### 3.1265 $$\int \frac{a+b x+c x^2}{(b d+2 c d x)^{3/2}} \, dx$$

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

[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)

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Rubi [A]  time = 0.0233488, antiderivative size = 55, normalized size of antiderivative = 1., 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} $\frac{b^2-4 a c}{4 c^2 d \sqrt{b d+2 c d x}}+\frac{(b d+2 c d x)^{3/2}}{12 c^2 d^3}$

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*}

Mathematica [A]  time = 0.0249194, size = 41, normalized size = 0.75 $\frac{c \left (c x^2-3 a\right )+b^2+b c x}{3 c^2 d \sqrt{d (b+2 c x)}}$

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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Maple [A]  time = 0.041, size = 46, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 2\,cx+b \right ) \left ( -{c}^{2}{x}^{2}-bcx+3\,ac-{b}^{2} \right ) }{3\,{c}^{2}} \left ( 2\,cdx+bd \right ) ^{-{\frac{3}{2}}}} \end{align*}

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 = 1.0142, size = 69, normalized size = 1.25 \begin{align*} \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{align*}

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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Fricas [A]  time = 1.95868, size = 112, normalized size = 2.04 \begin{align*} \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{align*}

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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Sympy [A]  time = 12.5489, size = 49, normalized size = 0.89 \begin{align*} - \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{align*}

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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Giac [A]  time = 1.11439, size = 63, normalized size = 1.15 \begin{align*} \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{align*}

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)