### 3.1235 $$\int \frac{(b d+2 c d x)^3}{\sqrt{a+b x+c x^2}} \, dx$$

Optimal. Leaf size=59 $\frac{4}{3} d^3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}+\frac{2}{3} d^3 (b+2 c x)^2 \sqrt{a+b x+c x^2}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0259978, antiderivative size = 59, 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 = {692, 629} $\frac{4}{3} d^3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}+\frac{2}{3} d^3 (b+2 c x)^2 \sqrt{a+b x+c x^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 692

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(2*d*(d + e*x)^(m -
1)*(a + b*x + c*x^2)^(p + 1))/(b*(m + 2*p + 1)), x] + Dist[(d^2*(m - 1)*(b^2 - 4*a*c))/(b^2*(m + 2*p + 1)), In
t[(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] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] &
& RationalQ[p]) || OddQ[m])

Rule 629

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

Rubi steps

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

Mathematica [A]  time = 0.0355574, size = 43, normalized size = 0.73 $\frac{2}{3} d^3 \sqrt{a+x (b+c x)} \left (4 c \left (c x^2-2 a\right )+3 b^2+4 b c x\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.046, size = 41, normalized size = 0.7 \begin{align*} -{\frac{2\,{d}^{3} \left ( -4\,{c}^{2}{x}^{2}-4\,bcx+8\,ac-3\,{b}^{2} \right ) }{3}\sqrt{c{x}^{2}+bx+a}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 2.84848, size = 107, normalized size = 1.81 \begin{align*} \frac{2}{3} \,{\left (4 \, c^{2} d^{3} x^{2} + 4 \, b c d^{3} x +{\left (3 \, b^{2} - 8 \, a c\right )} d^{3}\right )} \sqrt{c x^{2} + b x + a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.387048, size = 97, normalized size = 1.64 \begin{align*} - \frac{16 a c d^{3} \sqrt{a + b x + c x^{2}}}{3} + 2 b^{2} d^{3} \sqrt{a + b x + c x^{2}} + \frac{8 b c d^{3} x \sqrt{a + b x + c x^{2}}}{3} + \frac{8 c^{2} d^{3} x^{2} \sqrt{a + b x + c x^{2}}}{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16*a*c*d**3*sqrt(a + b*x + c*x**2)/3 + 2*b**2*d**3*sqrt(a + b*x + c*x**2) + 8*b*c*d**3*x*sqrt(a + b*x + c*x**
2)/3 + 8*c**2*d**3*x**2*sqrt(a + b*x + c*x**2)/3

________________________________________________________________________________________

Giac [A]  time = 1.18905, size = 78, normalized size = 1.32 \begin{align*} \frac{2}{3} \, \sqrt{c x^{2} + b x + a}{\left (4 \,{\left (c^{2} d^{3} x + b c d^{3}\right )} x + \frac{3 \, b^{2} c^{2} d^{3} - 8 \, a c^{3} d^{3}}{c^{2}}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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