### 3.1087 $$\int \frac{1}{(d+e x)^2 (c d^2+2 c d e x+c e^2 x^2)^{5/2}} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0206601, antiderivative size = 38, normalized size of antiderivative = 1., 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} $-\frac{1}{6 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}$

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 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]

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]

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*}

Mathematica [A]  time = 0.0177792, size = 26, normalized size = 0.68 $-\frac{c (d+e x)}{6 e \left (c (d+e x)^2\right )^{7/2}}$

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]

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

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Maple [A]  time = 0.042, size = 35, normalized size = 0.9 \begin{align*} -{\frac{1}{6\,e \left ( ex+d \right ) } \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

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 = 1.16159, size = 120, normalized size = 3.16 \begin{align*} -\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{align*}

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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Fricas [B]  time = 2.26976, size = 254, normalized size = 6.68 \begin{align*} -\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{align*}

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

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 [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out