3.2.7 \(\int \frac {(A+B x) (b x+c x^2)^{5/2}}{x^{11}} \, dx\)

Optimal. Leaf size=160 \[ \frac {32 c^3 \left (b x+c x^2\right )^{7/2} (15 b B-8 A c)}{45045 b^5 x^7}-\frac {16 c^2 \left (b x+c x^2\right )^{7/2} (15 b B-8 A c)}{6435 b^4 x^8}+\frac {4 c \left (b x+c x^2\right )^{7/2} (15 b B-8 A c)}{715 b^3 x^9}-\frac {2 \left (b x+c x^2\right )^{7/2} (15 b B-8 A c)}{195 b^2 x^{10}}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{15 b x^{11}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.16, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {792, 658, 650} \begin {gather*} \frac {32 c^3 \left (b x+c x^2\right )^{7/2} (15 b B-8 A c)}{45045 b^5 x^7}-\frac {16 c^2 \left (b x+c x^2\right )^{7/2} (15 b B-8 A c)}{6435 b^4 x^8}+\frac {4 c \left (b x+c x^2\right )^{7/2} (15 b B-8 A c)}{715 b^3 x^9}-\frac {2 \left (b x+c x^2\right )^{7/2} (15 b B-8 A c)}{195 b^2 x^{10}}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{15 b x^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^(5/2))/x^11,x]

[Out]

(-2*A*(b*x + c*x^2)^(7/2))/(15*b*x^11) - (2*(15*b*B - 8*A*c)*(b*x + c*x^2)^(7/2))/(195*b^2*x^10) + (4*c*(15*b*
B - 8*A*c)*(b*x + c*x^2)^(7/2))/(715*b^3*x^9) - (16*c^2*(15*b*B - 8*A*c)*(b*x + c*x^2)^(7/2))/(6435*b^4*x^8) +
(32*c^3*(15*b*B - 8*A*c)*(b*x + c*x^2)^(7/2))/(45045*b^5*x^7)

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) \left (b x+c x^2\right )^{5/2}}{x^{11}} \, dx &=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{15 b x^{11}}+\frac {\left (2 \left (-11 (-b B+A c)+\frac {7}{2} (-b B+2 A c)\right )\right ) \int \frac {\left (b x+c x^2\right )^{5/2}}{x^{10}} \, dx}{15 b}\\ &=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{15 b x^{11}}-\frac {2 (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{195 b^2 x^{10}}-\frac {(2 c (15 b B-8 A c)) \int \frac {\left (b x+c x^2\right )^{5/2}}{x^9} \, dx}{65 b^2}\\ &=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{15 b x^{11}}-\frac {2 (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{195 b^2 x^{10}}+\frac {4 c (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{715 b^3 x^9}+\frac {\left (8 c^2 (15 b B-8 A c)\right ) \int \frac {\left (b x+c x^2\right )^{5/2}}{x^8} \, dx}{715 b^3}\\ &=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{15 b x^{11}}-\frac {2 (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{195 b^2 x^{10}}+\frac {4 c (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{715 b^3 x^9}-\frac {16 c^2 (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{6435 b^4 x^8}-\frac {\left (16 c^3 (15 b B-8 A c)\right ) \int \frac {\left (b x+c x^2\right )^{5/2}}{x^7} \, dx}{6435 b^4}\\ &=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{15 b x^{11}}-\frac {2 (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{195 b^2 x^{10}}+\frac {4 c (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{715 b^3 x^9}-\frac {16 c^2 (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{6435 b^4 x^8}+\frac {32 c^3 (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{45045 b^5 x^7}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 107, normalized size = 0.67 \begin {gather*} \frac {2 (b+c x)^3 \sqrt {x (b+c x)} \left (A \left (-3003 b^4+1848 b^3 c x-1008 b^2 c^2 x^2+448 b c^3 x^3-128 c^4 x^4\right )+15 b B x \left (-231 b^3+126 b^2 c x-56 b c^2 x^2+16 c^3 x^3\right )\right )}{45045 b^5 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^(5/2))/x^11,x]

[Out]

