3.10 $$\int \frac{\sqrt{b x+c x^2}}{x^6} \, dx$$

Optimal. Leaf size=100 $\frac{32 c^3 \left (b x+c x^2\right )^{3/2}}{315 b^4 x^3}-\frac{16 c^2 \left (b x+c x^2\right )^{3/2}}{105 b^3 x^4}+\frac{4 c \left (b x+c x^2\right )^{3/2}}{21 b^2 x^5}-\frac{2 \left (b x+c x^2\right )^{3/2}}{9 b x^6}$

[Out]

(-2*(b*x + c*x^2)^(3/2))/(9*b*x^6) + (4*c*(b*x + c*x^2)^(3/2))/(21*b^2*x^5) - (16*c^2*(b*x + c*x^2)^(3/2))/(10
5*b^3*x^4) + (32*c^3*(b*x + c*x^2)^(3/2))/(315*b^4*x^3)

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Rubi [A]  time = 0.0407514, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.118, Rules used = {658, 650} $\frac{32 c^3 \left (b x+c x^2\right )^{3/2}}{315 b^4 x^3}-\frac{16 c^2 \left (b x+c x^2\right )^{3/2}}{105 b^3 x^4}+\frac{4 c \left (b x+c x^2\right )^{3/2}}{21 b^2 x^5}-\frac{2 \left (b x+c x^2\right )^{3/2}}{9 b x^6}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[b*x + c*x^2]/x^6,x]

[Out]

(-2*(b*x + c*x^2)^(3/2))/(9*b*x^6) + (4*c*(b*x + c*x^2)^(3/2))/(21*b^2*x^5) - (16*c^2*(b*x + c*x^2)^(3/2))/(10
5*b^3*x^4) + (32*c^3*(b*x + c*x^2)^(3/2))/(315*b^4*x^3)

Rule 658

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))/((m + p + 1)*(2*c*d - b*e)), x] + Dist[(c*Simplify[m + 2*p + 2])/((m + p + 1)*(2*c*d -
b*e)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], 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] && ILtQ[Simplify[m + 2*p + 2], 0]

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{\sqrt{b x+c x^2}}{x^6} \, dx &=-\frac{2 \left (b x+c x^2\right )^{3/2}}{9 b x^6}-\frac{(2 c) \int \frac{\sqrt{b x+c x^2}}{x^5} \, dx}{3 b}\\ &=-\frac{2 \left (b x+c x^2\right )^{3/2}}{9 b x^6}+\frac{4 c \left (b x+c x^2\right )^{3/2}}{21 b^2 x^5}+\frac{\left (8 c^2\right ) \int \frac{\sqrt{b x+c x^2}}{x^4} \, dx}{21 b^2}\\ &=-\frac{2 \left (b x+c x^2\right )^{3/2}}{9 b x^6}+\frac{4 c \left (b x+c x^2\right )^{3/2}}{21 b^2 x^5}-\frac{16 c^2 \left (b x+c x^2\right )^{3/2}}{105 b^3 x^4}-\frac{\left (16 c^3\right ) \int \frac{\sqrt{b x+c x^2}}{x^3} \, dx}{105 b^3}\\ &=-\frac{2 \left (b x+c x^2\right )^{3/2}}{9 b x^6}+\frac{4 c \left (b x+c x^2\right )^{3/2}}{21 b^2 x^5}-\frac{16 c^2 \left (b x+c x^2\right )^{3/2}}{105 b^3 x^4}+\frac{32 c^3 \left (b x+c x^2\right )^{3/2}}{315 b^4 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0125828, size = 62, normalized size = 0.62 $\frac{2 \sqrt{x (b+c x)} \left (6 b^2 c^2 x^2-5 b^3 c x-35 b^4-8 b c^3 x^3+16 c^4 x^4\right )}{315 b^4 x^5}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sqrt[b*x + c*x^2]/x^6,x]

[Out]

(2*Sqrt[x*(b + c*x)]*(-35*b^4 - 5*b^3*c*x + 6*b^2*c^2*x^2 - 8*b*c^3*x^3 + 16*c^4*x^4))/(315*b^4*x^5)

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Maple [A]  time = 0.047, size = 55, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -16\,{x}^{3}{c}^{3}+24\,b{x}^{2}{c}^{2}-30\,{b}^{2}xc+35\,{b}^{3} \right ) }{315\,{b}^{4}{x}^{5}}\sqrt{c{x}^{2}+bx}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x)^(1/2)/x^6,x)

[Out]

-2/315*(c*x+b)*(-16*c^3*x^3+24*b*c^2*x^2-30*b^2*c*x+35*b^3)*(c*x^2+b*x)^(1/2)/b^4/x^5

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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)^(1/2)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.08463, size = 134, normalized size = 1.34 \begin{align*} \frac{2 \,{\left (16 \, c^{4} x^{4} - 8 \, b c^{3} x^{3} + 6 \, b^{2} c^{2} x^{2} - 5 \, b^{3} c x - 35 \, b^{4}\right )} \sqrt{c x^{2} + b x}}{315 \, b^{4} x^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(1/2)/x^6,x, algorithm="fricas")

[Out]

2/315*(16*c^4*x^4 - 8*b*c^3*x^3 + 6*b^2*c^2*x^2 - 5*b^3*c*x - 35*b^4)*sqrt(c*x^2 + b*x)/(b^4*x^5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x \left (b + c x\right )}}{x^{6}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x)**(1/2)/x**6,x)

[Out]

Integral(sqrt(x*(b + c*x))/x**6, x)

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Giac [A]  time = 1.35586, size = 223, normalized size = 2.23 \begin{align*} \frac{2 \,{\left (630 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} c^{\frac{5}{2}} + 1764 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} b c^{2} + 1995 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} b^{2} c^{\frac{3}{2}} + 1125 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b^{3} c + 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{4} \sqrt{c} + 35 \, b^{5}\right )}}{315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{9}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x)^(1/2)/x^6,x, algorithm="giac")

[Out]

2/315*(630*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*c^(5/2) + 1764*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*b*c^2 + 1995*(sq
rt(c)*x - sqrt(c*x^2 + b*x))^3*b^2*c^(3/2) + 1125*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b^3*c + 315*(sqrt(c)*x - s
qrt(c*x^2 + b*x))*b^4*sqrt(c) + 35*b^5)/(sqrt(c)*x - sqrt(c*x^2 + b*x))^9