\(\int \frac {(d+e x)^5}{(b x+c x^2)^3} \, dx\) [277]

Optimal result

Integrand size = 19, antiderivative size = 171 \[ \int \frac {(d+e x)^5}{\left (b x+c x^2\right )^3} \, dx=-\frac {d^5}{2 b^3 x^2}+\frac {d^4 (3 c d-5 b e)}{b^4 x}+\frac {(c d-b e)^5}{2 b^3 c^3 (b+c x)^2}+\frac {(c d-b e)^4 (3 c d+2 b e)}{b^4 c^3 (b+c x)}+\frac {d^3 \left (6 c^2 d^2-15 b c d e+10 b^2 e^2\right ) \log (x)}{b^5}-\frac {(c d-b e)^3 \left (6 c^2 d^2+3 b c d e+b^2 e^2\right ) \log (b+c x)}{b^5 c^3} \]

[Out]

-1/2*d^5/b^3/x^2+d^4*(-5*b*e+3*c*d)/b^4/x+1/2*(-b*e+c*d)^5/b^3/c^3/(c*x+b)^2+(-b*e+c*d)^4*(2*b*e+3*c*d)/b^4/c^
3/(c*x+b)+d^3*(10*b^2*e^2-15*b*c*d*e+6*c^2*d^2)*ln(x)/b^5-(-b*e+c*d)^3*(b^2*e^2+3*b*c*d*e+6*c^2*d^2)*ln(c*x+b)
/b^5/c^3

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {712} \[ \int \frac {(d+e x)^5}{\left (b x+c x^2\right )^3} \, dx=\frac {(c d-b e)^4 (2 b e+3 c d)}{b^4 c^3 (b+c x)}+\frac {d^4 (3 c d-5 b e)}{b^4 x}+\frac {(c d-b e)^5}{2 b^3 c^3 (b+c x)^2}-\frac {d^5}{2 b^3 x^2}+\frac {d^3 \log (x) \left (10 b^2 e^2-15 b c d e+6 c^2 d^2\right )}{b^5}-\frac {(c d-b e)^3 \left (b^2 e^2+3 b c d e+6 c^2 d^2\right ) \log (b+c x)}{b^5 c^3} \]

[In]

Int[(d + e*x)^5/(b*x + c*x^2)^3,x]

[Out]

-1/2*d^5/(b^3*x^2) + (d^4*(3*c*d - 5*b*e))/(b^4*x) + (c*d - b*e)^5/(2*b^3*c^3*(b + c*x)^2) + ((c*d - b*e)^4*(3
*c*d + 2*b*e))/(b^4*c^3*(b + c*x)) + (d^3*(6*c^2*d^2 - 15*b*c*d*e + 10*b^2*e^2)*Log[x])/b^5 - ((c*d - b*e)^3*(
6*c^2*d^2 + 3*b*c*d*e + b^2*e^2)*Log[b + c*x])/(b^5*c^3)

Rule 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^5}{b^3 x^3}+\frac {d^4 (-3 c d+5 b e)}{b^4 x^2}+\frac {d^3 \left (6 c^2 d^2-15 b c d e+10 b^2 e^2\right )}{b^5 x}+\frac {(-c d+b e)^5}{b^3 c^2 (b+c x)^3}-\frac {(-c d+b e)^4 (3 c d+2 b e)}{b^4 c^2 (b+c x)^2}+\frac {(-c d+b e)^3 \left (6 c^2 d^2+3 b c d e+b^2 e^2\right )}{b^5 c^2 (b+c x)}\right ) \, dx \\ & = -\frac {d^5}{2 b^3 x^2}+\frac {d^4 (3 c d-5 b e)}{b^4 x}+\frac {(c d-b e)^5}{2 b^3 c^3 (b+c x)^2}+\frac {(c d-b e)^4 (3 c d+2 b e)}{b^4 c^3 (b+c x)}+\frac {d^3 \left (6 c^2 d^2-15 b c d e+10 b^2 e^2\right ) \log (x)}{b^5}-\frac {(c d-b e)^3 \left (6 c^2 d^2+3 b c d e+b^2 e^2\right ) \log (b+c x)}{b^5 c^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.96 \[ \int \frac {(d+e x)^5}{\left (b x+c x^2\right )^3} \, dx=-\frac {\frac {b^2 d^5}{x^2}+\frac {2 b d^4 (-3 c d+5 b e)}{x}+\frac {b^2 (-c d+b e)^5}{c^3 (b+c x)^2}-\frac {2 b (c d-b e)^4 (3 c d+2 b e)}{c^3 (b+c x)}-2 d^3 \left (6 c^2 d^2-15 b c d e+10 b^2 e^2\right ) \log (x)+\frac {2 (c d-b e)^3 \left (6 c^2 d^2+3 b c d e+b^2 e^2\right ) \log (b+c x)}{c^3}}{2 b^5} \]

