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

Optimal. Leaf size=55 \[ \frac {b^2-4 a c}{4 c^2 d \sqrt {b d+2 c d x}}+\frac {(b d+2 c d x)^{3/2}}{12 c^2 d^3} \]

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Rubi [A]  time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {683} \begin {gather*} \frac {b^2-4 a c}{4 c^2 d \sqrt {b d+2 c d x}}+\frac {(b d+2 c d x)^{3/2}}{12 c^2 d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps

\begin {align*} \int \frac {a+b x+c x^2}{(b d+2 c d x)^{3/2}} \, dx &=\int \left (\frac {-b^2+4 a c}{4 c (b d+2 c d x)^{3/2}}+\frac {\sqrt {b d+2 c d x}}{4 c d^2}\right ) \, dx\\ &=\frac {b^2-4 a c}{4 c^2 d \sqrt {b d+2 c d x}}+\frac {(b d+2 c d x)^{3/2}}{12 c^2 d^3}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 41, normalized size = 0.75 \begin {gather*} \frac {c \left (c x^2-3 a\right )+b^2+b c x}{3 c^2 d \sqrt {d (b+2 c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.08, size = 50, normalized size = 0.91 \begin {gather*} \frac {\left (-3 a c+b^2+b c x+c^2 x^2\right ) \sqrt {b d+2 c d x}}{3 c^2 d^2 (b+2 c x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.38, size = 52, normalized size = 0.95 \begin {gather*} \frac {{\left (c^{2} x^{2} + b c x + b^{2} - 3 \, a c\right )} \sqrt {2 \, c d x + b d}}{3 \, {\left (2 \, c^{3} d^{2} x + b c^{2} d^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.22, size = 47, normalized size = 0.85 \begin {gather*} \frac {b^{2} - 4 \, a c}{4 \, \sqrt {2 \, c d x + b d} c^{2} d} + \frac {{\left (2 \, c d x + b d\right )}^{\frac {3}{2}}}{12 \, c^{2} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(b^2 - 4*a*c)/(sqrt(2*c*d*x + b*d)*c^2*d) + 1/12*(2*c*d*x + b*d)^(3/2)/(c^2*d^3)

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maple [A]  time = 0.04, size = 46, normalized size = 0.84 \begin {gather*} -\frac {\left (2 c x +b \right ) \left (-c^{2} x^{2}-b c x +3 a c -b^{2}\right )}{3 \left (2 c d x +b d \right )^{\frac {3}{2}} c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.36, size = 51, normalized size = 0.93 \begin {gather*} \frac {\frac {3 \, {\left (b^{2} - 4 \, a c\right )}}{\sqrt {2 \, c d x + b d} c} + \frac {{\left (2 \, c d x + b d\right )}^{\frac {3}{2}}}{c d^{2}}}{12 \, c d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*(b^2 - 4*a*c)/(sqrt(2*c*d*x + b*d)*c) + (2*c*d*x + b*d)^(3/2)/(c*d^2))/(c*d)

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mupad [B]  time = 0.48, size = 37, normalized size = 0.67 \begin {gather*} \frac {{\left (b+2\,c\,x\right )}^2-12\,a\,c+3\,b^2}{12\,c^2\,d\,\sqrt {b\,d+2\,c\,d\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b + 2*c*x)^2 - 12*a*c + 3*b^2)/(12*c^2*d*(b*d + 2*c*d*x)^(1/2))

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sympy [A]  time = 13.08, size = 49, normalized size = 0.89 \begin {gather*} - \frac {4 a c - b^{2}}{4 c^{2} d \sqrt {b d + 2 c d x}} + \frac {\left (b d + 2 c d x\right )^{\frac {3}{2}}}{12 c^{2} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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