3.4.78 \(\int \frac {a+c x^2}{(d+e x)^4} \, dx\)

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

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Rubi [A]  time = 0.03, 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 = {697} \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]

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

Rule 697

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+c x^2}{(d+e x)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 63, normalized size = 1.17 \begin {gather*} -\frac {3 \, c e^{2} x^{2} + 3 \, c d e x + c d^{2} + a e^{2}}{3 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + 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*e^2*x^2 + 3*c*d*e*x + c*d^2 + a*e^2)/(e^6*x^3 + 3*d*e^5*x^2 + 3*d^2*e^4*x + d^3*e^3)

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giac [A]  time = 0.18, 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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maple [A]  time = 0.04, size = 51, normalized size = 0.94 \begin {gather*} \frac {c d}{\left (e x +d \right )^{2} e^{3}}-\frac {c}{\left (e x +d \right ) e^{3}}-\frac {a \,e^{2}+c \,d^{2}}{3 \left (e x +d \right )^{3} e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.39, size = 63, normalized size = 1.17 \begin {gather*} -\frac {3 \, c e^{2} x^{2} + 3 \, c d e x + c d^{2} + a e^{2}}{3 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + 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*e^2*x^2 + 3*c*d*e*x + c*d^2 + a*e^2)/(e^6*x^3 + 3*d*e^5*x^2 + 3*d^2*e^4*x + d^3*e^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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sympy [A]  time = 0.44, 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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