3.16.75 \(\int \frac {(d+e x)^5}{(a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx\)

Optimal. Leaf size=85 \[ \frac {e^2 \log (a e+c d x)}{c^3 d^3}-\frac {2 e \left (c d^2-a e^2\right )}{c^3 d^3 (a e+c d x)}-\frac {\left (c d^2-a e^2\right )^2}{2 c^3 d^3 (a e+c d x)^2} \]

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Rubi [A]  time = 0.06, antiderivative size = 85, 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 {2 e \left (c d^2-a e^2\right )}{c^3 d^3 (a e+c d x)}-\frac {\left (c d^2-a e^2\right )^2}{2 c^3 d^3 (a e+c d x)^2}+\frac {e^2 \log (a e+c d x)}{c^3 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

-(c*d^2 - a*e^2)^2/(2*c^3*d^3*(a*e + c*d*x)^2) - (2*e*(c*d^2 - a*e^2))/(c^3*d^3*(a*e + c*d*x)) + (e^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)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx &=\int \frac {(d+e x)^2}{(a e+c d x)^3} \, dx\\ &=\int \left (\frac {\left (c d^2-a e^2\right )^2}{c^2 d^2 (a e+c d x)^3}+\frac {2 \left (c d^2 e-a e^3\right )}{c^2 d^2 (a e+c d x)^2}+\frac {e^2}{c^2 d^2 (a e+c d x)}\right ) \, dx\\ &=-\frac {\left (c d^2-a e^2\right )^2}{2 c^3 d^3 (a e+c d x)^2}-\frac {2 e \left (c d^2-a e^2\right )}{c^3 d^3 (a e+c d x)}+\frac {e^2 \log (a e+c d x)}{c^3 d^3}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 65, normalized size = 0.76 \begin {gather*} \frac {2 e^2 \log (a e+c d x)-\frac {\left (c d^2-a e^2\right ) \left (3 a e^2+c d (d+4 e x)\right )}{(a e+c d x)^2}}{2 c^3 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

(-(((c*d^2 - a*e^2)*(3*a*e^2 + c*d*(d + 4*e*x)))/(a*e + c*d*x)^2) + 2*e^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)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

IntegrateAlgebraic[(d + e*x)^5/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3, x]

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fricas [A]  time = 0.40, size = 126, normalized size = 1.48 \begin {gather*} -\frac {c^{2} d^{4} + 2 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4} + 4 \, {\left (c^{2} d^{3} e - a c d e^{3}\right )} x - 2 \, {\left (c^{2} d^{2} e^{2} x^{2} + 2 \, a c d e^{3} x + a^{2} e^{4}\right )} \log \left (c d x + a e\right )}{2 \, {\left (c^{5} d^{5} x^{2} + 2 \, a c^{4} d^{4} e x + a^{2} c^{3} d^{3} e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.33, size = 601, normalized size = 7.07 \begin {gather*} \frac {{\left (c^{5} d^{10} e^{2} - 5 \, a c^{4} d^{8} e^{4} + 10 \, a^{2} c^{3} d^{6} e^{6} - 10 \, a^{3} c^{2} d^{4} e^{8} + 5 \, a^{4} c d^{2} e^{10} - a^{5} e^{12}\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^{7} d^{11} - 4 \, a c^{6} d^{9} e^{2} + 6 \, a^{2} c^{5} d^{7} e^{4} - 4 \, a^{3} c^{4} d^{5} e^{6} + a^{4} c^{3} d^{3} e^{8}\right )} \sqrt {-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} + \frac {e^{2} \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 {c^{6} d^{14} - 2 \, a c^{5} d^{12} e^{2} - 5 \, a^{2} c^{4} d^{10} e^{4} + 20 \, a^{3} c^{3} d^{8} e^{6} - 25 \, a^{4} c^{2} d^{6} e^{8} + 14 \, a^{5} c d^{4} e^{10} - 3 \, a^{6} d^{2} e^{12} + 4 \, {\left (c^{6} d^{11} e^{3} - 5 \, a c^{5} d^{9} e^{5} + 10 \, a^{2} c^{4} d^{7} e^{7} - 10 \, a^{3} c^{3} d^{5} e^{9} + 5 \, a^{4} c^{2} d^{3} e^{11} - a^{5} c d e^{13}\right )} x^{3} + 3 \, {\left (3 \, c^{6} d^{12} e^{2} - 14 \, a c^{5} d^{10} e^{4} + 25 \, a^{2} c^{4} d^{8} e^{6} - 20 \, a^{3} c^{3} d^{6} e^{8} + 5 \, a^{4} c^{2} d^{4} e^{10} + 2 \, a^{5} c d^{2} e^{12} - a^{6} e^{14}\right )} x^{2} + 6 \, {\left (c^{6} d^{13} e - 4 \, a c^{5} d^{11} e^{3} + 5 \, a^{2} c^{4} d^{9} e^{5} - 5 \, a^{4} c^{2} d^{5} e^{9} + 4 \, a^{5} c d^{3} e^{11} - a^{6} d e^{13}\right )} x}{2 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}^{2} {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{2} c^{3} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="giac")

