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]

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

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Rubi [A]  time = 0.20, antiderivative size = 195, normalized size of antiderivative = 1.00, 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 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]

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 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]

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*}

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Mathematica [A]  time = 0.04, size = 123, normalized size = 0.63 \[ -\frac {2 (x (b+c x))^{3/2} \left (5 A \left (693 b^5-630 b^4 c x+560 b^3 c^2 x^2-480 b^2 c^3 x^3+384 b c^4 x^4-256 c^5 x^5\right )+13 b B x \left (315 b^4-280 b^3 c x+240 b^2 c^2 x^2-192 b c^3 x^3+128 c^4 x^4\right )\right )}{45045 b^6 x^8} \]

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)

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fricas [A]  time = 0.94, size = 153, normalized size = 0.78 \[ -\frac {2 \, {\left (3465 \, A b^{6} + 128 \, {\left (13 \, B b c^{5} - 10 \, A c^{6}\right )} x^{6} - 64 \, {\left (13 \, B b^{2} c^{4} - 10 \, A b c^{5}\right )} x^{5} + 48 \, {\left (13 \, B b^{3} c^{3} - 10 \, A b^{2} c^{4}\right )} x^{4} - 40 \, {\left (13 \, B b^{4} c^{2} - 10 \, A b^{3} c^{3}\right )} x^{3} + 35 \, {\left (13 \, B b^{5} c - 10 \, A b^{4} c^{2}\right )} x^{2} + 315 \, {\left (13 \, B b^{6} + A b^{5} c\right )} x\right )} \sqrt {c x^{2} + b x}}{45045 \, b^{6} x^{7}} \]

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)

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giac [B]  time = 0.20, size = 431, normalized size = 2.21 \[ \frac {2 \, {\left (144144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{8} B c^{3} + 480480 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} B b c^{\frac {5}{2}} + 240240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} A c^{\frac {7}{2}} + 669240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} B b^{2} c^{2} + 926640 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} A b c^{3} + 495495 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B b^{3} c^{\frac {3}{2}} + 1531530 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} A b^{2} c^{\frac {5}{2}} + 205205 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B b^{4} c + 1401400 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A b^{3} c^{2} + 45045 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{5} \sqrt {c} + 765765 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{4} c^{\frac {3}{2}} + 4095 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{6} + 249795 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{5} c + 45045 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{6} \sqrt {c} + 3465 \, A b^{7}\right )}}{45045 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{13}} \]

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

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maple [A]  time = 0.04, size = 134, normalized size = 0.69 \[ -\frac {2 \left (c x +b \right ) \left (-1280 A \,c^{5} x^{5}+1664 B b \,c^{4} x^{5}+1920 A b \,c^{4} x^{4}-2496 B \,b^{2} c^{3} x^{4}-2400 A \,b^{2} c^{3} x^{3}+3120 B \,b^{3} c^{2} x^{3}+2800 A \,b^{3} c^{2} x^{2}-3640 B \,b^{4} c \,x^{2}-3150 A \,b^{4} c x +4095 B \,b^{5} x +3465 A \,b^{5}\right ) \sqrt {c \,x^{2}+b x}}{45045 b^{6} x^{7}} \]

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

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maxima [A]  time = 0.99, size = 284, normalized size = 1.46 \[ -\frac {256 \, \sqrt {c x^{2} + b x} B c^{5}}{3465 \, b^{5} x} + \frac {512 \, \sqrt {c x^{2} + b x} A c^{6}}{9009 \, b^{6} x} + \frac {128 \, \sqrt {c x^{2} + b x} B c^{4}}{3465 \, b^{4} x^{2}} - \frac {256 \, \sqrt {c x^{2} + b x} A c^{5}}{9009 \, b^{5} x^{2}} - \frac {32 \, \sqrt {c x^{2} + b x} B c^{3}}{1155 \, b^{3} x^{3}} + \frac {64 \, \sqrt {c x^{2} + b x} A c^{4}}{3003 \, b^{4} x^{3}} + \frac {16 \, \sqrt {c x^{2} + b x} B c^{2}}{693 \, b^{2} x^{4}} - \frac {160 \, \sqrt {c x^{2} + b x} A c^{3}}{9009 \, b^{3} x^{4}} - \frac {2 \, \sqrt {c x^{2} + b x} B c}{99 \, b x^{5}} + \frac {20 \, \sqrt {c x^{2} + b x} A c^{2}}{1287 \, b^{2} x^{5}} - \frac {2 \, \sqrt {c x^{2} + b x} B}{11 \, x^{6}} - \frac {2 \, \sqrt {c x^{2} + b x} A c}{143 \, b x^{6}} - \frac {2 \, \sqrt {c x^{2} + b x} A}{13 \, x^{7}} \]

