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

Optimal. Leaf size=48 \[ \frac {e \log (a e+c d x)}{c^2 d^2}-\frac {c d^2-a e^2}{c^2 d^2 (a e+c d x)} \]

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Rubi [A]  time = 0.04, antiderivative size = 48, 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 \log (a e+c d x)}{c^2 d^2}-\frac {c d^2-a e^2}{c^2 d^2 (a e+c d x)} \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)^2,x]

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 47, normalized size = 0.98 \begin {gather*} \frac {a e^2-c d^2}{c^2 d^2 (a e+c d x)}+\frac {e \log (a e+c d x)}{c^2 d^2} \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)^2,x]

[Out]

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

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

[Out]

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

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fricas [A]  time = 0.40, size = 56, normalized size = 1.17 \begin {gather*} -\frac {c d^{2} - a e^{2} - {\left (c d e x + a e^{2}\right )} \log \left (c d x + a e\right )}{c^{3} d^{3} x + a c^{2} d^{2} e} \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)^2,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.26, size = 322, normalized size = 6.71 \begin {gather*} \frac {{\left (c^{3} d^{6} e - 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} - a^{3} e^{7}\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 )}{{\left (c^{4} d^{6} - 2 \, a c^{3} d^{4} e^{2} + a^{2} c^{2} d^{2} e^{4}\right )} \sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} + \frac {e \log \left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{2 \, c^{2} d^{2}} - \frac {c^{3} d^{7} - 3 \, a c^{2} d^{5} e^{2} + 3 \, a^{2} c d^{3} e^{4} - a^{3} d e^{6} + {\left (c^{3} d^{6} e - 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} - a^{3} e^{7}\right )} x}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )} c^{2} d^{2}} \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)^2,x, algorithm="giac")

[Out]

(c^3*d^6*e - 3*a*c^2*d^4*e^3 + 3*a^2*c*d^2*e^5 - a^3*e^7)*arctan((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))/((c^4*d^6 - 2*a*c^3*d^4*e^2 + a^2*c^2*d^2*e^4)*sqrt(-c^2*d^4 + 2*a*c*d^2*e^2 - a^2*e^
4)) + 1/2*e*log(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)/(c^2*d^2) - (c^3*d^7 - 3*a*c^2*d^5*e^2 + 3*a^2*c*d^3*e^
4 - a^3*d*e^6 + (c^3*d^6*e - 3*a*c^2*d^4*e^3 + 3*a^2*c*d^2*e^5 - a^3*e^7)*x)/((c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e
^4)*(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*c^2*d^2)

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maple [A]  time = 0.07, size = 55, normalized size = 1.15 \begin {gather*} \frac {a \,e^{2}}{\left (c d x +a e \right ) c^{2} d^{2}}-\frac {1}{\left (c d x +a e \right ) c}+\frac {e \ln \left (c d x +a e \right )}{c^{2} d^{2}} \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)^2,x)

[Out]

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

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maxima [A]  time = 1.09, size = 52, normalized size = 1.08 \begin {gather*} -\frac {c d^{2} - a e^{2}}{c^{3} d^{3} x + a c^{2} d^{2} e} + \frac {e \log \left (c d x + a e\right )}{c^{2} d^{2}} \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)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.07, size = 47, normalized size = 0.98 \begin {gather*} \frac {a\,e^2-c\,d^2}{c^2\,d^2\,\left (a\,e+c\,d\,x\right )}+\frac {e\,\ln \left (a\,e+c\,d\,x\right )}{c^2\,d^2} \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)^2,x)

[Out]

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

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sympy [A]  time = 0.25, size = 46, normalized size = 0.96 \begin {gather*} \frac {a e^{2} - c d^{2}}{a c^{2} d^{2} e + c^{3} d^{3} x} + \frac {e \log {\left (a e + c d x \right )}}{c^{2} d^{2}} \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)**2,x)

[Out]

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

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