[Out]

(c^5*d^10*e^2 - 5*a*c^4*d^8*e^4 + 10*a^2*c^3*d^6*e^6 - 10*a^3*c^2*d^4*e^8 + 5*a^4*c*d^2*e^10 - a^5*e^12)*arcta
n((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^7*d^11 - 4*a*c^6*d^9*e^2 + 6*a^2*c
^5*d^7*e^4 - 4*a^3*c^4*d^5*e^6 + a^4*c^3*d^3*e^8)*sqrt(-c^2*d^4 + 2*a*c*d^2*e^2 - a^2*e^4)) + 1/2*e^2*log(c*d*
x^2*e + c*d^2*x + a*x*e^2 + a*d*e)/(c^3*d^3) - 1/2*(c^6*d^14 - 2*a*c^5*d^12*e^2 - 5*a^2*c^4*d^10*e^4 + 20*a^3*
c^3*d^8*e^6 - 25*a^4*c^2*d^6*e^8 + 14*a^5*c*d^4*e^10 - 3*a^6*d^2*e^12 + 4*(c^6*d^11*e^3 - 5*a*c^5*d^9*e^5 + 10
*a^2*c^4*d^7*e^7 - 10*a^3*c^3*d^5*e^9 + 5*a^4*c^2*d^3*e^11 - a^5*c*d*e^13)*x^3 + 3*(3*c^6*d^12*e^2 - 14*a*c^5*
d^10*e^4 + 25*a^2*c^4*d^8*e^6 - 20*a^3*c^3*d^6*e^8 + 5*a^4*c^2*d^4*e^10 + 2*a^5*c*d^2*e^12 - a^6*e^14)*x^2 + 6
*(c^6*d^13*e - 4*a*c^5*d^11*e^3 + 5*a^2*c^4*d^9*e^5 - 5*a^4*c^2*d^5*e^9 + 4*a^5*c*d^3*e^11 - a^6*d*e^13)*x)/((
c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)^2*(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)^2*c^3*d^3)

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maple [A]  time = 0.05, size = 123, normalized size = 1.45 \begin {gather*} -\frac {a^{2} e^{4}}{2 \left (c d x +a e \right )^{2} c^{3} d^{3}}+\frac {a \,e^{2}}{\left (c d x +a e \right )^{2} c^{2} d}-\frac {d}{2 \left (c d x +a e \right )^{2} c}+\frac {2 a \,e^{3}}{\left (c d x +a e \right ) c^{3} d^{3}}-\frac {2 e}{\left (c d x +a e \right ) c^{2} d}+\frac {e^{2} \ln \left (c d x +a e \right )}{c^{3} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x)

[Out]

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

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maxima [A]  time = 1.18, size = 105, normalized size = 1.24 \begin {gather*} -\frac {c^{2} d^{4} + 2 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4} + 4 \, {\left (c^{2} d^{3} e - a c d e^{3}\right )} x}{2 \, {\left (c^{5} d^{5} x^{2} + 2 \, a c^{4} d^{4} e x + a^{2} c^{3} d^{3} e^{2}\right )}} + \frac {e^{2} \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)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.09, size = 106, normalized size = 1.25 \begin {gather*} \frac {e^2\,\ln \left (a\,e+c\,d\,x\right )}{c^3\,d^3}-\frac {\frac {-3\,a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4}{2\,c^3\,d^3}-\frac {2\,e\,x\,\left (a\,e^2-c\,d^2\right )}{c^2\,d^2}}{a^2\,e^2+2\,a\,c\,d\,e\,x+c^2\,d^2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^5/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3,x)

[Out]

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

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sympy [A]  time = 0.61, size = 109, normalized size = 1.28 \begin {gather*} \frac {3 a^{2} e^{4} - 2 a c d^{2} e^{2} - c^{2} d^{4} + x \left (4 a c d e^{3} - 4 c^{2} d^{3} e\right )}{2 a^{2} c^{3} d^{3} e^{2} + 4 a c^{4} d^{4} e x + 2 c^{5} d^{5} x^{2}} + \frac {e^{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)**5/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

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

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