(2*(b + c*x)^3*Sqrt[x*(b + c*x)]*(15*b*B*x*(-231*b^3 + 126*b^2*c*x - 56*b*c^2*x^2 + 16*c^3*x^3) + A*(-3003*b^4
+ 1848*b^3*c*x - 1008*b^2*c^2*x^2 + 448*b*c^3*x^3 - 128*c^4*x^4)))/(45045*b^5*x^8)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.55, size = 180, normalized size = 1.12 \begin {gather*} -\frac {2 \sqrt {b x+c x^2} \left (3003 A b^7+7161 A b^6 c x+4473 A b^5 c^2 x^2+35 A b^4 c^3 x^3-40 A b^3 c^4 x^4+48 A b^2 c^5 x^5-64 A b c^6 x^6+128 A c^7 x^7+3465 b^7 B x+8505 b^6 B c x^2+5565 b^5 B c^2 x^3+75 b^4 B c^3 x^4-90 b^3 B c^4 x^5+120 b^2 B c^5 x^6-240 b B c^6 x^7\right )}{45045 b^5 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(b*x + c*x^2)^(5/2))/x^11,x]

[Out]

(-2*Sqrt[b*x + c*x^2]*(3003*A*b^7 + 3465*b^7*B*x + 7161*A*b^6*c*x + 8505*b^6*B*c*x^2 + 4473*A*b^5*c^2*x^2 + 55
65*b^5*B*c^2*x^3 + 35*A*b^4*c^3*x^3 + 75*b^4*B*c^3*x^4 - 40*A*b^3*c^4*x^4 - 90*b^3*B*c^4*x^5 + 48*A*b^2*c^5*x^
5 + 120*b^2*B*c^5*x^6 - 64*A*b*c^6*x^6 - 240*b*B*c^6*x^7 + 128*A*c^7*x^7))/(45045*b^5*x^8)

________________________________________________________________________________________

fricas [A]  time = 0.40, size = 177, normalized size = 1.11 \begin {gather*} -\frac {2 \, {\left (3003 \, A b^{7} - 16 \, {\left (15 \, B b c^{6} - 8 \, A c^{7}\right )} x^{7} + 8 \, {\left (15 \, B b^{2} c^{5} - 8 \, A b c^{6}\right )} x^{6} - 6 \, {\left (15 \, B b^{3} c^{4} - 8 \, A b^{2} c^{5}\right )} x^{5} + 5 \, {\left (15 \, B b^{4} c^{3} - 8 \, A b^{3} c^{4}\right )} x^{4} + 35 \, {\left (159 \, B b^{5} c^{2} + A b^{4} c^{3}\right )} x^{3} + 63 \, {\left (135 \, B b^{6} c + 71 \, A b^{5} c^{2}\right )} x^{2} + 231 \, {\left (15 \, B b^{7} + 31 \, A b^{6} c\right )} x\right )} \sqrt {c x^{2} + b x}}{45045 \, b^{5} x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(5/2)/x^11,x, algorithm="fricas")

[Out]

-2/45045*(3003*A*b^7 - 16*(15*B*b*c^6 - 8*A*c^7)*x^7 + 8*(15*B*b^2*c^5 - 8*A*b*c^6)*x^6 - 6*(15*B*b^3*c^4 - 8*
A*b^2*c^5)*x^5 + 5*(15*B*b^4*c^3 - 8*A*b^3*c^4)*x^4 + 35*(159*B*b^5*c^2 + A*b^4*c^3)*x^3 + 63*(135*B*b^6*c + 7
1*A*b^5*c^2)*x^2 + 231*(15*B*b^7 + 31*A*b^6*c)*x)*sqrt(c*x^2 + b*x)/(b^5*x^8)

________________________________________________________________________________________

giac [B]  time = 0.24, size = 611, normalized size = 3.82 \begin {gather*} \frac {2 \, {\left (90090 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{11} B c^{\frac {9}{2}} + 540540 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{10} B b c^{4} + 144144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{10} A c^{5} + 1486485 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{9} B b^{2} c^{\frac {7}{2}} + 960960 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{9} A b c^{\frac {9}{2}} + 2425995 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{8} B b^{3} c^{3} + 2934360 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{8} A b^{2} c^{4} + 2567565 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} B b^{4} c^{\frac {5}{2}} + 5360355 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} A b^{3} c^{\frac {7}{2}} + 1816815 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} B b^{5} c^{2} + 6451445 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} A b^{4} c^{3} + 855855 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B b^{6} c^{\frac {3}{2}} + 5324319 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} A b^{5} c^{\frac {5}{2}} + 257985 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B b^{7} c + 3042585 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A b^{6} c^{2} + 45045 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{8} \sqrt {c} + 1186185 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{7} c^{\frac {3}{2}} + 3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{9} + 301455 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{8} c + 45045 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{9} \sqrt {c} + 3003 \, A b^{10}\right )}}{45045 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(5/2)/x^11,x, algorithm="giac")

[Out]

