3.1844 \(\int \frac{(a d e+(c d^2+a e^2) x+c d e x^2)^2}{(d+e x)^3} \, dx\)

Optimal. Leaf size=62 \[ -\frac{c d x \left (c d^2-a e^2\right )}{e^2}+\frac{\left (c d^2-a e^2\right )^2 \log (d+e x)}{e^3}+\frac{(a e+c d x)^2}{2 e} \]

[Out]

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

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Rubi [A]  time = 0.0391704, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {626, 43} \[ -\frac{c d x \left (c d^2-a e^2\right )}{e^2}+\frac{\left (c d^2-a e^2\right )^2 \log (d+e x)}{e^3}+\frac{(a e+c d x)^2}{2 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 626

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

Rule 43

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

Mathematica [A]  time = 0.0215863, size = 52, normalized size = 0.84 \[ \frac{2 \left (c d^2-a e^2\right )^2 \log (d+e x)+c d e x \left (4 a e^2+c d (e x-2 d)\right )}{2 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.042, size = 77, normalized size = 1.2 \begin{align*}{\frac{{c}^{2}{d}^{2}{x}^{2}}{2\,e}}+2\,cdax-{\frac{{c}^{2}{d}^{3}x}{{e}^{2}}}+e\ln \left ( ex+d \right ){a}^{2}-2\,{\frac{\ln \left ( ex+d \right ) ac{d}^{2}}{e}}+{\frac{\ln \left ( ex+d \right ){c}^{2}{d}^{4}}{{e}^{3}}} \end{align*}

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

[Out]

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

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Maxima [A]  time = 1.11431, size = 97, normalized size = 1.56 \begin{align*} \frac{c^{2} d^{2} e x^{2} - 2 \,{\left (c^{2} d^{3} - 2 \, a c d e^{2}\right )} x}{2 \, e^{2}} + \frac{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \log \left (e x + d\right )}{e^{3}} \end{align*}

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)^2/(e*x+d)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.58243, size = 151, normalized size = 2.44 \begin{align*} \frac{c^{2} d^{2} e^{2} x^{2} - 2 \,{\left (c^{2} d^{3} e - 2 \, a c d e^{3}\right )} x + 2 \,{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \log \left (e x + d\right )}{2 \, e^{3}} \end{align*}

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)^2/(e*x+d)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.538764, size = 56, normalized size = 0.9 \begin{align*} \frac{c^{2} d^{2} x^{2}}{2 e} + \frac{x \left (2 a c d e^{2} - c^{2} d^{3}\right )}{e^{2}} + \frac{\left (a e^{2} - c d^{2}\right )^{2} \log{\left (d + e x \right )}}{e^{3}} \end{align*}

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)**2/(e*x+d)**3,x)

[Out]

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

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Giac [A]  time = 1.25246, size = 96, normalized size = 1.55 \begin{align*}{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} e^{\left (-3\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (c^{2} d^{2} x^{2} e^{5} - 2 \, c^{2} d^{3} x e^{4} + 4 \, a c d x e^{6}\right )} e^{\left (-6\right )} \end{align*}

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)^2/(e*x+d)^3,x, algorithm="giac")

[Out]

(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*e^(-3)*log(abs(x*e + d)) + 1/2*(c^2*d^2*x^2*e^5 - 2*c^2*d^3*x*e^4 + 4*a*c*
d*x*e^6)*e^(-6)