3.19.87 \(\int \frac {(d+e x)^7}{(a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx\) [1887]

Optimal. Leaf size=142 \[ \frac {e^3 \left (4 c d^2-3 a e^2\right ) x}{c^4 d^4}+\frac {e^4 x^2}{2 c^3 d^3}-\frac {\left (c d^2-a e^2\right )^4}{2 c^5 d^5 (a e+c d x)^2}-\frac {4 e \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)}+\frac {6 e^2 \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^5 d^5} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 142, 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 {4 e \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)}-\frac {\left (c d^2-a e^2\right )^4}{2 c^5 d^5 (a e+c d x)^2}+\frac {6 e^2 \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^5 d^5}+\frac {e^3 x \left (4 c d^2-3 a e^2\right )}{c^4 d^4}+\frac {e^4 x^2}{2 c^3 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^7/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

(e^3*(4*c*d^2 - 3*a*e^2)*x)/(c^4*d^4) + (e^4*x^2)/(2*c^3*d^3) - (c*d^2 - a*e^2)^4/(2*c^5*d^5*(a*e + c*d*x)^2)
- (4*e*(c*d^2 - a*e^2)^3)/(c^5*d^5*(a*e + c*d*x)) + (6*e^2*(c*d^2 - a*e^2)^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)^7}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx &=\int \frac {(d+e x)^4}{(a e+c d x)^3} \, dx\\ &=\int \left (\frac {4 c d^2 e^3-3 a e^5}{c^4 d^4}+\frac {e^4 x}{c^3 d^3}+\frac {\left (c d^2-a e^2\right )^4}{c^4 d^4 (a e+c d x)^3}+\frac {4 e \left (c d^2-a e^2\right )^3}{c^4 d^4 (a e+c d x)^2}+\frac {6 \left (c d^2 e-a e^3\right )^2}{c^4 d^4 (a e+c d x)}\right ) \, dx\\ &=\frac {e^3 \left (4 c d^2-3 a e^2\right ) x}{c^4 d^4}+\frac {e^4 x^2}{2 c^3 d^3}-\frac {\left (c d^2-a e^2\right )^4}{2 c^5 d^5 (a e+c d x)^2}-\frac {4 e \left (c d^2-a e^2\right )^3}{c^5 d^5 (a e+c d x)}+\frac {6 e^2 \left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^5 d^5}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 191, normalized size = 1.35 \begin {gather*} \frac {7 a^4 e^8+2 a^3 c d e^6 (-10 d+e x)+a^2 c^2 d^2 e^4 \left (18 d^2-16 d e x-11 e^2 x^2\right )-4 a c^3 d^3 e^2 \left (d^3-6 d^2 e x-4 d e^2 x^2+e^3 x^3\right )+c^4 d^4 \left (-d^4-8 d^3 e x+8 d e^3 x^3+e^4 x^4\right )+12 e^2 \left (c d^2-a e^2\right )^2 (a e+c d x)^2 \log (a e+c d x)}{2 c^5 d^5 (a e+c d x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^7/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

(7*a^4*e^8 + 2*a^3*c*d*e^6*(-10*d + e*x) + a^2*c^2*d^2*e^4*(18*d^2 - 16*d*e*x - 11*e^2*x^2) - 4*a*c^3*d^3*e^2*
(d^3 - 6*d^2*e*x - 4*d*e^2*x^2 + e^3*x^3) + c^4*d^4*(-d^4 - 8*d^3*e*x + 8*d*e^3*x^3 + e^4*x^4) + 12*e^2*(c*d^2
- a*e^2)^2*(a*e + c*d*x)^2*Log[a*e + c*d*x])/(2*c^5*d^5*(a*e + c*d*x)^2)

