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

Optimal. Leaf size=69 \[ \frac {\left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^3 d^3}+\frac {e x \left (c d^2-a e^2\right )}{c^2 d^2}+\frac {(d+e x)^2}{2 c d} \]

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Rubi [A]  time = 0.03, antiderivative size = 69, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {e x \left (c d^2-a e^2\right )}{c^2 d^2}+\frac {\left (c d^2-a e^2\right )^2 \log (a e+c d x)}{c^3 d^3}+\frac {(d+e x)^2}{2 c d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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])

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]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 58, normalized size = 0.84 \begin {gather*} \frac {2 \left (c d^2-a e^2\right )^2 \log (a e+c d x)+c d e x \left (c d (4 d+e x)-2 a e^2\right )}{2 c^3 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 79, normalized size = 1.14 \begin {gather*} \frac {c^{2} d^{2} e^{2} x^{2} + 2 \, {\left (2 \, c^{2} d^{3} e - 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 (c d x + a e\right )}{2 \, c^{3} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.19, size = 209, normalized size = 3.03 \begin {gather*} \frac {{\left (c d x^{2} e^{4} + 4 \, c d^{2} x e^{3} - 2 \, a x e^{5}\right )} e^{\left (-2\right )}}{2 \, c^{2} d^{2}} + \frac {{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \log \left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{2 \, c^{3} d^{3}} + \frac {{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \arctan \left (\frac {2 \, c d x e + c d^{2} + a e^{2}}{\sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{\sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}} c^{3} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 93, normalized size = 1.35 \begin {gather*} \frac {e^{2} x^{2}}{2 c d}+\frac {a^{2} e^{4} \ln \left (c d x +a e \right )}{c^{3} d^{3}}-\frac {2 a \,e^{2} \ln \left (c d x +a e \right )}{c^{2} d}-\frac {a \,e^{3} x}{c^{2} d^{2}}+\frac {d \ln \left (c d x +a e \right )}{c}+\frac {2 e x}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.28, size = 77, normalized size = 1.12 \begin {gather*} \frac {c d e^{2} x^{2} + 2 \, {\left (2 \, c d^{2} e - a e^{3}\right )} x}{2 \, c^{2} d^{2}} + \frac {{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \log \left (c d x + a e\right )}{c^{3} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.07, size = 77, normalized size = 1.12 \begin {gather*} x\,\left (\frac {2\,e}{c}-\frac {a\,e^3}{c^2\,d^2}\right )+\frac {\ln \left (a\,e+c\,d\,x\right )\,\left (a^2\,e^4-2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{c^3\,d^3}+\frac {e^2\,x^2}{2\,c\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.28, size = 58, normalized size = 0.84 \begin {gather*} x \left (- \frac {a e^{3}}{c^{2} d^{2}} + \frac {2 e}{c}\right ) + \frac {e^{2} x^{2}}{2 c d} + \frac {\left (a e^{2} - c d^{2}\right )^{2} \log {\left (a e + c d x \right )}}{c^{3} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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