\(\int \frac {(d+e x)^6}{(b x+c x^2)^3} \, dx\) [276]

Optimal result

Integrand size = 19, antiderivative size = 179 \[ \int \frac {(d+e x)^6}{\left (b x+c x^2\right )^3} \, dx=-\frac {d^6}{2 b^3 x^2}+\frac {3 d^5 (c d-2 b e)}{b^4 x}+\frac {e^6 x}{c^3}+\frac {(c d-b e)^6}{2 b^3 c^4 (b+c x)^2}+\frac {3 (c d-b e)^5 (c d+b e)}{b^4 c^4 (b+c x)}+\frac {3 d^4 \left (2 c^2 d^2-6 b c d e+5 b^2 e^2\right ) \log (x)}{b^5}-\frac {3 (c d-b e)^4 \left (2 c^2 d^2+2 b c d e+b^2 e^2\right ) \log (b+c x)}{b^5 c^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 179, 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} \[ \int \frac {(d+e x)^6}{\left (b x+c x^2\right )^3} \, dx=\frac {3 (c d-b e)^5 (b e+c d)}{b^4 c^4 (b+c x)}+\frac {3 d^5 (c d-2 b e)}{b^4 x}+\frac {(c d-b e)^6}{2 b^3 c^4 (b+c x)^2}-\frac {d^6}{2 b^3 x^2}+\frac {3 d^4 \log (x) \left (5 b^2 e^2-6 b c d e+2 c^2 d^2\right )}{b^5}-\frac {3 (c d-b e)^4 \left (b^2 e^2+2 b c d e+2 c^2 d^2\right ) \log (b+c x)}{b^5 c^4}+\frac {e^6 x}{c^3} \]

[In]

Int[(d + e*x)^6/(b*x + c*x^2)^3,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^6}{\left (b x+c x^2\right )^3} \, dx=-\frac {d^6}{2 b^3 x^2}+\frac {3 d^5 (c d-2 b e)}{b^4 x}+\frac {e^6 x}{c^3}+\frac {(c d-b e)^6}{2 b^3 c^4 (b+c x)^2}+\frac {3 (c d-b e)^5 (c d+b e)}{b^4 c^4 (b+c x)}+\frac {3 d^4 \left (2 c^2 d^2-6 b c d e+5 b^2 e^2\right ) \log (x)}{b^5}-\frac {3 (c d-b e)^4 \left (2 c^2 d^2+2 b c d e+b^2 e^2\right ) \log (b+c x)}{b^5 c^4} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.93 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.74

