3.1088 \(\int \frac {1}{(d+e x)^3 (c d^2+2 c d e x+c e^2 x^2)^{5/2}} \, dx\)

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

[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 = {643, 629} \[ -\frac {c}{7 e \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2}} \]

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]

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

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 {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 = 0.03, size = 21, normalized size = 0.66 \[ -\frac {c}{7 e \left (c (d+e x)^2\right )^{7/2}} \]

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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fricas [B]  time = 0.86, size = 139, normalized size = 4.34 \[ -\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{7 \, {\left (c^{3} e^{9} x^{8} + 8 \, c^{3} d e^{8} x^{7} + 28 \, c^{3} d^{2} e^{7} x^{6} + 56 \, c^{3} d^{3} e^{6} x^{5} + 70 \, c^{3} d^{4} e^{5} x^{4} + 56 \, c^{3} d^{5} e^{4} x^{3} + 28 \, c^{3} d^{6} e^{3} x^{2} + 8 \, c^{3} d^{7} e^{2} x + c^{3} d^{8} e\right )}} \]

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*e^2*x^2 + 2*c*d*e*x + c*d^2)/(c^3*e^9*x^8 + 8*c^3*d*e^8*x^7 + 28*c^3*d^2*e^7*x^6 + 56*c^3*d^3*e^6*
x^5 + 70*c^3*d^4*e^5*x^4 + 56*c^3*d^5*e^4*x^3 + 28*c^3*d^6*e^3*x^2 + 8*c^3*d^7*e^2*x + c^3*d^8*e)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to transpose Error: Bad Argument Value

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maple [A]  time = 0.05, size = 35, normalized size = 1.09 \[ -\frac {1}{7 \left (e x +d \right )^{2} \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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]  time = 1.35, size = 103, normalized size = 3.22 \[ -\frac {1}{7 \, {\left (c^{\frac {5}{2}} e^{8} x^{7} + 7 \, c^{\frac {5}{2}} d e^{7} x^{6} + 21 \, c^{\frac {5}{2}} d^{2} e^{6} x^{5} + 35 \, c^{\frac {5}{2}} d^{3} e^{5} x^{4} + 35 \, c^{\frac {5}{2}} d^{4} e^{4} x^{3} + 21 \, c^{\frac {5}{2}} d^{5} e^{3} x^{2} + 7 \, c^{\frac {5}{2}} d^{6} e^{2} x + c^{\frac {5}{2}} d^{7} e\right )}} \]

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

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

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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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \left (d + e x\right )^{2}\right )^{\frac {5}{2}} \left (d + e x\right )^{3}}\, dx \]

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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