3.7.8 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{3/2}}{x^{11}} \, dx\)

Optimal. Leaf size=210 \[ -\frac {a^2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+3 A b)}{9 x^9 (a+b x)}-\frac {3 a b \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{8 x^8 (a+b x)}-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (3 a B+A b)}{7 x^7 (a+b x)}-\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac {a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{10 x^{10} (a+b x)} \]

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Rubi [A]  time = 0.08, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {770, 76} \begin {gather*} -\frac {a^2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+3 A b)}{9 x^9 (a+b x)}-\frac {3 a b \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{8 x^8 (a+b x)}-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (3 a B+A b)}{7 x^7 (a+b x)}-\frac {a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{10 x^{10} (a+b x)}-\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/x^11,x]

[Out]

-(a^3*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(10*x^10*(a + b*x)) - (a^2*(3*A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(9*x^9*(a + b*x)) - (3*a*b*(A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*x^8*(a + b*x)) - (b^2*(A*b + 3*a*B)*
Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*x^7*(a + b*x)) - (b^3*B*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*x^6*(a + b*x))

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^{11}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^3 (A+B x)}{x^{11}} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a^3 A b^3}{x^{11}}+\frac {a^2 b^3 (3 A b+a B)}{x^{10}}+\frac {3 a b^4 (A b+a B)}{x^9}+\frac {b^5 (A b+3 a B)}{x^8}+\frac {b^6 B}{x^7}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac {a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{10 x^{10} (a+b x)}-\frac {a^2 (3 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac {3 a b (A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)}-\frac {b^2 (A b+3 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 87, normalized size = 0.41 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (28 a^3 (9 A+10 B x)+105 a^2 b x (8 A+9 B x)+135 a b^2 x^2 (7 A+8 B x)+60 b^3 x^3 (6 A+7 B x)\right )}{2520 x^{10} (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/x^11,x]

[Out]

-1/2520*(Sqrt[(a + b*x)^2]*(60*b^3*x^3*(6*A + 7*B*x) + 135*a*b^2*x^2*(7*A + 8*B*x) + 105*a^2*b*x*(8*A + 9*B*x)
+ 28*a^3*(9*A + 10*B*x)))/(x^10*(a + b*x))

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IntegrateAlgebraic [B]  time = 3.44, size = 894, normalized size = 4.26 \begin {gather*} \frac {64 \sqrt {a^2+2 b x a+b^2 x^2} \left (-420 B x^{13} b^{13}-360 A x^{12} b^{13}-4860 a B x^{12} b^{12}-4185 a A x^{11} b^{12}-25785 a^2 B x^{11} b^{11}-22305 a^2 A x^{10} b^{11}-82945 a^3 B x^{10} b^{10}-72072 a^3 A x^9 b^{10}-180180 a^4 B x^9 b^9-157248 a^4 A x^8 b^9-278460 a^5 B x^8 b^8-244062 a^5 A x^7 b^8-313950 a^6 B x^7 b^7-276318 a^6 A x^6 b^7-260190 a^7 B x^6 b^6-229932 a^7 A x^5 b^6-157320 a^8 B x^5 b^5-139572 a^8 A x^4 b^5-67680 a^9 B x^4 b^4-60273 a^9 A x^3 b^4-19665 a^{10} B x^3 b^3-17577 a^{10} A x^2 b^3-3465 a^{11} B x^2 b^2-3108 a^{11} A x b^2-252 a^{12} A b-280 a^{12} B x b\right ) b^9+64 \sqrt {b^2} \left (420 b^{13} B x^{14}+360 A b^{13} x^{13}+5280 a b^{12} B x^{13}+4545 a A b^{12} x^{12}+30645 a^2 b^{11} B x^{12}+26490 a^2 A b^{11} x^{11}+108730 a^3 b^{10} B x^{11}+94377 a^3 A b^{10} x^{10}+263125 a^4 b^9 B x^{10}+229320 a^4 A b^9 x^9+458640 a^5 b^8 B x^9+401310 a^5 A b^8 x^8+592410 a^6 b^7 B x^8+520380 a^6 A b^7 x^7+574140 a^7 b^6 B x^7+506250 a^7 A b^6 x^6+417510 a^8 b^5 B x^6+369504 a^8 A b^5 x^5+225000 a^9 b^4 B x^5+199845 a^9 A b^4 x^4+87345 a^{10} b^3 B x^4+77850 a^{10} A b^3 x^3+23130 a^{11} b^2 B x^3+20685 a^{11} A b^2 x^2+3745 a^{12} b B x^2+3360 a^{12} A b x+280 a^{13} B x+252 a^{13} A\right ) b^9}{315 \sqrt {b^2} \sqrt {a^2+2 b x a+b^2 x^2} \left (-512 x^9 b^{18}-4608 a x^8 b^{17}-18432 a^2 x^7 b^{16}-43008 a^3 x^6 b^{15}-64512 a^4 x^5 b^{14}-64512 a^5 x^4 b^{13}-43008 a^6 x^3 b^{12}-18432 a^7 x^2 b^{11}-4608 a^8 x b^{10}-512 a^9 b^9\right ) x^{10}+315 \left (512 x^{10} b^{20}+5120 a x^9 b^{19}+23040 a^2 x^8 b^{18}+61440 a^3 x^7 b^{17}+107520 a^4 x^6 b^{16}+129024 a^5 x^5 b^{15}+107520 a^6 x^4 b^{14}+61440 a^7 x^3 b^{13}+23040 a^8 x^2 b^{12}+5120 a^9 x b^{11}+512 a^{10} b^{10}\right ) x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/x^11,x]

