### 3.1634 $$\int \frac{(a^2+2 a b x+b^2 x^2)^2}{(d+e x)^{3/2}} \, dx$$

Optimal. Leaf size=123 $-\frac{8 b^3 (d+e x)^{5/2} (b d-a e)}{5 e^5}+\frac{4 b^2 (d+e x)^{3/2} (b d-a e)^2}{e^5}-\frac{8 b \sqrt{d+e x} (b d-a e)^3}{e^5}-\frac{2 (b d-a e)^4}{e^5 \sqrt{d+e x}}+\frac{2 b^4 (d+e x)^{7/2}}{7 e^5}$

[Out]

(-2*(b*d - a*e)^4)/(e^5*Sqrt[d + e*x]) - (8*b*(b*d - a*e)^3*Sqrt[d + e*x])/e^5 + (4*b^2*(b*d - a*e)^2*(d + e*x
)^(3/2))/e^5 - (8*b^3*(b*d - a*e)*(d + e*x)^(5/2))/(5*e^5) + (2*b^4*(d + e*x)^(7/2))/(7*e^5)

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Rubi [A]  time = 0.0406202, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.071, Rules used = {27, 43} $-\frac{8 b^3 (d+e x)^{5/2} (b d-a e)}{5 e^5}+\frac{4 b^2 (d+e x)^{3/2} (b d-a e)^2}{e^5}-\frac{8 b \sqrt{d+e x} (b d-a e)^3}{e^5}-\frac{2 (b d-a e)^4}{e^5 \sqrt{d+e x}}+\frac{2 b^4 (d+e x)^{7/2}}{7 e^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^(3/2),x]

[Out]

(-2*(b*d - a*e)^4)/(e^5*Sqrt[d + e*x]) - (8*b*(b*d - a*e)^3*Sqrt[d + e*x])/e^5 + (4*b^2*(b*d - a*e)^2*(d + e*x
)^(3/2))/e^5 - (8*b^3*(b*d - a*e)*(d + e*x)^(5/2))/(5*e^5) + (2*b^4*(d + e*x)^(7/2))/(7*e^5)

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{\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{3/2}} \, dx &=\int \frac{(a+b x)^4}{(d+e x)^{3/2}} \, dx\\ &=\int \left (\frac{(-b d+a e)^4}{e^4 (d+e x)^{3/2}}-\frac{4 b (b d-a e)^3}{e^4 \sqrt{d+e x}}+\frac{6 b^2 (b d-a e)^2 \sqrt{d+e x}}{e^4}-\frac{4 b^3 (b d-a e) (d+e x)^{3/2}}{e^4}+\frac{b^4 (d+e x)^{5/2}}{e^4}\right ) \, dx\\ &=-\frac{2 (b d-a e)^4}{e^5 \sqrt{d+e x}}-\frac{8 b (b d-a e)^3 \sqrt{d+e x}}{e^5}+\frac{4 b^2 (b d-a e)^2 (d+e x)^{3/2}}{e^5}-\frac{8 b^3 (b d-a e) (d+e x)^{5/2}}{5 e^5}+\frac{2 b^4 (d+e x)^{7/2}}{7 e^5}\\ \end{align*}

Mathematica [A]  time = 0.069786, size = 101, normalized size = 0.82 $\frac{2 \left (70 b^2 (d+e x)^2 (b d-a e)^2-28 b^3 (d+e x)^3 (b d-a e)-140 b (d+e x) (b d-a e)^3-35 (b d-a e)^4+5 b^4 (d+e x)^4\right )}{35 e^5 \sqrt{d+e x}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^(3/2),x]

[Out]

(2*(-35*(b*d - a*e)^4 - 140*b*(b*d - a*e)^3*(d + e*x) + 70*b^2*(b*d - a*e)^2*(d + e*x)^2 - 28*b^3*(b*d - a*e)*
(d + e*x)^3 + 5*b^4*(d + e*x)^4))/(35*e^5*Sqrt[d + e*x])

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Maple [A]  time = 0.046, size = 186, normalized size = 1.5 \begin{align*} -{\frac{-10\,{x}^{4}{b}^{4}{e}^{4}-56\,{x}^{3}a{b}^{3}{e}^{4}+16\,{x}^{3}{b}^{4}d{e}^{3}-140\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}+112\,{x}^{2}a{b}^{3}d{e}^{3}-32\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}-280\,x{a}^{3}b{e}^{4}+560\,x{a}^{2}{b}^{2}d{e}^{3}-448\,xa{b}^{3}{d}^{2}{e}^{2}+128\,x{b}^{4}{d}^{3}e+70\,{a}^{4}{e}^{4}-560\,{a}^{3}bd{e}^{3}+1120\,{d}^{2}{e}^{2}{a}^{2}{b}^{2}-896\,a{b}^{3}{d}^{3}e+256\,{b}^{4}{d}^{4}}{35\,{e}^{5}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^(3/2),x)

[Out]

-2/35*(-5*b^4*e^4*x^4-28*a*b^3*e^4*x^3+8*b^4*d*e^3*x^3-70*a^2*b^2*e^4*x^2+56*a*b^3*d*e^3*x^2-16*b^4*d^2*e^2*x^
2-140*a^3*b*e^4*x+280*a^2*b^2*d*e^3*x-224*a*b^3*d^2*e^2*x+64*b^4*d^3*e*x+35*a^4*e^4-280*a^3*b*d*e^3+560*a^2*b^
2*d^2*e^2-448*a*b^3*d^3*e+128*b^4*d^4)/(e*x+d)^(1/2)/e^5