2/45045*(90090*(sqrt(c)*x - sqrt(c*x^2 + b*x))^11*B*c^(9/2) + 540540*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*b*c^
4 + 144144*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*A*c^5 + 1486485*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b^2*c^(7/2)
+ 960960*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*b*c^(9/2) + 2425995*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b^3*c^3 +
2934360*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b^2*c^4 + 2567565*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^4*c^(5/2)
+ 5360355*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^3*c^(7/2) + 1816815*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^5*c
^2 + 6451445*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^4*c^3 + 855855*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^6*c^(3
/2) + 5324319*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^5*c^(5/2) + 257985*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^7
*c + 3042585*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^6*c^2 + 45045*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^8*sqrt(
c) + 1186185*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^7*c^(3/2) + 3465*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^9 +
301455*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^8*c + 45045*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^9*sqrt(c) + 3003*
A*b^10)/(sqrt(c)*x - sqrt(c*x^2 + b*x))^15

________________________________________________________________________________________

maple [A]  time = 0.05, size = 110, normalized size = 0.69 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (128 A \,c^{4} x^{4}-240 B b \,c^{3} x^{4}-448 A b \,c^{3} x^{3}+840 B \,b^{2} c^{2} x^{3}+1008 A \,b^{2} c^{2} x^{2}-1890 B \,b^{3} c \,x^{2}-1848 A \,b^{3} c x +3465 b^{4} B x +3003 A \,b^{4}\right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{45045 b^{5} x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^(5/2)/x^11,x)

[Out]

-2/45045*(c*x+b)*(128*A*c^4*x^4-240*B*b*c^3*x^4-448*A*b*c^3*x^3+840*B*b^2*c^2*x^3+1008*A*b^2*c^2*x^2-1890*B*b^
3*c*x^2-1848*A*b^3*c*x+3465*B*b^4*x+3003*A*b^4)*(c*x^2+b*x)^(5/2)/b^5/x^10

________________________________________________________________________________________

maxima [B]  time = 0.93, size = 396, normalized size = 2.48 \begin {gather*} \frac {32 \, \sqrt {c x^{2} + b x} B c^{6}}{3003 \, b^{4} x} - \frac {256 \, \sqrt {c x^{2} + b x} A c^{7}}{45045 \, b^{5} x} - \frac {16 \, \sqrt {c x^{2} + b x} B c^{5}}{3003 \, b^{3} x^{2}} + \frac {128 \, \sqrt {c x^{2} + b x} A c^{6}}{45045 \, b^{4} x^{2}} + \frac {4 \, \sqrt {c x^{2} + b x} B c^{4}}{1001 \, b^{2} x^{3}} - \frac {32 \, \sqrt {c x^{2} + b x} A c^{5}}{15015 \, b^{3} x^{3}} - \frac {10 \, \sqrt {c x^{2} + b x} B c^{3}}{3003 \, b x^{4}} + \frac {16 \, \sqrt {c x^{2} + b x} A c^{4}}{9009 \, b^{2} x^{4}} + \frac {5 \, \sqrt {c x^{2} + b x} B c^{2}}{1716 \, x^{5}} - \frac {2 \, \sqrt {c x^{2} + b x} A c^{3}}{1287 \, b x^{5}} - \frac {3 \, \sqrt {c x^{2} + b x} B b c}{1144 \, x^{6}} + \frac {\sqrt {c x^{2} + b x} A c^{2}}{715 \, x^{6}} - \frac {3 \, \sqrt {c x^{2} + b x} B b^{2}}{104 \, x^{7}} - \frac {\sqrt {c x^{2} + b x} A b c}{780 \, x^{7}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b}{8 \, x^{8}} - \frac {\sqrt {c x^{2} + b x} A b^{2}}{60 \, x^{8}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} B}{4 \, x^{9}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b}{12 \, x^{9}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} A}{5 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^(5/2)/x^11,x, algorithm="maxima")

[Out]

