3.78 \(\int \frac{(A+B x) \sqrt{b x+c x^2}}{x^8} \, dx\)

Optimal. Leaf size=195 \[ -\frac{256 c^4 \left (b x+c x^2\right )^{3/2} (13 b B-10 A c)}{45045 b^6 x^3}+\frac{128 c^3 \left (b x+c x^2\right )^{3/2} (13 b B-10 A c)}{15015 b^5 x^4}-\frac{32 c^2 \left (b x+c x^2\right )^{3/2} (13 b B-10 A c)}{3003 b^4 x^5}+\frac{16 c \left (b x+c x^2\right )^{3/2} (13 b B-10 A c)}{1287 b^3 x^6}-\frac{2 \left (b x+c x^2\right )^{3/2} (13 b B-10 A c)}{143 b^2 x^7}-\frac{2 A \left (b x+c x^2\right )^{3/2}}{13 b x^8} \]

[Out]

(-2*A*(b*x + c*x^2)^(3/2))/(13*b*x^8) - (2*(13*b*B - 10*A*c)*(b*x + c*x^2)^(3/2))/(143*b^2*x^7) + (16*c*(13*b*
B - 10*A*c)*(b*x + c*x^2)^(3/2))/(1287*b^3*x^6) - (32*c^2*(13*b*B - 10*A*c)*(b*x + c*x^2)^(3/2))/(3003*b^4*x^5
) + (128*c^3*(13*b*B - 10*A*c)*(b*x + c*x^2)^(3/2))/(15015*b^5*x^4) - (256*c^4*(13*b*B - 10*A*c)*(b*x + c*x^2)
^(3/2))/(45045*b^6*x^3)

________________________________________________________________________________________

Rubi [A]  time = 0.195372, 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} \[ -\frac{256 c^4 \left (b x+c x^2\right )^{3/2} (13 b B-10 A c)}{45045 b^6 x^3}+\frac{128 c^3 \left (b x+c x^2\right )^{3/2} (13 b B-10 A c)}{15015 b^5 x^4}-\frac{32 c^2 \left (b x+c x^2\right )^{3/2} (13 b B-10 A c)}{3003 b^4 x^5}+\frac{16 c \left (b x+c x^2\right )^{3/2} (13 b B-10 A c)}{1287 b^3 x^6}-\frac{2 \left (b x+c x^2\right )^{3/2} (13 b B-10 A c)}{143 b^2 x^7}-\frac{2 A \left (b x+c x^2\right )^{3/2}}{13 b x^8} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*Sqrt[b*x + c*x^2])/x^8,x]

[Out]

(-2*A*(b*x + c*x^2)^(3/2))/(13*b*x^8) - (2*(13*b*B - 10*A*c)*(b*x + c*x^2)^(3/2))/(143*b^2*x^7) + (16*c*(13*b*
B - 10*A*c)*(b*x + c*x^2)^(3/2))/(1287*b^3*x^6) - (32*c^2*(13*b*B - 10*A*c)*(b*x + c*x^2)^(3/2))/(3003*b^4*x^5
) + (128*c^3*(13*b*B - 10*A*c)*(b*x + c*x^2)^(3/2))/(15015*b^5*x^4) - (256*c^4*(13*b*B - 10*A*c)*(b*x + c*x^2)
^(3/2))/(45045*b^6*x^3)

