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*(2*c*x+b)^2/(c*x^2+b*x+a)^(1/2)+16*c*d^3*(c*x^2+b*x+a)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 48, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.03, 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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fricas [A]  time = 1.27, size = 47, normalized size = 0.98 \[ \frac {2 \, {\left (4 \, c^{2} d^{3} x^{2} + 4 \, b c d^{3} x - {\left (b^{2} - 8 \, a c\right )} d^{3}\right )}}{\sqrt {c x^{2} + b x + a}} \]

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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giac [A]  time = 0.18, size = 48, normalized size = 1.00 \[ 8 \, \sqrt {c x^{2} + b x + a} c d^{3} - \frac {2 \, {\left (b^{2} d^{3} - 4 \, a c d^{3}\right )}}{\sqrt {c x^{2} + b x + a}} \]

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]

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

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maple [A]  time = 0.04, size = 41, normalized size = 0.85 \[ \frac {2 \left (4 c^{2} x^{2}+4 b c x +8 a c -b^{2}\right ) d^{3}}{\sqrt {c \,x^{2}+b x +a}} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 zero or nonzero?

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mupad [B]  time = 0.67, size = 40, normalized size = 0.83 \[ \frac {2\,d^3\,\left (-b^2+4\,b\,c\,x+4\,c^2\,x^2+8\,a\,c\right )}{\sqrt {c\,x^2+b\,x+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 1.24, size = 92, normalized size = 1.92 \[ \frac {16 a c d^{3}}{\sqrt {a + b x + c x^{2}}} - \frac {2 b^{2} d^{3}}{\sqrt {a + b x + c x^{2}}} + \frac {8 b c d^{3} x}{\sqrt {a + b x + c x^{2}}} + \frac {8 c^{2} d^{3} x^{2}}{\sqrt {a + b x + c x^{2}}} \]

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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