### 3.76 $$\int \frac{\sqrt{b x+c x^2}}{\sqrt{x}} \, dx$$

Optimal. Leaf size=25 $\frac{2 \left (b x+c x^2\right )^{3/2}}{3 c x^{3/2}}$

[Out]

(2*(b*x + c*x^2)^(3/2))/(3*c*x^(3/2))

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Rubi [A]  time = 0.0062485, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.053, Rules used = {648} $\frac{2 \left (b x+c x^2\right )^{3/2}}{3 c x^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[b*x + c*x^2]/Sqrt[x],x]

[Out]

(2*(b*x + c*x^2)^(3/2))/(3*c*x^(3/2))

Rule 648

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

Rubi steps

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

Mathematica [A]  time = 0.0108567, size = 23, normalized size = 0.92 $\frac{2 (x (b+c x))^{3/2}}{3 c x^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sqrt[b*x + c*x^2]/Sqrt[x],x]

[Out]

(2*(x*(b + c*x))^(3/2))/(3*c*x^(3/2))

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Maple [A]  time = 0.042, size = 25, normalized size = 1. \begin{align*}{\frac{2\,cx+2\,b}{3\,c}\sqrt{c{x}^{2}+bx}{\frac{1}{\sqrt{x}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^(1/2)/x^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.16604, size = 16, normalized size = 0.64 \begin{align*} \frac{2 \,{\left (c x + b\right )}^{\frac{3}{2}}}{3 \, c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(1/2)/x^(1/2),x, algorithm="maxima")

[Out]

2/3*(c*x + b)^(3/2)/c

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Fricas [A]  time = 2.21826, size = 61, normalized size = 2.44 \begin{align*} \frac{2 \, \sqrt{c x^{2} + b x}{\left (c x + b\right )}}{3 \, c \sqrt{x}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(1/2)/x^(1/2),x, algorithm="fricas")

[Out]

2/3*sqrt(c*x^2 + b*x)*(c*x + b)/(c*sqrt(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x \left (b + c x\right )}}{\sqrt{x}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x)**(1/2)/x**(1/2),x)

[Out]

Integral(sqrt(x*(b + c*x))/sqrt(x), x)

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Giac [A]  time = 1.19745, size = 28, normalized size = 1.12 \begin{align*} \frac{2 \,{\left (c x + b\right )}^{\frac{3}{2}}}{3 \, c} - \frac{2 \, b^{\frac{3}{2}}}{3 \, c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(1/2)/x^(1/2),x, algorithm="giac")

[Out]

2/3*(c*x + b)^(3/2)/c - 2/3*b^(3/2)/c