3.119 \(\int \frac{A+B x}{x^6 \sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=195 \[ -\frac{256 c^4 \sqrt{b x+c x^2} (11 b B-10 A c)}{3465 b^6 x}+\frac{128 c^3 \sqrt{b x+c x^2} (11 b B-10 A c)}{3465 b^5 x^2}-\frac{32 c^2 \sqrt{b x+c x^2} (11 b B-10 A c)}{1155 b^4 x^3}+\frac{16 c \sqrt{b x+c x^2} (11 b B-10 A c)}{693 b^3 x^4}-\frac{2 \sqrt{b x+c x^2} (11 b B-10 A c)}{99 b^2 x^5}-\frac{2 A \sqrt{b x+c x^2}}{11 b x^6} \]

[Out]

(-2*A*Sqrt[b*x + c*x^2])/(11*b*x^6) - (2*(11*b*B - 10*A*c)*Sqrt[b*x + c*x^2])/(99*b^2*x^5) + (16*c*(11*b*B - 1
0*A*c)*Sqrt[b*x + c*x^2])/(693*b^3*x^4) - (32*c^2*(11*b*B - 10*A*c)*Sqrt[b*x + c*x^2])/(1155*b^4*x^3) + (128*c
^3*(11*b*B - 10*A*c)*Sqrt[b*x + c*x^2])/(3465*b^5*x^2) - (256*c^4*(11*b*B - 10*A*c)*Sqrt[b*x + c*x^2])/(3465*b
^6*x)

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Rubi [A]  time = 0.177203, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {792, 658, 650} \[ -\frac{256 c^4 \sqrt{b x+c x^2} (11 b B-10 A c)}{3465 b^6 x}+\frac{128 c^3 \sqrt{b x+c x^2} (11 b B-10 A c)}{3465 b^5 x^2}-\frac{32 c^2 \sqrt{b x+c x^2} (11 b B-10 A c)}{1155 b^4 x^3}+\frac{16 c \sqrt{b x+c x^2} (11 b B-10 A c)}{693 b^3 x^4}-\frac{2 \sqrt{b x+c x^2} (11 b B-10 A c)}{99 b^2 x^5}-\frac{2 A \sqrt{b x+c x^2}}{11 b x^6} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^6*Sqrt[b*x + c*x^2]),x]

[Out]

(-2*A*Sqrt[b*x + c*x^2])/(11*b*x^6) - (2*(11*b*B - 10*A*c)*Sqrt[b*x + c*x^2])/(99*b^2*x^5) + (16*c*(11*b*B - 1
0*A*c)*Sqrt[b*x + c*x^2])/(693*b^3*x^4) - (32*c^2*(11*b*B - 10*A*c)*Sqrt[b*x + c*x^2])/(1155*b^4*x^3) + (128*c
^3*(11*b*B - 10*A*c)*Sqrt[b*x + c*x^2])/(3465*b^5*x^2) - (256*c^4*(11*b*B - 10*A*c)*Sqrt[b*x + c*x^2])/(3465*b
^6*x)

Rule 792

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

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{A+B x}{x^6 \sqrt{b x+c x^2}} \, dx &=-\frac{2 A \sqrt{b x+c x^2}}{11 b x^6}+\frac{\left (2 \left (-6 (-b B+A c)+\frac{1}{2} (-b B+2 A c)\right )\right ) \int \frac{1}{x^5 \sqrt{b x+c x^2}} \, dx}{11 b}\\ &=-\frac{2 A \sqrt{b x+c x^2}}{11 b x^6}-\frac{2 (11 b B-10 A c) \sqrt{b x+c x^2}}{99 b^2 x^5}-\frac{(8 c (11 b B-10 A c)) \int \frac{1}{x^4 \sqrt{b x+c x^2}} \, dx}{99 b^2}\\ &=-\frac{2 A \sqrt{b x+c x^2}}{11 b x^6}-\frac{2 (11 b B-10 A c) \sqrt{b x+c x^2}}{99 b^2 x^5}+\frac{16 c (11 b B-10 A c) \sqrt{b x+c x^2}}{693 b^3 x^4}+\frac{\left (16 c^2 (11 b B-10 A c)\right ) \int \frac{1}{x^3 \sqrt{b x+c x^2}} \, dx}{231 b^3}\\ &=-\frac{2 A \sqrt{b x+c x^2}}{11 b x^6}-\frac{2 (11 b B-10 A c) \sqrt{b x+c x^2}}{99 b^2 x^5}+\frac{16 c (11 b B-10 A c) \sqrt{b x+c x^2}}{693 b^3 x^4}-\frac{32 c^2 (11 b B-10 A c) \sqrt{b x+c x^2}}{1155 b^4 x^3}-\frac{\left (64 c^3 (11 b B-10 A c)\right ) \int \frac{1}{x^2 \sqrt{b x+c x^2}} \, dx}{1155 b^4}\\ &=-\frac{2 A \sqrt{b x+c x^2}}{11 b x^6}-\frac{2 (11 b B-10 A c) \sqrt{b x+c x^2}}{99 b^2 x^5}+\frac{16 c (11 b B-10 A c) \sqrt{b x+c x^2}}{693 b^3 x^4}-\frac{32 c^2 (11 b B-10 A c) \sqrt{b x+c x^2}}{1155 b^4 x^3}+\frac{128 c^3 (11 b B-10 A c) \sqrt{b x+c x^2}}{3465 b^5 x^2}+\frac{\left (128 c^4 (11 b B-10 A c)\right ) \int \frac{1}{x \sqrt{b x+c x^2}} \, dx}{3465 b^5}\\ &=-\frac{2 A \sqrt{b x+c x^2}}{11 b x^6}-\frac{2 (11 b B-10 A c) \sqrt{b x+c x^2}}{99 b^2 x^5}+\frac{16 c (11 b B-10 A c) \sqrt{b x+c x^2}}{693 b^3 x^4}-\frac{32 c^2 (11 b B-10 A c) \sqrt{b x+c x^2}}{1155 b^4 x^3}+\frac{128 c^3 (11 b B-10 A c) \sqrt{b x+c x^2}}{3465 b^5 x^2}-\frac{256 c^4 (11 b B-10 A c) \sqrt{b x+c x^2}}{3465 b^6 x}\\ \end{align*}

