### 3.1275 $$\int \frac{(a+b x+c x^2)^2}{(b d+2 c d x)^{9/2}} \, dx$$

Optimal. Leaf size=88 $\frac{b^2-4 a c}{24 c^3 d^3 (b d+2 c d x)^{3/2}}-\frac{\left (b^2-4 a c\right )^2}{112 c^3 d (b d+2 c d x)^{7/2}}+\frac{\sqrt{b d+2 c d x}}{16 c^3 d^5}$

[Out]

-(b^2 - 4*a*c)^2/(112*c^3*d*(b*d + 2*c*d*x)^(7/2)) + (b^2 - 4*a*c)/(24*c^3*d^3*(b*d + 2*c*d*x)^(3/2)) + Sqrt[b
*d + 2*c*d*x]/(16*c^3*d^5)

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Rubi [A]  time = 0.0374583, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.038, Rules used = {683} $\frac{b^2-4 a c}{24 c^3 d^3 (b d+2 c d x)^{3/2}}-\frac{\left (b^2-4 a c\right )^2}{112 c^3 d (b d+2 c d x)^{7/2}}+\frac{\sqrt{b d+2 c d x}}{16 c^3 d^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^(9/2),x]

[Out]

-(b^2 - 4*a*c)^2/(112*c^3*d*(b*d + 2*c*d*x)^(7/2)) + (b^2 - 4*a*c)/(24*c^3*d^3*(b*d + 2*c*d*x)^(3/2)) + Sqrt[b
*d + 2*c*d*x]/(16*c^3*d^5)

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps

\begin{align*} \int \frac{\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{9/2}} \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^2}{16 c^2 (b d+2 c d x)^{9/2}}+\frac{-b^2+4 a c}{8 c^2 d^2 (b d+2 c d x)^{5/2}}+\frac{1}{16 c^2 d^4 \sqrt{b d+2 c d x}}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right )^2}{112 c^3 d (b d+2 c d x)^{7/2}}+\frac{b^2-4 a c}{24 c^3 d^3 (b d+2 c d x)^{3/2}}+\frac{\sqrt{b d+2 c d x}}{16 c^3 d^5}\\ \end{align*}

Mathematica [A]  time = 0.0476852, size = 63, normalized size = 0.72 $\frac{14 \left (b^2-4 a c\right ) (b+2 c x)^2-3 \left (b^2-4 a c\right )^2+21 (b+2 c x)^4}{336 c^3 d (d (b+2 c x))^{7/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^(9/2),x]

[Out]

(-3*(b^2 - 4*a*c)^2 + 14*(b^2 - 4*a*c)*(b + 2*c*x)^2 + 21*(b + 2*c*x)^4)/(336*c^3*d*(d*(b + 2*c*x))^(7/2))

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Maple [A]  time = 0.049, size = 96, normalized size = 1.1 \begin{align*} -{\frac{ \left ( 2\,cx+b \right ) \left ( -21\,{c}^{4}{x}^{4}-42\,b{x}^{3}{c}^{3}+14\,a{c}^{3}{x}^{2}-35\,{b}^{2}{c}^{2}{x}^{2}+14\,ab{c}^{2}x-14\,{b}^{3}cx+3\,{a}^{2}{c}^{2}+2\,ac{b}^{2}-2\,{b}^{4} \right ) }{21\,{c}^{3}} \left ( 2\,cdx+bd \right ) ^{-{\frac{9}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^(9/2),x)

[Out]

-1/21*(2*c*x+b)*(-21*c^4*x^4-42*b*c^3*x^3+14*a*c^3*x^2-35*b^2*c^2*x^2+14*a*b*c^2*x-14*b^3*c*x+3*a^2*c^2+2*a*b^
2*c-2*b^4)/c^3/(2*c*d*x+b*d)^(9/2)

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Maxima [A]  time = 0.993463, size = 124, normalized size = 1.41 \begin{align*} \frac{\frac{21 \, \sqrt{2 \, c d x + b d}}{c^{2} d^{4}} + \frac{14 \,{\left (2 \, c d x + b d\right )}^{2}{\left (b^{2} - 4 \, a c\right )} - 3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{2}}{{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} c^{2} d^{2}}}{336 \, c d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^(9/2),x, algorithm="maxima")

[Out]

1/336*(21*sqrt(2*c*d*x + b*d)/(c^2*d^4) + (14*(2*c*d*x + b*d)^2*(b^2 - 4*a*c) - 3*(b^4 - 8*a*b^2*c + 16*a^2*c^
2)*d^2)/((2*c*d*x + b*d)^(7/2)*c^2*d^2))/(c*d)

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Fricas [A]  time = 2.01866, size = 311, normalized size = 3.53 \begin{align*} \frac{{\left (21 \, c^{4} x^{4} + 42 \, b c^{3} x^{3} + 2 \, b^{4} - 2 \, a b^{2} c - 3 \, a^{2} c^{2} + 7 \,{\left (5 \, b^{2} c^{2} - 2 \, a c^{3}\right )} x^{2} + 14 \,{\left (b^{3} c - a b c^{2}\right )} x\right )} \sqrt{2 \, c d x + b d}}{21 \,{\left (16 \, c^{7} d^{5} x^{4} + 32 \, b c^{6} d^{5} x^{3} + 24 \, b^{2} c^{5} d^{5} x^{2} + 8 \, b^{3} c^{4} d^{5} x + b^{4} c^{3} d^{5}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^(9/2),x, algorithm="fricas")