[In]

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

[Out]

-1/2*((b^2*d^5)/x^2 + (2*b*d^4*(-3*c*d + 5*b*e))/x + (b^2*(-(c*d) + b*e)^5)/(c^3*(b + c*x)^2) - (2*b*(c*d - b*
e)^4*(3*c*d + 2*b*e))/(c^3*(b + c*x)) - 2*d^3*(6*c^2*d^2 - 15*b*c*d*e + 10*b^2*e^2)*Log[x] + (2*(c*d - b*e)^3*
(6*c^2*d^2 + 3*b*c*d*e + b^2*e^2)*Log[b + c*x])/c^3)/b^5

Maple [A] (verified)

Time = 2.01 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.53

method	result	size
norman	\(\frac {\frac {\left (2 b^{5} e^{5}-5 b^{4} c d \,e^{4}+10 b^{2} c^{3} d^{3} e^{2}-15 b \,c^{4} d^{4} e +6 c^{5} d^{5}\right ) x^{3}}{c^{2} b^{4}}-\frac {d^{5}}{2 b}-\frac {d^{4} \left (5 b e -2 c d \right ) x}{b^{2}}+\frac {\left (3 b^{5} e^{5}-5 b^{4} c d \,e^{4}-10 b^{3} c^{2} d^{2} e^{3}+30 b^{2} c^{3} d^{3} e^{2}-45 b \,c^{4} d^{4} e +18 c^{5} d^{5}\right ) x^{2}}{2 b^{3} c^{3}}}{x^{2} \left (c x +b \right )^{2}}+\frac {d^{3} \left (10 b^{2} e^{2}-15 b c d e +6 c^{2} d^{2}\right ) \ln \left (x \right )}{b^{5}}+\frac {\left (b^{5} e^{5}-10 b^{2} c^{3} d^{3} e^{2}+15 b \,c^{4} d^{4} e -6 c^{5} d^{5}\right ) \ln \left (c x +b \right )}{b^{5} c^{3}}\)	\(262\)
default	\(-\frac {d^{5}}{2 b^{3} x^{2}}+\frac {d^{3} \left (10 b^{2} e^{2}-15 b c d e +6 c^{2} d^{2}\right ) \ln \left (x \right )}{b^{5}}-\frac {d^{4} \left (5 b e -3 c d \right )}{b^{4} x}+\frac {\left (b^{5} e^{5}-10 b^{2} c^{3} d^{3} e^{2}+15 b \,c^{4} d^{4} e -6 c^{5} d^{5}\right ) \ln \left (c x +b \right )}{b^{5} c^{3}}-\frac {-2 b^{5} e^{5}+5 b^{4} c d \,e^{4}-10 b^{2} c^{3} d^{3} e^{2}+10 b \,c^{4} d^{4} e -3 c^{5} d^{5}}{c^{3} b^{4} \left (c x +b \right )}-\frac {b^{5} e^{5}-5 b^{4} c d \,e^{4}+10 b^{3} c^{2} d^{2} e^{3}-10 b^{2} c^{3} d^{3} e^{2}+5 b \,c^{4} d^{4} e -c^{5} d^{5}}{2 c^{3} b^{3} \left (c x +b \right )^{2}}\)	\(263\)
risch	\(\frac {\frac {\left (2 b^{5} e^{5}-5 b^{4} c d \,e^{4}+10 b^{2} c^{3} d^{3} e^{2}-15 b \,c^{4} d^{4} e +6 c^{5} d^{5}\right ) x^{3}}{c^{2} b^{4}}-\frac {d^{5}}{2 b}-\frac {d^{4} \left (5 b e -2 c d \right ) x}{b^{2}}+\frac {\left (3 b^{5} e^{5}-5 b^{4} c d \,e^{4}-10 b^{3} c^{2} d^{2} e^{3}+30 b^{2} c^{3} d^{3} e^{2}-45 b \,c^{4} d^{4} e +18 c^{5} d^{5}\right ) x^{2}}{2 b^{3} c^{3}}}{x^{2} \left (c x +b \right )^{2}}+\frac {10 d^{3} \ln \left (x \right ) e^{2}}{b^{3}}-\frac {15 d^{4} \ln \left (x \right ) c e}{b^{4}}+\frac {6 d^{5} \ln \left (x \right ) c^{2}}{b^{5}}+\frac {\ln \left (-c x -b \right ) e^{5}}{c^{3}}-\frac {10 \ln \left (-c x -b \right ) d^{3} e^{2}}{b^{3}}+\frac {15 c \ln \left (-c x -b \right ) d^{4} e}{b^{4}}-\frac {6 c^{2} \ln \left (-c x -b \right ) d^{5}}{b^{5}}\)	\(290\)
parallelrisch	\(\frac {20 \ln \left (x \right ) x^{4} b^{2} c^{5} d^{3} e^{2}-30 \ln \left (x \right ) x^{4} b \,c^{6} d^{4} e -20 \ln \left (c x +b \right ) x^{4} b^{2} c^{5} d^{3} e^{2}+30 \ln \left (c x +b \right ) x^{4} b \,c^{6} d^{4} e +40 \ln \left (x \right ) x^{3} b^{3} c^{4} d^{3} e^{2}-60 \ln \left (x \right ) x^{3} b^{2} c^{5} d^{4} e -40 \ln \left (c x +b \right ) x^{3} b^{3} c^{4} d^{3} e^{2}-b^{4} c^{3} d^{5}+60 \ln \left (c x +b \right ) x^{3} b^{2} c^{5} d^{4} e +20 \ln \left (x \right ) x^{2} b^{4} c^{3} d^{3} e^{2}-30 \ln \left (x \right ) x^{2} b^{3} c^{4} d^{4} e -20 \ln \left (c x +b \right ) x^{2} b^{4} c^{3} d^{3} e^{2}+30 \ln \left (c x +b \right ) x^{2} b^{3} c^{4} d^{4} e -10 x^{3} b^{5} c^{2} d \,e^{4}+20 x^{3} b^{3} c^{4} d^{3} e^{2}-30 x^{3} b^{2} c^{5} d^{4} e -5 x^{2} b^{6} c d \,e^{4}+12 \ln \left (x \right ) x^{4} c^{7} d^{5}-12 \ln \left (c x +b \right ) x^{4} c^{7} d^{5}+4 x^{3} b^{6} c \,e^{5}+12 x^{3} b \,c^{6} d^{5}+18 x^{2} b^{2} c^{5} d^{5}+4 x \,b^{3} c^{4} d^{5}+2 \ln \left (c x +b \right ) x^{2} b^{7} e^{5}+3 x^{2} b^{7} e^{5}-10 x^{2} b^{5} c^{2} d^{2} e^{3}+30 x^{2} b^{4} c^{3} d^{3} e^{2}-45 x^{2} b^{3} c^{4} d^{4} e -10 x \,b^{4} c^{3} d^{4} e +12 \ln \left (x \right ) x^{2} b^{2} c^{5} d^{5}-12 \ln \left (c x +b \right ) x^{2} b^{2} c^{5} d^{5}+2 \ln \left (c x +b \right ) x^{4} b^{5} c^{2} e^{5}+24 \ln \left (x \right ) x^{3} b \,c^{6} d^{5}+4 \ln \left (c x +b \right ) x^{3} b^{6} c \,e^{5}-24 \ln \left (c x +b \right ) x^{3} b \,c^{6} d^{5}}{2 b^{5} c^{3} x^{2} \left (c x +b \right )^{2}}\)	\(603\)

[In]

int((e*x+d)^5/(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)

[Out]