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Maple [A]
time = 0.73, size = 211, normalized size = 1.49

method	result	size
default	\(-\frac {e^{3} \left (-\frac {1}{2} c d e \,x^{2}+3 a \,e^{2} x -4 c \,d^{2} x \right )}{c^{4} d^{4}}+\frac {4 e \left (e^{6} a^{3}-3 e^{4} d^{2} a^{2} c +3 d^{4} e^{2} c^{2} a -d^{6} c^{3}\right )}{c^{5} d^{5} \left (c d x +a e \right )}+\frac {6 e^{2} \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \ln \left (c d x +a e \right )}{c^{5} d^{5}}-\frac {a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}{2 c^{5} d^{5} \left (c d x +a e \right )^{2}}\)	\(211\)
risch	\(\frac {e^{4} x^{2}}{2 c^{3} d^{3}}-\frac {3 e^{5} a x}{c^{4} d^{4}}+\frac {4 e^{3} x}{c^{3} d^{2}}+\frac {\left (4 e^{7} a^{3}-12 d^{2} e^{5} a^{2} c +12 c^{2} d^{4} a \,e^{3}-4 c^{3} d^{6} e \right ) x +\frac {7 a^{4} e^{8}-20 a^{3} c \,d^{2} e^{6}+18 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}-c^{4} d^{8}}{2 c d}}{c^{4} d^{4} \left (c d x +a e \right )^{2}}+\frac {6 e^{6} \ln \left (c d x +a e \right ) a^{2}}{c^{5} d^{5}}-\frac {12 e^{4} \ln \left (c d x +a e \right ) a}{c^{4} d^{3}}+\frac {6 e^{2} \ln \left (c d x +a e \right )}{c^{3} d}\)	\(230\)
norman	\(\frac {\frac {\left (18 a^{4} e^{10}-16 a^{3} c \,d^{2} e^{8}-15 a^{2} c^{2} d^{4} e^{6}-9 a \,c^{3} d^{6} e^{4}-5 c^{4} d^{8} e^{2}\right ) x}{c^{5} d^{4} e}+\frac {\left (12 a^{3} e^{10}-16 a^{2} c \,d^{2} e^{8}+d^{4} c^{2} a \,e^{6}-17 c^{3} d^{6} e^{4}\right ) x^{3}}{c^{4} d^{4} e}+\frac {18 a^{4} e^{8}-28 a^{3} c \,d^{2} e^{6}+a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}-c^{4} d^{8}}{2 d^{3} c^{5}}+\frac {e^{6} x^{6}}{2 c d}+\frac {\left (18 a^{4} e^{12}+20 a^{3} c \,d^{2} e^{10}-63 d^{4} a^{2} c^{2} e^{8}-16 a \,c^{3} d^{6} e^{6}-34 c^{4} d^{8} e^{4}\right ) x^{2}}{2 c^{5} d^{5} e^{2}}-\frac {e^{5} \left (2 e^{2} a -5 c \,d^{2}\right ) x^{5}}{c^{2} d^{2}}}{\left (c d x +a e \right )^{2} \left (e x +d \right )^{2}}+\frac {6 e^{2} \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \ln \left (c d x +a e \right )}{c^{5} d^{5}}\)	\(366\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^7/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 213, normalized size = 1.50 \begin {gather*} -\frac {c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} - 18 \, a^{2} c^{2} d^{4} e^{4} + 20 \, a^{3} c d^{2} e^{6} - 7 \, a^{4} e^{8} + 8 \, {\left (c^{4} d^{7} e - 3 \, a c^{3} d^{5} e^{3} + 3 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x}{2 \, {\left (c^{7} d^{7} x^{2} + 2 \, a c^{6} d^{6} x e + a^{2} c^{5} d^{5} e^{2}\right )}} + \frac {c d x^{2} e^{4} + 2 \, {\left (4 \, c d^{2} e^{3} - 3 \, a e^{5}\right )} x}{2 \, c^{4} d^{4}} + \frac {6 \, {\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\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)^7/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="maxima")

[Out]