[Out]

1/21*(21*c^4*x^4 + 42*b*c^3*x^3 + 2*b^4 - 2*a*b^2*c - 3*a^2*c^2 + 7*(5*b^2*c^2 - 2*a*c^3)*x^2 + 14*(b^3*c - a*
b*c^2)*x)*sqrt(2*c*d*x + b*d)/(16*c^7*d^5*x^4 + 32*b*c^6*d^5*x^3 + 24*b^2*c^5*d^5*x^2 + 8*b^3*c^4*d^5*x + b^4*
c^3*d^5)

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Sympy [A]  time = 9.5047, size = 826, normalized size = 9.39 \begin{align*} \begin{cases} - \frac{3 a^{2} c^{2} \sqrt{b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} - \frac{2 a b^{2} c \sqrt{b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} - \frac{14 a b c^{2} x \sqrt{b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} - \frac{14 a c^{3} x^{2} \sqrt{b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} + \frac{2 b^{4} \sqrt{b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} + \frac{14 b^{3} c x \sqrt{b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} + \frac{35 b^{2} c^{2} x^{2} \sqrt{b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} + \frac{42 b c^{3} x^{3} \sqrt{b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} + \frac{21 c^{4} x^{4} \sqrt{b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} & \text{for}\: c \neq 0 \\\frac{a^{2} x + a b x^{2} + \frac{b^{2} x^{3}}{3}}{\left (b d\right )^{\frac{9}{2}}} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**2/(2*c*d*x+b*d)**(9/2),x)

[Out]

Piecewise((-3*a**2*c**2*sqrt(b*d + 2*c*d*x)/(21*b**4*c**3*d**5 + 168*b**3*c**4*d**5*x + 504*b**2*c**5*d**5*x**
2 + 672*b*c**6*d**5*x**3 + 336*c**7*d**5*x**4) - 2*a*b**2*c*sqrt(b*d + 2*c*d*x)/(21*b**4*c**3*d**5 + 168*b**3*
c**4*d**5*x + 504*b**2*c**5*d**5*x**2 + 672*b*c**6*d**5*x**3 + 336*c**7*d**5*x**4) - 14*a*b*c**2*x*sqrt(b*d +
2*c*d*x)/(21*b**4*c**3*d**5 + 168*b**3*c**4*d**5*x + 504*b**2*c**5*d**5*x**2 + 672*b*c**6*d**5*x**3 + 336*c**7
*d**5*x**4) - 14*a*c**3*x**2*sqrt(b*d + 2*c*d*x)/(21*b**4*c**3*d**5 + 168*b**3*c**4*d**5*x + 504*b**2*c**5*d**
5*x**2 + 672*b*c**6*d**5*x**3 + 336*c**7*d**5*x**4) + 2*b**4*sqrt(b*d + 2*c*d*x)/(21*b**4*c**3*d**5 + 168*b**3
*c**4*d**5*x + 504*b**2*c**5*d**5*x**2 + 672*b*c**6*d**5*x**3 + 336*c**7*d**5*x**4) + 14*b**3*c*x*sqrt(b*d + 2
*c*d*x)/(21*b**4*c**3*d**5 + 168*b**3*c**4*d**5*x + 504*b**2*c**5*d**5*x**2 + 672*b*c**6*d**5*x**3 + 336*c**7*
d**5*x**4) + 35*b**2*c**2*x**2*sqrt(b*d + 2*c*d*x)/(21*b**4*c**3*d**5 + 168*b**3*c**4*d**5*x + 504*b**2*c**5*d
**5*x**2 + 672*b*c**6*d**5*x**3 + 336*c**7*d**5*x**4) + 42*b*c**3*x**3*sqrt(b*d + 2*c*d*x)/(21*b**4*c**3*d**5
+ 168*b**3*c**4*d**5*x + 504*b**2*c**5*d**5*x**2 + 672*b*c**6*d**5*x**3 + 336*c**7*d**5*x**4) + 21*c**4*x**4*s
qrt(b*d + 2*c*d*x)/(21*b**4*c**3*d**5 + 168*b**3*c**4*d**5*x + 504*b**2*c**5*d**5*x**2 + 672*b*c**6*d**5*x**3
+ 336*c**7*d**5*x**4), Ne(c, 0)), ((a**2*x + a*b*x**2 + b**2*x**3/3)/(b*d)**(9/2), True))

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Giac [A]  time = 1.16398, size = 135, normalized size = 1.53 \begin{align*} \frac{\sqrt{2 \, c d x + b d}}{16 \, c^{3} d^{5}} - \frac{3 \, b^{4} d^{2} - 24 \, a b^{2} c d^{2} + 48 \, a^{2} c^{2} d^{2} - 14 \,{\left (2 \, c d x + b d\right )}^{2} b^{2} + 56 \,{\left (2 \, c d x + b d\right )}^{2} a c}{336 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} c^{3} d^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^(9/2),x, algorithm="giac")

[Out]

1/16*sqrt(2*c*d*x + b*d)/(c^3*d^5) - 1/336*(3*b^4*d^2 - 24*a*b^2*c*d^2 + 48*a^2*c^2*d^2 - 14*(2*c*d*x + b*d)^2
*b^2 + 56*(2*c*d*x + b*d)^2*a*c)/((2*c*d*x + b*d)^(7/2)*c^3*d^3)