((2*b^5*e^5-5*b^4*c*d*e^4+10*b^2*c^3*d^3*e^2-15*b*c^4*d^4*e+6*c^5*d^5)/c^2/b^4*x^3-1/2*d^5/b-d^4*(5*b*e-2*c*d)
/b^2*x+1/2*(3*b^5*e^5-5*b^4*c*d*e^4-10*b^3*c^2*d^2*e^3+30*b^2*c^3*d^3*e^2-45*b*c^4*d^4*e+18*c^5*d^5)/b^3/c^3*x
^2)/x^2/(c*x+b)^2+d^3*(10*b^2*e^2-15*b*c*d*e+6*c^2*d^2)*ln(x)/b^5+(b^5*e^5-10*b^2*c^3*d^3*e^2+15*b*c^4*d^4*e-6
*c^5*d^5)/b^5/c^3*ln(c*x+b)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 493 vs. \(2 (167) = 334\).

Time = 0.27 (sec) , antiderivative size = 493, normalized size of antiderivative = 2.88 \[ \int \frac {(d+e x)^5}{\left (b x+c x^2\right )^3} \, dx=-\frac {b^{4} c^{3} d^{5} - 2 \, {\left (6 \, b c^{6} d^{5} - 15 \, b^{2} c^{5} d^{4} e + 10 \, b^{3} c^{4} d^{3} e^{2} - 5 \, b^{5} c^{2} d e^{4} + 2 \, b^{6} c e^{5}\right )} x^{3} - {\left (18 \, b^{2} c^{5} d^{5} - 45 \, b^{3} c^{4} d^{4} e + 30 \, b^{4} c^{3} d^{3} e^{2} - 10 \, b^{5} c^{2} d^{2} e^{3} - 5 \, b^{6} c d e^{4} + 3 \, b^{7} e^{5}\right )} x^{2} - 2 \, {\left (2 \, b^{3} c^{4} d^{5} - 5 \, b^{4} c^{3} d^{4} e\right )} x + 2 \, {\left ({\left (6 \, c^{7} d^{5} - 15 \, b c^{6} d^{4} e + 10 \, b^{2} c^{5} d^{3} e^{2} - b^{5} c^{2} e^{5}\right )} x^{4} + 2 \, {\left (6 \, b c^{6} d^{5} - 15 \, b^{2} c^{5} d^{4} e + 10 \, b^{3} c^{4} d^{3} e^{2} - b^{6} c e^{5}\right )} x^{3} + {\left (6 \, b^{2} c^{5} d^{5} - 15 \, b^{3} c^{4} d^{4} e + 10 \, b^{4} c^{3} d^{3} e^{2} - b^{7} e^{5}\right )} x^{2}\right )} \log \left (c x + b\right ) - 2 \, {\left ({\left (6 \, c^{7} d^{5} - 15 \, b c^{6} d^{4} e + 10 \, b^{2} c^{5} d^{3} e^{2}\right )} x^{4} + 2 \, {\left (6 \, b c^{6} d^{5} - 15 \, b^{2} c^{5} d^{4} e + 10 \, b^{3} c^{4} d^{3} e^{2}\right )} x^{3} + {\left (6 \, b^{2} c^{5} d^{5} - 15 \, b^{3} c^{4} d^{4} e + 10 \, b^{4} c^{3} d^{3} e^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (b^{5} c^{5} x^{4} + 2 \, b^{6} c^{4} x^{3} + b^{7} c^{3} x^{2}\right )}} \]

[In]

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

[Out]

-1/2*(b^4*c^3*d^5 - 2*(6*b*c^6*d^5 - 15*b^2*c^5*d^4*e + 10*b^3*c^4*d^3*e^2 - 5*b^5*c^2*d*e^4 + 2*b^6*c*e^5)*x^
3 - (18*b^2*c^5*d^5 - 45*b^3*c^4*d^4*e + 30*b^4*c^3*d^3*e^2 - 10*b^5*c^2*d^2*e^3 - 5*b^6*c*d*e^4 + 3*b^7*e^5)*
x^2 - 2*(2*b^3*c^4*d^5 - 5*b^4*c^3*d^4*e)*x + 2*((6*c^7*d^5 - 15*b*c^6*d^4*e + 10*b^2*c^5*d^3*e^2 - b^5*c^2*e^
5)*x^4 + 2*(6*b*c^6*d^5 - 15*b^2*c^5*d^4*e + 10*b^3*c^4*d^3*e^2 - b^6*c*e^5)*x^3 + (6*b^2*c^5*d^5 - 15*b^3*c^4
*d^4*e + 10*b^4*c^3*d^3*e^2 - b^7*e^5)*x^2)*log(c*x + b) - 2*((6*c^7*d^5 - 15*b*c^6*d^4*e + 10*b^2*c^5*d^3*e^2
)*x^4 + 2*(6*b*c^6*d^5 - 15*b^2*c^5*d^4*e + 10*b^3*c^4*d^3*e^2)*x^3 + (6*b^2*c^5*d^5 - 15*b^3*c^4*d^4*e + 10*b
^4*c^3*d^3*e^2)*x^2)*log(x))/(b^5*c^5*x^4 + 2*b^6*c^4*x^3 + b^7*c^3*x^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 524 vs. \(2 (165) = 330\).

