3.5.58 \(\int \frac {a+c x^2}{(d+e x)^4} \, dx\) [458]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 52, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {711} \begin {gather*} -\frac {a e^2+c d^2}{3 e^3 (d+e x)^3}-\frac {c}{e^3 (d+e x)}+\frac {c d}{e^3 (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)/(d + e*x)^4,x]

[Out]

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

Rule 711

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 39, normalized size = 0.72 \begin {gather*} -\frac {a e^2+c \left (d^2+3 d e x+3 e^2 x^2\right )}{3 e^3 (d+e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)/(d + e*x)^4,x]

[Out]

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

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Maple [A]
time = 0.39, size = 51, normalized size = 0.94

method	result	size
gosper	\(-\frac {3 c \,x^{2} e^{2}+3 c d e x +e^{2} a +c \,d^{2}}{3 \left (e x +d \right )^{3} e^{3}}\)	\(39\)
norman	\(\frac {-\frac {c \,x^{2}}{e}-\frac {c d x}{e^{2}}-\frac {e^{2} a +c \,d^{2}}{3 e^{3}}}{\left (e x +d \right )^{3}}\)	\(43\)
risch	\(\frac {-\frac {c \,x^{2}}{e}-\frac {c d x}{e^{2}}-\frac {e^{2} a +c \,d^{2}}{3 e^{3}}}{\left (e x +d \right )^{3}}\)	\(43\)
default	\(-\frac {e^{2} a +c \,d^{2}}{3 e^{3} \left (e x +d \right )^{3}}-\frac {c}{e^{3} \left (e x +d \right )}+\frac {c d}{e^{3} \left (e x +d \right )^{2}}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 58, normalized size = 1.07 \begin {gather*} -\frac {3 \, c x^{2} e^{2} + 3 \, c d x e + c d^{2} + a e^{2}}{3 \, {\left (x^{3} e^{6} + 3 \, d x^{2} e^{5} + 3 \, d^{2} x e^{4} + d^{3} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 1.44, size = 57, normalized size = 1.06 \begin {gather*} -\frac {3 \, c d x e + c d^{2} + {\left (3 \, c x^{2} + a\right )} e^{2}}{3 \, {\left (x^{3} e^{6} + 3 \, d x^{2} e^{5} + 3 \, d^{2} x e^{4} + d^{3} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.21, size = 66, normalized size = 1.22 \begin {gather*} \frac {- a e^{2} - c d^{2} - 3 c d e x - 3 c e^{2} x^{2}}{3 d^{3} e^{3} + 9 d^{2} e^{4} x + 9 d e^{5} x^{2} + 3 e^{6} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-a*e**2 - c*d**2 - 3*c*d*e*x - 3*c*e**2*x**2)/(3*d**3*e**3 + 9*d**2*e**4*x + 9*d*e**5*x**2 + 3*e**6*x**3)

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Giac [A]
time = 1.34, size = 37, normalized size = 0.69 \begin {gather*} -\frac {{\left (3 \, c x^{2} e^{2} + 3 \, c d x e + c d^{2} + a e^{2}\right )} e^{\left (-3\right )}}{3 \, {\left (x e + d\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.04, size = 63, normalized size = 1.17 \begin {gather*} -\frac {\frac {c\,d^2+a\,e^2}{3\,e^3}+\frac {c\,x^2}{e}+\frac {c\,d\,x}{e^2}}{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((a + c*x^2)/(d + e*x)^4,x)

[Out]

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

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