### 3.1253 $$\int \frac{(b d+2 c d x)^3}{(a+b x+c x^2)^{5/2}} \, dx$$

Optimal. Leaf size=52 $-\frac{16 c d^3}{3 \sqrt{a+b x+c x^2}}-\frac{2 d^3 (b+2 c x)^2}{3 \left (a+b x+c x^2\right )^{3/2}}$

[Out]

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

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Rubi [A]  time = 0.0233337, antiderivative size = 52, 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} $-\frac{16 c d^3}{3 \sqrt{a+b x+c x^2}}-\frac{2 d^3 (b+2 c x)^2}{3 \left (a+b x+c x^2\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*d + 2*c*d*x)^3/(a + b*x + c*x^2)^(5/2),x]

[Out]

(-2*d^3*(b + 2*c*x)^2)/(3*(a + b*x + c*x^2)^(3/2)) - (16*c*d^3)/(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 )^{5/2}} \, dx &=-\frac{2 d^3 (b+2 c x)^2}{3 \left (a+b x+c x^2\right )^{3/2}}+\frac{1}{3} \left (8 c d^2\right ) \int \frac{b d+2 c d x}{\left (a+b x+c x^2\right )^{3/2}} \, dx\\ &=-\frac{2 d^3 (b+2 c x)^2}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac{16 c d^3}{3 \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0292933, size = 42, normalized size = 0.81 $-\frac{2 d^3 \left (4 c \left (2 a+3 c x^2\right )+b^2+12 b c x\right )}{3 (a+x (b+c x))^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(b*d + 2*c*d*x)^3/(a + b*x + c*x^2)^(5/2),x]

[Out]

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

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Maple [A]  time = 0.044, size = 39, normalized size = 0.8 \begin{align*} -{\frac{2\,{d}^{3} \left ( 12\,{c}^{2}{x}^{2}+12\,bcx+8\,ac+{b}^{2} \right ) }{3} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}} \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)^(5/2),x)

[Out]

-2/3*d^3*(12*c^2*x^2+12*b*c*x+8*a*c+b^2)/(c*x^2+b*x+a)^(3/2)

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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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 5.68305, size = 186, normalized size = 3.58 \begin{align*} -\frac{2 \,{\left (12 \, c^{2} d^{3} x^{2} + 12 \, b c d^{3} x +{\left (b^{2} + 8 \, a c\right )} d^{3}\right )} \sqrt{c x^{2} + b x + a}}{3 \,{\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 )}} \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)^(5/2),x, algorithm="fricas")

[Out]

-2/3*(12*c^2*d^3*x^2 + 12*b*c*d^3*x + (b^2 + 8*a*c)*d^3)*sqrt(c*x^2 + b*x + a)/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x
+ (b^2 + 2*a*c)*x^2 + a^2)

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Sympy [B]  time = 1.99567, size = 264, normalized size = 5.08 \begin{align*} - \frac{16 a c d^{3}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} - \frac{2 b^{2} d^{3}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} - \frac{24 b c d^{3} x}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} - \frac{24 c^{2} d^{3} x^{2}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c 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)**(5/2),x)

[Out]

-16*a*c*d**3/(3*a*sqrt(a + b*x + c*x**2) + 3*b*x*sqrt(a + b*x + c*x**2) + 3*c*x**2*sqrt(a + b*x + c*x**2)) - 2
*b**2*d**3/(3*a*sqrt(a + b*x + c*x**2) + 3*b*x*sqrt(a + b*x + c*x**2) + 3*c*x**2*sqrt(a + b*x + c*x**2)) - 24*
b*c*d**3*x/(3*a*sqrt(a + b*x + c*x**2) + 3*b*x*sqrt(a + b*x + c*x**2) + 3*c*x**2*sqrt(a + b*x + c*x**2)) - 24*
c**2*d**3*x**2/(3*a*sqrt(a + b*x + c*x**2) + 3*b*x*sqrt(a + b*x + c*x**2) + 3*c*x**2*sqrt(a + b*x + c*x**2))

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Giac [B]  time = 1.14325, size = 275, normalized size = 5.29 \begin{align*} -\frac{12 \,{\left (\frac{{\left (b^{4} c^{2} d^{3} - 8 \, a b^{2} c^{3} d^{3} + 16 \, a^{2} c^{4} d^{3}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{b^{5} c d^{3} - 8 \, a b^{3} c^{2} d^{3} + 16 \, a^{2} b c^{3} d^{3}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{b^{6} d^{3} - 48 \, a^{2} b^{2} c^{2} d^{3} + 128 \, a^{3} c^{3} d^{3}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{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)^(5/2),x, algorithm="giac")

[Out]

-1/3*(12*((b^4*c^2*d^3 - 8*a*b^2*c^3*d^3 + 16*a^2*c^4*d^3)*x/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4) + (b^5*c*d^3
- 8*a*b^3*c^2*d^3 + 16*a^2*b*c^3*d^3)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))*x + (b^6*d^3 - 48*a^2*b^2*c^2*d^3
+ 128*a^3*c^3*d^3)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))/(c*x^2 + b*x + a)^(3/2)