Time = 12.13 (sec) , antiderivative size = 524, normalized size of antiderivative = 3.06 \[ \int \frac {(d+e x)^5}{\left (b x+c x^2\right )^3} \, dx=\frac {- b^{3} c^{3} d^{5} + x^{3} \cdot \left (4 b^{5} c e^{5} - 10 b^{4} c^{2} d e^{4} + 20 b^{2} c^{4} d^{3} e^{2} - 30 b c^{5} d^{4} e + 12 c^{6} d^{5}\right ) + x^{2} \cdot \left (3 b^{6} e^{5} - 5 b^{5} c d e^{4} - 10 b^{4} c^{2} d^{2} e^{3} + 30 b^{3} c^{3} d^{3} e^{2} - 45 b^{2} c^{4} d^{4} e + 18 b c^{5} d^{5}\right ) + x \left (- 10 b^{3} c^{3} d^{4} e + 4 b^{2} c^{4} d^{5}\right )}{2 b^{6} c^{3} x^{2} + 4 b^{5} c^{4} x^{3} + 2 b^{4} c^{5} x^{4}} + \frac {d^{3} \cdot \left (10 b^{2} e^{2} - 15 b c d e + 6 c^{2} d^{2}\right ) \log {\left (x + \frac {- 10 b^{3} c^{2} d^{3} e^{2} + 15 b^{2} c^{3} d^{4} e - 6 b c^{4} d^{5} + b c^{2} d^{3} \cdot \left (10 b^{2} e^{2} - 15 b c d e + 6 c^{2} d^{2}\right )}{b^{5} e^{5} - 20 b^{2} c^{3} d^{3} e^{2} + 30 b c^{4} d^{4} e - 12 c^{5} d^{5}} \right )}}{b^{5}} + \frac {\left (b e - c d\right )^{3} \left (b^{2} e^{2} + 3 b c d e + 6 c^{2} d^{2}\right ) \log {\left (x + \frac {- 10 b^{3} c^{2} d^{3} e^{2} + 15 b^{2} c^{3} d^{4} e - 6 b c^{4} d^{5} + \frac {b \left (b e - c d\right )^{3} \left (b^{2} e^{2} + 3 b c d e + 6 c^{2} d^{2}\right )}{c}}{b^{5} e^{5} - 20 b^{2} c^{3} d^{3} e^{2} + 30 b c^{4} d^{4} e - 12 c^{5} d^{5}} \right )}}{b^{5} c^{3}} \]

[In]

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

[Out]

