3.19.37 \(\int \frac {a d e+(c d^2+a e^2) x+c d e x^2}{(d+e x)^5} \, dx\) [1837]

Optimal. Leaf size=39 \[ -\frac {a-\frac {c d^2}{e^2}}{3 (d+e x)^3}-\frac {c d}{2 e^2 (d+e x)^2} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {24, 45} \begin {gather*} -\frac {a-\frac {c d^2}{e^2}}{3 (d+e x)^3}-\frac {c d}{2 e^2 (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 24

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((A_.) + (B_.)*(v_) + (C_.)*(v_)^2), x_Symbol] :> Dist[1/b^2, Int[u*(a + b*
v)^(m + 1)*Simp[b*B - a*C + b*C*v, x], x], x] /; FreeQ[{a, b, A, B, C}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0] &&
LeQ[m, -1]

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])

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 30, normalized size = 0.77 \begin {gather*} -\frac {2 a e^2+c d (d+3 e x)}{6 e^2 (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.49, size = 40, normalized size = 1.03

method	result	size
gosper	\(-\frac {3 c d e x +2 e^{2} a +c \,d^{2}}{6 e^{2} \left (e x +d \right )^{3}}\)	\(31\)
risch	\(\frac {-\frac {c d x}{2 e}-\frac {2 e^{2} a +c \,d^{2}}{6 e^{2}}}{\left (e x +d \right )^{3}}\)	\(35\)
default	\(-\frac {e^{2} a -c \,d^{2}}{3 e^{2} \left (e x +d \right )^{3}}-\frac {c d}{2 e^{2} \left (e x +d \right )^{2}}\)	\(40\)
norman	\(\frac {-\frac {d \left (2 e^{4} a +d^{2} e^{2} c \right )}{6 e^{4}}-\frac {\left (e^{4} a +2 d^{2} e^{2} c \right ) x}{3 e^{3}}-\frac {c d \,x^{2}}{2}}{\left (e x +d \right )^{4}}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 51, normalized size = 1.31 \begin {gather*} -\frac {3 \, c d x e + c d^{2} + 2 \, a e^{2}}{6 \, {\left (x^{3} e^{5} + 3 \, d x^{2} e^{4} + 3 \, d^{2} x e^{3} + d^{3} e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 2.03, size = 51, normalized size = 1.31 \begin {gather*} -\frac {3 \, c d x e + c d^{2} + 2 \, a e^{2}}{6 \, {\left (x^{3} e^{5} + 3 \, d x^{2} e^{4} + 3 \, d^{2} x e^{3} + d^{3} e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.19, size = 58, normalized size = 1.49 \begin {gather*} \frac {- 2 a e^{2} - c d^{2} - 3 c d e x}{6 d^{3} e^{2} + 18 d^{2} e^{3} x + 18 d e^{4} x^{2} + 6 e^{5} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.75, size = 42, normalized size = 1.08 \begin {gather*} -\frac {c d e^{\left (-2\right )}}{2 \, {\left (x e + d\right )}^{2}} + \frac {c d^{2} e^{\left (-2\right )}}{3 \, {\left (x e + d\right )}^{3}} - \frac {a}{3 \, {\left (x e + d\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.04, size = 57, normalized size = 1.46 \begin {gather*} -\frac {\frac {c\,d^2+2\,a\,e^2}{6\,e^2}+\frac {c\,d\,x}{2\,e}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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