Mathematica [A]  time = 0.0398415, size = 123, normalized size = 0.63 \[ -\frac{2 \sqrt{x (b+c x)} \left (5 A \left (80 b^3 c^2 x^2-96 b^2 c^3 x^3-70 b^4 c x+63 b^5+128 b c^4 x^4-256 c^5 x^5\right )+11 b B x \left (48 b^2 c^2 x^2-40 b^3 c x+35 b^4-64 b c^3 x^3+128 c^4 x^4\right )\right )}{3465 b^6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^6*Sqrt[b*x + c*x^2]),x]

[Out]

(-2*Sqrt[x*(b + c*x)]*(11*b*B*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) + 5*A*(63*
b^5 - 70*b^4*c*x + 80*b^3*c^2*x^2 - 96*b^2*c^3*x^3 + 128*b*c^4*x^4 - 256*c^5*x^5)))/(3465*b^6*x^6)

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Maple [A]  time = 0.006, size = 134, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -1280\,A{c}^{5}{x}^{5}+1408\,Bb{c}^{4}{x}^{5}+640\,Ab{c}^{4}{x}^{4}-704\,B{b}^{2}{c}^{3}{x}^{4}-480\,A{b}^{2}{c}^{3}{x}^{3}+528\,B{b}^{3}{c}^{2}{x}^{3}+400\,A{b}^{3}{c}^{2}{x}^{2}-440\,B{b}^{4}c{x}^{2}-350\,A{b}^{4}cx+385\,B{b}^{5}x+315\,A{b}^{5} \right ) }{3465\,{x}^{5}{b}^{6}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3465*(c*x+b)*(-1280*A*c^5*x^5+1408*B*b*c^4*x^5+640*A*b*c^4*x^4-704*B*b^2*c^3*x^4-480*A*b^2*c^3*x^3+528*B*b^
3*c^2*x^3+400*A*b^3*c^2*x^2-440*B*b^4*c*x^2-350*A*b^4*c*x+385*B*b^5*x+315*A*b^5)/x^5/b^6/(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((B*x+A)/x^6/(c*x^2+b*x)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.83564, size = 306, normalized size = 1.57 \begin{align*} -\frac{2 \,{\left (315 \, A b^{5} + 128 \,{\left (11 \, B b c^{4} - 10 \, A c^{5}\right )} x^{5} - 64 \,{\left (11 \, B b^{2} c^{3} - 10 \, A b c^{4}\right )} x^{4} + 48 \,{\left (11 \, B b^{3} c^{2} - 10 \, A b^{2} c^{3}\right )} x^{3} - 40 \,{\left (11 \, B b^{4} c - 10 \, A b^{3} c^{2}\right )} x^{2} + 35 \,{\left (11 \, B b^{5} - 10 \, A b^{4} c\right )} x\right )} \sqrt{c x^{2} + b x}}{3465 \, b^{6} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3465*(315*A*b^5 + 128*(11*B*b*c^4 - 10*A*c^5)*x^5 - 64*(11*B*b^2*c^3 - 10*A*b*c^4)*x^4 + 48*(11*B*b^3*c^2 -
10*A*b^2*c^3)*x^3 - 40*(11*B*b^4*c - 10*A*b^3*c^2)*x^2 + 35*(11*B*b^5 - 10*A*b^4*c)*x)*sqrt(c*x^2 + b*x)/(b^6
*x^6)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((A + B*x)/(x**6*sqrt(x*(b + c*x))), x)

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Giac [A]  time = 1.15057, size = 420, normalized size = 2.15 \begin{align*} \frac{2 \,{\left (11088 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} B c^{2} + 18480 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} B b c^{\frac{3}{2}} + 18480 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} A c^{\frac{5}{2}} + 11880 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} B b^{2} c + 39600 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} A b c^{2} + 3465 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} B b^{3} \sqrt{c} + 34650 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} A b^{2} c^{\frac{3}{2}} + 385 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B b^{4} + 15400 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} A b^{3} c + 3465 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A b^{4} \sqrt{c} + 315 \, A b^{5}\right )}}{3465 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(11088*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*c^2 + 18480*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b*c^(3/2) +
18480*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*c^(5/2) + 11880*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^2*c + 39600*(s
qrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b*c^2 + 3465*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^3*sqrt(c) + 34650*(sqrt(c
)*x - sqrt(c*x^2 + b*x))^3*A*b^2*c^(3/2) + 385*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^4 + 15400*(sqrt(c)*x - sq
rt(c*x^2 + b*x))^2*A*b^3*c + 3465*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^4*sqrt(c) + 315*A*b^5)/(sqrt(c)*x - sqrt
(c*x^2 + b*x))^11