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]

-256/3465*sqrt(c*x^2 + b*x)*B*c^5/(b^5*x) + 512/9009*sqrt(c*x^2 + b*x)*A*c^6/(b^6*x) + 128/3465*sqrt(c*x^2 + b
*x)*B*c^4/(b^4*x^2) - 256/9009*sqrt(c*x^2 + b*x)*A*c^5/(b^5*x^2) - 32/1155*sqrt(c*x^2 + b*x)*B*c^3/(b^3*x^3) +
64/3003*sqrt(c*x^2 + b*x)*A*c^4/(b^4*x^3) + 16/693*sqrt(c*x^2 + b*x)*B*c^2/(b^2*x^4) - 160/9009*sqrt(c*x^2 +
b*x)*A*c^3/(b^3*x^4) - 2/99*sqrt(c*x^2 + b*x)*B*c/(b*x^5) + 20/1287*sqrt(c*x^2 + b*x)*A*c^2/(b^2*x^5) - 2/11*s
qrt(c*x^2 + b*x)*B/x^6 - 2/143*sqrt(c*x^2 + b*x)*A*c/(b*x^6) - 2/13*sqrt(c*x^2 + b*x)*A/x^7

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mupad [B]  time = 2.49, size = 284, normalized size = 1.46 \[ \frac {20\,A\,c^2\,\sqrt {c\,x^2+b\,x}}{1287\,b^2\,x^5}-\frac {2\,B\,\sqrt {c\,x^2+b\,x}}{11\,x^6}-\frac {2\,A\,c\,\sqrt {c\,x^2+b\,x}}{143\,b\,x^6}-\frac {2\,B\,c\,\sqrt {c\,x^2+b\,x}}{99\,b\,x^5}-\frac {2\,A\,\sqrt {c\,x^2+b\,x}}{13\,x^7}-\frac {160\,A\,c^3\,\sqrt {c\,x^2+b\,x}}{9009\,b^3\,x^4}+\frac {64\,A\,c^4\,\sqrt {c\,x^2+b\,x}}{3003\,b^4\,x^3}-\frac {256\,A\,c^5\,\sqrt {c\,x^2+b\,x}}{9009\,b^5\,x^2}+\frac {512\,A\,c^6\,\sqrt {c\,x^2+b\,x}}{9009\,b^6\,x}+\frac {16\,B\,c^2\,\sqrt {c\,x^2+b\,x}}{693\,b^2\,x^4}-\frac {32\,B\,c^3\,\sqrt {c\,x^2+b\,x}}{1155\,b^3\,x^3}+\frac {128\,B\,c^4\,\sqrt {c\,x^2+b\,x}}{3465\,b^4\,x^2}-\frac {256\,B\,c^5\,\sqrt {c\,x^2+b\,x}}{3465\,b^5\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(20*A*c^2*(b*x + c*x^2)^(1/2))/(1287*b^2*x^5) - (2*B*(b*x + c*x^2)^(1/2))/(11*x^6) - (2*A*c*(b*x + c*x^2)^(1/2
))/(143*b*x^6) - (2*B*c*(b*x + c*x^2)^(1/2))/(99*b*x^5) - (2*A*(b*x + c*x^2)^(1/2))/(13*x^7) - (160*A*c^3*(b*x
+ c*x^2)^(1/2))/(9009*b^3*x^4) + (64*A*c^4*(b*x + c*x^2)^(1/2))/(3003*b^4*x^3) - (256*A*c^5*(b*x + c*x^2)^(1/
2))/(9009*b^5*x^2) + (512*A*c^6*(b*x + c*x^2)^(1/2))/(9009*b^6*x) + (16*B*c^2*(b*x + c*x^2)^(1/2))/(693*b^2*x^
4) - (32*B*c^3*(b*x + c*x^2)^(1/2))/(1155*b^3*x^3) + (128*B*c^4*(b*x + c*x^2)^(1/2))/(3465*b^4*x^2) - (256*B*c
^5*(b*x + c*x^2)^(1/2))/(3465*b^5*x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x \left (b + c x\right )} \left (A + B x\right )}{x^{8}}\, dx \]

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)

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