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Maxima [A]  time = 1.04781, size = 255, normalized size = 2.07 \begin{align*} \frac{2 \,{\left (\frac{5 \,{\left (e x + d\right )}^{\frac{7}{2}} b^{4} - 28 \,{\left (b^{4} d - a b^{3} e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 70 \,{\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 140 \,{\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )} \sqrt{e x + d}}{e^{4}} - \frac{35 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )}}{\sqrt{e x + d} e^{4}}\right )}}{35 \, e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

2/35*((5*(e*x + d)^(7/2)*b^4 - 28*(b^4*d - a*b^3*e)*(e*x + d)^(5/2) + 70*(b^4*d^2 - 2*a*b^3*d*e + a^2*b^2*e^2)
*(e*x + d)^(3/2) - 140*(b^4*d^3 - 3*a*b^3*d^2*e + 3*a^2*b^2*d*e^2 - a^3*b*e^3)*sqrt(e*x + d))/e^4 - 35*(b^4*d^
4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)/(sqrt(e*x + d)*e^4))/e

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Fricas [A]  time = 1.50403, size = 412, normalized size = 3.35 \begin{align*} \frac{2 \,{\left (5 \, b^{4} e^{4} x^{4} - 128 \, b^{4} d^{4} + 448 \, a b^{3} d^{3} e - 560 \, a^{2} b^{2} d^{2} e^{2} + 280 \, a^{3} b d e^{3} - 35 \, a^{4} e^{4} - 4 \,{\left (2 \, b^{4} d e^{3} - 7 \, a b^{3} e^{4}\right )} x^{3} + 2 \,{\left (8 \, b^{4} d^{2} e^{2} - 28 \, a b^{3} d e^{3} + 35 \, a^{2} b^{2} e^{4}\right )} x^{2} - 4 \,{\left (16 \, b^{4} d^{3} e - 56 \, a b^{3} d^{2} e^{2} + 70 \, a^{2} b^{2} d e^{3} - 35 \, a^{3} b e^{4}\right )} x\right )} \sqrt{e x + d}}{35 \,{\left (e^{6} x + d e^{5}\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)^2/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

2/35*(5*b^4*e^4*x^4 - 128*b^4*d^4 + 448*a*b^3*d^3*e - 560*a^2*b^2*d^2*e^2 + 280*a^3*b*d*e^3 - 35*a^4*e^4 - 4*(
2*b^4*d*e^3 - 7*a*b^3*e^4)*x^3 + 2*(8*b^4*d^2*e^2 - 28*a*b^3*d*e^3 + 35*a^2*b^2*e^4)*x^2 - 4*(16*b^4*d^3*e - 5
6*a*b^3*d^2*e^2 + 70*a^2*b^2*d*e^3 - 35*a^3*b*e^4)*x)*sqrt(e*x + d)/(e^6*x + d*e^5)

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Sympy [A]  time = 40.107, size = 168, normalized size = 1.37 \begin{align*} \frac{2 b^{4} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{5}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (8 a b^{3} e - 8 b^{4} d\right )}{5 e^{5}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (12 a^{2} b^{2} e^{2} - 24 a b^{3} d e + 12 b^{4} d^{2}\right )}{3 e^{5}} + \frac{\sqrt{d + e x} \left (8 a^{3} b e^{3} - 24 a^{2} b^{2} d e^{2} + 24 a b^{3} d^{2} e - 8 b^{4} d^{3}\right )}{e^{5}} - \frac{2 \left (a e - b d\right )^{4}}{e^{5} \sqrt{d + e x}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**(3/2),x)

[Out]

2*b**4*(d + e*x)**(7/2)/(7*e**5) + (d + e*x)**(5/2)*(8*a*b**3*e - 8*b**4*d)/(5*e**5) + (d + e*x)**(3/2)*(12*a*
*2*b**2*e**2 - 24*a*b**3*d*e + 12*b**4*d**2)/(3*e**5) + sqrt(d + e*x)*(8*a**3*b*e**3 - 24*a**2*b**2*d*e**2 + 2
4*a*b**3*d**2*e - 8*b**4*d**3)/e**5 - 2*(a*e - b*d)**4/(e**5*sqrt(d + e*x))

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Giac [B]  time = 1.18195, size = 320, normalized size = 2.6 \begin{align*} \frac{2}{35} \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} b^{4} e^{30} - 28 \,{\left (x e + d\right )}^{\frac{5}{2}} b^{4} d e^{30} + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{4} d^{2} e^{30} - 140 \, \sqrt{x e + d} b^{4} d^{3} e^{30} + 28 \,{\left (x e + d\right )}^{\frac{5}{2}} a b^{3} e^{31} - 140 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{3} d e^{31} + 420 \, \sqrt{x e + d} a b^{3} d^{2} e^{31} + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} b^{2} e^{32} - 420 \, \sqrt{x e + d} a^{2} b^{2} d e^{32} + 140 \, \sqrt{x e + d} a^{3} b e^{33}\right )} e^{\left (-35\right )} - \frac{2 \,{\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} e^{\left (-5\right )}}{\sqrt{x e + d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

2/35*(5*(x*e + d)^(7/2)*b^4*e^30 - 28*(x*e + d)^(5/2)*b^4*d*e^30 + 70*(x*e + d)^(3/2)*b^4*d^2*e^30 - 140*sqrt(
x*e + d)*b^4*d^3*e^30 + 28*(x*e + d)^(5/2)*a*b^3*e^31 - 140*(x*e + d)^(3/2)*a*b^3*d*e^31 + 420*sqrt(x*e + d)*a
*b^3*d^2*e^31 + 70*(x*e + d)^(3/2)*a^2*b^2*e^32 - 420*sqrt(x*e + d)*a^2*b^2*d*e^32 + 140*sqrt(x*e + d)*a^3*b*e
^33)*e^(-35) - 2*(b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*e^(-5)/sqrt(x*e + d)