[Out]

(64*b^9*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-252*a^12*A*b - 3108*a^11*A*b^2*x - 280*a^12*b*B*x - 17577*a^10*A*b^3*x
^2 - 3465*a^11*b^2*B*x^2 - 60273*a^9*A*b^4*x^3 - 19665*a^10*b^3*B*x^3 - 139572*a^8*A*b^5*x^4 - 67680*a^9*b^4*B
*x^4 - 229932*a^7*A*b^6*x^5 - 157320*a^8*b^5*B*x^5 - 276318*a^6*A*b^7*x^6 - 260190*a^7*b^6*B*x^6 - 244062*a^5*
A*b^8*x^7 - 313950*a^6*b^7*B*x^7 - 157248*a^4*A*b^9*x^8 - 278460*a^5*b^8*B*x^8 - 72072*a^3*A*b^10*x^9 - 180180
*a^4*b^9*B*x^9 - 22305*a^2*A*b^11*x^10 - 82945*a^3*b^10*B*x^10 - 4185*a*A*b^12*x^11 - 25785*a^2*b^11*B*x^11 -
360*A*b^13*x^12 - 4860*a*b^12*B*x^12 - 420*b^13*B*x^13) + 64*b^9*Sqrt[b^2]*(252*a^13*A + 3360*a^12*A*b*x + 280
*a^13*B*x + 20685*a^11*A*b^2*x^2 + 3745*a^12*b*B*x^2 + 77850*a^10*A*b^3*x^3 + 23130*a^11*b^2*B*x^3 + 199845*a^
9*A*b^4*x^4 + 87345*a^10*b^3*B*x^4 + 369504*a^8*A*b^5*x^5 + 225000*a^9*b^4*B*x^5 + 506250*a^7*A*b^6*x^6 + 4175
10*a^8*b^5*B*x^6 + 520380*a^6*A*b^7*x^7 + 574140*a^7*b^6*B*x^7 + 401310*a^5*A*b^8*x^8 + 592410*a^6*b^7*B*x^8 +
229320*a^4*A*b^9*x^9 + 458640*a^5*b^8*B*x^9 + 94377*a^3*A*b^10*x^10 + 263125*a^4*b^9*B*x^10 + 26490*a^2*A*b^1
1*x^11 + 108730*a^3*b^10*B*x^11 + 4545*a*A*b^12*x^12 + 30645*a^2*b^11*B*x^12 + 360*A*b^13*x^13 + 5280*a*b^12*B
*x^13 + 420*b^13*B*x^14))/(315*Sqrt[b^2]*x^10*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-512*a^9*b^9 - 4608*a^8*b^10*x -
18432*a^7*b^11*x^2 - 43008*a^6*b^12*x^3 - 64512*a^5*b^13*x^4 - 64512*a^4*b^14*x^5 - 43008*a^3*b^15*x^6 - 18432
*a^2*b^16*x^7 - 4608*a*b^17*x^8 - 512*b^18*x^9) + 315*x^10*(512*a^10*b^10 + 5120*a^9*b^11*x + 23040*a^8*b^12*x
^2 + 61440*a^7*b^13*x^3 + 107520*a^6*b^14*x^4 + 129024*a^5*b^15*x^5 + 107520*a^4*b^16*x^6 + 61440*a^3*b^17*x^7
+ 23040*a^2*b^18*x^8 + 5120*a*b^19*x^9 + 512*b^20*x^10))