method	result	size
default	\(\frac {e^{6} x}{c^{3}}-\frac {d^{6}}{2 b^{3} x^{2}}+\frac {3 d^{4} \left (5 b^{2} e^{2}-6 b c d e +2 c^{2} d^{2}\right ) \ln \left (x \right )}{b^{5}}-\frac {3 d^{5} \left (2 b e -c d \right )}{b^{4} x}+\frac {\left (-3 b^{6} e^{6}+6 b^{5} d \,e^{5} c -15 b^{2} c^{4} d^{4} e^{2}+18 b \,c^{5} d^{5} e -6 c^{6} d^{6}\right ) \ln \left (c x +b \right )}{b^{5} c^{4}}-\frac {3 b^{6} e^{6}-12 b^{5} d \,e^{5} c +15 b^{4} d^{2} e^{4} c^{2}-15 b^{2} c^{4} d^{4} e^{2}+12 b \,c^{5} d^{5} e -3 c^{6} d^{6}}{b^{4} c^{4} \left (c x +b \right )}-\frac {-b^{6} e^{6}+6 b^{5} d \,e^{5} c -15 b^{4} d^{2} e^{4} c^{2}+20 b^{3} d^{3} e^{3} c^{3}-15 b^{2} c^{4} d^{4} e^{2}+6 b \,c^{5} d^{5} e -c^{6} d^{6}}{2 b^{3} c^{4} \left (c x +b \right )^{2}}\)	\(312\)
norman	\(\frac {\frac {e^{6} x^{5}}{c}-\frac {d^{6}}{2 b}-\frac {2 d^{5} \left (3 b e -c d \right ) x}{b^{2}}-\frac {\left (6 b^{6} e^{6}-12 b^{5} d \,e^{5} c +15 b^{4} d^{2} e^{4} c^{2}-15 b^{2} c^{4} d^{4} e^{2}+18 b \,c^{5} d^{5} e -6 c^{6} d^{6}\right ) x^{3}}{b^{4} c^{3}}-\frac {\left (9 b^{6} e^{6}-18 b^{5} d \,e^{5} c +15 b^{4} d^{2} e^{4} c^{2}+20 b^{3} d^{3} e^{3} c^{3}-45 b^{2} c^{4} d^{4} e^{2}+54 b \,c^{5} d^{5} e -18 c^{6} d^{6}\right ) x^{2}}{2 c^{4} b^{3}}}{x^{2} \left (c x +b \right )^{2}}+\frac {3 d^{4} \left (5 b^{2} e^{2}-6 b c d e +2 c^{2} d^{2}\right ) \ln \left (x \right )}{b^{5}}-\frac {3 \left (b^{6} e^{6}-2 b^{5} d \,e^{5} c +5 b^{2} c^{4} d^{4} e^{2}-6 b \,c^{5} d^{5} e +2 c^{6} d^{6}\right ) \ln \left (c x +b \right )}{b^{5} c^{4}}\)	\(313\)
risch	\(\frac {e^{6} x}{c^{3}}+\frac {-\frac {3 \left (b^{6} e^{6}-4 b^{5} d \,e^{5} c +5 b^{4} d^{2} e^{4} c^{2}-5 b^{2} c^{4} d^{4} e^{2}+6 b \,c^{5} d^{5} e -2 c^{6} d^{6}\right ) x^{3}}{b^{4}}-\frac {\left (5 b^{6} e^{6}-18 b^{5} d \,e^{5} c +15 b^{4} d^{2} e^{4} c^{2}+20 b^{3} d^{3} e^{3} c^{3}-45 b^{2} c^{4} d^{4} e^{2}+54 b \,c^{5} d^{5} e -18 c^{6} d^{6}\right ) x^{2}}{2 b^{3} c}-\frac {2 c^{3} d^{5} \left (3 b e -c d \right ) x}{b^{2}}-\frac {c^{3} d^{6}}{2 b}}{c^{3} x^{2} \left (c x +b \right )^{2}}+\frac {15 d^{4} \ln \left (-x \right ) e^{2}}{b^{3}}-\frac {18 d^{5} \ln \left (-x \right ) c e}{b^{4}}+\frac {6 d^{6} \ln \left (-x \right ) c^{2}}{b^{5}}-\frac {3 b \ln \left (c x +b \right ) e^{6}}{c^{4}}+\frac {6 \ln \left (c x +b \right ) d \,e^{5}}{c^{3}}-\frac {15 \ln \left (c x +b \right ) d^{4} e^{2}}{b^{3}}+\frac {18 c \ln \left (c x +b \right ) d^{5} e}{b^{4}}-\frac {6 c^{2} \ln \left (c x +b \right ) d^{6}}{b^{5}}\)	\(343\)
parallelrisch	\(\frac {12 x^{3} b \,c^{7} d^{6}+18 x^{2} b^{2} c^{6} d^{6}+4 x \,b^{3} c^{5} d^{6}+2 x^{5} b^{5} c^{3} e^{6}+12 \ln \left (x \right ) x^{4} c^{8} d^{6}+30 \ln \left (x \right ) x^{4} b^{2} c^{6} d^{4} e^{2}-36 \ln \left (x \right ) x^{4} b \,c^{7} d^{5} e +12 \ln \left (c x +b \right ) x^{4} b^{5} c^{3} d \,e^{5}-30 \ln \left (c x +b \right ) x^{4} b^{2} c^{6} d^{4} e^{2}+36 \ln \left (c x +b \right ) x^{4} b \,c^{7} d^{5} e +60 \ln \left (x \right ) x^{3} b^{3} c^{5} d^{4} e^{2}-72 \ln \left (x \right ) x^{3} b^{2} c^{6} d^{5} e +24 \ln \left (c x +b \right ) x^{3} b^{6} c^{2} d \,e^{5}-60 \ln \left (c x +b \right ) x^{3} b^{3} c^{5} d^{4} e^{2}+72 \ln \left (c x +b \right ) x^{3} b^{2} c^{6} d^{5} e +24 x^{3} b^{6} c^{2} d \,e^{5}-30 x^{3} b^{5} c^{3} d^{2} e^{4}+30 x^{3} b^{3} c^{5} d^{4} e^{2}-36 x^{3} b^{2} c^{6} d^{5} e +18 x^{2} b^{7} c d \,e^{5}-15 x^{2} b^{6} c^{2} d^{2} e^{4}-20 x^{2} b^{5} c^{3} d^{3} e^{3}+45 x^{2} b^{4} c^{4} d^{4} e^{2}-54 x^{2} b^{3} c^{5} d^{5} e -12 x \,b^{4} c^{4} d^{5} e -6 \ln \left (c x +b \right ) x^{4} b^{6} c^{2} e^{6}+24 \ln \left (x \right ) x^{3} b \,c^{7} d^{6}-12 \ln \left (c x +b \right ) x^{3} b^{7} c \,e^{6}-24 \ln \left (c x +b \right ) x^{3} b \,c^{7} d^{6}+12 \ln \left (x \right ) x^{2} b^{2} c^{6} d^{6}+30 \ln \left (x \right ) x^{2} b^{4} c^{4} d^{4} e^{2}-36 \ln \left (x \right ) x^{2} b^{3} c^{5} d^{5} e +12 \ln \left (c x +b \right ) x^{2} b^{7} c d \,e^{5}-30 \ln \left (c x +b \right ) x^{2} b^{4} c^{4} d^{4} e^{2}+36 \ln \left (c x +b \right ) x^{2} b^{3} c^{5} d^{5} e -12 \ln \left (c x +b \right ) x^{4} c^{8} d^{6}-6 \ln \left (c x +b \right ) x^{2} b^{8} e^{6}-12 x^{3} b^{7} c \,e^{6}-b^{4} c^{4} d^{6}-9 x^{2} b^{8} e^{6}-12 \ln \left (c x +b \right ) x^{2} b^{2} c^{6} d^{6}}{2 b^{5} c^{4} x^{2} \left (c x +b \right )^{2}}\)	\(712\)

