3.4.77 \(\int \frac {a+c x^2}{(d+e x)^3} \, dx\)

Optimal. Leaf size=53 \[ \frac {-a e^2-c d^2}{2 e^3 (d+e x)^2}+\frac {2 c d}{e^3 (d+e x)}+\frac {c \log (d+e x)}{e^3} \]

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 51, 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}{2 e^3 (d+e x)^2}+\frac {2 c d}{e^3 (d+e x)}+\frac {c \log (d+e x)}{e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)/(d + e*x)^3,x]

[Out]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 48, normalized size = 0.91 \begin {gather*} \frac {-a e^2+c d (3 d+4 e x)+2 c (d+e x)^2 \log (d+e x)}{2 e^3 (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)/(d + e*x)^3,x]

[Out]

(-(a*e^2) + c*d*(3*d + 4*e*x) + 2*c*(d + e*x)^2*Log[d + e*x])/(2*e^3*(d + e*x)^2)

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+c x^2}{(d+e x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + c*x^2)/(d + e*x)^3,x]

[Out]

IntegrateAlgebraic[(a + c*x^2)/(d + e*x)^3, x]

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 73, normalized size = 1.38 \begin {gather*} \frac {4 \, c d e x + 3 \, c d^{2} - a e^{2} + 2 \, {\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )} \log \left (e x + d\right )}{2 \, {\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)/(e*x+d)^3,x, algorithm="fricas")

[Out]

1/2*(4*c*d*e*x + 3*c*d^2 - a*e^2 + 2*(c*e^2*x^2 + 2*c*d*e*x + c*d^2)*log(e*x + d))/(e^5*x^2 + 2*d*e^4*x + d^2*
e^3)

________________________________________________________________________________________

giac [A]  time = 0.16, size = 46, normalized size = 0.87 \begin {gather*} c e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {{\left (4 \, c d x + {\left (3 \, c d^{2} - a e^{2}\right )} e^{\left (-1\right )}\right )} e^{\left (-2\right )}}{2 \, {\left (x e + d\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)/(e*x+d)^3,x, algorithm="giac")

[Out]

c*e^(-3)*log(abs(x*e + d)) + 1/2*(4*c*d*x + (3*c*d^2 - a*e^2)*e^(-1))*e^(-2)/(x*e + d)^2

________________________________________________________________________________________

maple [A]  time = 0.05, size = 56, normalized size = 1.06 \begin {gather*} -\frac {a}{2 \left (e x +d \right )^{2} e}-\frac {c \,d^{2}}{2 \left (e x +d \right )^{2} e^{3}}+\frac {2 c d}{\left (e x +d \right ) e^{3}}+\frac {c \ln \left (e x +d \right )}{e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.32, size = 57, normalized size = 1.08 \begin {gather*} \frac {4 \, c d e x + 3 \, c d^{2} - a e^{2}}{2 \, {\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} + \frac {c \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)^3,x, algorithm="maxima")

[Out]

1/2*(4*c*d*e*x + 3*c*d^2 - a*e^2)/(e^5*x^2 + 2*d*e^4*x + d^2*e^3) + c*log(e*x + d)/e^3

________________________________________________________________________________________

mupad [B]  time = 0.28, size = 58, normalized size = 1.09 \begin {gather*} \frac {c\,\ln \left (d+e\,x\right )}{e^3}-\frac {\frac {a\,e^2-3\,c\,d^2}{2\,e^3}-\frac {2\,c\,d\,x}{e^2}}{d^2+2\,d\,e\,x+e^2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)/(d + e*x)^3,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.33, size = 56, normalized size = 1.06 \begin {gather*} \frac {c \log {\left (d + e x \right )}}{e^{3}} + \frac {- a e^{2} + 3 c d^{2} + 4 c d e x}{2 d^{2} e^{3} + 4 d e^{4} x + 2 e^{5} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)/(e*x+d)**3,x)

[Out]

c*log(d + e*x)/e**3 + (-a*e**2 + 3*c*d**2 + 4*c*d*e*x)/(2*d**2*e**3 + 4*d*e**4*x + 2*e**5*x**2)

________________________________________________________________________________________