-1/2*(c^4*d^8 + 4*a*c^3*d^6*e^2 - 18*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2*e^6 - 7*a^4*e^8 + 8*(c^4*d^7*e - 3*a*c^3*d
^5*e^3 + 3*a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*x)/(c^7*d^7*x^2 + 2*a*c^6*d^6*x*e + a^2*c^5*d^5*e^2) + 1/2*(c*d*x^2*
e^4 + 2*(4*c*d^2*e^3 - 3*a*e^5)*x)/(c^4*d^4) + 6*(c^2*d^4*e^2 - 2*a*c*d^2*e^4 + a^2*e^6)*log(c*d*x + a*e)/(c^5
*d^5)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (135) = 270\).
time = 3.96, size = 325, normalized size = 2.29 \begin {gather*} -\frac {8 \, c^{4} d^{7} x e + c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} - 2 \, a^{3} c d x e^{7} - 7 \, a^{4} e^{8} + {\left (11 \, a^{2} c^{2} d^{2} x^{2} + 20 \, a^{3} c d^{2}\right )} e^{6} + 4 \, {\left (a c^{3} d^{3} x^{3} + 4 \, a^{2} c^{2} d^{3} x\right )} e^{5} - {\left (c^{4} d^{4} x^{4} + 16 \, a c^{3} d^{4} x^{2} + 18 \, a^{2} c^{2} d^{4}\right )} e^{4} - 8 \, {\left (c^{4} d^{5} x^{3} + 3 \, a c^{3} d^{5} x\right )} e^{3} - 12 \, {\left (c^{4} d^{6} x^{2} e^{2} + 2 \, a c^{3} d^{5} x e^{3} - 4 \, a^{2} c^{2} d^{3} x e^{5} + 2 \, a^{3} c d x e^{7} + a^{4} e^{8} + {\left (a^{2} c^{2} d^{2} x^{2} - 2 \, a^{3} c d^{2}\right )} e^{6} - {\left (2 \, a c^{3} d^{4} x^{2} - a^{2} c^{2} d^{4}\right )} e^{4}\right )} \log \left (c d x + a e\right )}{2 \, {\left (c^{7} d^{7} x^{2} + 2 \, a c^{6} d^{6} x e + a^{2} c^{5} d^{5} e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^7/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="fricas")

[Out]

-1/2*(8*c^4*d^7*x*e + c^4*d^8 + 4*a*c^3*d^6*e^2 - 2*a^3*c*d*x*e^7 - 7*a^4*e^8 + (11*a^2*c^2*d^2*x^2 + 20*a^3*c
*d^2)*e^6 + 4*(a*c^3*d^3*x^3 + 4*a^2*c^2*d^3*x)*e^5 - (c^4*d^4*x^4 + 16*a*c^3*d^4*x^2 + 18*a^2*c^2*d^4)*e^4 -
8*(c^4*d^5*x^3 + 3*a*c^3*d^5*x)*e^3 - 12*(c^4*d^6*x^2*e^2 + 2*a*c^3*d^5*x*e^3 - 4*a^2*c^2*d^3*x*e^5 + 2*a^3*c*
d*x*e^7 + a^4*e^8 + (a^2*c^2*d^2*x^2 - 2*a^3*c*d^2)*e^6 - (2*a*c^3*d^4*x^2 - a^2*c^2*d^4)*e^4)*log(c*d*x + a*e
))/(c^7*d^7*x^2 + 2*a*c^6*d^6*x*e + a^2*c^5*d^5*e^2)

