3.19.98 \(\int \frac {(d+e x)^8}{(a d e+(c d^2+a e^2) x+c d e x^2)^4} \, dx\) [1898]

Optimal. Leaf size=146 \[ \frac {e^4 x}{c^4 d^4}-\frac {\left (c d^2-a e^2\right )^4}{3 c^5 d^5 (a e+c d x)^3}-\frac {2 e \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)^2}-\frac {6 e^2 \left (c d^2-a e^2\right )^2}{c^5 d^5 (a e+c d x)}+\frac {4 e^3 \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^5 d^5} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.09, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 45} \begin {gather*} -\frac {6 e^2 \left (c d^2-a e^2\right )^2}{c^5 d^5 (a e+c d x)}-\frac {2 e \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)^2}-\frac {\left (c d^2-a e^2\right )^4}{3 c^5 d^5 (a e+c d x)^3}+\frac {4 e^3 \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^5 d^5}+\frac {e^4 x}{c^4 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^8/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

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

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 640

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c/e)*x)^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 {(d+e x)^8}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx &=\int \frac {(d+e x)^4}{(a e+c d x)^4} \, dx\\ &=\int \left (\frac {e^4}{c^4 d^4}+\frac {\left (c d^2-a e^2\right )^4}{c^4 d^4 (a e+c d x)^4}+\frac {4 e \left (c d^2-a e^2\right )^3}{c^4 d^4 (a e+c d x)^3}+\frac {6 \left (c d^2 e-a e^3\right )^2}{c^4 d^4 (a e+c d x)^2}+\frac {4 \left (c d^2 e^3-a e^5\right )}{c^4 d^4 (a e+c d x)}\right ) \, dx\\ &=\frac {e^4 x}{c^4 d^4}-\frac {\left (c d^2-a e^2\right )^4}{3 c^5 d^5 (a e+c d x)^3}-\frac {2 e \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)^2}-\frac {6 e^2 \left (c d^2-a e^2\right )^2}{c^5 d^5 (a e+c d x)}+\frac {4 e^3 \left (c d^2-a e^2\right ) \log (a e+c d x)}{c^5 d^5}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.05, size = 194, normalized size = 1.33 \begin {gather*} \frac {-13 a^4 e^8+a^3 c d e^6 (22 d-27 e x)-3 a^2 c^2 d^2 e^4 \left (2 d^2-18 d e x+3 e^2 x^2\right )+a c^3 d^3 e^2 \left (-2 d^3-18 d^2 e x+36 d e^2 x^2+9 e^3 x^3\right )-c^4 \left (d^8+6 d^7 e x+18 d^6 e^2 x^2-3 d^4 e^4 x^4\right )-12 e^3 \left (-c d^2+a e^2\right ) (a e+c d x)^3 \log (a e+c d x)}{3 c^5 d^5 (a e+c d x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^8/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

(-13*a^4*e^8 + a^3*c*d*e^6*(22*d - 27*e*x) - 3*a^2*c^2*d^2*e^4*(2*d^2 - 18*d*e*x + 3*e^2*x^2) + a*c^3*d^3*e^2*
(-2*d^3 - 18*d^2*e*x + 36*d*e^2*x^2 + 9*e^3*x^3) - c^4*(d^8 + 6*d^7*e*x + 18*d^6*e^2*x^2 - 3*d^4*e^4*x^4) - 12
*e^3*(-(c*d^2) + a*e^2)*(a*e + c*d*x)^3*Log[a*e + c*d*x])/(3*c^5*d^5*(a*e + c*d*x)^3)

