3.16.48 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^3}{(d+e x)^9} \, dx\)

Optimal. Leaf size=73 \[ \frac {c d (a e+c d x)^4}{20 (d+e x)^4 \left (c d^2-a e^2\right )^2}+\frac {(a e+c d x)^4}{5 (d+e x)^5 \left (c d^2-a e^2\right )} \]

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Rubi [A]  time = 0.03, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {626, 45, 37} \begin {gather*} \frac {c d (a e+c d x)^4}{20 (d+e x)^4 \left (c d^2-a e^2\right )^2}+\frac {(a e+c d x)^4}{5 (d+e x)^5 \left (c d^2-a e^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^9,x]

[Out]

(a*e + c*d*x)^4/(5*(c*d^2 - a*e^2)*(d + e*x)^5) + (c*d*(a*e + c*d*x)^4)/(20*(c*d^2 - a*e^2)^2*(d + e*x)^4)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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 {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^9} \, dx &=\int \frac {(a e+c d x)^3}{(d+e x)^6} \, dx\\ &=\frac {(a e+c d x)^4}{5 \left (c d^2-a e^2\right ) (d+e x)^5}+\frac {(c d) \int \frac {(a e+c d x)^3}{(d+e x)^5} \, dx}{5 \left (c d^2-a e^2\right )}\\ &=\frac {(a e+c d x)^4}{5 \left (c d^2-a e^2\right ) (d+e x)^5}+\frac {c d (a e+c d x)^4}{20 \left (c d^2-a e^2\right )^2 (d+e x)^4}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 103, normalized size = 1.41 \begin {gather*} -\frac {4 a^3 e^6+3 a^2 c d e^4 (d+5 e x)+2 a c^2 d^2 e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+c^3 d^3 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )}{20 e^4 (d+e x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^9,x]

[Out]

-1/20*(4*a^3*e^6 + 3*a^2*c*d*e^4*(d + 5*e*x) + 2*a*c^2*d^2*e^2*(d^2 + 5*d*e*x + 10*e^2*x^2) + c^3*d^3*(d^3 + 5
*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3))/(e^4*(d + e*x)^5)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^9} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^9,x]

[Out]

IntegrateAlgebraic[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^9, x]

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fricas [B]  time = 0.39, size = 175, normalized size = 2.40 \begin {gather*} -\frac {10 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + 2 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + 4 \, a^{3} e^{6} + 10 \, {\left (c^{3} d^{4} e^{2} + 2 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 5 \, {\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x}{20 \, {\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\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)^3/(e*x+d)^9,x, algorithm="fricas")

[Out]

-1/20*(10*c^3*d^3*e^3*x^3 + c^3*d^6 + 2*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 4*a^3*e^6 + 10*(c^3*d^4*e^2 + 2*a*c^
2*d^2*e^4)*x^2 + 5*(c^3*d^5*e + 2*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*x)/(e^9*x^5 + 5*d*e^8*x^4 + 10*d^2*e^7*x^3 +
10*d^3*e^6*x^2 + 5*d^4*e^5*x + d^5*e^4)

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giac [B]  time = 0.17, size = 280, normalized size = 3.84 \begin {gather*} -\frac {{\left (10 \, c^{3} d^{3} x^{6} e^{6} + 40 \, c^{3} d^{4} x^{5} e^{5} + 65 \, c^{3} d^{5} x^{4} e^{4} + 56 \, c^{3} d^{6} x^{3} e^{3} + 28 \, c^{3} d^{7} x^{2} e^{2} + 8 \, c^{3} d^{8} x e + c^{3} d^{9} + 20 \, a c^{2} d^{2} x^{5} e^{7} + 70 \, a c^{2} d^{3} x^{4} e^{6} + 92 \, a c^{2} d^{4} x^{3} e^{5} + 56 \, a c^{2} d^{5} x^{2} e^{4} + 16 \, a c^{2} d^{6} x e^{3} + 2 \, a c^{2} d^{7} e^{2} + 15 \, a^{2} c d x^{4} e^{8} + 48 \, a^{2} c d^{2} x^{3} e^{7} + 54 \, a^{2} c d^{3} x^{2} e^{6} + 24 \, a^{2} c d^{4} x e^{5} + 3 \, a^{2} c d^{5} e^{4} + 4 \, a^{3} x^{3} e^{9} + 12 \, a^{3} d x^{2} e^{8} + 12 \, a^{3} d^{2} x e^{7} + 4 \, a^{3} d^{3} e^{6}\right )} e^{\left (-4\right )}}{20 \, {\left (x e + d\right )}^{8}} \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)^3/(e*x+d)^9,x, algorithm="giac")

[Out]

