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

Optimal. Leaf size=41 \[ \frac {\left (a e^2+c d^2\right ) \log (d+e x)}{e^3}-\frac {c d x}{e^2}+\frac {c x^2}{2 e} \]

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Rubi [A]  time = 0.03, antiderivative size = 41, 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 {\left (a e^2+c d^2\right ) \log (d+e x)}{e^3}-\frac {c d x}{e^2}+\frac {c x^2}{2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)/(d + e*x),x]

[Out]

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

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Mathematica [A]  time = 0.01, size = 38, normalized size = 0.93 \begin {gather*} \frac {2 \left (a e^2+c d^2\right ) \log (d+e x)+c e x (e x-2 d)}{2 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)/(d + e*x),x]

[Out]

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 39, normalized size = 0.95 \begin {gather*} \frac {c e^{2} x^{2} - 2 \, c d e x + 2 \, {\left (c d^{2} + a e^{2}\right )} \log \left (e x + d\right )}{2 \, e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.15, size = 39, normalized size = 0.95 \begin {gather*} {\left (c d^{2} + a e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{2} \, {\left (c x^{2} e - 2 \, c d x\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 44, normalized size = 1.07 \begin {gather*} \frac {c \,x^{2}}{2 e}+\frac {a \ln \left (e x +d \right )}{e}+\frac {c \,d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {c d x}{e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.35, size = 39, normalized size = 0.95 \begin {gather*} \frac {c e x^{2} - 2 \, c d x}{2 \, e^{2}} + \frac {{\left (c d^{2} + a e^{2}\right )} \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),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.05, size = 39, normalized size = 0.95 \begin {gather*} \frac {c\,x^2}{2\,e}+\frac {\ln \left (d+e\,x\right )\,\left (c\,d^2+a\,e^2\right )}{e^3}-\frac {c\,d\,x}{e^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)/(d + e*x),x)

[Out]

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

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sympy [A]  time = 0.17, size = 36, normalized size = 0.88 \begin {gather*} - \frac {c d x}{e^{2}} + \frac {c x^{2}}{2 e} + \frac {\left (a e^{2} + c d^{2}\right ) \log {\left (d + e x \right )}}{e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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