[In]

int((e*x+d)^6/(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)

[Out]

e^6*x/c^3-1/2*d^6/b^3/x^2+3*d^4*(5*b^2*e^2-6*b*c*d*e+2*c^2*d^2)*ln(x)/b^5-3*d^5*(2*b*e-c*d)/b^4/x+(-3*b^6*e^6+
6*b^5*c*d*e^5-15*b^2*c^4*d^4*e^2+18*b*c^5*d^5*e-6*c^6*d^6)/b^5/c^4*ln(c*x+b)-(3*b^6*e^6-12*b^5*c*d*e^5+15*b^4*
c^2*d^2*e^4-15*b^2*c^4*d^4*e^2+12*b*c^5*d^5*e-3*c^6*d^6)/b^4/c^4/(c*x+b)-1/2*(-b^6*e^6+6*b^5*c*d*e^5-15*b^4*c^
2*d^2*e^4+20*b^3*c^3*d^3*e^3-15*b^2*c^4*d^4*e^2+6*b*c^5*d^5*e-c^6*d^6)/b^3/c^4/(c*x+b)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 579 vs. \(2 (175) = 350\).

Time = 0.28 (sec) , antiderivative size = 579, normalized size of antiderivative = 3.23 \[ \int \frac {(d+e x)^6}{\left (b x+c x^2\right )^3} \, dx=\frac {2 \, b^{5} c^{3} e^{6} x^{5} + 4 \, b^{6} c^{2} e^{6} x^{4} - b^{4} c^{4} d^{6} + 2 \, {\left (6 \, b c^{7} d^{6} - 18 \, b^{2} c^{6} d^{5} e + 15 \, b^{3} c^{5} d^{4} e^{2} - 15 \, b^{5} c^{3} d^{2} e^{4} + 12 \, b^{6} c^{2} d e^{5} - 2 \, b^{7} c e^{6}\right )} x^{3} + {\left (18 \, b^{2} c^{6} d^{6} - 54 \, b^{3} c^{5} d^{5} e + 45 \, b^{4} c^{4} d^{4} e^{2} - 20 \, b^{5} c^{3} d^{3} e^{3} - 15 \, b^{6} c^{2} d^{2} e^{4} + 18 \, b^{7} c d e^{5} - 5 \, b^{8} e^{6}\right )} x^{2} + 4 \, {\left (b^{3} c^{5} d^{6} - 3 \, b^{4} c^{4} d^{5} e\right )} x - 6 \, {\left ({\left (2 \, c^{8} d^{6} - 6 \, b c^{7} d^{5} e + 5 \, b^{2} c^{6} d^{4} e^{2} - 2 \, b^{5} c^{3} d e^{5} + b^{6} c^{2} e^{6}\right )} x^{4} + 2 \, {\left (2 \, b c^{7} d^{6} - 6 \, b^{2} c^{6} d^{5} e + 5 \, b^{3} c^{5} d^{4} e^{2} - 2 \, b^{6} c^{2} d e^{5} + b^{7} c e^{6}\right )} x^{3} + {\left (2 \, b^{2} c^{6} d^{6} - 6 \, b^{3} c^{5} d^{5} e + 5 \, b^{4} c^{4} d^{4} e^{2} - 2 \, b^{7} c d e^{5} + b^{8} e^{6}\right )} x^{2}\right )} \log \left (c x + b\right ) + 6 \, {\left ({\left (2 \, c^{8} d^{6} - 6 \, b c^{7} d^{5} e + 5 \, b^{2} c^{6} d^{4} e^{2}\right )} x^{4} + 2 \, {\left (2 \, b c^{7} d^{6} - 6 \, b^{2} c^{6} d^{5} e + 5 \, b^{3} c^{5} d^{4} e^{2}\right )} x^{3} + {\left (2 \, b^{2} c^{6} d^{6} - 6 \, b^{3} c^{5} d^{5} e + 5 \, b^{4} c^{4} d^{4} e^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (b^{5} c^{6} x^{4} + 2 \, b^{6} c^{5} x^{3} + b^{7} c^{4} x^{2}\right )}} \]

[In]

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

[Out]

1/2*(2*b^5*c^3*e^6*x^5 + 4*b^6*c^2*e^6*x^4 - b^4*c^4*d^6 + 2*(6*b*c^7*d^6 - 18*b^2*c^6*d^5*e + 15*b^3*c^5*d^4*
e^2 - 15*b^5*c^3*d^2*e^4 + 12*b^6*c^2*d*e^5 - 2*b^7*c*e^6)*x^3 + (18*b^2*c^6*d^6 - 54*b^3*c^5*d^5*e + 45*b^4*c
^4*d^4*e^2 - 20*b^5*c^3*d^3*e^3 - 15*b^6*c^2*d^2*e^4 + 18*b^7*c*d*e^5 - 5*b^8*e^6)*x^2 + 4*(b^3*c^5*d^6 - 3*b^
4*c^4*d^5*e)*x - 6*((2*c^8*d^6 - 6*b*c^7*d^5*e + 5*b^2*c^6*d^4*e^2 - 2*b^5*c^3*d*e^5 + b^6*c^2*e^6)*x^4 + 2*(2
*b*c^7*d^6 - 6*b^2*c^6*d^5*e + 5*b^3*c^5*d^4*e^2 - 2*b^6*c^2*d*e^5 + b^7*c*e^6)*x^3 + (2*b^2*c^6*d^6 - 6*b^3*c
^5*d^5*e + 5*b^4*c^4*d^4*e^2 - 2*b^7*c*d*e^5 + b^8*e^6)*x^2)*log(c*x + b) + 6*((2*c^8*d^6 - 6*b*c^7*d^5*e + 5*
b^2*c^6*d^4*e^2)*x^4 + 2*(2*b*c^7*d^6 - 6*b^2*c^6*d^5*e + 5*b^3*c^5*d^4*e^2)*x^3 + (2*b^2*c^6*d^6 - 6*b^3*c^5*
d^5*e + 5*b^4*c^4*d^4*e^2)*x^2)*log(x))/(b^5*c^6*x^4 + 2*b^6*c^5*x^3 + b^7*c^4*x^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 597 vs. \(2 (175) = 350\).

