### 3.1212 $$\int \frac{(a+b x+c x^2)^{3/2}}{(b d+2 c d x)^6} \, dx$$

Optimal. Leaf size=39 $\frac{2 \left (a+b x+c x^2\right )^{5/2}}{5 d^6 \left (b^2-4 a c\right ) (b+2 c x)^5}$

[Out]

(2*(a + b*x + c*x^2)^(5/2))/(5*(b^2 - 4*a*c)*d^6*(b + 2*c*x)^5)

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Rubi [A]  time = 0.0164574, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.038, Rules used = {682} $\frac{2 \left (a+b x+c x^2\right )^{5/2}}{5 d^6 \left (b^2-4 a c\right ) (b+2 c x)^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^6,x]

[Out]

(2*(a + b*x + c*x^2)^(5/2))/(5*(b^2 - 4*a*c)*d^6*(b + 2*c*x)^5)

Rule 682

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

Rubi steps

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

Mathematica [A]  time = 0.025451, size = 38, normalized size = 0.97 $\frac{2 (a+x (b+c x))^{5/2}}{5 d^6 \left (b^2-4 a c\right ) (b+2 c x)^5}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^6,x]

[Out]

(2*(a + x*(b + c*x))^(5/2))/(5*(b^2 - 4*a*c)*d^6*(b + 2*c*x)^5)

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Maple [A]  time = 0.043, size = 38, normalized size = 1. \begin{align*} -{\frac{2}{5\, \left ( 2\,cx+b \right ) ^{5}{d}^{6} \left ( 4\,ac-{b}^{2} \right ) } \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^6,x)

[Out]

-2/5*(c*x^2+b*x+a)^(5/2)/(2*c*x+b)^5/d^6/(4*a*c-b^2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 24.2278, size = 383, normalized size = 9.82 \begin{align*} \frac{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt{c x^{2} + b x + a}}{5 \,{\left (32 \,{\left (b^{2} c^{5} - 4 \, a c^{6}\right )} d^{6} x^{5} + 80 \,{\left (b^{3} c^{4} - 4 \, a b c^{5}\right )} d^{6} x^{4} + 80 \,{\left (b^{4} c^{3} - 4 \, a b^{2} c^{4}\right )} d^{6} x^{3} + 40 \,{\left (b^{5} c^{2} - 4 \, a b^{3} c^{3}\right )} d^{6} x^{2} + 10 \,{\left (b^{6} c - 4 \, a b^{4} c^{2}\right )} d^{6} x +{\left (b^{7} - 4 \, a b^{5} c\right )} d^{6}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^6,x, algorithm="fricas")

[Out]

2/5*(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*sqrt(c*x^2 + b*x + a)/(32*(b^2*c^5 - 4*a*c^6)*d^
6*x^5 + 80*(b^3*c^4 - 4*a*b*c^5)*d^6*x^4 + 80*(b^4*c^3 - 4*a*b^2*c^4)*d^6*x^3 + 40*(b^5*c^2 - 4*a*b^3*c^3)*d^6
*x^2 + 10*(b^6*c - 4*a*b^4*c^2)*d^6*x + (b^7 - 4*a*b^5*c)*d^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a \sqrt{a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx + \int \frac{b x \sqrt{a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx + \int \frac{c x^{2} \sqrt{a + b x + c x^{2}}}{b^{6} + 12 b^{5} c x + 60 b^{4} c^{2} x^{2} + 160 b^{3} c^{3} x^{3} + 240 b^{2} c^{4} x^{4} + 192 b c^{5} x^{5} + 64 c^{6} x^{6}}\, dx}{d^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**(3/2)/(2*c*d*x+b*d)**6,x)

[Out]

(Integral(a*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*x + 60*b**4*c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*c**
4*x**4 + 192*b*c**5*x**5 + 64*c**6*x**6), x) + Integral(b*x*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*x + 60*b*
*4*c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*c**4*x**4 + 192*b*c**5*x**5 + 64*c**6*x**6), x) + Integral(c*x**2
*sqrt(a + b*x + c*x**2)/(b**6 + 12*b**5*c*x + 60*b**4*c**2*x**2 + 160*b**3*c**3*x**3 + 240*b**2*c**4*x**4 + 19
2*b*c**5*x**5 + 64*c**6*x**6), x))/d**6

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Giac [B]  time = 1.84344, size = 802, normalized size = 20.56 \begin{align*} \frac{80 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{8} c^{\frac{9}{2}} + 320 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{7} b c^{4} + 560 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{6} b^{2} c^{\frac{7}{2}} + 560 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{5} b^{3} c^{3} + 360 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{4} b^{4} c^{\frac{5}{2}} - 80 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{4} a b^{2} c^{\frac{7}{2}} + 160 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{4} a^{2} c^{\frac{9}{2}} + 160 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} b^{5} c^{2} - 160 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} a b^{3} c^{3} + 320 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} a^{2} b c^{4} + 50 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} b^{6} c^{\frac{3}{2}} - 120 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} a b^{4} c^{\frac{5}{2}} + 240 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} a^{2} b^{2} c^{\frac{7}{2}} + 10 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} b^{7} c - 40 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} a b^{5} c^{2} + 80 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} a^{2} b^{3} c^{3} + b^{8} \sqrt{c} - 6 \, a b^{6} c^{\frac{3}{2}} + 16 \, a^{2} b^{4} c^{\frac{5}{2}} - 16 \, a^{3} b^{2} c^{\frac{7}{2}} + 16 \, a^{4} c^{\frac{9}{2}}}{80 \,{\left (2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} c + 2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} b \sqrt{c} + b^{2} - 2 \, a c\right )}^{5} c^{3} d^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^6,x, algorithm="giac")

[Out]

1/80*(80*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*c^(9/2) + 320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b*c^4 + 560
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b^2*c^(7/2) + 560*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^3*c^3 + 360*(
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^4*c^(5/2) - 80*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^2*c^(7/2) + 16
0*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*c^(9/2) + 160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^5*c^2 - 160*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b^3*c^3 + 320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b*c^4 + 50*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^6*c^(3/2) - 120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^4*c^(5/2) + 240
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^2*c^(7/2) + 10*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^7*c - 40*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^5*c^2 + 80*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^3*c^3 + b^8*sqrt(c)
- 6*a*b^6*c^(3/2) + 16*a^2*b^4*c^(5/2) - 16*a^3*b^2*c^(7/2) + 16*a^4*c^(9/2))/((2*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^2*c + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*sqrt(c) + b^2 - 2*a*c)^5*c^3*d^6)