### 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]

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

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Rubi [A]  time = 0.0231345, antiderivative size = 32, 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 = {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 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]

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]

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*}

Mathematica [A]  time = 0.0272238, 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]

-c/(7*e*(c*(d + e*x)^2)^(7/2))

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Maple [A]  time = 0.042, size = 35, normalized size = 1.1 \begin{align*} -{\frac{1}{7\, \left ( ex+d \right ) ^{2}e} \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)^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 = 1.14404, size = 139, normalized size = 4.34 \begin{align*} -\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 )}} \end{align*}

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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Fricas [B]  time = 2.3652, size = 282, normalized size = 8.81 \begin{align*} -\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 )}} \end{align*}

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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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 )^{3}}\, dx \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \end{align*}

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]

[undef, undef, undef, undef, undef, 1]