________________________________________________________________________________________

Maple [A]
time = 0.72, size = 221, normalized size = 1.51

method	result	size
risch	\(\frac {e^{4} x}{c^{4} d^{4}}+\frac {\left (-6 d \,e^{6} a^{2} c +12 c^{2} d^{3} a \,e^{4}-6 c^{3} d^{5} e^{2}\right ) x^{2}-2 e \left (5 e^{6} a^{3}-9 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a +d^{6} c^{3}\right ) x -\frac {13 a^{4} e^{8}-22 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}+2 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}{3 c d}}{c^{4} d^{4} \left (c d x +a e \right )^{3}}-\frac {4 e^{5} \ln \left (c d x +a e \right ) a}{c^{5} d^{5}}+\frac {4 e^{3} \ln \left (c d x +a e \right )}{c^{4} d^{3}}\)	\(216\)
default	\(\frac {e^{4} x}{c^{4} d^{4}}-\frac {6 e^{2} \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{c^{5} d^{5} \left (c d x +a e \right )}-\frac {a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}{3 c^{5} d^{5} \left (c d x +a e \right )^{3}}-\frac {4 e^{3} \left (e^{2} a -c \,d^{2}\right ) \ln \left (c d x +a e \right )}{c^{5} d^{5}}+\frac {2 e \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right )}{c^{5} d^{5} \left (c d x +a e \right )^{2}}\)	\(221\)
norman	\(\frac {\frac {e^{7} x^{7}}{c d}-\frac {22 a^{4} e^{8}-13 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}+2 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}{3 c^{5} d^{2}}-\frac {\left (22 a^{4} e^{14}+149 a^{3} c \,d^{2} e^{12}+33 d^{4} a^{2} c^{2} e^{10}+29 a \,c^{3} d^{6} e^{8}+82 c^{4} d^{8} e^{6}\right ) x^{3}}{3 c^{5} d^{5} e^{3}}-\frac {\left (22 a^{4} e^{12}+41 a^{3} c \,d^{2} e^{10}-9 a^{2} e^{8} d^{4} c^{2}+17 a \,c^{3} d^{6} e^{6}+13 e^{4} d^{8} c^{4}\right ) x^{2}}{c^{5} d^{4} e^{2}}-\frac {\left (22 a^{4} e^{10}+5 a^{3} c \,d^{2} e^{8}-3 a^{2} c^{2} d^{4} e^{6}+8 a \,c^{3} d^{6} e^{4}+3 e^{2} d^{8} c^{4}\right ) x}{c^{5} d^{3} e}-\frac {\left (18 a^{3} e^{12}+27 a^{2} c \,d^{2} e^{10}-3 d^{4} c^{2} a \,e^{8}+28 d^{6} e^{6} c^{3}\right ) x^{4}}{c^{4} d^{4} e^{2}}-\frac {3 \left (4 a^{2} e^{10}-a c \,d^{2} e^{8}+4 c^{2} d^{4} e^{6}\right ) x^{5}}{c^{3} d^{3} e}}{\left (c d x +a e \right )^{3} \left (e x +d \right )^{3}}-\frac {4 e^{3} \left (e^{2} a -c \,d^{2}\right ) \ln \left (c d x +a e \right )}{c^{5} d^{5}}\)	\(444\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^8/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x,method=_RETURNVERBOSE)

[Out]

e^4*x/c^4/d^4-6/c^5/d^5*e^2*(a^2*e^4-2*a*c*d^2*e^2+c^2*d^4)/(c*d*x+a*e)-1/3/c^5/d^5*(a^4*e^8-4*a^3*c*d^2*e^6+6
*a^2*c^2*d^4*e^4-4*a*c^3*d^6*e^2+c^4*d^8)/(c*d*x+a*e)^3-4/c^5/d^5*e^3*(a*e^2-c*d^2)*ln(c*d*x+a*e)+2/c^5/d^5*e*
(a^3*e^6-3*a^2*c*d^2*e^4+3*a*c^2*d^4*e^2-c^3*d^6)/(c*d*x+a*e)^2

________________________________________________________________________________________

Maxima [A]
time = 0.30, size = 231, normalized size = 1.58 \begin {gather*} -\frac {c^{4} d^{8} + 2 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 22 \, a^{3} c d^{2} e^{6} + 13 \, a^{4} e^{8} + 18 \, {\left (c^{4} d^{6} e^{2} - 2 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 6 \, {\left (c^{4} d^{7} e + 3 \, a c^{3} d^{5} e^{3} - 9 \, a^{2} c^{2} d^{3} e^{5} + 5 \, a^{3} c d e^{7}\right )} x}{3 \, {\left (c^{8} d^{8} x^{3} + 3 \, a c^{7} d^{7} x^{2} e + 3 \, a^{2} c^{6} d^{6} x e^{2} + a^{3} c^{5} d^{5} e^{3}\right )}} + \frac {x e^{4}}{c^{4} d^{4}} + \frac {4 \, {\left (c d^{2} e^{3} - a e^{5}\right )} \log \left (c d x + a e\right )}{c^{5} d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^8/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="maxima")

[Out]

