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

Optimal. Leaf size=43 \[ -\frac {a e^2+c d^2}{e^3 (d+e x)}-\frac {2 c d \log (d+e x)}{e^3}+\frac {c x}{e^2} \]

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Rubi [A]  time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, 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}{e^3 (d+e x)}-\frac {2 c d \log (d+e x)}{e^3}+\frac {c x}{e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)/(d + e*x)^2,x]

[Out]

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

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

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Mathematica [A]  time = 0.02, size = 39, normalized size = 0.91 \begin {gather*} \frac {-\frac {a e^2+c d^2}{d+e x}-2 c d \log (d+e x)+c e x}{e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)/(d + e*x)^2,x]

[Out]

(c*e*x - (c*d^2 + a*e^2)/(d + e*x) - 2*c*d*Log[d + e*x])/e^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)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 59, normalized size = 1.37 \begin {gather*} \frac {c e^{2} x^{2} + c d e x - c d^{2} - a e^{2} - 2 \, {\left (c d e x + c d^{2}\right )} \log \left (e x + d\right )}{e^{4} x + d e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*e^2*x^2 + c*d*e*x - c*d^2 - a*e^2 - 2*(c*d*e*x + c*d^2)*log(e*x + d))/(e^4*x + d*e^3)

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giac [A]  time = 0.15, size = 65, normalized size = 1.51 \begin {gather*} {\left (2 \, d e^{\left (-3\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + {\left (x e + d\right )} e^{\left (-3\right )} - \frac {d^{2} e^{\left (-3\right )}}{x e + d}\right )} c - \frac {a e^{\left (-1\right )}}{x e + d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 50, normalized size = 1.16 \begin {gather*} -\frac {a}{\left (e x +d \right ) e}-\frac {c \,d^{2}}{\left (e x +d \right ) e^{3}}-\frac {2 c d \ln \left (e x +d \right )}{e^{3}}+\frac {c x}{e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.34, size = 46, normalized size = 1.07 \begin {gather*} -\frac {c d^{2} + a e^{2}}{e^{4} x + d e^{3}} + \frac {c x}{e^{2}} - \frac {2 \, c d \log \left (e x + d\right )}{e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(c*d^2 + a*e^2)/(e^4*x + d*e^3) + c*x/e^2 - 2*c*d*log(e*x + d)/e^3

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mupad [B]  time = 0.06, size = 49, normalized size = 1.14 \begin {gather*} \frac {c\,x}{e^2}-\frac {c\,d^2+a\,e^2}{e\,\left (x\,e^3+d\,e^2\right )}-\frac {2\,c\,d\,\ln \left (d+e\,x\right )}{e^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)/(d + e*x)^2,x)

[Out]

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

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sympy [A]  time = 0.24, size = 42, normalized size = 0.98 \begin {gather*} - \frac {2 c d \log {\left (d + e x \right )}}{e^{3}} + \frac {c x}{e^{2}} + \frac {- a e^{2} - c d^{2}}{d e^{3} + e^{4} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*c*d*log(d + e*x)/e**3 + c*x/e**2 + (-a*e**2 - c*d**2)/(d*e**3 + e**4*x)

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