3.11.87 \(\int \frac {1}{(d+e x)^2 (c d^2+2 c d e x+c e^2 x^2)^{5/2}} \, dx\) [1087]

Optimal. Leaf size=38 \[ -\frac {1}{6 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 38, 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 = {656, 621} \begin {gather*} -\frac {1}{6 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^2*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)),x]

[Out]

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

Rule 621

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[2*((a + b*x + c*x^2)^(p + 1)/((2*p + 1)*(b + 2*
c*x))), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && LtQ[p, -1]

Rule 656

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[e^m/c^(m/2), Int[(a +
b*x + c*x^2)^(p + m/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/2]

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx &=c \int \frac {1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}} \, dx\\ &=-\frac {1}{6 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 26, normalized size = 0.68 \begin {gather*} -\frac {c (d+e x)}{6 e \left (c (d+e x)^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^2*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)),x]

[Out]

-1/6*(c*(d + e*x))/(e*(c*(d + e*x)^2)^(7/2))

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Maple [A]
time = 0.60, size = 35, normalized size = 0.92

method	result	size
risch	\(-\frac {1}{6 c^{2} \left (e x +d \right )^{5} \sqrt {\left (e x +d \right )^{2} c}\, e}\)	\(27\)
gosper	\(-\frac {1}{6 e \left (e x +d \right ) \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}}}\)	\(35\)
default	\(-\frac {1}{6 e \left (e x +d \right ) \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}}}\)	\(35\)
trager	\(\frac {\left (e^{5} x^{5}+6 d \,e^{4} x^{4}+15 d^{2} e^{3} x^{3}+20 d^{3} e^{2} x^{2}+15 d^{4} e x +6 d^{5}\right ) x \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{6 c^{3} d^{6} \left (e x +d \right )^{7}}\)	\(90\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^2/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (34) = 68\).
time = 0.28, size = 84, normalized size = 2.21 \begin {gather*} -\frac {1}{6 \, {\left (c^{\frac {5}{2}} x^{6} e^{7} + 6 \, c^{\frac {5}{2}} d x^{5} e^{6} + 15 \, c^{\frac {5}{2}} d^{2} x^{4} e^{5} + 20 \, c^{\frac {5}{2}} d^{3} x^{3} e^{4} + 15 \, c^{\frac {5}{2}} d^{4} x^{2} e^{3} + 6 \, c^{\frac {5}{2}} d^{5} x e^{2} + c^{\frac {5}{2}} d^{6} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2),x, algorithm="maxima")

[Out]

-1/6/(c^(5/2)*x^6*e^7 + 6*c^(5/2)*d*x^5*e^6 + 15*c^(5/2)*d^2*x^4*e^5 + 20*c^(5/2)*d^3*x^3*e^4 + 15*c^(5/2)*d^4
*x^2*e^3 + 6*c^(5/2)*d^5*x*e^2 + c^(5/2)*d^6*e)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (34) = 68\).
time = 1.98, size = 119, normalized size = 3.13 \begin {gather*} -\frac {\sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}}}{6 \, {\left (c^{3} x^{7} e^{8} + 7 \, c^{3} d x^{6} e^{7} + 21 \, c^{3} d^{2} x^{5} e^{6} + 35 \, c^{3} d^{3} x^{4} e^{5} + 35 \, c^{3} d^{4} x^{3} e^{4} + 21 \, c^{3} d^{5} x^{2} e^{3} + 7 \, c^{3} d^{6} x e^{2} + c^{3} d^{7} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2),x, algorithm="fricas")

[Out]

-1/6*sqrt(c*x^2*e^2 + 2*c*d*x*e + c*d^2)/(c^3*x^7*e^8 + 7*c^3*d*x^6*e^7 + 21*c^3*d^2*x^5*e^6 + 35*c^3*d^3*x^4*
e^5 + 35*c^3*d^4*x^3*e^4 + 21*c^3*d^5*x^2*e^3 + 7*c^3*d^6*x*e^2 + c^3*d^7*e)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (c \left (d + e x\right )^{2}\right )^{\frac {5}{2}} \left (d + e x\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**2/(c*e**2*x**2+2*c*d*e*x+c*d**2)**(5/2),x)

[Out]

Integral(1/((c*(d + e*x)**2)**(5/2)*(d + e*x)**2), x)

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Giac [A]
time = 1.12, size = 24, normalized size = 0.63 \begin {gather*} -\frac {e^{\left (-1\right )}}{6 \, {\left (x e + d\right )}^{6} c^{\frac {5}{2}} \mathrm {sgn}\left (x e + d\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2),x, algorithm="giac")

[Out]

-1/6*e^(-1)/((x*e + d)^6*c^(5/2)*sgn(x*e + d))

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Mupad [B]
time = 0.56, size = 37, normalized size = 0.97 \begin {gather*} -\frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{6\,c^3\,e\,{\left (d+e\,x\right )}^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d + e*x)^2*(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)/(6*c^3*e*(d + e*x)^7)

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