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

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

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Rubi [A]  time = 0.02, 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 = {642, 607} \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*e*(d + e*x)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2))

Rule 607

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 642

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 = 0.02, 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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IntegrateAlgebraic [B]  time = 1.59, size = 323, normalized size = 8.50 \begin {gather*} \frac {-16 c^3 \left (-c d^6 e-c e^7 x^6\right )-16 c^3 \sqrt {c e^2} \left (-d^5+d^4 e x-d^3 e^2 x^2+d^2 e^3 x^3-d e^4 x^4+e^5 x^5\right ) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{3 e x^6 \sqrt {c d^2+2 c d e x+c e^2 x^2} \left (-32 c^6 d^5 e^7-160 c^6 d^4 e^8 x-320 c^6 d^3 e^9 x^2-320 c^6 d^2 e^{10} x^3-160 c^6 d e^{11} x^4-32 c^6 e^{12} x^5\right )+3 e x^6 \sqrt {c e^2} \left (32 c^6 d^6 e^6+192 c^6 d^5 e^7 x+480 c^6 d^4 e^8 x^2+640 c^6 d^3 e^9 x^3+480 c^6 d^2 e^{10} x^4+192 c^6 d e^{11} x^5+32 c^6 e^{12} x^6\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-16*c^3*Sqrt[c*e^2]*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]*(-d^5 + d^4*e*x - d^3*e^2*x^2 + d^2*e^3*x^3 - d*e^4*x
^4 + e^5*x^5) - 16*c^3*(-(c*d^6*e) - c*e^7*x^6))/(3*e*x^6*Sqrt[c*d^2 + 2*c*d*e*x + c*e^2*x^2]*(-32*c^6*d^5*e^7
- 160*c^6*d^4*e^8*x - 320*c^6*d^3*e^9*x^2 - 320*c^6*d^2*e^10*x^3 - 160*c^6*d*e^11*x^4 - 32*c^6*e^12*x^5) + 3*
e*Sqrt[c*e^2]*x^6*(32*c^6*d^6*e^6 + 192*c^6*d^5*e^7*x + 480*c^6*d^4*e^8*x^2 + 640*c^6*d^3*e^9*x^3 + 480*c^6*d^
2*e^10*x^4 + 192*c^6*d*e^11*x^5 + 32*c^6*e^12*x^6))

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fricas [B]  time = 0.40, size = 125, normalized size = 3.29 \begin {gather*} -\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{6 \, {\left (c^{3} e^{8} x^{7} + 7 \, c^{3} d e^{7} x^{6} + 21 \, c^{3} d^{2} e^{6} x^{5} + 35 \, c^{3} d^{3} e^{5} x^{4} + 35 \, c^{3} d^{4} e^{4} x^{3} + 21 \, c^{3} d^{5} e^{3} x^{2} + 7 \, c^{3} d^{6} e^{2} x + 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*e^2*x^2 + 2*c*d*e*x + c*d^2)/(c^3*e^8*x^7 + 7*c^3*d*e^7*x^6 + 21*c^3*d^2*e^6*x^5 + 35*c^3*d^3*e^5*
x^4 + 35*c^3*d^4*e^4*x^3 + 21*c^3*d^5*e^3*x^2 + 7*c^3*d^6*e^2*x + c^3*d^7*e)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \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]

sage0*x

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maple [A]  time = 0.05, size = 35, normalized size = 0.92 \begin {gather*} -\frac {1}{6 \left (e x +d \right ) \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}} e} \end {gather*}

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)

[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]  time = 1.41, size = 89, normalized size = 2.34 \begin {gather*} -\frac {1}{6 \, {\left (c^{\frac {5}{2}} e^{7} x^{6} + 6 \, c^{\frac {5}{2}} d e^{6} x^{5} + 15 \, c^{\frac {5}{2}} d^{2} e^{5} x^{4} + 20 \, c^{\frac {5}{2}} d^{3} e^{4} x^{3} + 15 \, c^{\frac {5}{2}} d^{4} e^{3} x^{2} + 6 \, c^{\frac {5}{2}} d^{5} e^{2} x + 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)*e^7*x^6 + 6*c^(5/2)*d*e^6*x^5 + 15*c^(5/2)*d^2*e^5*x^4 + 20*c^(5/2)*d^3*e^4*x^3 + 15*c^(5/2)*d^4
*e^3*x^2 + 6*c^(5/2)*d^5*e^2*x + c^(5/2)*d^6*e)

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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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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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