### 3.1274 $$\int \frac{(a+b x+c x^2)^2}{(b d+2 c d x)^{7/2}} \, dx$$

Optimal. Leaf size=88 $\frac{b^2-4 a c}{8 c^3 d^3 \sqrt{b d+2 c d x}}-\frac{\left (b^2-4 a c\right )^2}{80 c^3 d (b d+2 c d x)^{5/2}}+\frac{(b d+2 c d x)^{3/2}}{48 c^3 d^5}$

[Out]

-(b^2 - 4*a*c)^2/(80*c^3*d*(b*d + 2*c*d*x)^(5/2)) + (b^2 - 4*a*c)/(8*c^3*d^3*Sqrt[b*d + 2*c*d*x]) + (b*d + 2*c
*d*x)^(3/2)/(48*c^3*d^5)

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Rubi [A]  time = 0.0383003, 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} $\frac{b^2-4 a c}{8 c^3 d^3 \sqrt{b d+2 c d x}}-\frac{\left (b^2-4 a c\right )^2}{80 c^3 d (b d+2 c d x)^{5/2}}+\frac{(b d+2 c d x)^{3/2}}{48 c^3 d^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(b^2 - 4*a*c)^2/(80*c^3*d*(b*d + 2*c*d*x)^(5/2)) + (b^2 - 4*a*c)/(8*c^3*d^3*Sqrt[b*d + 2*c*d*x]) + (b*d + 2*c
*d*x)^(3/2)/(48*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)^{7/2}} \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^2}{16 c^2 (b d+2 c d x)^{7/2}}+\frac{-b^2+4 a c}{8 c^2 d^2 (b d+2 c d x)^{3/2}}+\frac{\sqrt{b d+2 c d x}}{16 c^2 d^4}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right )^2}{80 c^3 d (b d+2 c d x)^{5/2}}+\frac{b^2-4 a c}{8 c^3 d^3 \sqrt{b d+2 c d x}}+\frac{(b d+2 c d x)^{3/2}}{48 c^3 d^5}\\ \end{align*}

Mathematica [A]  time = 0.0431497, size = 92, normalized size = 1.05 $\frac{c^2 \left (-3 a^2-30 a c x^2+5 c^2 x^4\right )+3 b^2 c \left (5 c x^2-2 a\right )+10 b c^2 x \left (c x^2-3 a\right )+10 b^3 c x+2 b^4}{15 c^3 d (d (b+2 c x))^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*b^4 + 10*b^3*c*x + 10*b*c^2*x*(-3*a + c*x^2) + 3*b^2*c*(-2*a + 5*c*x^2) + c^2*(-3*a^2 - 30*a*c*x^2 + 5*c^2*
x^4))/(15*c^3*d*(d*(b + 2*c*x))^(5/2))

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Maple [A]  time = 0.046, size = 96, normalized size = 1.1 \begin{align*} -{\frac{ \left ( 2\,cx+b \right ) \left ( -5\,{c}^{4}{x}^{4}-10\,b{x}^{3}{c}^{3}+30\,a{c}^{3}{x}^{2}-15\,{b}^{2}{c}^{2}{x}^{2}+30\,ab{c}^{2}x-10\,{b}^{3}cx+3\,{a}^{2}{c}^{2}+6\,ac{b}^{2}-2\,{b}^{4} \right ) }{15\,{c}^{3}} \left ( 2\,cdx+bd \right ) ^{-{\frac{7}{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)^(7/2),x)

[Out]

-1/15*(2*c*x+b)*(-5*c^4*x^4-10*b*c^3*x^3+30*a*c^3*x^2-15*b^2*c^2*x^2+30*a*b*c^2*x-10*b^3*c*x+3*a^2*c^2+6*a*b^2
*c-2*b^4)/c^3/(2*c*d*x+b*d)^(7/2)

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Maxima [A]  time = 1.00077, size = 126, normalized size = 1.43 \begin{align*} \frac{\frac{5 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}}{c^{2} d^{4}} + \frac{3 \,{\left (10 \,{\left (2 \, c d x + b d\right )}^{2}{\left (b^{2} - 4 \, a c\right )} -{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{2}\right )}}{{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} c^{2} d^{2}}}{240 \, 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)^(7/2),x, algorithm="maxima")

[Out]

1/240*(5*(2*c*d*x + b*d)^(3/2)/(c^2*d^4) + 3*(10*(2*c*d*x + b*d)^2*(b^2 - 4*a*c) - (b^4 - 8*a*b^2*c + 16*a^2*c
^2)*d^2)/((2*c*d*x + b*d)^(5/2)*c^2*d^2))/(c*d)

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Fricas [A]  time = 2.01388, size = 281, normalized size = 3.19 \begin{align*} \frac{{\left (5 \, c^{4} x^{4} + 10 \, b c^{3} x^{3} + 2 \, b^{4} - 6 \, a b^{2} c - 3 \, a^{2} c^{2} + 15 \,{\left (b^{2} c^{2} - 2 \, a c^{3}\right )} x^{2} + 10 \,{\left (b^{3} c - 3 \, a b c^{2}\right )} x\right )} \sqrt{2 \, c d x + b d}}{15 \,{\left (8 \, c^{6} d^{4} x^{3} + 12 \, b c^{5} d^{4} x^{2} + 6 \, b^{2} c^{4} d^{4} x + b^{3} c^{3} d^{4}\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)^(7/2),x, algorithm="fricas")