Time = 147.21 (sec) , antiderivative size = 597, normalized size of antiderivative = 3.34 \[ \int \frac {(d+e x)^6}{\left (b x+c x^2\right )^3} \, dx=\frac {- b^{3} c^{4} d^{6} + x^{3} \left (- 6 b^{6} c e^{6} + 24 b^{5} c^{2} d e^{5} - 30 b^{4} c^{3} d^{2} e^{4} + 30 b^{2} c^{5} d^{4} e^{2} - 36 b c^{6} d^{5} e + 12 c^{7} d^{6}\right ) + x^{2} \left (- 5 b^{7} e^{6} + 18 b^{6} c d e^{5} - 15 b^{5} c^{2} d^{2} e^{4} - 20 b^{4} c^{3} d^{3} e^{3} + 45 b^{3} c^{4} d^{4} e^{2} - 54 b^{2} c^{5} d^{5} e + 18 b c^{6} d^{6}\right ) + x \left (- 12 b^{3} c^{4} d^{5} e + 4 b^{2} c^{5} d^{6}\right )}{2 b^{6} c^{4} x^{2} + 4 b^{5} c^{5} x^{3} + 2 b^{4} c^{6} x^{4}} + \frac {e^{6} x}{c^{3}} + \frac {3 d^{4} \cdot \left (5 b^{2} e^{2} - 6 b c d e + 2 c^{2} d^{2}\right ) \log {\left (x + \frac {15 b^{3} c^{3} d^{4} e^{2} - 18 b^{2} c^{4} d^{5} e + 6 b c^{5} d^{6} - 3 b c^{3} d^{4} \cdot \left (5 b^{2} e^{2} - 6 b c d e + 2 c^{2} d^{2}\right )}{3 b^{6} e^{6} - 6 b^{5} c d e^{5} + 30 b^{2} c^{4} d^{4} e^{2} - 36 b c^{5} d^{5} e + 12 c^{6} d^{6}} \right )}}{b^{5}} - \frac {3 \left (b e - c d\right )^{4} \left (b^{2} e^{2} + 2 b c d e + 2 c^{2} d^{2}\right ) \log {\left (x + \frac {15 b^{3} c^{3} d^{4} e^{2} - 18 b^{2} c^{4} d^{5} e + 6 b c^{5} d^{6} + \frac {3 b \left (b e - c d\right )^{4} \left (b^{2} e^{2} + 2 b c d e + 2 c^{2} d^{2}\right )}{c}}{3 b^{6} e^{6} - 6 b^{5} c d e^{5} + 30 b^{2} c^{4} d^{4} e^{2} - 36 b c^{5} d^{5} e + 12 c^{6} d^{6}} \right )}}{b^{5} c^{4}} \]

[In]

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

[Out]

(-b**3*c**4*d**6 + x**3*(-6*b**6*c*e**6 + 24*b**5*c**2*d*e**5 - 30*b**4*c**3*d**2*e**4 + 30*b**2*c**5*d**4*e**
2 - 36*b*c**6*d**5*e + 12*c**7*d**6) + x**2*(-5*b**7*e**6 + 18*b**6*c*d*e**5 - 15*b**5*c**2*d**2*e**4 - 20*b**
4*c**3*d**3*e**3 + 45*b**3*c**4*d**4*e**2 - 54*b**2*c**5*d**5*e + 18*b*c**6*d**6) + x*(-12*b**3*c**4*d**5*e +
4*b**2*c**5*d**6))/(2*b**6*c**4*x**2 + 4*b**5*c**5*x**3 + 2*b**4*c**6*x**4) + e**6*x/c**3 + 3*d**4*(5*b**2*e**
2 - 6*b*c*d*e + 2*c**2*d**2)*log(x + (15*b**3*c**3*d**4*e**2 - 18*b**2*c**4*d**5*e + 6*b*c**5*d**6 - 3*b*c**3*
d**4*(5*b**2*e**2 - 6*b*c*d*e + 2*c**2*d**2))/(3*b**6*e**6 - 6*b**5*c*d*e**5 + 30*b**2*c**4*d**4*e**2 - 36*b*c
**5*d**5*e + 12*c**6*d**6))/b**5 - 3*(b*e - c*d)**4*(b**2*e**2 + 2*b*c*d*e + 2*c**2*d**2)*log(x + (15*b**3*c**
3*d**4*e**2 - 18*b**2*c**4*d**5*e + 6*b*c**5*d**6 + 3*b*(b*e - c*d)**4*(b**2*e**2 + 2*b*c*d*e + 2*c**2*d**2)/c
)/(3*b**6*e**6 - 6*b**5*c*d*e**5 + 30*b**2*c**4*d**4*e**2 - 36*b*c**5*d**5*e + 12*c**6*d**6))/(b**5*c**4)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.91 \[ \int \frac {(d+e x)^6}{\left (b x+c x^2\right )^3} \, dx=\frac {e^{6} x}{c^{3}} - \frac {b^{3} c^{4} d^{6} - 6 \, {\left (2 \, c^{7} d^{6} - 6 \, b c^{6} d^{5} e + 5 \, b^{2} c^{5} d^{4} e^{2} - 5 \, b^{4} c^{3} d^{2} e^{4} + 4 \, b^{5} c^{2} d e^{5} - b^{6} c e^{6}\right )} x^{3} - {\left (18 \, b c^{6} d^{6} - 54 \, b^{2} c^{5} d^{5} e + 45 \, b^{3} c^{4} d^{4} e^{2} - 20 \, b^{4} c^{3} d^{3} e^{3} - 15 \, b^{5} c^{2} d^{2} e^{4} + 18 \, b^{6} c d e^{5} - 5 \, b^{7} e^{6}\right )} x^{2} - 4 \, {\left (b^{2} c^{5} d^{6} - 3 \, b^{3} c^{4} d^{5} e\right )} x}{2 \, {\left (b^{4} c^{6} x^{4} + 2 \, b^{5} c^{5} x^{3} + b^{6} c^{4} x^{2}\right )}} + \frac {3 \, {\left (2 \, c^{2} d^{6} - 6 \, b c d^{5} e + 5 \, b^{2} d^{4} e^{2}\right )} \log \left (x\right )}{b^{5}} - \frac {3 \, {\left (2 \, c^{6} d^{6} - 6 \, b c^{5} d^{5} e + 5 \, b^{2} c^{4} d^{4} e^{2} - 2 \, b^{5} c d e^{5} + b^{6} e^{6}\right )} \log \left (c x + b\right )}{b^{5} c^{4}} \]

