )}}{{\left (b x + a\right )}^{2} b^{2}}\right )} {\left (b x + a\right )}}{2 \, {\left (b c - a d\right )}^{4} b^{3} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}^{2} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)^2/(d*x+c)^3,x, algorithm="giac")

[Out]

a^5*b^4/((b^10*c^3 - 3*a*b^9*c^2*d + 3*a^2*b^8*c*d^2 - a^3*b^7*d^3)*(b*x + a)) - (3*b^3*c^5 - 10*a*b^2*c^4*d +
10*a^2*b*c^3*d^2)*log(abs(b*c/(b*x + a) - a*d/(b*x + a) + d))/(b^5*c^4*d^4 - 4*a*b^4*c^3*d^5 + 6*a^2*b^3*c^2*
d^6 - 4*a^3*b^2*c*d^7 + a^4*b*d^8) + (3*b*c + 2*a*d)*log(abs(b*x + a)/((b*x + a)^2*abs(b)))/(b^3*d^4) + 1/2*(2
*b^4*c^4*d^3 - 8*a*b^3*c^3*d^4 + 12*a^2*b^2*c^2*d^5 - 8*a^3*b*c*d^6 + 2*a^4*d^7 + (9*b^6*c^5*d^2 - 30*a*b^5*c^
4*d^3 + 40*a^2*b^4*c^3*d^4 - 40*a^3*b^3*c^2*d^5 + 20*a^4*b^2*c*d^6 - 4*a^5*b*d^7)/((b*x + a)*b) + 2*(3*b^8*c^6
*d - 13*a*b^7*c^5*d^2 + 20*a^2*b^6*c^4*d^3 - 20*a^3*b^5*c^3*d^4 + 15*a^4*b^4*c^2*d^5 - 6*a^5*b^3*c*d^6 + a^6*b
^2*d^7)/((b*x + a)^2*b^2))*(b*x + a)/((b*c - a*d)^4*b^3*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*d^4)

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maple [A]  time = 0.02, size = 242, normalized size = 1.40 \begin {gather*} -\frac {2 a^{5} d \ln \left (b x +a \right )}{\left (a d -b c \right )^{4} b^{3}}+\frac {5 a^{4} c \ln \left (b x +a \right )}{\left (a d -b c \right )^{4} b^{2}}-\frac {10 a^{2} c^{3} \ln \left (d x +c \right )}{\left (a d -b c \right )^{4} d^{2}}+\frac {10 a b \,c^{4} \ln \left (d x +c \right )}{\left (a d -b c \right )^{4} d^{3}}-\frac {3 b^{2} c^{5} \ln \left (d x +c \right )}{\left (a d -b c \right )^{4} d^{4}}-\frac {a^{5}}{\left (a d -b c \right )^{3} \left (b x +a \right ) b^{3}}-\frac {5 a \,c^{4}}{\left (a d -b c \right )^{3} \left (d x +c \right ) d^{3}}+\frac {3 b \,c^{5}}{\left (a d -b c \right )^{3} \left (d x +c \right ) d^{4}}+\frac {c^{5}}{2 \left (a d -b c \right )^{2} \left (d x +c \right )^{2} d^{4}}+\frac {x}{b^{2} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b*x+a)^2/(d*x+c)^3,x)

[Out]

x/b^2/d^3-5/d^3*c^4/(a*d-b*c)^3/(d*x+c)*a+3/d^4*c^5/(a*d-b*c)^3/(d*x+c)*b+1/2/d^4*c^5/(a*d-b*c)^2/(d*x+c)^2-10
/d^2*c^3/(a*d-b*c)^4*ln(d*x+c)*a^2+10/d^3*c^4/(a*d-b*c)^4*ln(d*x+c)*a*b-3/d^4*c^5/(a*d-b*c)^4*ln(d*x+c)*b^2-1/
b^3*a^5/(a*d-b*c)^3/(b*x+a)-2/b^3*a^5/(a*d-b*c)^4*ln(b*x+a)*d+5/b^2*a^4/(a*d-b*c)^4*ln(b*x+a)*c

