3.51 \(\int \frac{1}{x^5 \sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=126 \[ -\frac{256 c^4 \sqrt{b x+c x^2}}{315 b^5 x}+\frac{128 c^3 \sqrt{b x+c x^2}}{315 b^4 x^2}-\frac{32 c^2 \sqrt{b x+c x^2}}{105 b^3 x^3}+\frac{16 c \sqrt{b x+c x^2}}{63 b^2 x^4}-\frac{2 \sqrt{b x+c x^2}}{9 b x^5} \]

[Out]

(-2*Sqrt[b*x + c*x^2])/(9*b*x^5) + (16*c*Sqrt[b*x + c*x^2])/(63*b^2*x^4) - (32*c^2*Sqrt[b*x + c*x^2])/(105*b^3
*x^3) + (128*c^3*Sqrt[b*x + c*x^2])/(315*b^4*x^2) - (256*c^4*Sqrt[b*x + c*x^2])/(315*b^5*x)

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Rubi [A]  time = 0.0547222, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {658, 650} \[ -\frac{256 c^4 \sqrt{b x+c x^2}}{315 b^5 x}+\frac{128 c^3 \sqrt{b x+c x^2}}{315 b^4 x^2}-\frac{32 c^2 \sqrt{b x+c x^2}}{105 b^3 x^3}+\frac{16 c \sqrt{b x+c x^2}}{63 b^2 x^4}-\frac{2 \sqrt{b x+c x^2}}{9 b x^5} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*Sqrt[b*x + c*x^2]),x]

[Out]

(-2*Sqrt[b*x + c*x^2])/(9*b*x^5) + (16*c*Sqrt[b*x + c*x^2])/(63*b^2*x^4) - (32*c^2*Sqrt[b*x + c*x^2])/(105*b^3
*x^3) + (128*c^3*Sqrt[b*x + c*x^2])/(315*b^4*x^2) - (256*c^4*Sqrt[b*x + c*x^2])/(315*b^5*x)

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{1}{x^5 \sqrt{b x+c x^2}} \, dx &=-\frac{2 \sqrt{b x+c x^2}}{9 b x^5}-\frac{(8 c) \int \frac{1}{x^4 \sqrt{b x+c x^2}} \, dx}{9 b}\\ &=-\frac{2 \sqrt{b x+c x^2}}{9 b x^5}+\frac{16 c \sqrt{b x+c x^2}}{63 b^2 x^4}+\frac{\left (16 c^2\right ) \int \frac{1}{x^3 \sqrt{b x+c x^2}} \, dx}{21 b^2}\\ &=-\frac{2 \sqrt{b x+c x^2}}{9 b x^5}+\frac{16 c \sqrt{b x+c x^2}}{63 b^2 x^4}-\frac{32 c^2 \sqrt{b x+c x^2}}{105 b^3 x^3}-\frac{\left (64 c^3\right ) \int \frac{1}{x^2 \sqrt{b x+c x^2}} \, dx}{105 b^3}\\ &=-\frac{2 \sqrt{b x+c x^2}}{9 b x^5}+\frac{16 c \sqrt{b x+c x^2}}{63 b^2 x^4}-\frac{32 c^2 \sqrt{b x+c x^2}}{105 b^3 x^3}+\frac{128 c^3 \sqrt{b x+c x^2}}{315 b^4 x^2}+\frac{\left (128 c^4\right ) \int \frac{1}{x \sqrt{b x+c x^2}} \, dx}{315 b^4}\\ &=-\frac{2 \sqrt{b x+c x^2}}{9 b x^5}+\frac{16 c \sqrt{b x+c x^2}}{63 b^2 x^4}-\frac{32 c^2 \sqrt{b x+c x^2}}{105 b^3 x^3}+\frac{128 c^3 \sqrt{b x+c x^2}}{315 b^4 x^2}-\frac{256 c^4 \sqrt{b x+c x^2}}{315 b^5 x}\\ \end{align*}

Mathematica [A]  time = 0.01652, size = 62, normalized size = 0.49 \[ -\frac{2 \sqrt{x (b+c x)} \left (48 b^2 c^2 x^2-40 b^3 c x+35 b^4-64 b c^3 x^3+128 c^4 x^4\right )}{315 b^5 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*Sqrt[b*x + c*x^2]),x]

[Out]

(-2*Sqrt[x*(b + c*x)]*(35*b^4 - 40*b^3*c*x + 48*b^2*c^2*x^2 - 64*b*c^3*x^3 + 128*c^4*x^4))/(315*b^5*x^5)

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Maple [A]  time = 0.051, size = 66, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 128\,{c}^{4}{x}^{4}-64\,{x}^{3}{c}^{3}b+48\,{c}^{2}{x}^{2}{b}^{2}-40\,cx{b}^{3}+35\,{b}^{4} \right ) }{315\,{x}^{4}{b}^{5}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(c*x^2+b*x)^(1/2),x)

[Out]

-2/315*(c*x+b)*(128*c^4*x^4-64*b*c^3*x^3+48*b^2*c^2*x^2-40*b^3*c*x+35*b^4)/x^4/b^5/(c*x^2+b*x)^(1/2)

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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(1/x^5/(c*x^2+b*x)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.946, size = 140, normalized size = 1.11 \begin{align*} -\frac{2 \,{\left (128 \, c^{4} x^{4} - 64 \, b c^{3} x^{3} + 48 \, b^{2} c^{2} x^{2} - 40 \, b^{3} c x + 35 \, b^{4}\right )} \sqrt{c x^{2} + b x}}{315 \, b^{5} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(c*x^2+b*x)^(1/2),x, algorithm="fricas")

[Out]

-2/315*(128*c^4*x^4 - 64*b*c^3*x^3 + 48*b^2*c^2*x^2 - 40*b^3*c*x + 35*b^4)*sqrt(c*x^2 + b*x)/(b^5*x^5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{5} \sqrt{x \left (b + c x\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(c*x**2+b*x)**(1/2),x)

[Out]

Integral(1/(x**5*sqrt(x*(b + c*x))), x)

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Giac [A]  time = 1.18812, size = 184, normalized size = 1.46 \begin{align*} \frac{2 \,{\left (1008 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} c^{2} + 1680 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} b c^{\frac{3}{2}} + 1080 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b^{2} c + 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{3} \sqrt{c} + 35 \, b^{4}\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(1/x^5/(c*x^2+b*x)^(1/2),x, algorithm="giac")

[Out]

2/315*(1008*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*c^2 + 1680*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b*c^(3/2) + 1080*(s
qrt(c)*x - sqrt(c*x^2 + b*x))^2*b^2*c + 315*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^3*sqrt(c) + 35*b^4)/(sqrt(c)*x -
sqrt(c*x^2 + b*x))^9