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

Optimal. Leaf size=113 \[ -\frac {2 c^2 d^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )}{e^4}+\frac {6 c d \sqrt {d+e x} \left (c d^2-a e^2\right )^2}{e^4}+\frac {2 \left (c d^2-a e^2\right )^3}{e^4 \sqrt {d+e x}}+\frac {2 c^3 d^3 (d+e x)^{5/2}}{5 e^4} \]

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Rubi [A]  time = 0.05, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {626, 43} \begin {gather*} -\frac {2 c^2 d^2 (d+e x)^{3/2} \left (c d^2-a e^2\right )}{e^4}+\frac {6 c d \sqrt {d+e x} \left (c d^2-a e^2\right )^2}{e^4}+\frac {2 \left (c d^2-a e^2\right )^3}{e^4 \sqrt {d+e x}}+\frac {2 c^3 d^3 (d+e x)^{5/2}}{5 e^4} \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)^(9/2),x]

[Out]

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

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

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Mathematica [A]  time = 0.05, size = 109, normalized size = 0.96 \begin {gather*} \frac {2 \left (-5 a^3 e^6+15 a^2 c d e^4 (2 d+e x)-5 a c^2 d^2 e^2 \left (8 d^2+4 d e x-e^2 x^2\right )+c^3 d^3 \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )}{5 e^4 \sqrt {d+e x}} \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)^(9/2),x]

[Out]

(2*(-5*a^3*e^6 + 15*a^2*c*d*e^4*(2*d + e*x) - 5*a*c^2*d^2*e^2*(8*d^2 + 4*d*e*x - e^2*x^2) + c^3*d^3*(16*d^3 +
8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3)))/(5*e^4*Sqrt[d + e*x])

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IntegrateAlgebraic [A]  time = 0.13, size = 150, normalized size = 1.33 \begin {gather*} \frac {2 \left (-5 a^3 e^6+15 a^2 c d^2 e^4+15 a^2 c d e^4 (d+e x)-15 a c^2 d^4 e^2-30 a c^2 d^3 e^2 (d+e x)+5 a c^2 d^2 e^2 (d+e x)^2+5 c^3 d^6+15 c^3 d^5 (d+e x)-5 c^3 d^4 (d+e x)^2+c^3 d^3 (d+e x)^3\right )}{5 e^4 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(5*c^3*d^6 - 15*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 - 5*a^3*e^6 + 15*c^3*d^5*(d + e*x) - 30*a*c^2*d^3*e^2*(d +
e*x) + 15*a^2*c*d*e^4*(d + e*x) - 5*c^3*d^4*(d + e*x)^2 + 5*a*c^2*d^2*e^2*(d + e*x)^2 + c^3*d^3*(d + e*x)^3))
/(5*e^4*Sqrt[d + e*x])

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fricas [A]  time = 0.41, size = 139, normalized size = 1.23 \begin {gather*} \frac {2 \, {\left (c^{3} d^{3} e^{3} x^{3} + 16 \, c^{3} d^{6} - 40 \, a c^{2} d^{4} e^{2} + 30 \, a^{2} c d^{2} e^{4} - 5 \, a^{3} e^{6} - {\left (2 \, c^{3} d^{4} e^{2} - 5 \, a c^{2} d^{2} e^{4}\right )} x^{2} + {\left (8 \, c^{3} d^{5} e - 20 \, a c^{2} d^{3} e^{3} + 15 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {e x + d}}{5 \, {\left (e^{5} x + d 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)^(9/2),x, algorithm="fricas")

[Out]

2/5*(c^3*d^3*e^3*x^3 + 16*c^3*d^6 - 40*a*c^2*d^4*e^2 + 30*a^2*c*d^2*e^4 - 5*a^3*e^6 - (2*c^3*d^4*e^2 - 5*a*c^2
*d^2*e^4)*x^2 + (8*c^3*d^5*e - 20*a*c^2*d^3*e^3 + 15*a^2*c*d*e^5)*x)*sqrt(e*x + d)/(e^5*x + d*e^4)

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giac [A]  time = 0.31, size = 195, normalized size = 1.73 \begin {gather*} \frac {2}{5} \, {\left ({\left (x e + d\right )}^{\frac {5}{2}} c^{3} d^{3} e^{16} - 5 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{4} e^{16} + 15 \, \sqrt {x e + d} c^{3} d^{5} e^{16} + 5 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{2} d^{2} e^{18} - 30 \, \sqrt {x e + d} a c^{2} d^{3} e^{18} + 15 \, \sqrt {x e + d} a^{2} c d e^{20}\right )} e^{\left (-20\right )} + \frac {2 \, {\left ({\left (x e + d\right )}^{3} c^{3} d^{6} - 3 \, {\left (x e + d\right )}^{3} a c^{2} d^{4} e^{2} + 3 \, {\left (x e + d\right )}^{3} a^{2} c d^{2} e^{4} - {\left (x e + d\right )}^{3} a^{3} e^{6}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{\frac {7}{2}}} \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)^(9/2),x, algorithm="giac")

[Out]