[In]

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

[Out]

e^6*x/c^3 - 1/2*(b^3*c^4*d^6 - 6*(2*c^7*d^6 - 6*b*c^6*d^5*e + 5*b^2*c^5*d^4*e^2 - 5*b^4*c^3*d^2*e^4 + 4*b^5*c^
2*d*e^5 - b^6*c*e^6)*x^3 - (18*b*c^6*d^6 - 54*b^2*c^5*d^5*e + 45*b^3*c^4*d^4*e^2 - 20*b^4*c^3*d^3*e^3 - 15*b^5
*c^2*d^2*e^4 + 18*b^6*c*d*e^5 - 5*b^7*e^6)*x^2 - 4*(b^2*c^5*d^6 - 3*b^3*c^4*d^5*e)*x)/(b^4*c^6*x^4 + 2*b^5*c^5
*x^3 + b^6*c^4*x^2) + 3*(2*c^2*d^6 - 6*b*c*d^5*e + 5*b^2*d^4*e^2)*log(x)/b^5 - 3*(2*c^6*d^6 - 6*b*c^5*d^5*e +
5*b^2*c^4*d^4*e^2 - 2*b^5*c*d*e^5 + b^6*e^6)*log(c*x + b)/(b^5*c^4)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.82 \[ \int \frac {(d+e x)^6}{\left (b x+c x^2\right )^3} \, dx=\frac {e^{6} x}{c^{3}} + \frac {3 \, {\left (2 \, c^{2} d^{6} - 6 \, b c d^{5} e + 5 \, b^{2} d^{4} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} - \frac {3 \, {\left (2 \, c^{6} d^{6} - 6 \, b c^{5} d^{5} e + 5 \, b^{2} c^{4} d^{4} e^{2} - 2 \, b^{5} c d e^{5} + b^{6} e^{6}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c^{4}} - \frac {b^{3} c^{4} d^{6} - 6 \, {\left (2 \, c^{7} d^{6} - 6 \, b c^{6} d^{5} e + 5 \, b^{2} c^{5} d^{4} e^{2} - 5 \, b^{4} c^{3} d^{2} e^{4} + 4 \, b^{5} c^{2} d e^{5} - b^{6} c e^{6}\right )} x^{3} - {\left (18 \, b c^{6} d^{6} - 54 \, b^{2} c^{5} d^{5} e + 45 \, b^{3} c^{4} d^{4} e^{2} - 20 \, b^{4} c^{3} d^{3} e^{3} - 15 \, b^{5} c^{2} d^{2} e^{4} + 18 \, b^{6} c d e^{5} - 5 \, b^{7} e^{6}\right )} x^{2} - 4 \, {\left (b^{2} c^{5} d^{6} - 3 \, b^{3} c^{4} d^{5} e\right )} x}{2 \, {\left (c x + b\right )}^{2} b^{4} c^{4} x^{2}} \]