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) \sqrt{b x+c x^2}}{x^8} \, dx &=-\frac{2 A \left (b x+c x^2\right )^{3/2}}{13 b x^8}+\frac{\left (2 \left (-8 (-b B+A c)+\frac{3}{2} (-b B+2 A c)\right )\right ) \int \frac{\sqrt{b x+c x^2}}{x^7} \, dx}{13 b}\\ &=-\frac{2 A \left (b x+c x^2\right )^{3/2}}{13 b x^8}-\frac{2 (13 b B-10 A c) \left (b x+c x^2\right )^{3/2}}{143 b^2 x^7}-\frac{(8 c (13 b B-10 A c)) \int \frac{\sqrt{b x+c x^2}}{x^6} \, dx}{143 b^2}\\ &=-\frac{2 A \left (b x+c x^2\right )^{3/2}}{13 b x^8}-\frac{2 (13 b B-10 A c) \left (b x+c x^2\right )^{3/2}}{143 b^2 x^7}+\frac{16 c (13 b B-10 A c) \left (b x+c x^2\right )^{3/2}}{1287 b^3 x^6}+\frac{\left (16 c^2 (13 b B-10 A c)\right ) \int \frac{\sqrt{b x+c x^2}}{x^5} \, dx}{429 b^3}\\ &=-\frac{2 A \left (b x+c x^2\right )^{3/2}}{13 b x^8}-\frac{2 (13 b B-10 A c) \left (b x+c x^2\right )^{3/2}}{143 b^2 x^7}+\frac{16 c (13 b B-10 A c) \left (b x+c x^2\right )^{3/2}}{1287 b^3 x^6}-\frac{32 c^2 (13 b B-10 A c) \left (b x+c x^2\right )^{3/2}}{3003 b^4 x^5}-\frac{\left (64 c^3 (13 b B-10 A c)\right ) \int \frac{\sqrt{b x+c x^2}}{x^4} \, dx}{3003 b^4}\\ &=-\frac{2 A \left (b x+c x^2\right )^{3/2}}{13 b x^8}-\frac{2 (13 b B-10 A c) \left (b x+c x^2\right )^{3/2}}{143 b^2 x^7}+\frac{16 c (13 b B-10 A c) \left (b x+c x^2\right )^{3/2}}{1287 b^3 x^6}-\frac{32 c^2 (13 b B-10 A c) \left (b x+c x^2\right )^{3/2}}{3003 b^4 x^5}+\frac{128 c^3 (13 b B-10 A c) \left (b x+c x^2\right )^{3/2}}{15015 b^5 x^4}+\frac{\left (128 c^4 (13 b B-10 A c)\right ) \int \frac{\sqrt{b x+c x^2}}{x^3} \, dx}{15015 b^5}\\ &=-\frac{2 A \left (b x+c x^2\right )^{3/2}}{13 b x^8}-\frac{2 (13 b B-10 A c) \left (b x+c x^2\right )^{3/2}}{143 b^2 x^7}+\frac{16 c (13 b B-10 A c) \left (b x+c x^2\right )^{3/2}}{1287 b^3 x^6}-\frac{32 c^2 (13 b B-10 A c) \left (b x+c x^2\right )^{3/2}}{3003 b^4 x^5}+\frac{128 c^3 (13 b B-10 A c) \left (b x+c x^2\right )^{3/2}}{15015 b^5 x^4}-\frac{256 c^4 (13 b B-10 A c) \left (b x+c x^2\right )^{3/2}}{45045 b^6 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0448725, size = 123, normalized size = 0.63 \[ -\frac{2 (x (b+c x))^{3/2} \left (5 A \left (560 b^3 c^2 x^2-480 b^2 c^3 x^3-630 b^4 c x+693 b^5+384 b c^4 x^4-256 c^5 x^5\right )+13 b B x \left (240 b^2 c^2 x^2-280 b^3 c x+315 b^4-192 b c^3 x^3+128 c^4 x^4\right )\right )}{45045 b^6 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*Sqrt[b*x + c*x^2])/x^8,x]

[Out]

(-2*(x*(b + c*x))^(3/2)*(13*b*B*x*(315*b^4 - 280*b^3*c*x + 240*b^2*c^2*x^2 - 192*b*c^3*x^3 + 128*c^4*x^4) + 5*
A*(693*b^5 - 630*b^4*c*x + 560*b^3*c^2*x^2 - 480*b^2*c^3*x^3 + 384*b*c^4*x^4 - 256*c^5*x^5)))/(45045*b^6*x^8)

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 134, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -1280\,A{c}^{5}{x}^{5}+1664\,Bb{c}^{4}{x}^{5}+1920\,Ab{c}^{4}{x}^{4}-2496\,B{b}^{2}{c}^{3}{x}^{4}-2400\,A{b}^{2}{c}^{3}{x}^{3}+3120\,B{b}^{3}{c}^{2}{x}^{3}+2800\,A{b}^{3}{c}^{2}{x}^{2}-3640\,B{b}^{4}c{x}^{2}-3150\,A{b}^{4}cx+4095\,B{b}^{5}x+3465\,A{b}^{5} \right ) }{45045\,{x}^{7}{b}^{6}}\sqrt{c{x}^{2}+bx}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^(1/2)/x^8,x)

