3.1243 \(\int \frac{(b d+2 c d x)^3}{(a+b x+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=48 \[ 16 c d^3 \sqrt{a+b x+c x^2}-\frac{2 d^3 (b+2 c x)^2}{\sqrt{a+b x+c x^2}} \]

[Out]

(-2*d^3*(b + 2*c*x)^2)/Sqrt[a + b*x + c*x^2] + 16*c*d^3*Sqrt[a + b*x + c*x^2]

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Rubi [A]  time = 0.0229013, antiderivative size = 48, 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 = {686, 629} \[ 16 c d^3 \sqrt{a+b x+c x^2}-\frac{2 d^3 (b+2 c x)^2}{\sqrt{a+b x+c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d^3*(b + 2*c*x)^2)/Sqrt[a + b*x + c*x^2] + 16*c*d^3*Sqrt[a + b*x + c*x^2]

Rule 686

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

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

Mathematica [A]  time = 0.0287474, size = 40, normalized size = 0.83 \[ \frac{d^3 \left (8 c \left (2 a+c x^2\right )-2 b^2+8 b c x\right )}{\sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.044, size = 41, normalized size = 0.9 \begin{align*} 2\,{\frac{{d}^{3} \left ( 4\,{c}^{2}{x}^{2}+4\,bcx+8\,ac-{b}^{2} \right ) }{\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)^(3/2),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.55734, size = 101, normalized size = 2.1 \begin{align*} \frac{2 \,{\left (4 \, c^{2} d^{3} x^{2} + 4 \, b c d^{3} x -{\left (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)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.36838, size = 92, normalized size = 1.92 \begin{align*} \frac{16 a c d^{3}}{\sqrt{a + b x + c x^{2}}} - \frac{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}}} + \frac{8 c^{2} d^{3} x^{2}}{\sqrt{a + b x + c x^{2}}} \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)**(3/2),x)

[Out]

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

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Giac [B]  time = 1.17413, size = 188, normalized size = 3.92 \begin{align*} \frac{2 \,{\left (4 \,{\left (\frac{{\left (b^{2} c^{3} d^{3} - 4 \, a c^{4} d^{3}\right )} x}{b^{2} c - 4 \, a c^{2}} + \frac{b^{3} c^{2} d^{3} - 4 \, a b c^{3} d^{3}}{b^{2} c - 4 \, a c^{2}}\right )} x - \frac{b^{4} c d^{3} - 12 \, a b^{2} c^{2} d^{3} + 32 \, a^{2} c^{3} d^{3}}{b^{2} c - 4 \, a c^{2}}\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)^(3/2),x, algorithm="giac")

[Out]

2*(4*((b^2*c^3*d^3 - 4*a*c^4*d^3)*x/(b^2*c - 4*a*c^2) + (b^3*c^2*d^3 - 4*a*b*c^3*d^3)/(b^2*c - 4*a*c^2))*x - (
b^4*c*d^3 - 12*a*b^2*c^2*d^3 + 32*a^2*c^3*d^3)/(b^2*c - 4*a*c^2))/sqrt(c*x^2 + b*x + a)