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fricas [A]  time = 0.42, size = 73, normalized size = 0.35 \begin {gather*} -\frac {420 \, B b^{3} x^{4} + 252 \, A a^{3} + 360 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 945 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} + 280 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2520 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/x^11,x, algorithm="fricas")

[Out]

-1/2520*(420*B*b^3*x^4 + 252*A*a^3 + 360*(3*B*a*b^2 + A*b^3)*x^3 + 945*(B*a^2*b + A*a*b^2)*x^2 + 280*(B*a^3 +
3*A*a^2*b)*x)/x^10

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giac [A]  time = 0.16, size = 149, normalized size = 0.71 \begin {gather*} \frac {{\left (5 \, B a b^{9} - 3 \, A b^{10}\right )} \mathrm {sgn}\left (b x + a\right )}{2520 \, a^{7}} - \frac {420 \, B b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 1080 \, B a b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 360 \, A b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 945 \, B a^{2} b x^{2} \mathrm {sgn}\left (b x + a\right ) + 945 \, A a b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 280 \, B a^{3} x \mathrm {sgn}\left (b x + a\right ) + 840 \, A a^{2} b x \mathrm {sgn}\left (b x + a\right ) + 252 \, A a^{3} \mathrm {sgn}\left (b x + a\right )}{2520 \, x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/x^11,x, algorithm="giac")

[Out]

1/2520*(5*B*a*b^9 - 3*A*b^10)*sgn(b*x + a)/a^7 - 1/2520*(420*B*b^3*x^4*sgn(b*x + a) + 1080*B*a*b^2*x^3*sgn(b*x
+ a) + 360*A*b^3*x^3*sgn(b*x + a) + 945*B*a^2*b*x^2*sgn(b*x + a) + 945*A*a*b^2*x^2*sgn(b*x + a) + 280*B*a^3*x
*sgn(b*x + a) + 840*A*a^2*b*x*sgn(b*x + a) + 252*A*a^3*sgn(b*x + a))/x^10

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maple [A]  time = 0.05, size = 92, normalized size = 0.44 \begin {gather*} -\frac {\left (420 B \,b^{3} x^{4}+360 A \,b^{3} x^{3}+1080 B a \,b^{2} x^{3}+945 A a \,b^{2} x^{2}+945 B \,a^{2} b \,x^{2}+840 A \,a^{2} b x +280 B \,a^{3} x +252 A \,a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{2520 \left (b x +a \right )^{3} x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/x^11,x)

[Out]

-1/2520*(420*B*b^3*x^4+360*A*b^3*x^3+1080*B*a*b^2*x^3+945*A*a*b^2*x^2+945*B*a^2*b*x^2+840*A*a^2*b*x+280*B*a^3*
x+252*A*a^3)*((b*x+a)^2)^(3/2)/x^10/(b*x+a)^3