2/5*((x*e + d)^(5/2)*c^3*d^3*e^16 - 5*(x*e + d)^(3/2)*c^3*d^4*e^16 + 15*sqrt(x*e + d)*c^3*d^5*e^16 + 5*(x*e +
d)^(3/2)*a*c^2*d^2*e^18 - 30*sqrt(x*e + d)*a*c^2*d^3*e^18 + 15*sqrt(x*e + d)*a^2*c*d*e^20)*e^(-20) + 2*((x*e +
d)^3*c^3*d^6 - 3*(x*e + d)^3*a*c^2*d^4*e^2 + 3*(x*e + d)^3*a^2*c*d^2*e^4 - (x*e + d)^3*a^3*e^6)*e^(-4)/(x*e +
d)^(7/2)

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maple [A]  time = 0.04, size = 131, normalized size = 1.16 \begin {gather*} -\frac {2 \left (-c^{3} d^{3} e^{3} x^{3}-5 a \,c^{2} d^{2} e^{4} x^{2}+2 c^{3} d^{4} e^{2} x^{2}-15 a^{2} c d \,e^{5} x +20 a \,c^{2} d^{3} e^{3} x -8 c^{3} d^{5} e x +5 a^{3} e^{6}-30 a^{2} c \,d^{2} e^{4}+40 a \,c^{2} d^{4} e^{2}-16 c^{3} d^{6}\right )}{5 \sqrt {e x +d}\, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^3/(e*x+d)^(9/2),x)

[Out]

-2/5/(e*x+d)^(1/2)*(-c^3*d^3*e^3*x^3-5*a*c^2*d^2*e^4*x^2+2*c^3*d^4*e^2*x^2-15*a^2*c*d*e^5*x+20*a*c^2*d^3*e^3*x
-8*c^3*d^5*e*x+5*a^3*e^6-30*a^2*c*d^2*e^4+40*a*c^2*d^4*e^2-16*c^3*d^6)/e^4

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maxima [A]  time = 1.03, size = 144, normalized size = 1.27 \begin {gather*} \frac {2 \, {\left (\frac {{\left (e x + d\right )}^{\frac {5}{2}} c^{3} d^{3} - 5 \, {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 15 \, {\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} \sqrt {e x + d}}{e^{3}} + \frac {5 \, {\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 )}}{\sqrt {e x + d} e^{3}}\right )}}{5 \, e} \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)^(9/2),x, algorithm="maxima")

[Out]

2/5*(((e*x + d)^(5/2)*c^3*d^3 - 5*(c^3*d^4 - a*c^2*d^2*e^2)*(e*x + d)^(3/2) + 15*(c^3*d^5 - 2*a*c^2*d^3*e^2 +
a^2*c*d*e^4)*sqrt(e*x + d))/e^3 + 5*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)/(sqrt(e*x + d)*e^3
))/e

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mupad [B]  time = 0.07, size = 133, normalized size = 1.18 \begin {gather*} \frac {2\,c^3\,d^3\,{\left (d+e\,x\right )}^{5/2}}{5\,e^4}-\frac {\left (6\,c^3\,d^4-6\,a\,c^2\,d^2\,e^2\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,e^4}-\frac {2\,a^3\,e^6-6\,a^2\,c\,d^2\,e^4+6\,a\,c^2\,d^4\,e^2-2\,c^3\,d^6}{e^4\,\sqrt {d+e\,x}}+\frac {6\,c\,d\,{\left (a\,e^2-c\,d^2\right )}^2\,\sqrt {d+e\,x}}{e^4} \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)^(9/2),x)

[Out]

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

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sympy [A]  time = 15.48, size = 230, normalized size = 2.04 \begin {gather*} \begin {cases} - \frac {2 a^{3} e^{2}}{\sqrt {d + e x}} + \frac {12 a^{2} c d^{2}}{\sqrt {d + e x}} + \frac {6 a^{2} c d e x}{\sqrt {d + e x}} - \frac {16 a c^{2} d^{4}}{e^{2} \sqrt {d + e x}} - \frac {8 a c^{2} d^{3} x}{e \sqrt {d + e x}} + \frac {2 a c^{2} d^{2} x^{2}}{\sqrt {d + e x}} + \frac {32 c^{3} d^{6}}{5 e^{4} \sqrt {d + e x}} + \frac {16 c^{3} d^{5} x}{5 e^{3} \sqrt {d + e x}} - \frac {4 c^{3} d^{4} x^{2}}{5 e^{2} \sqrt {d + e x}} + \frac {2 c^{3} d^{3} x^{3}}{5 e \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {c^{3} d^{\frac {3}{2}} x^{4}}{4} & \text {otherwise} \end {cases} \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)**(9/2),x)

[Out]

Piecewise((-2*a**3*e**2/sqrt(d + e*x) + 12*a**2*c*d**2/sqrt(d + e*x) + 6*a**2*c*d*e*x/sqrt(d + e*x) - 16*a*c**
2*d**4/(e**2*sqrt(d + e*x)) - 8*a*c**2*d**3*x/(e*sqrt(d + e*x)) + 2*a*c**2*d**2*x**2/sqrt(d + e*x) + 32*c**3*d
**6/(5*e**4*sqrt(d + e*x)) + 16*c**3*d**5*x/(5*e**3*sqrt(d + e*x)) - 4*c**3*d**4*x**2/(5*e**2*sqrt(d + e*x)) +
2*c**3*d**3*x**3/(5*e*sqrt(d + e*x)), Ne(e, 0)), (c**3*d**(3/2)*x**4/4, True))

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