### 3.1628 $$\int \frac{a^2+2 a b x+b^2 x^2}{(d+e x)^{7/2}} \, dx$$

Optimal. Leaf size=69 $\frac{4 b (b d-a e)}{3 e^3 (d+e x)^{3/2}}-\frac{2 (b d-a e)^2}{5 e^3 (d+e x)^{5/2}}-\frac{2 b^2}{e^3 \sqrt{d+e x}}$

[Out]

(-2*(b*d - a*e)^2)/(5*e^3*(d + e*x)^(5/2)) + (4*b*(b*d - a*e))/(3*e^3*(d + e*x)^(3/2)) - (2*b^2)/(e^3*Sqrt[d +
e*x])

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Rubi [A]  time = 0.0260152, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.077, Rules used = {27, 43} $\frac{4 b (b d-a e)}{3 e^3 (d+e x)^{3/2}}-\frac{2 (b d-a e)^2}{5 e^3 (d+e x)^{5/2}}-\frac{2 b^2}{e^3 \sqrt{d+e x}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^(7/2),x]

[Out]

(-2*(b*d - a*e)^2)/(5*e^3*(d + e*x)^(5/2)) + (4*b*(b*d - a*e))/(3*e^3*(d + e*x)^(3/2)) - (2*b^2)/(e^3*Sqrt[d +
e*x])

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{a^2+2 a b x+b^2 x^2}{(d+e x)^{7/2}} \, dx &=\int \frac{(a+b x)^2}{(d+e x)^{7/2}} \, dx\\ &=\int \left (\frac{(-b d+a e)^2}{e^2 (d+e x)^{7/2}}-\frac{2 b (b d-a e)}{e^2 (d+e x)^{5/2}}+\frac{b^2}{e^2 (d+e x)^{3/2}}\right ) \, dx\\ &=-\frac{2 (b d-a e)^2}{5 e^3 (d+e x)^{5/2}}+\frac{4 b (b d-a e)}{3 e^3 (d+e x)^{3/2}}-\frac{2 b^2}{e^3 \sqrt{d+e x}}\\ \end{align*}

Mathematica [A]  time = 0.0326638, size = 61, normalized size = 0.88 $-\frac{2 \left (3 a^2 e^2+2 a b e (2 d+5 e x)+b^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )\right )}{15 e^3 (d+e x)^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^(7/2),x]

[Out]

(-2*(3*a^2*e^2 + 2*a*b*e*(2*d + 5*e*x) + b^2*(8*d^2 + 20*d*e*x + 15*e^2*x^2)))/(15*e^3*(d + e*x)^(5/2))

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Maple [A]  time = 0.044, size = 63, normalized size = 0.9 \begin{align*} -{\frac{30\,{b}^{2}{x}^{2}{e}^{2}+20\,xab{e}^{2}+40\,x{b}^{2}de+6\,{a}^{2}{e}^{2}+8\,abde+16\,{b}^{2}{d}^{2}}{15\,{e}^{3}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^2+2*a*b*x+a^2)/(e*x+d)^(7/2),x)

[Out]

-2/15*(15*b^2*e^2*x^2+10*a*b*e^2*x+20*b^2*d*e*x+3*a^2*e^2+4*a*b*d*e+8*b^2*d^2)/(e*x+d)^(5/2)/e^3

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Maxima [A]  time = 1.03004, size = 88, normalized size = 1.28 \begin{align*} -\frac{2 \,{\left (15 \,{\left (e x + d\right )}^{2} b^{2} + 3 \, b^{2} d^{2} - 6 \, a b d e + 3 \, a^{2} e^{2} - 10 \,{\left (b^{2} d - a b e\right )}{\left (e x + d\right )}\right )}}{15 \,{\left (e x + d\right )}^{\frac{5}{2}} e^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

-2/15*(15*(e*x + d)^2*b^2 + 3*b^2*d^2 - 6*a*b*d*e + 3*a^2*e^2 - 10*(b^2*d - a*b*e)*(e*x + d))/((e*x + d)^(5/2)
*e^3)

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Fricas [A]  time = 1.47153, size = 204, normalized size = 2.96 \begin{align*} -\frac{2 \,{\left (15 \, b^{2} e^{2} x^{2} + 8 \, b^{2} d^{2} + 4 \, a b d e + 3 \, a^{2} e^{2} + 10 \,{\left (2 \, b^{2} d e + a b e^{2}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

-2/15*(15*b^2*e^2*x^2 + 8*b^2*d^2 + 4*a*b*d*e + 3*a^2*e^2 + 10*(2*b^2*d*e + a*b*e^2)*x)*sqrt(e*x + d)/(e^6*x^3
+ 3*d*e^5*x^2 + 3*d^2*e^4*x + d^3*e^3)

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Sympy [A]  time = 3.13579, size = 389, normalized size = 5.64 \begin{align*} \begin{cases} - \frac{6 a^{2} e^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{8 a b d e}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{20 a b e^{2} x}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{16 b^{2} d^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{40 b^{2} d e x}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} - \frac{30 b^{2} e^{2} x^{2}}{15 d^{2} e^{3} \sqrt{d + e x} + 30 d e^{4} x \sqrt{d + e x} + 15 e^{5} x^{2} \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{a^{2} x + a b x^{2} + \frac{b^{2} x^{3}}{3}}{d^{\frac{7}{2}}} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**2+2*a*b*x+a**2)/(e*x+d)**(7/2),x)

[Out]

Piecewise((-6*a**2*e**2/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x))
- 8*a*b*d*e/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 20*a*b*e**
2*x/(15*d**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 16*b**2*d**2/(15*d
**2*e**3*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 40*b**2*d*e*x/(15*d**2*e**3
*sqrt(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)) - 30*b**2*e**2*x**2/(15*d**2*e**3*sqr
t(d + e*x) + 30*d*e**4*x*sqrt(d + e*x) + 15*e**5*x**2*sqrt(d + e*x)), Ne(e, 0)), ((a**2*x + a*b*x**2 + b**2*x*
*3/3)/d**(7/2), True))

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Giac [A]  time = 1.19207, size = 97, normalized size = 1.41 \begin{align*} -\frac{2 \,{\left (15 \,{\left (x e + d\right )}^{2} b^{2} - 10 \,{\left (x e + d\right )} b^{2} d + 3 \, b^{2} d^{2} + 10 \,{\left (x e + d\right )} a b e - 6 \, a b d e + 3 \, a^{2} e^{2}\right )} e^{\left (-3\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

-2/15*(15*(x*e + d)^2*b^2 - 10*(x*e + d)*b^2*d + 3*b^2*d^2 + 10*(x*e + d)*a*b*e - 6*a*b*d*e + 3*a^2*e^2)*e^(-3
)/(x*e + d)^(5/2)