3.20.1 \(\int \frac {(d+e x)^5}{(a d e+(c d^2+a e^2) x+c d e x^2)^4} \, dx\) [1901]

Optimal. Leaf size=54 \[ -\frac {c d^2-a e^2}{3 c^2 d^2 (a e+c d x)^3}-\frac {e}{2 c^2 d^2 (a e+c d x)^2} \]

[Out]

1/3*(a*e^2-c*d^2)/c^2/d^2/(c*d*x+a*e)^3-1/2*e/c^2/d^2/(c*d*x+a*e)^2

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Rubi [A]
time = 0.03, antiderivative size = 54, 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 {c d^2-a e^2}{3 c^2 d^2 (a e+c d x)^3}-\frac {e}{2 c^2 d^2 (a e+c d x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

-1/3*(c*d^2 - a*e^2)/(c^2*d^2*(a*e + c*d*x)^3) - e/(2*c^2*d^2*(a*e + c*d*x)^2)

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

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Mathematica [A]
time = 0.01, size = 37, normalized size = 0.69 \begin {gather*} -\frac {a e^2+c d (2 d+3 e x)}{6 c^2 d^2 (a e+c d x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^4,x]

[Out]

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

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Maple [A]
time = 0.73, size = 51, normalized size = 0.94

method	result	size
gosper	\(-\frac {3 c d e x +e^{2} a +2 c \,d^{2}}{6 c^{2} d^{2} \left (c d x +a e \right )^{3}}\)	\(37\)
risch	\(\frac {-\frac {e x}{2 c d}-\frac {e^{2} a +2 c \,d^{2}}{6 c^{2} d^{2}}}{\left (c d x +a e \right )^{3}}\)	\(43\)
default	\(-\frac {-e^{2} a +c \,d^{2}}{3 c^{2} d^{2} \left (c d x +a e \right )^{3}}-\frac {e}{2 c^{2} d^{2} \left (c d x +a e \right )^{2}}\)	\(51\)
norman	\(\frac {\frac {-a d \,e^{2} c -2 c^{2} d^{3}}{6 c^{3}}+\frac {\left (-a c d \,e^{4}-3 c^{2} d^{3} e^{2}\right ) x}{2 c^{3} d e}-\frac {e^{4} x^{4}}{2 c d}+\frac {\left (-a c d \,e^{6}-5 e^{4} d^{3} c^{2}\right ) x^{2}}{2 c^{3} e^{2} d^{2}}+\frac {\left (-a c d \,e^{8}-11 e^{6} d^{3} c^{2}\right ) x^{3}}{6 c^{3} d^{3} e^{3}}}{\left (c d x +a e \right )^{3} \left (e x +d \right )^{3}}\)	\(156\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x,method=_RETURNVERBOSE)

[Out]

-1/3*(-a*e^2+c*d^2)/c^2/d^2/(c*d*x+a*e)^3-1/2*e/c^2/d^2/(c*d*x+a*e)^2

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Maxima [A]
time = 0.30, size = 73, normalized size = 1.35 \begin {gather*} -\frac {3 \, c d x e + 2 \, c d^{2} + a e^{2}}{6 \, {\left (c^{5} d^{5} x^{3} + 3 \, a c^{4} d^{4} x^{2} e + 3 \, a^{2} c^{3} d^{3} x e^{2} + a^{3} c^{2} d^{2} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 3.23, size = 73, normalized size = 1.35 \begin {gather*} -\frac {3 \, c d x e + 2 \, c d^{2} + a e^{2}}{6 \, {\left (c^{5} d^{5} x^{3} + 3 \, a c^{4} d^{4} x^{2} e + 3 \, a^{2} c^{3} d^{3} x e^{2} + a^{3} c^{2} d^{2} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.25, size = 80, normalized size = 1.48 \begin {gather*} \frac {- a e^{2} - 2 c d^{2} - 3 c d e x}{6 a^{3} c^{2} d^{2} e^{3} + 18 a^{2} c^{3} d^{3} e^{2} x + 18 a c^{4} d^{4} e x^{2} + 6 c^{5} d^{5} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**5/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)

[Out]

(-a*e**2 - 2*c*d**2 - 3*c*d*e*x)/(6*a**3*c**2*d**2*e**3 + 18*a**2*c**3*d**3*e**2*x + 18*a*c**4*d**4*e*x**2 + 6
*c**5*d**5*x**3)

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Giac [A]
time = 0.68, size = 37, normalized size = 0.69 \begin {gather*} -\frac {3 \, c d x e + 2 \, c d^{2} + a e^{2}}{6 \, {\left (c d x + a e\right )}^{3} c^{2} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.57, size = 77, normalized size = 1.43 \begin {gather*} -\frac {\frac {2\,c\,d^2+a\,e^2}{6\,c^2\,d^2}+\frac {e\,x}{2\,c\,d}}{a^3\,e^3+3\,a^2\,c\,d\,e^2\,x+3\,a\,c^2\,d^2\,e\,x^2+c^3\,d^3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^5/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^4,x)

[Out]

-((a*e^2 + 2*c*d^2)/(6*c^2*d^2) + (e*x)/(2*c*d))/(a^3*e^3 + c^3*d^3*x^3 + 3*a^2*c*d*e^2*x + 3*a*c^2*d^2*e*x^2)

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