3.11.88 \(\int \frac {1}{(d+e x)^3 (c d^2+2 c d e x+c e^2 x^2)^{5/2}} \, dx\) [1088]

Optimal. Leaf size=32 \[ -\frac {c}{7 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}} \]

[Out]

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

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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 = {657, 643} \begin {gather*} -\frac {c}{7 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^3*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)),x]

[Out]

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

Rule 643

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 657

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 {1}{(d+e x)^3 \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx &=c^2 \int \frac {d+e x}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{9/2}} \, dx\\ &=-\frac {c}{7 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 21, normalized size = 0.66 \begin {gather*} -\frac {c}{7 e \left (c (d+e x)^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^3*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)),x]

[Out]

-1/7*c/(e*(c*(d + e*x)^2)^(7/2))

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Maple [A]
time = 0.67, size = 35, normalized size = 1.09

method	result	size
risch	\(-\frac {1}{7 c^{2} \left (e x +d \right )^{6} \sqrt {\left (e x +d \right )^{2} c}\, e}\)	\(27\)
gosper	\(-\frac {1}{7 \left (e x +d \right )^{2} e \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}}}\)	\(35\)
default	\(-\frac {1}{7 \left (e x +d \right )^{2} e \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}}}\)	\(35\)
trager	\(\frac {\left (e^{6} x^{6}+7 d \,e^{5} x^{5}+21 d^{2} e^{4} x^{4}+35 d^{3} e^{3} x^{3}+35 d^{4} e^{2} x^{2}+21 d^{5} e x +7 d^{6}\right ) x \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{7 c^{3} d^{7} \left (e x +d \right )^{8}}\)	\(101\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^3/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/7/(e*x+d)^2/e/(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. 97 vs. \(2 (27) = 54\).
time = 0.28, size = 97, normalized size = 3.03 \begin {gather*} -\frac {1}{7 \, {\left (c^{\frac {5}{2}} x^{7} e^{8} + 7 \, c^{\frac {5}{2}} d x^{6} e^{7} + 21 \, c^{\frac {5}{2}} d^{2} x^{5} e^{6} + 35 \, c^{\frac {5}{2}} d^{3} x^{4} e^{5} + 35 \, c^{\frac {5}{2}} d^{4} x^{3} e^{4} + 21 \, c^{\frac {5}{2}} d^{5} x^{2} e^{3} + 7 \, c^{\frac {5}{2}} d^{6} x e^{2} + c^{\frac {5}{2}} d^{7} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (27) = 54\).
time = 1.45, size = 132, normalized size = 4.12 \begin {gather*} -\frac {\sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}}}{7 \, {\left (c^{3} x^{8} e^{9} + 8 \, c^{3} d x^{7} e^{8} + 28 \, c^{3} d^{2} x^{6} e^{7} + 56 \, c^{3} d^{3} x^{5} e^{6} + 70 \, c^{3} d^{4} x^{4} e^{5} + 56 \, c^{3} d^{5} x^{3} e^{4} + 28 \, c^{3} d^{6} x^{2} e^{3} + 8 \, c^{3} d^{7} x e^{2} + c^{3} d^{8} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2),x, algorithm="fricas")

[Out]

-1/7*sqrt(c*x^2*e^2 + 2*c*d*x*e + c*d^2)/(c^3*x^8*e^9 + 8*c^3*d*x^7*e^8 + 28*c^3*d^2*x^6*e^7 + 56*c^3*d^3*x^5*
e^6 + 70*c^3*d^4*x^4*e^5 + 56*c^3*d^5*x^3*e^4 + 28*c^3*d^6*x^2*e^3 + 8*c^3*d^7*x*e^2 + c^3*d^8*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 )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**3/(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)**3), x)

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Giac [A]
time = 1.89, size = 24, normalized size = 0.75 \begin {gather*} -\frac {e^{\left (-1\right )}}{7 \, {\left (x e + d\right )}^{7} 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)^3/(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2),x, algorithm="giac")

[Out]

-1/7*e^(-1)/((x*e + d)^7*c^(5/2)*sgn(x*e + d))

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Mupad [B]
time = 0.57, size = 37, normalized size = 1.16 \begin {gather*} -\frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{7\,c^3\,e\,{\left (d+e\,x\right )}^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((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)/(7*c^3*e*(d + e*x)^8)

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