[Out]

-2/45045*(c*x+b)*(-1280*A*c^5*x^5+1664*B*b*c^4*x^5+1920*A*b*c^4*x^4-2496*B*b^2*c^3*x^4-2400*A*b^2*c^3*x^3+3120
*B*b^3*c^2*x^3+2800*A*b^3*c^2*x^2-3640*B*b^4*c*x^2-3150*A*b^4*c*x+4095*B*b^5*x+3465*A*b^5)*(c*x^2+b*x)^(1/2)/x
^7/b^6

________________________________________________________________________________________

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)*(c*x^2+b*x)^(1/2)/x^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.77453, size = 359, normalized size = 1.84 \begin{align*} -\frac{2 \,{\left (3465 \, A b^{6} + 128 \,{\left (13 \, B b c^{5} - 10 \, A c^{6}\right )} x^{6} - 64 \,{\left (13 \, B b^{2} c^{4} - 10 \, A b c^{5}\right )} x^{5} + 48 \,{\left (13 \, B b^{3} c^{3} - 10 \, A b^{2} c^{4}\right )} x^{4} - 40 \,{\left (13 \, B b^{4} c^{2} - 10 \, A b^{3} c^{3}\right )} x^{3} + 35 \,{\left (13 \, B b^{5} c - 10 \, A b^{4} c^{2}\right )} x^{2} + 315 \,{\left (13 \, B b^{6} + A b^{5} c\right )} x\right )} \sqrt{c x^{2} + b x}}{45045 \, b^{6} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(1/2)/x^8,x, algorithm="fricas")

[Out]

-2/45045*(3465*A*b^6 + 128*(13*B*b*c^5 - 10*A*c^6)*x^6 - 64*(13*B*b^2*c^4 - 10*A*b*c^5)*x^5 + 48*(13*B*b^3*c^3
- 10*A*b^2*c^4)*x^4 - 40*(13*B*b^4*c^2 - 10*A*b^3*c^3)*x^3 + 35*(13*B*b^5*c - 10*A*b^4*c^2)*x^2 + 315*(13*B*b
^6 + A*b^5*c)*x)*sqrt(c*x^2 + b*x)/(b^6*x^7)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x \left (b + c x\right )} \left (A + B x\right )}{x^{8}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**(1/2)/x**8,x)

[Out]

Integral(sqrt(x*(b + c*x))*(A + B*x)/x**8, x)

________________________________________________________________________________________

Giac [B]  time = 1.16219, size = 582, normalized size = 2.98 \begin{align*} \frac{2 \,{\left (144144 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{8} B c^{3} + 480480 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7} B b c^{\frac{5}{2}} + 240240 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7} A c^{\frac{7}{2}} + 669240 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} B b^{2} c^{2} + 926640 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{6} A b c^{3} + 495495 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} B b^{3} c^{\frac{3}{2}} + 1531530 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} A b^{2} c^{\frac{5}{2}} + 205205 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} B b^{4} c + 1401400 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} A b^{3} c^{2} + 45045 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} B b^{5} \sqrt{c} + 765765 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} A b^{4} c^{\frac{3}{2}} + 4095 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B b^{6} + 249795 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} A b^{5} c + 45045 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A b^{6} \sqrt{c} + 3465 \, A b^{7}\right )}}{45045 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(1/2)/x^8,x, algorithm="giac")

[Out]

2/45045*(144144*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*c^3 + 480480*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b*c^(5/2)
+ 240240*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*c^(7/2) + 669240*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^2*c^2 + 9
26640*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b*c^3 + 495495*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^3*c^(3/2) + 153
1530*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^2*c^(5/2) + 205205*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^4*c + 1401
400*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^3*c^2 + 45045*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^5*sqrt(c) + 7657
65*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^4*c^(3/2) + 4095*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^6 + 249795*(sq
rt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^5*c + 45045*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^6*sqrt(c) + 3465*A*b^7)/(sq
rt(c)*x - sqrt(c*x^2 + b*x))^13