3.1082 \(\int \frac {(d+e x)^3}{(c d^2+2 c d e x+c e^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=32 \[ -\frac {1}{c^2 e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {643, 629} \[ -\frac {1}{c^2 e \sqrt {c d^2+2 c d e x+c e^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(c^2*e*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]))

Rule 629

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

Rule 643

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[e^(m - 1)/c^((m - 1)/2
), Int[(d + e*x)*(a + b*x + c*x^2)^(p + (m - 1)/2), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[b^2 - 4*a*c,
0] &&  !IntegerQ[p] && EqQ[2*c*d - b*e, 0] && IntegerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \frac {(d+e x)^3}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx &=\frac {\int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx}{c}\\ &=-\frac {1}{c^2 e \sqrt {c d^2+2 c d e x+c e^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 0.66 \[ -\frac {1}{c^2 e \sqrt {c (d+e x)^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(c^2*e*Sqrt[c*(d + e*x)^2]))

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fricas [A]  time = 1.19, size = 55, normalized size = 1.72 \[ -\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{c^{3} e^{3} x^{2} + 2 \, c^{3} d e^{2} x + c^{3} d^{2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.50, size = 79, normalized size = 2.47 \[ \frac {2 \, C_{0} d^{3} e^{\left (-3\right )} + {\left (6 \, C_{0} d^{2} e^{\left (-2\right )} + {\left (6 \, C_{0} d e^{\left (-1\right )} + 2 \, C_{0} x - \frac {e}{c}\right )} x - \frac {2 \, d}{c}\right )} x - \frac {d^{2} e^{\left (-1\right )}}{c}}{{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*C_0*d^3*e^(-3) + (6*C_0*d^2*e^(-2) + (6*C_0*d*e^(-1) + 2*C_0*x - e/c)*x - 2*d/c)*x - d^2*e^(-1)/c)/(c*x^2*e
^2 + 2*c*d*x*e + c*d^2)^(3/2)

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maple [A]  time = 0.05, size = 35, normalized size = 1.09 \[ -\frac {\left (e x +d \right )^{4}}{\left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.43, size = 103, normalized size = 3.22 \[ -\frac {e x^{2}}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}} c} - \frac {5 \, d^{2}}{3 \, {\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{\frac {3}{2}} c e} - \frac {2 \, d}{c^{\frac {5}{2}} e^{3} {\left (x + \frac {d}{e}\right )}^{2}} + \frac {8 \, d^{2}}{3 \, c^{\frac {5}{2}} e^{4} {\left (x + \frac {d}{e}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e*x^2/((c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2)*c) - 5/3*d^2/((c*e^2*x^2 + 2*c*d*e*x + c*d^2)^(3/2)*c*e) - 2*d/(
c^(5/2)*e^3*(x + d/e)^2) + 8/3*d^2/(c^(5/2)*e^4*(x + d/e)^3)

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mupad [B]  time = 0.48, size = 37, normalized size = 1.16 \[ -\frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{c^3\,e\,{\left (d+e\,x\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 1.46, size = 70, normalized size = 2.19 \[ \begin {cases} - \frac {\sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{c^{3} d^{2} e + 2 c^{3} d e^{2} x + c^{3} e^{3} x^{2}} & \text {for}\: e \neq 0 \\\frac {d^{3} x}{\left (c d^{2}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(5/2),x)

[Out]

Piecewise((-sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2)/(c**3*d**2*e + 2*c**3*d*e**2*x + c**3*e**3*x**2), Ne(e, 0))
, (d**3*x/(c*d**2)**(5/2), True))

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