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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.18, 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} \[ -\frac {256 c^4 \sqrt {b x+c x^2} (11 b B-10 A c)}{3465 b^6 x}+\frac {128 c^3 \sqrt {b x+c x^2} (11 b B-10 A c)}{3465 b^5 x^2}-\frac {32 c^2 \sqrt {b x+c x^2} (11 b B-10 A c)}{1155 b^4 x^3}+\frac {16 c \sqrt {b x+c x^2} (11 b B-10 A c)}{693 b^3 x^4}-\frac {2 \sqrt {b x+c x^2} (11 b B-10 A c)}{99 b^2 x^5}-\frac {2 A \sqrt {b x+c x^2}}{11 b x^6} \]

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 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}{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 = 0.04, size = 123, normalized size = 0.63 \[ -\frac {2 \sqrt {x (b+c x)} \left (5 A \left (63 b^5-70 b^4 c x+80 b^3 c^2 x^2-96 b^2 c^3 x^3+128 b c^4 x^4-256 c^5 x^5\right )+11 b B x \left (35 b^4-40 b^3 c x+48 b^2 c^2 x^2-64 b c^3 x^3+128 c^4 x^4\right )\right )}{3465 b^6 x^6} \]

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)

________________________________________________________________________________________

fricas [A]  time = 0.89, size = 130, normalized size = 0.67 \[ -\frac {2 \, {\left (315 \, A b^{5} + 128 \, {\left (11 \, B b c^{4} - 10 \, A c^{5}\right )} x^{5} - 64 \, {\left (11 \, B b^{2} c^{3} - 10 \, A b c^{4}\right )} x^{4} + 48 \, {\left (11 \, B b^{3} c^{2} - 10 \, A b^{2} c^{3}\right )} x^{3} - 40 \, {\left (11 \, B b^{4} c - 10 \, A b^{3} c^{2}\right )} x^{2} + 35 \, {\left (11 \, B b^{5} - 10 \, A b^{4} c\right )} x\right )} \sqrt {c x^{2} + b x}}{3465 \, b^{6} x^{6}} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.22, size = 311, normalized size = 1.59 \[ \frac {2 \, {\left (11088 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} B c^{2} + 18480 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B b c^{\frac {3}{2}} + 18480 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} A c^{\frac {5}{2}} + 11880 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B b^{2} c + 39600 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A b c^{2} + 3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{3} \sqrt {c} + 34650 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{2} c^{\frac {3}{2}} + 385 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{4} + 15400 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{3} c + 3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{4} \sqrt {c} + 315 \, A b^{5}\right )}}{3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{11}} \]

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

________________________________________________________________________________________

maple [A]  time = 0.05, size = 134, normalized size = 0.69 \[ -\frac {2 \left (c x +b \right ) \left (-1280 A \,c^{5} x^{5}+1408 B b \,c^{4} x^{5}+640 A b \,c^{4} x^{4}-704 B \,b^{2} c^{3} x^{4}-480 A \,b^{2} c^{3} x^{3}+528 B \,b^{3} c^{2} x^{3}+400 A \,b^{3} c^{2} x^{2}-440 B \,b^{4} c \,x^{2}-350 A \,b^{4} c x +385 B \,b^{5} x +315 A \,b^{5}\right )}{3465 \sqrt {c \,x^{2}+b x}\, b^{6} x^{5}} \]

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)

________________________________________________________________________________________

maxima [A]  time = 0.95, size = 244, normalized size = 1.25 \[ -\frac {256 \, \sqrt {c x^{2} + b x} B c^{4}}{315 \, b^{5} x} + \frac {512 \, \sqrt {c x^{2} + b x} A c^{5}}{693 \, b^{6} x} + \frac {128 \, \sqrt {c x^{2} + b x} B c^{3}}{315 \, b^{4} x^{2}} - \frac {256 \, \sqrt {c x^{2} + b x} A c^{4}}{693 \, b^{5} x^{2}} - \frac {32 \, \sqrt {c x^{2} + b x} B c^{2}}{105 \, b^{3} x^{3}} + \frac {64 \, \sqrt {c x^{2} + b x} A c^{3}}{231 \, b^{4} x^{3}} + \frac {16 \, \sqrt {c x^{2} + b x} B c}{63 \, b^{2} x^{4}} - \frac {160 \, \sqrt {c x^{2} + b x} A c^{2}}{693 \, b^{3} x^{4}} - \frac {2 \, \sqrt {c x^{2} + b x} B}{9 \, b x^{5}} + \frac {20 \, \sqrt {c x^{2} + b x} A c}{99 \, b^{2} x^{5}} - \frac {2 \, \sqrt {c x^{2} + b x} A}{11 \, b x^{6}} \]

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]

-256/315*sqrt(c*x^2 + b*x)*B*c^4/(b^5*x) + 512/693*sqrt(c*x^2 + b*x)*A*c^5/(b^6*x) + 128/315*sqrt(c*x^2 + b*x)
*B*c^3/(b^4*x^2) - 256/693*sqrt(c*x^2 + b*x)*A*c^4/(b^5*x^2) - 32/105*sqrt(c*x^2 + b*x)*B*c^2/(b^3*x^3) + 64/2
31*sqrt(c*x^2 + b*x)*A*c^3/(b^4*x^3) + 16/63*sqrt(c*x^2 + b*x)*B*c/(b^2*x^4) - 160/693*sqrt(c*x^2 + b*x)*A*c^2
/(b^3*x^4) - 2/9*sqrt(c*x^2 + b*x)*B/(b*x^5) + 20/99*sqrt(c*x^2 + b*x)*A*c/(b^2*x^5) - 2/11*sqrt(c*x^2 + b*x)*
A/(b*x^6)

________________________________________________________________________________________

mupad [B]  time = 1.12, size = 177, normalized size = 0.91 \[ \frac {\sqrt {c\,x^2+b\,x}\,\left (320\,A\,c^3-352\,B\,b\,c^2\right )}{1155\,b^4\,x^3}-\frac {\sqrt {c\,x^2+b\,x}\,\left (1280\,A\,c^4-1408\,B\,b\,c^3\right )}{3465\,b^5\,x^2}-\frac {\left (160\,A\,c^2-176\,B\,b\,c\right )\,\sqrt {c\,x^2+b\,x}}{693\,b^3\,x^4}-\frac {2\,A\,\sqrt {c\,x^2+b\,x}}{11\,b\,x^6}+\frac {\sqrt {c\,x^2+b\,x}\,\left (20\,A\,c-22\,B\,b\right )}{99\,b^2\,x^5}+\frac {256\,c^4\,\sqrt {c\,x^2+b\,x}\,\left (10\,A\,c-11\,B\,b\right )}{3465\,b^6\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*x + c*x^2)^(1/2)*(320*A*c^3 - 352*B*b*c^2))/(1155*b^4*x^3) - ((b*x + c*x^2)^(1/2)*(1280*A*c^4 - 1408*B*b*c
^3))/(3465*b^5*x^2) - ((160*A*c^2 - 176*B*b*c)*(b*x + c*x^2)^(1/2))/(693*b^3*x^4) - (2*A*(b*x + c*x^2)^(1/2))/
(11*b*x^6) + ((b*x + c*x^2)^(1/2)*(20*A*c - 22*B*b))/(99*b^2*x^5) + (256*c^4*(b*x + c*x^2)^(1/2)*(10*A*c - 11*
B*b))/(3465*b^6*x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {A + B x}{x^{6} \sqrt {x \left (b + c x\right )}}\, dx \]

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)

________________________________________________________________________________________