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maxima [B]  time = 0.69, size = 615, normalized size = 2.93 \begin {gather*} -\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b^{9}}{4 \, a^{9}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{10}}{4 \, a^{10}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b^{8}}{4 \, a^{8} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{9}}{4 \, a^{9} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{7}}{4 \, a^{9} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{8}}{4 \, a^{10} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{6}}{4 \, a^{8} x^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{7}}{4 \, a^{9} x^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{5}}{4 \, a^{7} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{6}}{4 \, a^{8} x^{4}} - \frac {125 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{4}}{504 \, a^{6} x^{5}} + \frac {209 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{5}}{840 \, a^{7} x^{5}} + \frac {121 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{3}}{504 \, a^{5} x^{6}} - \frac {41 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{4}}{168 \, a^{6} x^{6}} - \frac {37 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{2}}{168 \, a^{4} x^{7}} + \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{3}}{56 \, a^{5} x^{7}} + \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b}{72 \, a^{3} x^{8}} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{2}}{24 \, a^{4} x^{8}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B}{9 \, a^{2} x^{9}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b}{6 \, a^{3} x^{9}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A}{10 \, a^{2} x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(3/2)/x^11,x, algorithm="maxima")

[Out]

-1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*B*b^9/a^9 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*A*b^10/a^10 - 1/4*(b^2*x^
2 + 2*a*b*x + a^2)^(3/2)*B*b^8/(a^8*x) + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*A*b^9/(a^9*x) + 1/4*(b^2*x^2 + 2*
a*b*x + a^2)^(5/2)*B*b^7/(a^9*x^2) - 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^8/(a^10*x^2) - 1/4*(b^2*x^2 + 2*a
*b*x + a^2)^(5/2)*B*b^6/(a^8*x^3) + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^7/(a^9*x^3) + 1/4*(b^2*x^2 + 2*a*b
*x + a^2)^(5/2)*B*b^5/(a^7*x^4) - 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^6/(a^8*x^4) - 125/504*(b^2*x^2 + 2*a
*b*x + a^2)^(5/2)*B*b^4/(a^6*x^5) + 209/840*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^5/(a^7*x^5) + 121/504*(b^2*x^2
+ 2*a*b*x + a^2)^(5/2)*B*b^3/(a^5*x^6) - 41/168*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^4/(a^6*x^6) - 37/168*(b^2
*x^2 + 2*a*b*x + a^2)^(5/2)*B*b^2/(a^4*x^7) + 13/56*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^3/(a^5*x^7) + 13/72*(b
^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*b/(a^3*x^8) - 5/24*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^2/(a^4*x^8) - 1/9*(b^2*
x^2 + 2*a*b*x + a^2)^(5/2)*B/(a^2*x^9) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b/(a^3*x^9) - 1/10*(b^2*x^2 + 2
*a*b*x + a^2)^(5/2)*A/(a^2*x^10)

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mupad [B]  time = 1.18, size = 196, normalized size = 0.93 \begin {gather*} -\frac {\left (\frac {B\,a^3}{9}+\frac {A\,b\,a^2}{3}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^9\,\left (a+b\,x\right )}-\frac {\left (\frac {A\,b^3}{7}+\frac {3\,B\,a\,b^2}{7}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^7\,\left (a+b\,x\right )}-\frac {A\,a^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{10\,x^{10}\,\left (a+b\,x\right )}-\frac {B\,b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{6\,x^6\,\left (a+b\,x\right )}-\frac {3\,a\,b\,\left (A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{8\,x^8\,\left (a+b\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2))/x^11,x)

[Out]

- (((B*a^3)/9 + (A*a^2*b)/3)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(x^9*(a + b*x)) - (((A*b^3)/7 + (3*B*a*b^2)/7)*(
a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(x^7*(a + b*x)) - (A*a^3*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(10*x^10*(a + b*x))
- (B*b^3*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(6*x^6*(a + b*x)) - (3*a*b*(A*b + B*a)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/
2))/(8*x^8*(a + b*x))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{x^{11}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(3/2)/x**11,x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(3/2)/x**11, x)

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