### 3.1246 $$\int \frac{1}{(b d+2 c d x) (a+b x+c x^2)^{3/2}} \, dx$$

Optimal. Leaf size=86 $-\frac{2}{d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{4 \sqrt{c} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{d \left (b^2-4 a c\right )^{3/2}}$

[Out]

-2/((b^2 - 4*a*c)*d*Sqrt[a + b*x + c*x^2]) - (4*Sqrt[c]*ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*
a*c]])/((b^2 - 4*a*c)^(3/2)*d)

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Rubi [A]  time = 0.0582978, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.115, Rules used = {687, 688, 205} $-\frac{2}{d \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{4 \sqrt{c} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{d \left (b^2-4 a c\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((b*d + 2*c*d*x)*(a + b*x + c*x^2)^(3/2)),x]

[Out]

-2/((b^2 - 4*a*c)*d*Sqrt[a + b*x + c*x^2]) - (4*Sqrt[c]*ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*
a*c]])/((b^2 - 4*a*c)^(3/2)*d)

Rule 687

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] - Dist[(2*c*e*(m + 2*p + 3))/(e*(p + 1)*(b^2 - 4*a
*c)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0]
&& EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[p, -1] &&  !GtQ[m, 1] && RationalQ[m] && IntegerQ[2*p]

Rule 688

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

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

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

Mathematica [C]  time = 0.0284685, size = 60, normalized size = 0.7 $-\frac{2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{4 c (a+x (b+c x))}{4 a c-b^2}\right )}{d \left (b^2-4 a c\right ) \sqrt{a+x (b+c x)}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((b*d + 2*c*d*x)*(a + b*x + c*x^2)^(3/2)),x]

[Out]

(-2*Hypergeometric2F1[-1/2, 1, 1/2, (4*c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])/((b^2 - 4*a*c)*d*Sqrt[a + x*(b +
c*x)])

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Maple [B]  time = 0.196, size = 158, normalized size = 1.8 \begin{align*} 2\,{\frac{1}{d \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{ \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+1/4\,{\frac{4\,ac-{b}^{2}}{c}}}}}}-4\,{\frac{1}{d \left ( 4\,ac-{b}^{2} \right ) }\ln \left ({ \left ( 1/2\,{\frac{4\,ac-{b}^{2}}{c}}+1/2\,\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}} \right ) \left ( x+1/2\,{\frac{b}{c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*c*d*x+b*d)/(c*x^2+b*x+a)^(3/2),x)

[Out]

2/d/(4*a*c-b^2)/((x+1/2*b/c)^2*c+1/4*(4*a*c-b^2)/c)^(1/2)-4/d/(4*a*c-b^2)/((4*a*c-b^2)/c)^(1/2)*ln((1/2*(4*a*c
-b^2)/c+1/2*((4*a*c-b^2)/c)^(1/2)*(4*(x+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2))/(x+1/2*b/c))

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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(1/(2*c*d*x+b*d)/(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 4.43152, size = 695, normalized size = 8.08 \begin{align*} \left [-\frac{2 \,{\left ({\left (c x^{2} + b x + a\right )} \sqrt{-\frac{c}{b^{2} - 4 \, a c}} \log \left (-\frac{4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c + 4 \, \sqrt{c x^{2} + b x + a}{\left (b^{2} - 4 \, a c\right )} \sqrt{-\frac{c}{b^{2} - 4 \, a c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) + \sqrt{c x^{2} + b x + a}\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} d x^{2} +{\left (b^{3} - 4 \, a b c\right )} d x +{\left (a b^{2} - 4 \, a^{2} c\right )} d}, -\frac{2 \,{\left (2 \,{\left (c x^{2} + b x + a\right )} \sqrt{\frac{c}{b^{2} - 4 \, a c}} \arctan \left (-\frac{\sqrt{c x^{2} + b x + a}{\left (b^{2} - 4 \, a c\right )} \sqrt{\frac{c}{b^{2} - 4 \, a c}}}{2 \,{\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + \sqrt{c x^{2} + b x + a}\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} d x^{2} +{\left (b^{3} - 4 \, a b c\right )} d x +{\left (a b^{2} - 4 \, a^{2} c\right )} d}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*c*d*x+b*d)/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")

[Out]

[-2*((c*x^2 + b*x + a)*sqrt(-c/(b^2 - 4*a*c))*log(-(4*c^2*x^2 + 4*b*c*x - b^2 + 8*a*c + 4*sqrt(c*x^2 + b*x + a
)*(b^2 - 4*a*c)*sqrt(-c/(b^2 - 4*a*c)))/(4*c^2*x^2 + 4*b*c*x + b^2)) + sqrt(c*x^2 + b*x + a))/((b^2*c - 4*a*c^
2)*d*x^2 + (b^3 - 4*a*b*c)*d*x + (a*b^2 - 4*a^2*c)*d), -2*(2*(c*x^2 + b*x + a)*sqrt(c/(b^2 - 4*a*c))*arctan(-1
/2*sqrt(c*x^2 + b*x + a)*(b^2 - 4*a*c)*sqrt(c/(b^2 - 4*a*c))/(c^2*x^2 + b*c*x + a*c)) + sqrt(c*x^2 + b*x + a))
/((b^2*c - 4*a*c^2)*d*x^2 + (b^3 - 4*a*b*c)*d*x + (a*b^2 - 4*a^2*c)*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{a b \sqrt{a + b x + c x^{2}} + 2 a c x \sqrt{a + b x + c x^{2}} + b^{2} x \sqrt{a + b x + c x^{2}} + 3 b c x^{2} \sqrt{a + b x + c x^{2}} + 2 c^{2} x^{3} \sqrt{a + b x + c x^{2}}}\, dx}{d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*c*d*x+b*d)/(c*x**2+b*x+a)**(3/2),x)

[Out]

Integral(1/(a*b*sqrt(a + b*x + c*x**2) + 2*a*c*x*sqrt(a + b*x + c*x**2) + b**2*x*sqrt(a + b*x + c*x**2) + 3*b*
c*x**2*sqrt(a + b*x + c*x**2) + 2*c**2*x**3*sqrt(a + b*x + c*x**2)), x)/d

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Giac [B]  time = 1.15744, size = 213, normalized size = 2.48 \begin{align*} \frac{8 \, c \arctan \left (\frac{2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} c + b \sqrt{c}}{\sqrt{b^{2} c - 4 \, a c^{2}}}\right )}{\sqrt{b^{2} c - 4 \, a c^{2}}{\left (b^{2} d - 4 \, a c d\right )}} - \frac{2 \,{\left (b^{4} d - 8 \, a b^{2} c d + 16 \, a^{2} c^{2} d\right )}}{{\left (b^{6} d^{2} - 12 \, a b^{4} c d^{2} + 48 \, a^{2} b^{2} c^{2} d^{2} - 64 \, a^{3} c^{3} d^{2}\right )} \sqrt{c x^{2} + b x + a}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*c*d*x+b*d)/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")

[Out]

8*c*arctan((2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*c + b*sqrt(c))/sqrt(b^2*c - 4*a*c^2))/(sqrt(b^2*c - 4*a*c^2)
*(b^2*d - 4*a*c*d)) - 2*(b^4*d - 8*a*b^2*c*d + 16*a^2*c^2*d)/((b^6*d^2 - 12*a*b^4*c*d^2 + 48*a^2*b^2*c^2*d^2 -
64*a^3*c^3*d^2)*sqrt(c*x^2 + b*x + a))