3.5.55 \(\int \frac {a+c x^2}{d+e x} \, dx\) [455]

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

[Out]

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

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Rubi [A]
time = 0.02, 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 = {711} \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 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} \, 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 {c e x (-2 d+e x)+2 \left (c d^2+a e^2\right ) \log (d+e x)}{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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Maple [A]
time = 0.39, size = 39, normalized size = 0.95

method	result	size
default	\(-\frac {c \left (-\frac {1}{2} e \,x^{2}+d x \right )}{e^{2}}+\frac {\left (e^{2} a +c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{3}}\)	\(39\)
norman	\(-\frac {c d x}{e^{2}}+\frac {c \,x^{2}}{2 e}+\frac {\left (e^{2} a +c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{3}}\)	\(40\)
risch	\(\frac {c \,x^{2}}{2 e}-\frac {c d x}{e^{2}}+\frac {\ln \left (e x +d \right ) a}{e}+\frac {\ln \left (e x +d \right ) c \,d^{2}}{e^{3}}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.33, size = 38, normalized size = 0.93 \begin {gather*} {\left (c d^{2} + a e^{2}\right )} e^{\left (-3\right )} \log \left (x e + d\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="maxima")

[Out]

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

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Fricas [A]
time = 1.10, size = 38, normalized size = 0.93 \begin {gather*} \frac {1}{2} \, {\left (c x^{2} e^{2} - 2 \, c d x e + 2 \, {\left (c d^{2} + a e^{2}\right )} \log \left (x e + d\right )\right )} e^{\left (-3\right )} \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*x^2*e^2 - 2*c*d*x*e + 2*(c*d^2 + a*e^2)*log(x*e + d))*e^(-3)

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Sympy [A]
time = 0.06, 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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Giac [A]
time = 2.45, 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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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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