-1/3*(c^4*d^8 + 2*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 22*a^3*c*d^2*e^6 + 13*a^4*e^8 + 18*(c^4*d^6*e^2 - 2*a*c^
3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^2 + 6*(c^4*d^7*e + 3*a*c^3*d^5*e^3 - 9*a^2*c^2*d^3*e^5 + 5*a^3*c*d*e^7)*x)/(c^8
*d^8*x^3 + 3*a*c^7*d^7*x^2*e + 3*a^2*c^6*d^6*x*e^2 + a^3*c^5*d^5*e^3) + x*e^4/(c^4*d^4) + 4*(c*d^2*e^3 - a*e^5
)*log(c*d*x + a*e)/(c^5*d^5)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 332 vs. \(2 (142) = 284\).
time = 4.29, size = 332, normalized size = 2.27 \begin {gather*} -\frac {6 \, c^{4} d^{7} x e + c^{4} d^{8} + 18 \, a c^{3} d^{5} x e^{3} + 27 \, a^{3} c d x e^{7} + 13 \, a^{4} e^{8} + {\left (9 \, a^{2} c^{2} d^{2} x^{2} - 22 \, a^{3} c d^{2}\right )} e^{6} - 9 \, {\left (a c^{3} d^{3} x^{3} + 6 \, a^{2} c^{2} d^{3} x\right )} e^{5} - 3 \, {\left (c^{4} d^{4} x^{4} + 12 \, a c^{3} d^{4} x^{2} - 2 \, a^{2} c^{2} d^{4}\right )} e^{4} + 2 \, {\left (9 \, c^{4} d^{6} x^{2} + a c^{3} d^{6}\right )} e^{2} - 12 \, {\left (c^{4} d^{5} x^{3} e^{3} + 3 \, a c^{3} d^{4} x^{2} e^{4} - 3 \, a^{3} c d x e^{7} - a^{4} e^{8} - {\left (3 \, a^{2} c^{2} d^{2} x^{2} - a^{3} c d^{2}\right )} e^{6} - {\left (a c^{3} d^{3} x^{3} - 3 \, a^{2} c^{2} d^{3} x\right )} e^{5}\right )} \log \left (c d x + a e\right )}{3 \, {\left (c^{8} d^{8} x^{3} + 3 \, a c^{7} d^{7} x^{2} e + 3 \, a^{2} c^{6} d^{6} x e^{2} + a^{3} c^{5} d^{5} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^8/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="fricas")

[Out]

-1/3*(6*c^4*d^7*x*e + c^4*d^8 + 18*a*c^3*d^5*x*e^3 + 27*a^3*c*d*x*e^7 + 13*a^4*e^8 + (9*a^2*c^2*d^2*x^2 - 22*a
^3*c*d^2)*e^6 - 9*(a*c^3*d^3*x^3 + 6*a^2*c^2*d^3*x)*e^5 - 3*(c^4*d^4*x^4 + 12*a*c^3*d^4*x^2 - 2*a^2*c^2*d^4)*e
^4 + 2*(9*c^4*d^6*x^2 + a*c^3*d^6)*e^2 - 12*(c^4*d^5*x^3*e^3 + 3*a*c^3*d^4*x^2*e^4 - 3*a^3*c*d*x*e^7 - a^4*e^8
- (3*a^2*c^2*d^2*x^2 - a^3*c*d^2)*e^6 - (a*c^3*d^3*x^3 - 3*a^2*c^2*d^3*x)*e^5)*log(c*d*x + a*e))/(c^8*d^8*x^3
+ 3*a*c^7*d^7*x^2*e + 3*a^2*c^6*d^6*x*e^2 + a^3*c^5*d^5*e^3)

________________________________________________________________________________________

Sympy [A]
time = 129.67, size = 257, normalized size = 1.76 \begin {gather*} \frac {- 13 a^{4} e^{8} + 22 a^{3} c d^{2} e^{6} - 6 a^{2} c^{2} d^{4} e^{4} - 2 a c^{3} d^{6} e^{2} - c^{4} d^{8} + x^{2} \left (- 18 a^{2} c^{2} d^{2} e^{6} + 36 a c^{3} d^{4} e^{4} - 18 c^{4} d^{6} e^{2}\right ) + x \left (- 30 a^{3} c d e^{7} + 54 a^{2} c^{2} d^{3} e^{5} - 18 a c^{3} d^{5} e^{3} - 6 c^{4} d^{7} e\right )}{3 a^{3} c^{5} d^{5} e^{3} + 9 a^{2} c^{6} d^{6} e^{2} x + 9 a c^{7} d^{7} e x^{2} + 3 c^{8} d^{8} x^{3}} + \frac {e^{4} x}{c^{4} d^{4}} - \frac {4 e^{3} \left (a e^{2} - c d^{2}\right ) \log {\left (a e + c d x \right )}}{c^{5} d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**8/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)