32/3003*sqrt(c*x^2 + b*x)*B*c^6/(b^4*x) - 256/45045*sqrt(c*x^2 + b*x)*A*c^7/(b^5*x) - 16/3003*sqrt(c*x^2 + b*x
)*B*c^5/(b^3*x^2) + 128/45045*sqrt(c*x^2 + b*x)*A*c^6/(b^4*x^2) + 4/1001*sqrt(c*x^2 + b*x)*B*c^4/(b^2*x^3) - 3
2/15015*sqrt(c*x^2 + b*x)*A*c^5/(b^3*x^3) - 10/3003*sqrt(c*x^2 + b*x)*B*c^3/(b*x^4) + 16/9009*sqrt(c*x^2 + b*x
)*A*c^4/(b^2*x^4) + 5/1716*sqrt(c*x^2 + b*x)*B*c^2/x^5 - 2/1287*sqrt(c*x^2 + b*x)*A*c^3/(b*x^5) - 3/1144*sqrt(
c*x^2 + b*x)*B*b*c/x^6 + 1/715*sqrt(c*x^2 + b*x)*A*c^2/x^6 - 3/104*sqrt(c*x^2 + b*x)*B*b^2/x^7 - 1/780*sqrt(c*
x^2 + b*x)*A*b*c/x^7 + 1/8*(c*x^2 + b*x)^(3/2)*B*b/x^8 - 1/60*sqrt(c*x^2 + b*x)*A*b^2/x^8 - 1/4*(c*x^2 + b*x)^
(5/2)*B/x^9 + 1/12*(c*x^2 + b*x)^(3/2)*A*b/x^9 - 1/5*(c*x^2 + b*x)^(5/2)*A/x^10

________________________________________________________________________________________

mupad [B]  time = 4.76, size = 326, normalized size = 2.04 \begin {gather*} \frac {16\,A\,c^4\,\sqrt {c\,x^2+b\,x}}{9009\,b^2\,x^4}-\frac {142\,A\,c^2\,\sqrt {c\,x^2+b\,x}}{715\,x^6}-\frac {2\,B\,b^2\,\sqrt {c\,x^2+b\,x}}{13\,x^7}-\frac {106\,B\,c^2\,\sqrt {c\,x^2+b\,x}}{429\,x^5}-\frac {2\,A\,c^3\,\sqrt {c\,x^2+b\,x}}{1287\,b\,x^5}-\frac {2\,A\,b^2\,\sqrt {c\,x^2+b\,x}}{15\,x^8}-\frac {32\,A\,c^5\,\sqrt {c\,x^2+b\,x}}{15015\,b^3\,x^3}+\frac {128\,A\,c^6\,\sqrt {c\,x^2+b\,x}}{45045\,b^4\,x^2}-\frac {256\,A\,c^7\,\sqrt {c\,x^2+b\,x}}{45045\,b^5\,x}-\frac {10\,B\,c^3\,\sqrt {c\,x^2+b\,x}}{3003\,b\,x^4}+\frac {4\,B\,c^4\,\sqrt {c\,x^2+b\,x}}{1001\,b^2\,x^3}-\frac {16\,B\,c^5\,\sqrt {c\,x^2+b\,x}}{3003\,b^3\,x^2}+\frac {32\,B\,c^6\,\sqrt {c\,x^2+b\,x}}{3003\,b^4\,x}-\frac {62\,A\,b\,c\,\sqrt {c\,x^2+b\,x}}{195\,x^7}-\frac {54\,B\,b\,c\,\sqrt {c\,x^2+b\,x}}{143\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x + c*x^2)^(5/2)*(A + B*x))/x^11,x)

[Out]

(16*A*c^4*(b*x + c*x^2)^(1/2))/(9009*b^2*x^4) - (142*A*c^2*(b*x + c*x^2)^(1/2))/(715*x^6) - (2*B*b^2*(b*x + c*
x^2)^(1/2))/(13*x^7) - (106*B*c^2*(b*x + c*x^2)^(1/2))/(429*x^5) - (2*A*c^3*(b*x + c*x^2)^(1/2))/(1287*b*x^5)
- (2*A*b^2*(b*x + c*x^2)^(1/2))/(15*x^8) - (32*A*c^5*(b*x + c*x^2)^(1/2))/(15015*b^3*x^3) + (128*A*c^6*(b*x +
c*x^2)^(1/2))/(45045*b^4*x^2) - (256*A*c^7*(b*x + c*x^2)^(1/2))/(45045*b^5*x) - (10*B*c^3*(b*x + c*x^2)^(1/2))
/(3003*b*x^4) + (4*B*c^4*(b*x + c*x^2)^(1/2))/(1001*b^2*x^3) - (16*B*c^5*(b*x + c*x^2)^(1/2))/(3003*b^3*x^2) +
(32*B*c^6*(b*x + c*x^2)^(1/2))/(3003*b^4*x) - (62*A*b*c*(b*x + c*x^2)^(1/2))/(195*x^7) - (54*B*b*c*(b*x + c*x
^2)^(1/2))/(143*x^6)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (A + B x\right )}{x^{11}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**(5/2)/x**11,x)

[Out]

Integral((x*(b + c*x))**(5/2)*(A + B*x)/x**11, x)

________________________________________________________________________________________