3.19.34 \(\int \frac {a d e+(c d^2+a e^2) x+c d e x^2}{(d+e x)^2} \, dx\) [1834]

Optimal. Leaf size=26 \[ \frac {c d x}{e}+\left (a-\frac {c d^2}{e^2}\right ) \log (d+e x) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {24, 45} \begin {gather*} \left (a-\frac {c d^2}{e^2}\right ) \log (d+e x)+\frac {c d x}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 24

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((A_.) + (B_.)*(v_) + (C_.)*(v_)^2), x_Symbol] :> Dist[1/b^2, Int[u*(a + b*
v)^(m + 1)*Simp[b*B - a*C + b*C*v, x], x], x] /; FreeQ[{a, b, A, B, C}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0] &&
LeQ[m, -1]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 30, normalized size = 1.15 \begin {gather*} \frac {c d x}{e}+\frac {\left (-c d^2+a e^2\right ) \log (d+e x)}{e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.43, size = 31, normalized size = 1.19

method	result	size
default	\(\frac {c d x}{e}+\frac {\left (e^{2} a -c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{2}}\)	\(31\)
risch	\(\frac {c d x}{e}+\ln \left (e x +d \right ) a -\frac {\ln \left (e x +d \right ) c \,d^{2}}{e^{2}}\)	\(32\)
norman	\(\frac {c d \,x^{2}-\frac {d^{3} c}{e^{2}}}{e x +d}+\frac {\left (e^{2} a -c \,d^{2}\right ) \ln \left (e x +d \right )}{e^{2}}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 29, normalized size = 1.12 \begin {gather*} c d x e^{\left (-1\right )} - {\left (c d^{2} - a e^{2}\right )} e^{\left (-2\right )} \log \left (x e + d\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 3.14, size = 30, normalized size = 1.15 \begin {gather*} {\left (c d x e - {\left (c d^{2} - a e^{2}\right )} \log \left (x e + d\right )\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.07, size = 26, normalized size = 1.00 \begin {gather*} \frac {c d x}{e} + \frac {\left (a e^{2} - c d^{2}\right ) \log {\left (d + e x \right )}}{e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (27) = 54\).
time = 0.91, size = 117, normalized size = 4.50 \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 d e - {\left (c d^{2} + a e^{2}\right )} {\left (e^{\left (-1\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - \frac {d e^{\left (-1\right )}}{x e + d}\right )} e^{\left (-1\right )} - \frac {a d}{x e + d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

Mupad [B]
time = 0.05, size = 30, normalized size = 1.15 \begin {gather*} \frac {\ln \left (d+e\,x\right )\,\left (a\,e^2-c\,d^2\right )}{e^2}+\frac {c\,d\,x}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________