3.3.26 \(\int \frac {b x+c x^2}{d+e x} \, dx\) [226]

Optimal. Leaf size=45 \[ -\frac {(c d-b e) x}{e^2}+\frac {c x^2}{2 e}+\frac {d (c d-b e) \log (d+e x)}{e^3} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {712} \begin {gather*} \frac {d (c d-b e) \log (d+e x)}{e^3}-\frac {x (c d-b e)}{e^2}+\frac {c x^2}{2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)/(d + e*x),x]

[Out]

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

Rule 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 41, normalized size = 0.91 \begin {gather*} \frac {e x (-2 c d+2 b e+c e x)+2 d (c d-b e) \log (d+e x)}{2 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x + c*x^2)/(d + e*x),x]

[Out]

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

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Maple [A]
time = 0.40, size = 43, normalized size = 0.96

method	result	size
default	\(\frac {\frac {1}{2} c e \,x^{2}+b e x -c d x}{e^{2}}-\frac {d \left (b e -c d \right ) \ln \left (e x +d \right )}{e^{3}}\)	\(43\)
norman	\(\frac {\left (b e -c d \right ) x}{e^{2}}+\frac {c \,x^{2}}{2 e}-\frac {d \left (b e -c d \right ) \ln \left (e x +d \right )}{e^{3}}\)	\(44\)
risch	\(\frac {c \,x^{2}}{2 e}+\frac {b x}{e}-\frac {c d x}{e^{2}}-\frac {d \ln \left (e x +d \right ) b}{e^{2}}+\frac {d^{2} \ln \left (e x +d \right ) c}{e^{3}}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 47, normalized size = 1.04 \begin {gather*} {\left (c d^{2} - b d e\right )} e^{\left (-3\right )} \log \left (x e + d\right ) + \frac {1}{2} \, {\left (c x^{2} e - 2 \, {\left (c d - b e\right )} x\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 2.07, size = 47, normalized size = 1.04 \begin {gather*} -\frac {1}{2} \, {\left (2 \, c d x e - {\left (c x^{2} + 2 \, b x\right )} e^{2} - 2 \, {\left (c d^{2} - b d e\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+b*x)/(e*x+d),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.08, size = 37, normalized size = 0.82 \begin {gather*} \frac {c x^{2}}{2 e} - \frac {d \left (b e - c d\right ) \log {\left (d + e x \right )}}{e^{3}} + x \left (\frac {b}{e} - \frac {c d}{e^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.68, size = 47, normalized size = 1.04 \begin {gather*} {\left (c d^{2} - b d e\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 + 2 \, b x e\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.07, size = 46, normalized size = 1.02 \begin {gather*} x\,\left (\frac {b}{e}-\frac {c\,d}{e^2}\right )+\frac {c\,x^2}{2\,e}+\frac {\ln \left (d+e\,x\right )\,\left (c\,d^2-b\,d\,e\right )}{e^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)/(d + e*x),x)

[Out]

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

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