-1/20*(10*c^3*d^3*x^6*e^6 + 40*c^3*d^4*x^5*e^5 + 65*c^3*d^5*x^4*e^4 + 56*c^3*d^6*x^3*e^3 + 28*c^3*d^7*x^2*e^2
+ 8*c^3*d^8*x*e + c^3*d^9 + 20*a*c^2*d^2*x^5*e^7 + 70*a*c^2*d^3*x^4*e^6 + 92*a*c^2*d^4*x^3*e^5 + 56*a*c^2*d^5*
x^2*e^4 + 16*a*c^2*d^6*x*e^3 + 2*a*c^2*d^7*e^2 + 15*a^2*c*d*x^4*e^8 + 48*a^2*c*d^2*x^3*e^7 + 54*a^2*c*d^3*x^2*
e^6 + 24*a^2*c*d^4*x*e^5 + 3*a^2*c*d^5*e^4 + 4*a^3*x^3*e^9 + 12*a^3*d*x^2*e^8 + 12*a^3*d^2*x*e^7 + 4*a^3*d^3*e
^6)*e^(-4)/(x*e + d)^8

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maple [B]  time = 0.06, size = 141, normalized size = 1.93 \begin {gather*} -\frac {c^{3} d^{3}}{2 \left (e x +d \right )^{2} e^{4}}-\frac {\left (a \,e^{2}-c \,d^{2}\right ) c^{2} d^{2}}{\left (e x +d \right )^{3} e^{4}}-\frac {3 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) c d}{4 \left (e x +d \right )^{4} e^{4}}-\frac {a^{3} e^{6}-3 a^{2} c \,d^{2} e^{4}+3 a \,c^{2} d^{4} e^{2}-c^{3} d^{6}}{5 \left (e x +d \right )^{5} e^{4}} \end {gather*}

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)^3/(e*x+d)^9,x)

[Out]

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

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maxima [B]  time = 1.09, size = 175, normalized size = 2.40 \begin {gather*} -\frac {10 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + 2 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + 4 \, a^{3} e^{6} + 10 \, {\left (c^{3} d^{4} e^{2} + 2 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 5 \, {\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x}{20 \, {\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\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)^3/(e*x+d)^9,x, algorithm="maxima")

[Out]

-1/20*(10*c^3*d^3*e^3*x^3 + c^3*d^6 + 2*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 4*a^3*e^6 + 10*(c^3*d^4*e^2 + 2*a*c^
2*d^2*e^4)*x^2 + 5*(c^3*d^5*e + 2*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*x)/(e^9*x^5 + 5*d*e^8*x^4 + 10*d^2*e^7*x^3 +
10*d^3*e^6*x^2 + 5*d^4*e^5*x + d^5*e^4)

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mupad [B]  time = 0.60, size = 135, normalized size = 1.85 \begin {gather*} -\frac {d^2\,\left (\frac {3\,a^2\,c}{20}+a\,c^2\,x^2-\frac {c^3\,x^4}{4}\right )-d\,\left (\frac {c^3\,e\,x^5}{20}-\frac {3\,a^2\,c\,e\,x}{4}\right )+\frac {a^3\,e^2}{5}+\frac {a\,c^2\,d^4}{10\,e^2}+\frac {a\,c^2\,d^3\,x}{2\,e}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \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)^3/(d + e*x)^9,x)

[Out]

-(d^2*((3*a^2*c)/20 - (c^3*x^4)/4 + a*c^2*x^2) - d*((c^3*e*x^5)/20 - (3*a^2*c*e*x)/4) + (a^3*e^2)/5 + (a*c^2*d
^4)/(10*e^2) + (a*c^2*d^3*x)/(2*e))/(d^5 + e^5*x^5 + 5*d*e^4*x^4 + 10*d^3*e^2*x^2 + 10*d^2*e^3*x^3 + 5*d^4*e*x
)

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sympy [B]  time = 7.26, size = 187, normalized size = 2.56 \begin {gather*} \frac {- 4 a^{3} e^{6} - 3 a^{2} c d^{2} e^{4} - 2 a c^{2} d^{4} e^{2} - c^{3} d^{6} - 10 c^{3} d^{3} e^{3} x^{3} + x^{2} \left (- 20 a c^{2} d^{2} e^{4} - 10 c^{3} d^{4} e^{2}\right ) + x \left (- 15 a^{2} c d e^{5} - 10 a c^{2} d^{3} e^{3} - 5 c^{3} d^{5} e\right )}{20 d^{5} e^{4} + 100 d^{4} e^{5} x + 200 d^{3} e^{6} x^{2} + 200 d^{2} e^{7} x^{3} + 100 d e^{8} x^{4} + 20 e^{9} x^{5}} \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)**3/(e*x+d)**9,x)

[Out]

(-4*a**3*e**6 - 3*a**2*c*d**2*e**4 - 2*a*c**2*d**4*e**2 - c**3*d**6 - 10*c**3*d**3*e**3*x**3 + x**2*(-20*a*c**
2*d**2*e**4 - 10*c**3*d**4*e**2) + x*(-15*a**2*c*d*e**5 - 10*a*c**2*d**3*e**3 - 5*c**3*d**5*e))/(20*d**5*e**4
+ 100*d**4*e**5*x + 200*d**3*e**6*x**2 + 200*d**2*e**7*x**3 + 100*d*e**8*x**4 + 20*e**9*x**5)

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