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maxima [B]  time = 1.33, size = 527, normalized size = 3.05 \begin {gather*} \frac {{\left (5 \, a^{4} b c - 2 \, a^{5} d\right )} \log \left (b x + a\right )}{b^{7} c^{4} - 4 \, a b^{6} c^{3} d + 6 \, a^{2} b^{5} c^{2} d^{2} - 4 \, a^{3} b^{4} c d^{3} + a^{4} b^{3} d^{4}} - \frac {{\left (3 \, b^{2} c^{5} - 10 \, a b c^{4} d + 10 \, a^{2} c^{3} d^{2}\right )} \log \left (d x + c\right )}{b^{4} c^{4} d^{4} - 4 \, a b^{3} c^{3} d^{5} + 6 \, a^{2} b^{2} c^{2} d^{6} - 4 \, a^{3} b c d^{7} + a^{4} d^{8}} - \frac {5 \, a b^{4} c^{6} - 9 \, a^{2} b^{3} c^{5} d - 2 \, a^{5} c^{2} d^{4} + 2 \, {\left (3 \, b^{5} c^{5} d - 5 \, a b^{4} c^{4} d^{2} - a^{5} d^{6}\right )} x^{2} + {\left (5 \, b^{5} c^{6} - 3 \, a b^{4} c^{5} d - 10 \, a^{2} b^{3} c^{4} d^{2} - 4 \, a^{5} c d^{5}\right )} x}{2 \, {\left (a b^{6} c^{5} d^{4} - 3 \, a^{2} b^{5} c^{4} d^{5} + 3 \, a^{3} b^{4} c^{3} d^{6} - a^{4} b^{3} c^{2} d^{7} + {\left (b^{7} c^{3} d^{6} - 3 \, a b^{6} c^{2} d^{7} + 3 \, a^{2} b^{5} c d^{8} - a^{3} b^{4} d^{9}\right )} x^{3} + {\left (2 \, b^{7} c^{4} d^{5} - 5 \, a b^{6} c^{3} d^{6} + 3 \, a^{2} b^{5} c^{2} d^{7} + a^{3} b^{4} c d^{8} - a^{4} b^{3} d^{9}\right )} x^{2} + {\left (b^{7} c^{5} d^{4} - a b^{6} c^{4} d^{5} - 3 \, a^{2} b^{5} c^{3} d^{6} + 5 \, a^{3} b^{4} c^{2} d^{7} - 2 \, a^{4} b^{3} c d^{8}\right )} x\right )}} + \frac {x}{b^{2} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)^2/(d*x+c)^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.75, size = 488, normalized size = 2.82 \begin {gather*} \frac {x}{b^2\,d^3}-\frac {\ln \left (c+d\,x\right )\,\left (10\,a^2\,c^3\,d^2-10\,a\,b\,c^4\,d+3\,b^2\,c^5\right )}{a^4\,d^8-4\,a^3\,b\,c\,d^7+6\,a^2\,b^2\,c^2\,d^6-4\,a\,b^3\,c^3\,d^5+b^4\,c^4\,d^4}-\frac {\frac {x^2\,\left (a^5\,d^5+5\,a\,b^4\,c^4\,d-3\,b^5\,c^5\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 {a\,c^2\,\left (2\,a^4\,d^4+9\,a\,b^3\,c^3\,d-5\,b^4\,c^4\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 )}+\frac {c\,x\,\left (4\,a^5\,d^5+10\,a^2\,b^3\,c^3\,d^2+3\,a\,b^4\,c^4\,d-5\,b^5\,c^5\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^3\,d^4+a\,b^2\,d^5\right )+x\,\left (b^3\,c^2\,d^3+2\,a\,b^2\,c\,d^4\right )+b^3\,d^5\,x^3+a\,b^2\,c^2\,d^3}-\frac {\ln \left (a+b\,x\right )\,\left (2\,a^5\,d-5\,a^4\,b\,c\right )}{a^4\,b^3\,d^4-4\,a^3\,b^4\,c\,d^3+6\,a^2\,b^5\,c^2\,d^2-4\,a\,b^6\,c^3\,d+b^7\,c^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/((a + b*x)^2*(c + d*x)^3),x)

[Out]