[In]

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

[Out]

e^6*x/c^3 + 3*(2*c^2*d^6 - 6*b*c*d^5*e + 5*b^2*d^4*e^2)*log(abs(x))/b^5 - 3*(2*c^6*d^6 - 6*b*c^5*d^5*e + 5*b^2
*c^4*d^4*e^2 - 2*b^5*c*d*e^5 + b^6*e^6)*log(abs(c*x + b))/(b^5*c^4) - 1/2*(b^3*c^4*d^6 - 6*(2*c^7*d^6 - 6*b*c^
6*d^5*e + 5*b^2*c^5*d^4*e^2 - 5*b^4*c^3*d^2*e^4 + 4*b^5*c^2*d*e^5 - b^6*c*e^6)*x^3 - (18*b*c^6*d^6 - 54*b^2*c^
5*d^5*e + 45*b^3*c^4*d^4*e^2 - 20*b^4*c^3*d^3*e^3 - 15*b^5*c^2*d^2*e^4 + 18*b^6*c*d*e^5 - 5*b^7*e^6)*x^2 - 4*(
b^2*c^5*d^6 - 3*b^3*c^4*d^5*e)*x)/((c*x + b)^2*b^4*c^4*x^2)

Mupad [B] (verification not implemented)

Time = 9.72 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.86 \[ \int \frac {(d+e x)^6}{\left (b x+c x^2\right )^3} \, dx=\frac {e^6\,x}{c^3}-\frac {\frac {3\,x^3\,\left (b^6\,e^6-4\,b^5\,c\,d\,e^5+5\,b^4\,c^2\,d^2\,e^4-5\,b^2\,c^4\,d^4\,e^2+6\,b\,c^5\,d^5\,e-2\,c^6\,d^6\right )}{b^4}+\frac {c^3\,d^6}{2\,b}+\frac {x^2\,\left (5\,b^6\,e^6-18\,b^5\,c\,d\,e^5+15\,b^4\,c^2\,d^2\,e^4+20\,b^3\,c^3\,d^3\,e^3-45\,b^2\,c^4\,d^4\,e^2+54\,b\,c^5\,d^5\,e-18\,c^6\,d^6\right )}{2\,b^3\,c}+\frac {2\,c^3\,d^5\,x\,\left (3\,b\,e-c\,d\right )}{b^2}}{b^2\,c^3\,x^2+2\,b\,c^4\,x^3+c^5\,x^4}-\frac {\ln \left (b+c\,x\right )\,\left (3\,b^6\,e^6-6\,b^5\,c\,d\,e^5+15\,b^2\,c^4\,d^4\,e^2-18\,b\,c^5\,d^5\,e+6\,c^6\,d^6\right )}{b^5\,c^4}+\frac {3\,d^4\,\ln \left (x\right )\,\left (5\,b^2\,e^2-6\,b\,c\,d\,e+2\,c^2\,d^2\right )}{b^5} \]

[In]

int((d + e*x)^6/(b*x + c*x^2)^3,x)

[Out]

(e^6*x)/c^3 - ((3*x^3*(b^6*e^6 - 2*c^6*d^6 - 5*b^2*c^4*d^4*e^2 + 5*b^4*c^2*d^2*e^4 + 6*b*c^5*d^5*e - 4*b^5*c*d
*e^5))/b^4 + (c^3*d^6)/(2*b) + (x^2*(5*b^6*e^6 - 18*c^6*d^6 - 45*b^2*c^4*d^4*e^2 + 20*b^3*c^3*d^3*e^3 + 15*b^4
*c^2*d^2*e^4 + 54*b*c^5*d^5*e - 18*b^5*c*d*e^5))/(2*b^3*c) + (2*c^3*d^5*x*(3*b*e - c*d))/b^2)/(c^5*x^4 + 2*b*c
^4*x^3 + b^2*c^3*x^2) - (log(b + c*x)*(3*b^6*e^6 + 6*c^6*d^6 + 15*b^2*c^4*d^4*e^2 - 18*b*c^5*d^5*e - 6*b^5*c*d
*e^5))/(b^5*c^4) + (3*d^4*log(x)*(5*b^2*e^2 + 2*c^2*d^2 - 6*b*c*d*e))/b^5