### 3.19 $$\int \frac{(b x+c x^2)^{3/2}}{x^5} \, dx$$

Optimal. Leaf size=23 $-\frac{2 \left (b x+c x^2\right )^{5/2}}{5 b x^5}$

[Out]

(-2*(b*x + c*x^2)^(5/2))/(5*b*x^5)

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Rubi [A]  time = 0.0075107, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.059, Rules used = {650} $-\frac{2 \left (b x+c x^2\right )^{5/2}}{5 b x^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b*x + c*x^2)^(3/2)/x^5,x]

[Out]

(-2*(b*x + c*x^2)^(5/2))/(5*b*x^5)

Rule 650

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((p + 1)*(2*c*d - b*e)), 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 + 2*p + 2, 0]

Rubi steps

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

Mathematica [A]  time = 0.0120201, size = 21, normalized size = 0.91 $-\frac{2 (x (b+c x))^{5/2}}{5 b x^5}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(b*x + c*x^2)^(3/2)/x^5,x]

[Out]

(-2*(x*(b + c*x))^(5/2))/(5*b*x^5)

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Maple [A]  time = 0.044, size = 25, normalized size = 1.1 \begin{align*} -{\frac{2\,cx+2\,b}{5\,b{x}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\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^5,x)

[Out]

-2/5/x^4*(c*x+b)/b*(c*x^2+b*x)^(3/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(3/2)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.98778, size = 78, normalized size = 3.39 \begin{align*} -\frac{2 \,{\left (c^{2} x^{2} + 2 \, b c x + b^{2}\right )} \sqrt{c x^{2} + b x}}{5 \, b x^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(3/2)/x^5,x, algorithm="fricas")

[Out]

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

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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^{5}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x)**(3/2)/x**5,x)

[Out]

Integral((x*(b + c*x))**(3/2)/x**5, x)

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Giac [B]  time = 1.25282, size = 181, normalized size = 7.87 \begin{align*} \frac{2 \,{\left (5 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} c^{2} + 10 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} b c^{\frac{3}{2}} + 10 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b^{2} c + 5 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{3} \sqrt{c} + b^{4}\right )}}{5 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(3/2)/x^5,x, algorithm="giac")

[Out]

2/5*(5*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*c^2 + 10*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b*c^(3/2) + 10*(sqrt(c)*x
- sqrt(c*x^2 + b*x))^2*b^2*c + 5*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^3*sqrt(c) + b^4)/(sqrt(c)*x - sqrt(c*x^2 +
b*x))^5