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

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sympy [B]  time = 65.68, size = 1161, normalized size = 6.71 \begin {gather*} - \frac {a^{4} \left (2 a d - 5 b c\right ) \log {\left (x + \frac {\frac {a^{9} d^{8} \left (2 a d - 5 b c\right )}{b \left (a d - b c\right )^{4}} - \frac {5 a^{8} c d^{7} \left (2 a d - 5 b c\right )}{\left (a d - b c\right )^{4}} + \frac {10 a^{7} b c^{2} d^{6} \left (2 a d - 5 b c\right )}{\left (a d - b c\right )^{4}} - \frac {10 a^{6} b^{2} c^{3} d^{5} \left (2 a d - 5 b c\right )}{\left (a d - b c\right )^{4}} + \frac {5 a^{5} b^{3} c^{4} d^{4} \left (2 a d - 5 b c\right )}{\left (a d - b c\right )^{4}} + 2 a^{5} c d^{4} - \frac {a^{4} b^{4} c^{5} d^{3} \left (2 a d - 5 b c\right )}{\left (a d - b c\right )^{4}} - 5 a^{4} b c^{2} d^{3} - 10 a^{3} b^{2} c^{3} d^{2} + 10 a^{2} b^{3} c^{4} d - 3 a b^{4} c^{5}}{2 a^{5} d^{5} - 5 a^{4} b c d^{4} - 10 a^{2} b^{3} c^{3} d^{2} + 10 a b^{4} c^{4} d - 3 b^{5} c^{5}} \right )}}{b^{3} \left (a d - b c\right )^{4}} - \frac {c^{3} \left (10 a^{2} d^{2} - 10 a b c d + 3 b^{2} c^{2}\right ) \log {\left (x + \frac {\frac {a^{5} b^{2} c^{3} d^{4} \left (10 a^{2} d^{2} - 10 a b c d + 3 b^{2} c^{2}\right )}{\left (a d - b c\right )^{4}} + 2 a^{5} c d^{4} - \frac {5 a^{4} b^{3} c^{4} d^{3} \left (10 a^{2} d^{2} - 10 a b c d + 3 b^{2} c^{2}\right )}{\left (a d - b c\right )^{4}} - 5 a^{4} b c^{2} d^{3} + \frac {10 a^{3} b^{4} c^{5} d^{2} \left (10 a^{2} d^{2} - 10 a b c d + 3 b^{2} c^{2}\right )}{\left (a d - b c\right )^{4}} - 10 a^{3} b^{2} c^{3} d^{2} - \frac {10 a^{2} b^{5} c^{6} d \left (10 a^{2} d^{2} - 10 a b c d + 3 b^{2} c^{2}\right )}{\left (a d - b c\right )^{4}} + 10 a^{2} b^{3} c^{4} d + \frac {5 a b^{6} c^{7} \left (10 a^{2} d^{2} - 10 a b c d + 3 b^{2} c^{2}\right )}{\left (a d - b c\right )^{4}} - 3 a b^{4} c^{5} - \frac {b^{7} c^{8} \left (10 a^{2} d^{2} - 10 a b c d + 3 b^{2} c^{2}\right )}{d \left (a d - b c\right )^{4}}}{2 a^{5} d^{5} - 5 a^{4} b c d^{4} - 10 a^{2} b^{3} c^{3} d^{2} + 10 a b^{4} c^{4} d - 3 b^{5} c^{5}} \right )}}{d^{4} \left (a d - b c\right )^{4}} + \frac {- 2 a^{5} c^{2} d^{4} - 9 a^{2} b^{3} c^{5} d + 5 a b^{4} c^{6} + x^{2} \left (- 2 a^{5} d^{6} - 10 a b^{4} c^{4} d^{2} + 6 b^{5} c^{5} d\right ) + x \left (- 4 a^{5} c d^{5} - 10 a^{2} b^{3} c^{4} d^{2} - 3 a b^{4} c^{5} d + 5 b^{5} c^{6}\right )}{2 a^{4} b^{3} c^{2} d^{7} - 6 a^{3} b^{4} c^{3} d^{6} + 6 a^{2} b^{5} c^{4} d^{5} - 2 a b^{6} c^{5} d^{4} + x^{3} \left (2 a^{3} b^{4} d^{9} - 6 a^{2} b^{5} c d^{8} + 6 a b^{6} c^{2} d^{7} - 2 b^{7} c^{3} d^{6}\right ) + x^{2} \left (2 a^{4} b^{3} d^{9} - 2 a^{3} b^{4} c d^{8} - 6 a^{2} b^{5} c^{2} d^{7} + 10 a b^{6} c^{3} d^{6} - 4 b^{7} c^{4} d^{5}\right ) + x \left (4 a^{4} b^{3} c d^{8} - 10 a^{3} b^{4} c^{2} d^{7} + 6 a^{2} b^{5} c^{3} d^{6} + 2 a b^{6} c^{4} d^{5} - 2 b^{7} c^{5} d^{4}\right )} + \frac {x}{b^{2} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(b*x+a)**2/(d*x+c)**3,x)

[Out]

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

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