[Out]

(-13*a**4*e**8 + 22*a**3*c*d**2*e**6 - 6*a**2*c**2*d**4*e**4 - 2*a*c**3*d**6*e**2 - c**4*d**8 + x**2*(-18*a**2
*c**2*d**2*e**6 + 36*a*c**3*d**4*e**4 - 18*c**4*d**6*e**2) + x*(-30*a**3*c*d*e**7 + 54*a**2*c**2*d**3*e**5 - 1
8*a*c**3*d**5*e**3 - 6*c**4*d**7*e))/(3*a**3*c**5*d**5*e**3 + 9*a**2*c**6*d**6*e**2*x + 9*a*c**7*d**7*e*x**2 +
3*c**8*d**8*x**3) + e**4*x/(c**4*d**4) - 4*e**3*(a*e**2 - c*d**2)*log(a*e + c*d*x)/(c**5*d**5)

________________________________________________________________________________________

Giac [A]
time = 0.82, size = 196, normalized size = 1.34 \begin {gather*} \frac {x e^{4}}{c^{4} d^{4}} + \frac {4 \, {\left (c d^{2} e^{3} - a e^{5}\right )} \log \left ({\left | c d x + a e \right |}\right )}{c^{5} d^{5}} - \frac {c^{4} d^{8} + 2 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 22 \, a^{3} c d^{2} e^{6} + 13 \, a^{4} e^{8} + 18 \, {\left (c^{4} d^{6} e^{2} - 2 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 6 \, {\left (c^{4} d^{7} e + 3 \, a c^{3} d^{5} e^{3} - 9 \, a^{2} c^{2} d^{3} e^{5} + 5 \, a^{3} c d e^{7}\right )} x}{3 \, {\left (c d x + a e\right )}^{3} c^{5} d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^8/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="giac")

[Out]

x*e^4/(c^4*d^4) + 4*(c*d^2*e^3 - a*e^5)*log(abs(c*d*x + a*e))/(c^5*d^5) - 1/3*(c^4*d^8 + 2*a*c^3*d^6*e^2 + 6*a
^2*c^2*d^4*e^4 - 22*a^3*c*d^2*e^6 + 13*a^4*e^8 + 18*(c^4*d^6*e^2 - 2*a*c^3*d^4*e^4 + a^2*c^2*d^2*e^6)*x^2 + 6*
(c^4*d^7*e + 3*a*c^3*d^5*e^3 - 9*a^2*c^2*d^3*e^5 + 5*a^3*c*d*e^7)*x)/((c*d*x + a*e)^3*c^5*d^5)

________________________________________________________________________________________

Mupad [B]
time = 0.17, size = 246, normalized size = 1.68 \begin {gather*} \frac {e^4\,x}{c^4\,d^4}-\frac {x\,\left (10\,a^3\,e^7-18\,a^2\,c\,d^2\,e^5+6\,a\,c^2\,d^4\,e^3+2\,c^3\,d^6\,e\right )+x^2\,\left (6\,a^2\,c\,d\,e^6-12\,a\,c^2\,d^3\,e^4+6\,c^3\,d^5\,e^2\right )+\frac {13\,a^4\,e^8-22\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4+2\,a\,c^3\,d^6\,e^2+c^4\,d^8}{3\,c\,d}}{a^3\,c^4\,d^4\,e^3+3\,a^2\,c^5\,d^5\,e^2\,x+3\,a\,c^6\,d^6\,e\,x^2+c^7\,d^7\,x^3}-\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (4\,a\,e^5-4\,c\,d^2\,e^3\right )}{c^5\,d^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^8/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^4,x)

[Out]

(e^4*x)/(c^4*d^4) - (x*(10*a^3*e^7 + 2*c^3*d^6*e + 6*a*c^2*d^4*e^3 - 18*a^2*c*d^2*e^5) + x^2*(6*c^3*d^5*e^2 -
12*a*c^2*d^3*e^4 + 6*a^2*c*d*e^6) + (13*a^4*e^8 + c^4*d^8 + 2*a*c^3*d^6*e^2 - 22*a^3*c*d^2*e^6 + 6*a^2*c^2*d^4
*e^4)/(3*c*d))/(c^7*d^7*x^3 + a^3*c^4*d^4*e^3 + 3*a*c^6*d^6*e*x^2 + 3*a^2*c^5*d^5*e^2*x) - (log(a*e + c*d*x)*(
4*a*e^5 - 4*c*d^2*e^3))/(c^5*d^5)

________________________________________________________________________________________