(-b**3*c**3*d**5 + x**3*(4*b**5*c*e**5 - 10*b**4*c**2*d*e**4 + 20*b**2*c**4*d**3*e**2 - 30*b*c**5*d**4*e + 12*
c**6*d**5) + x**2*(3*b**6*e**5 - 5*b**5*c*d*e**4 - 10*b**4*c**2*d**2*e**3 + 30*b**3*c**3*d**3*e**2 - 45*b**2*c
**4*d**4*e + 18*b*c**5*d**5) + x*(-10*b**3*c**3*d**4*e + 4*b**2*c**4*d**5))/(2*b**6*c**3*x**2 + 4*b**5*c**4*x*
*3 + 2*b**4*c**5*x**4) + d**3*(10*b**2*e**2 - 15*b*c*d*e + 6*c**2*d**2)*log(x + (-10*b**3*c**2*d**3*e**2 + 15*
b**2*c**3*d**4*e - 6*b*c**4*d**5 + b*c**2*d**3*(10*b**2*e**2 - 15*b*c*d*e + 6*c**2*d**2))/(b**5*e**5 - 20*b**2
*c**3*d**3*e**2 + 30*b*c**4*d**4*e - 12*c**5*d**5))/b**5 + (b*e - c*d)**3*(b**2*e**2 + 3*b*c*d*e + 6*c**2*d**2
)*log(x + (-10*b**3*c**2*d**3*e**2 + 15*b**2*c**3*d**4*e - 6*b*c**4*d**5 + b*(b*e - c*d)**3*(b**2*e**2 + 3*b*c
*d*e + 6*c**2*d**2)/c)/(b**5*e**5 - 20*b**2*c**3*d**3*e**2 + 30*b*c**4*d**4*e - 12*c**5*d**5))/(b**5*c**3)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.73 \[ \int \frac {(d+e x)^5}{\left (b x+c x^2\right )^3} \, dx=-\frac {b^{3} c^{3} d^{5} - 2 \, {\left (6 \, c^{6} d^{5} - 15 \, b c^{5} d^{4} e + 10 \, b^{2} c^{4} d^{3} e^{2} - 5 \, b^{4} c^{2} d e^{4} + 2 \, b^{5} c e^{5}\right )} x^{3} - {\left (18 \, b c^{5} d^{5} - 45 \, b^{2} c^{4} d^{4} e + 30 \, b^{3} c^{3} d^{3} e^{2} - 10 \, b^{4} c^{2} d^{2} e^{3} - 5 \, b^{5} c d e^{4} + 3 \, b^{6} e^{5}\right )} x^{2} - 2 \, {\left (2 \, b^{2} c^{4} d^{5} - 5 \, b^{3} c^{3} d^{4} e\right )} x}{2 \, {\left (b^{4} c^{5} x^{4} + 2 \, b^{5} c^{4} x^{3} + b^{6} c^{3} x^{2}\right )}} + \frac {{\left (6 \, c^{2} d^{5} - 15 \, b c d^{4} e + 10 \, b^{2} d^{3} e^{2}\right )} \log \left (x\right )}{b^{5}} - \frac {{\left (6 \, c^{5} d^{5} - 15 \, b c^{4} d^{4} e + 10 \, b^{2} c^{3} d^{3} e^{2} - b^{5} e^{5}\right )} \log \left (c x + b\right )}{b^{5} c^{3}} \]

[In]

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

[Out]

-1/2*(b^3*c^3*d^5 - 2*(6*c^6*d^5 - 15*b*c^5*d^4*e + 10*b^2*c^4*d^3*e^2 - 5*b^4*c^2*d*e^4 + 2*b^5*c*e^5)*x^3 -
(18*b*c^5*d^5 - 45*b^2*c^4*d^4*e + 30*b^3*c^3*d^3*e^2 - 10*b^4*c^2*d^2*e^3 - 5*b^5*c*d*e^4 + 3*b^6*e^5)*x^2 -
2*(2*b^2*c^4*d^5 - 5*b^3*c^3*d^4*e)*x)/(b^4*c^5*x^4 + 2*b^5*c^4*x^3 + b^6*c^3*x^2) + (6*c^2*d^5 - 15*b*c*d^4*e
+ 10*b^2*d^3*e^2)*log(x)/b^5 - (6*c^5*d^5 - 15*b*c^4*d^4*e + 10*b^2*c^3*d^3*e^2 - b^5*e^5)*log(c*x + b)/(b^5*
c^3)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.64 \[ \int \frac {(d+e x)^5}{\left (b x+c x^2\right )^3} \, dx=\frac {{\left (6 \, c^{2} d^{5} - 15 \, b c d^{4} e + 10 \, b^{2} d^{3} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} - \frac {{\left (6 \, c^{5} d^{5} - 15 \, b c^{4} d^{4} e + 10 \, b^{2} c^{3} d^{3} e^{2} - b^{5} e^{5}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c^{3}} - \frac {b^{3} c^{3} d^{5} - 2 \, {\left (6 \, c^{6} d^{5} - 15 \, b c^{5} d^{4} e + 10 \, b^{2} c^{4} d^{3} e^{2} - 5 \, b^{4} c^{2} d e^{4} + 2 \, b^{5} c e^{5}\right )} x^{3} - {\left (18 \, b c^{5} d^{5} - 45 \, b^{2} c^{4} d^{4} e + 30 \, b^{3} c^{3} d^{3} e^{2} - 10 \, b^{4} c^{2} d^{2} e^{3} - 5 \, b^{5} c d e^{4} + 3 \, b^{6} e^{5}\right )} x^{2} - 2 \, {\left (2 \, b^{2} c^{4} d^{5} - 5 \, b^{3} c^{3} d^{4} e\right )} x}{2 \, {\left (c x + b\right )}^{2} b^{4} c^{3} x^{2}} \]

