### 3.87 $$\int \frac{(b x+c x^2)^{3/2}}{x^{3/2}} \, dx$$

Optimal. Leaf size=25 $\frac{2 \left (b x+c x^2\right )^{5/2}}{5 c x^{5/2}}$

[Out]

(2*(b*x + c*x^2)^(5/2))/(5*c*x^(5/2))

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Rubi [A]  time = 0.0069109, 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} $\frac{2 \left (b x+c x^2\right )^{5/2}}{5 c x^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*x + c*x^2)^(3/2)/x^(3/2),x]

[Out]

(2*(b*x + c*x^2)^(5/2))/(5*c*x^(5/2))

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{\left (b x+c x^2\right )^{3/2}}{x^{3/2}} \, dx &=\frac{2 \left (b x+c x^2\right )^{5/2}}{5 c x^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0138718, size = 23, normalized size = 0.92 $\frac{2 (x (b+c x))^{5/2}}{5 c x^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(x*(b + c*x))^(5/2))/(5*c*x^(5/2))

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Maple [A]  time = 0.044, size = 25, normalized size = 1. \begin{align*}{\frac{2\,cx+2\,b}{5\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(c*x+b)*(c*x^2+b*x)^(3/2)/c/x^(3/2)

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Maxima [B]  time = 1.13455, size = 66, normalized size = 2.64 \begin{align*} \frac{2 \,{\left (5 \, b c x^{2} + 5 \, b^{2} x +{\left (3 \, c^{2} x^{2} + b c x - 2 \, b^{2}\right )} x\right )} \sqrt{c x + b}}{15 \, c x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(5*b*c*x^2 + 5*b^2*x + (3*c^2*x^2 + b*c*x - 2*b^2)*x)*sqrt(c*x + b)/(c*x)

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Fricas [A]  time = 1.97336, size = 82, normalized size = 3.28 \begin{align*} \frac{2 \,{\left (c^{2} x^{2} + 2 \, b c x + b^{2}\right )} \sqrt{c x^{2} + b x}}{5 \, c \sqrt{x}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(c^2*x^2 + 2*b*c*x + b^2)*sqrt(c*x^2 + b*x)/(c*sqrt(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}{x^{\frac{3}{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((x*(b + c*x))**(3/2)/x**(3/2), x)

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Giac [B]  time = 1.32269, size = 81, normalized size = 3.24 \begin{align*} \frac{2}{15} \, c{\left (\frac{2 \, b^{\frac{5}{2}}}{c^{2}} + \frac{3 \,{\left (c x + b\right )}^{\frac{5}{2}} - 5 \,{\left (c x + b\right )}^{\frac{3}{2}} b}{c^{2}}\right )} + \frac{2}{3} \, b{\left (\frac{{\left (c x + b\right )}^{\frac{3}{2}}}{c} - \frac{b^{\frac{3}{2}}}{c}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*c*(2*b^(5/2)/c^2 + (3*(c*x + b)^(5/2) - 5*(c*x + b)^(3/2)*b)/c^2) + 2/3*b*((c*x + b)^(3/2)/c - b^(3/2)/c)