[Out]

1/15*(5*c^4*x^4 + 10*b*c^3*x^3 + 2*b^4 - 6*a*b^2*c - 3*a^2*c^2 + 15*(b^2*c^2 - 2*a*c^3)*x^2 + 10*(b^3*c - 3*a*
b*c^2)*x)*sqrt(2*c*d*x + b*d)/(8*c^6*d^4*x^3 + 12*b*c^5*d^4*x^2 + 6*b^2*c^4*d^4*x + b^3*c^3*d^4)

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Sympy [A]  time = 5.93285, size = 688, normalized size = 7.82 \begin{align*} \begin{cases} - \frac{3 a^{2} c^{2} \sqrt{b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} - \frac{6 a b^{2} c \sqrt{b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} - \frac{30 a b c^{2} x \sqrt{b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} - \frac{30 a c^{3} x^{2} \sqrt{b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} + \frac{2 b^{4} \sqrt{b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} + \frac{10 b^{3} c x \sqrt{b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} + \frac{15 b^{2} c^{2} x^{2} \sqrt{b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} + \frac{10 b c^{3} x^{3} \sqrt{b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} + \frac{5 c^{4} x^{4} \sqrt{b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} & \text{for}\: c \neq 0 \\\frac{a^{2} x + a b x^{2} + \frac{b^{2} x^{3}}{3}}{\left (b d\right )^{\frac{7}{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)**(7/2),x)

[Out]

Piecewise((-3*a**2*c**2*sqrt(b*d + 2*c*d*x)/(15*b**3*c**3*d**4 + 90*b**2*c**4*d**4*x + 180*b*c**5*d**4*x**2 +
120*c**6*d**4*x**3) - 6*a*b**2*c*sqrt(b*d + 2*c*d*x)/(15*b**3*c**3*d**4 + 90*b**2*c**4*d**4*x + 180*b*c**5*d**
4*x**2 + 120*c**6*d**4*x**3) - 30*a*b*c**2*x*sqrt(b*d + 2*c*d*x)/(15*b**3*c**3*d**4 + 90*b**2*c**4*d**4*x + 18
0*b*c**5*d**4*x**2 + 120*c**6*d**4*x**3) - 30*a*c**3*x**2*sqrt(b*d + 2*c*d*x)/(15*b**3*c**3*d**4 + 90*b**2*c**
4*d**4*x + 180*b*c**5*d**4*x**2 + 120*c**6*d**4*x**3) + 2*b**4*sqrt(b*d + 2*c*d*x)/(15*b**3*c**3*d**4 + 90*b**
2*c**4*d**4*x + 180*b*c**5*d**4*x**2 + 120*c**6*d**4*x**3) + 10*b**3*c*x*sqrt(b*d + 2*c*d*x)/(15*b**3*c**3*d**
4 + 90*b**2*c**4*d**4*x + 180*b*c**5*d**4*x**2 + 120*c**6*d**4*x**3) + 15*b**2*c**2*x**2*sqrt(b*d + 2*c*d*x)/(
15*b**3*c**3*d**4 + 90*b**2*c**4*d**4*x + 180*b*c**5*d**4*x**2 + 120*c**6*d**4*x**3) + 10*b*c**3*x**3*sqrt(b*d
+ 2*c*d*x)/(15*b**3*c**3*d**4 + 90*b**2*c**4*d**4*x + 180*b*c**5*d**4*x**2 + 120*c**6*d**4*x**3) + 5*c**4*x**
4*sqrt(b*d + 2*c*d*x)/(15*b**3*c**3*d**4 + 90*b**2*c**4*d**4*x + 180*b*c**5*d**4*x**2 + 120*c**6*d**4*x**3), N
e(c, 0)), ((a**2*x + a*b*x**2 + b**2*x**3/3)/(b*d)**(7/2), True))

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Giac [A]  time = 1.18467, size = 134, normalized size = 1.52 \begin{align*} \frac{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}}{48 \, c^{3} d^{5}} - \frac{b^{4} d^{2} - 8 \, a b^{2} c d^{2} + 16 \, a^{2} c^{2} d^{2} - 10 \,{\left (2 \, c d x + b d\right )}^{2} b^{2} + 40 \,{\left (2 \, c d x + b d\right )}^{2} a c}{80 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{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)^(7/2),x, algorithm="giac")

[Out]

1/48*(2*c*d*x + b*d)^(3/2)/(c^3*d^5) - 1/80*(b^4*d^2 - 8*a*b^2*c*d^2 + 16*a^2*c^2*d^2 - 10*(2*c*d*x + b*d)^2*b
^2 + 40*(2*c*d*x + b*d)^2*a*c)/((2*c*d*x + b*d)^(5/2)*c^3*d^3)