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Sympy [A]
time = 7.12, size = 226, normalized size = 1.59 \begin {gather*} x \left (- \frac {3 a e^{5}}{c^{4} d^{4}} + \frac {4 e^{3}}{c^{3} d^{2}}\right ) + \frac {7 a^{4} e^{8} - 20 a^{3} c d^{2} e^{6} + 18 a^{2} c^{2} d^{4} e^{4} - 4 a c^{3} d^{6} e^{2} - c^{4} d^{8} + x \left (8 a^{3} c d e^{7} - 24 a^{2} c^{2} d^{3} e^{5} + 24 a c^{3} d^{5} e^{3} - 8 c^{4} d^{7} e\right )}{2 a^{2} c^{5} d^{5} e^{2} + 4 a c^{6} d^{6} e x + 2 c^{7} d^{7} x^{2}} + \frac {e^{4} x^{2}}{2 c^{3} d^{3}} + \frac {6 e^{2} \left (a e^{2} - c d^{2}\right )^{2} \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)**7/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

x*(-3*a*e**5/(c**4*d**4) + 4*e**3/(c**3*d**2)) + (7*a**4*e**8 - 20*a**3*c*d**2*e**6 + 18*a**2*c**2*d**4*e**4 -
4*a*c**3*d**6*e**2 - c**4*d**8 + x*(8*a**3*c*d*e**7 - 24*a**2*c**2*d**3*e**5 + 24*a*c**3*d**5*e**3 - 8*c**4*d
**7*e))/(2*a**2*c**5*d**5*e**2 + 4*a*c**6*d**6*e*x + 2*c**7*d**7*x**2) + e**4*x**2/(2*c**3*d**3) + 6*e**2*(a*e
**2 - c*d**2)**2*log(a*e + c*d*x)/(c**5*d**5)

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Giac [A]
time = 1.74, size = 204, normalized size = 1.44 \begin {gather*} \frac {6 \, {\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} \log \left ({\left | c d x + a e \right |}\right )}{c^{5} d^{5}} + \frac {c^{3} d^{3} x^{2} e^{4} + 8 \, c^{3} d^{4} x e^{3} - 6 \, a c^{2} d^{2} x e^{5}}{2 \, c^{6} d^{6}} - \frac {c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} - 18 \, a^{2} c^{2} d^{4} e^{4} + 20 \, a^{3} c d^{2} e^{6} - 7 \, a^{4} e^{8} + 8 \, {\left (c^{4} d^{7} e - 3 \, a c^{3} d^{5} e^{3} + 3 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x}{2 \, {\left (c d x + a e\right )}^{2} c^{5} d^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^7/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="giac")

[Out]

6*(c^2*d^4*e^2 - 2*a*c*d^2*e^4 + a^2*e^6)*log(abs(c*d*x + a*e))/(c^5*d^5) + 1/2*(c^3*d^3*x^2*e^4 + 8*c^3*d^4*x
*e^3 - 6*a*c^2*d^2*x*e^5)/(c^6*d^6) - 1/2*(c^4*d^8 + 4*a*c^3*d^6*e^2 - 18*a^2*c^2*d^4*e^4 + 20*a^3*c*d^2*e^6 -
7*a^4*e^8 + 8*(c^4*d^7*e - 3*a*c^3*d^5*e^3 + 3*a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*x)/((c*d*x + a*e)^2*c^5*d^5)

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Mupad [B]
time = 0.11, size = 232, normalized size = 1.63 \begin {gather*} \frac {x\,\left (4\,a^3\,e^7-12\,a^2\,c\,d^2\,e^5+12\,a\,c^2\,d^4\,e^3-4\,c^3\,d^6\,e\right )-\frac {-7\,a^4\,e^8+20\,a^3\,c\,d^2\,e^6-18\,a^2\,c^2\,d^4\,e^4+4\,a\,c^3\,d^6\,e^2+c^4\,d^8}{2\,c\,d}}{a^2\,c^4\,d^4\,e^2+2\,a\,c^5\,d^5\,e\,x+c^6\,d^6\,x^2}+x\,\left (\frac {4\,e^3}{c^3\,d^2}-\frac {3\,a\,e^5}{c^4\,d^4}\right )+\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (6\,a^2\,e^6-12\,a\,c\,d^2\,e^4+6\,c^2\,d^4\,e^2\right )}{c^5\,d^5}+\frac {e^4\,x^2}{2\,c^3\,d^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^7/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3,x)

[Out]

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

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