3.1200 \(\int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^6} \, dx\)

Optimal. Leaf size=79 \[ \frac{4 \left (a+b x+c x^2\right )^{3/2}}{15 d^6 \left (b^2-4 a c\right )^2 (b+2 c x)^3}+\frac{2 \left (a+b x+c x^2\right )^{3/2}}{5 d^6 \left (b^2-4 a c\right ) (b+2 c x)^5} \]

[Out]

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

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Rubi [A]  time = 0.0348507, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {693, 682} \[ \frac{4 \left (a+b x+c x^2\right )^{3/2}}{15 d^6 \left (b^2-4 a c\right )^2 (b+2 c x)^3}+\frac{2 \left (a+b x+c x^2\right )^{3/2}}{5 d^6 \left (b^2-4 a c\right ) (b+2 c x)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 693

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

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{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^6} \, dx &=\frac{2 \left (a+b x+c x^2\right )^{3/2}}{5 \left (b^2-4 a c\right ) d^6 (b+2 c x)^5}+\frac{2 \int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^4} \, dx}{5 \left (b^2-4 a c\right ) d^2}\\ &=\frac{2 \left (a+b x+c x^2\right )^{3/2}}{5 \left (b^2-4 a c\right ) d^6 (b+2 c x)^5}+\frac{4 \left (a+b x+c x^2\right )^{3/2}}{15 \left (b^2-4 a c\right )^2 d^6 (b+2 c x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0293914, size = 62, normalized size = 0.78 \[ \frac{2 (a+x (b+c x))^{3/2} \left (4 c \left (2 c x^2-3 a\right )+5 b^2+8 b c x\right )}{15 d^6 \left (b^2-4 a c\right )^2 (b+2 c x)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.043, size = 70, normalized size = 0.9 \begin{align*} -{\frac{-16\,{c}^{2}{x}^{2}-16\,bcx+24\,ac-10\,{b}^{2}}{15\, \left ( 2\,cx+b \right ) ^{5}{d}^{6} \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) } \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(1/2)/(2*c*d*x+b*d)^6,x)

[Out]

-2/15*(-8*c^2*x^2-8*b*c*x+12*a*c-5*b^2)*(c*x^2+b*x+a)^(3/2)/(2*c*x+b)^5/d^6/(16*a^2*c^2-8*a*b^2*c+b^4)

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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)^(1/2)/(2*c*d*x+b*d)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 21.128, size = 575, normalized size = 7.28 \begin{align*} \frac{2 \,{\left (8 \, c^{3} x^{4} + 16 \, b c^{2} x^{3} + 5 \, a b^{2} - 12 \, a^{2} c +{\left (13 \, b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (5 \, b^{3} - 4 \, a b c\right )} x\right )} \sqrt{c x^{2} + b x + a}}{15 \,{\left (32 \,{\left (b^{4} c^{5} - 8 \, a b^{2} c^{6} + 16 \, a^{2} c^{7}\right )} d^{6} x^{5} + 80 \,{\left (b^{5} c^{4} - 8 \, a b^{3} c^{5} + 16 \, a^{2} b c^{6}\right )} d^{6} x^{4} + 80 \,{\left (b^{6} c^{3} - 8 \, a b^{4} c^{4} + 16 \, a^{2} b^{2} c^{5}\right )} d^{6} x^{3} + 40 \,{\left (b^{7} c^{2} - 8 \, a b^{5} c^{3} + 16 \, a^{2} b^{3} c^{4}\right )} d^{6} x^{2} + 10 \,{\left (b^{8} c - 8 \, a b^{6} c^{2} + 16 \, a^{2} b^{4} c^{3}\right )} d^{6} x +{\left (b^{9} - 8 \, a b^{7} c + 16 \, a^{2} b^{5} c^{2}\right )} d^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(1/2)/(2*c*d*x+b*d)^6,x, algorithm="fricas")

[Out]

2/15*(8*c^3*x^4 + 16*b*c^2*x^3 + 5*a*b^2 - 12*a^2*c + (13*b^2*c - 4*a*c^2)*x^2 + (5*b^3 - 4*a*b*c)*x)*sqrt(c*x
^2 + b*x + a)/(32*(b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*d^6*x^5 + 80*(b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*d^6
*x^4 + 80*(b^6*c^3 - 8*a*b^4*c^4 + 16*a^2*b^2*c^5)*d^6*x^3 + 40*(b^7*c^2 - 8*a*b^5*c^3 + 16*a^2*b^3*c^4)*d^6*x
^2 + 10*(b^8*c - 8*a*b^6*c^2 + 16*a^2*b^4*c^3)*d^6*x + (b^9 - 8*a*b^7*c + 16*a^2*b^5*c^2)*d^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\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)**(1/2)/(2*c*d*x+b*d)**6,x)

[Out]

Integral(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)/d**6

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Giac [B]  time = 1.48839, size = 571, normalized size = 7.23 \begin{align*} \frac{60 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{6} c^{\frac{7}{2}} + 180 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{5} b c^{3} + 220 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{4} b^{2} c^{\frac{5}{2}} + 20 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{4} a c^{\frac{7}{2}} + 140 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} b^{3} c^{2} + 40 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} a b c^{3} + 50 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} b^{4} c^{\frac{3}{2}} + 20 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} a b^{2} c^{\frac{5}{2}} + 20 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} a^{2} c^{\frac{7}{2}} + 10 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} b^{5} c + 20 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} a^{2} b c^{3} + b^{6} \sqrt{c} - 2 \, a b^{4} c^{\frac{3}{2}} + 8 \, a^{2} b^{2} c^{\frac{5}{2}} - 4 \, a^{3} c^{\frac{7}{2}}}{30 \,{\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^{2} d^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(1/2)/(2*c*d*x+b*d)^6,x, algorithm="giac")

[Out]

1/30*(60*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*c^(7/2) + 180*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b*c^3 + 220
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^2*c^(5/2) + 20*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*c^(7/2) + 140*
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^3*c^2 + 40*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*c^3 + 50*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))^2*b^4*c^(3/2) + 20*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b^2*c^(5/2) + 20*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*c^(7/2) + 10*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^5*c + 20*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))*a^2*b*c^3 + b^6*sqrt(c) - 2*a*b^4*c^(3/2) + 8*a^2*b^2*c^(5/2) - 4*a^3*c^(7/2))/((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^2*d^6)