[In]

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

[Out]

(6*c^2*d^5 - 15*b*c*d^4*e + 10*b^2*d^3*e^2)*log(abs(x))/b^5 - (6*c^5*d^5 - 15*b*c^4*d^4*e + 10*b^2*c^3*d^3*e^2
- b^5*e^5)*log(abs(c*x + b))/(b^5*c^3) - 1/2*(b^3*c^3*d^5 - 2*(6*c^6*d^5 - 15*b*c^5*d^4*e + 10*b^2*c^4*d^3*e^
2 - 5*b^4*c^2*d*e^4 + 2*b^5*c*e^5)*x^3 - (18*b*c^5*d^5 - 45*b^2*c^4*d^4*e + 30*b^3*c^3*d^3*e^2 - 10*b^4*c^2*d^
2*e^3 - 5*b^5*c*d*e^4 + 3*b^6*e^5)*x^2 - 2*(2*b^2*c^4*d^5 - 5*b^3*c^3*d^4*e)*x)/((c*x + b)^2*b^4*c^3*x^2)

Mupad [B] (verification not implemented)

Time = 9.79 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.57 \[ \int \frac {(d+e x)^5}{\left (b x+c x^2\right )^3} \, dx=\frac {d^3\,\ln \left (x\right )\,\left (10\,b^2\,e^2-15\,b\,c\,d\,e+6\,c^2\,d^2\right )}{b^5}-\frac {\frac {d^5}{2\,b}+\frac {d^4\,x\,\left (5\,b\,e-2\,c\,d\right )}{b^2}-\frac {x^2\,\left (3\,b^5\,e^5-5\,b^4\,c\,d\,e^4-10\,b^3\,c^2\,d^2\,e^3+30\,b^2\,c^3\,d^3\,e^2-45\,b\,c^4\,d^4\,e+18\,c^5\,d^5\right )}{2\,b^3\,c^3}-\frac {x^3\,\left (2\,b^5\,e^5-5\,b^4\,c\,d\,e^4+10\,b^2\,c^3\,d^3\,e^2-15\,b\,c^4\,d^4\,e+6\,c^5\,d^5\right )}{b^4\,c^2}}{b^2\,x^2+2\,b\,c\,x^3+c^2\,x^4}+\frac {\ln \left (b+c\,x\right )\,{\left (b\,e-c\,d\right )}^3\,\left (b^2\,e^2+3\,b\,c\,d\,e+6\,c^2\,d^2\right )}{b^5\,c^3} \]

[In]

int((d + e*x)^5/(b*x + c*x^2)^3,x)

[Out]

(d^3*log(x)*(10*b^2*e^2 + 6*c^2*d^2 - 15*b*c*d*e))/b^5 - (d^5/(2*b) + (d^4*x*(5*b*e - 2*c*d))/b^2 - (x^2*(3*b^
5*e^5 + 18*c^5*d^5 + 30*b^2*c^3*d^3*e^2 - 10*b^3*c^2*d^2*e^3 - 45*b*c^4*d^4*e - 5*b^4*c*d*e^4))/(2*b^3*c^3) -
(x^3*(2*b^5*e^5 + 6*c^5*d^5 + 10*b^2*c^3*d^3*e^2 - 15*b*c^4*d^4*e - 5*b^4*c*d*e^4))/(b^4*c^2))/(b^2*x^2 + c^2*
x^4 + 2*b*c*x^3) + (log(b + c*x)*(b*e - c*d)^3*(b^2*e^2 + 6*c^2*d^2 + 3*b*c*d*e))/(b^5*c^3)