3.7.28 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{x^{12}} \, dx\)

Optimal. Leaf size=306 \[ -\frac {5 a^2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{4 x^8 (a+b x)}-\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{6 x^6 (a+b x)}-\frac {5 a b^3 \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{7 x^7 (a+b x)}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac {a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac {a^4 \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{10 x^{10} (a+b x)}-\frac {5 a^3 b \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{9 x^9 (a+b x)} \]

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Rubi [A]  time = 0.12, antiderivative size = 306, 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^4 \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{10 x^{10} (a+b x)}-\frac {5 a^3 b \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{9 x^9 (a+b x)}-\frac {5 a^2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{4 x^8 (a+b x)}-\frac {5 a b^3 \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{7 x^7 (a+b x)}-\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{6 x^6 (a+b x)}-\frac {a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^12,x]

[Out]

-(a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*x^11*(a + b*x)) - (a^4*(5*A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(10*x^10*(a + b*x)) - (5*a^3*b*(2*A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*x^9*(a + b*x)) - (5*a^2*b^2*(A
*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*x^8*(a + b*x)) - (5*a*b^3*(A*b + 2*a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(7*x^7*(a + b*x)) - (b^4*(A*b + 5*a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*x^6*(a + b*x)) - (b^5*B*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(5*x^5*(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 )^{5/2}}{x^{12}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5 (A+B x)}{x^{12}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a^5 A b^5}{x^{12}}+\frac {a^4 b^5 (5 A b+a B)}{x^{11}}+\frac {5 a^3 b^6 (2 A b+a B)}{x^{10}}+\frac {10 a^2 b^7 (A b+a B)}{x^9}+\frac {5 a b^8 (A b+2 a B)}{x^8}+\frac {b^9 (A b+5 a B)}{x^7}+\frac {b^{10} B}{x^6}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac {a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac {a^4 (5 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{10 x^{10} (a+b x)}-\frac {5 a^3 b (2 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac {5 a^2 b^2 (A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^8 (a+b x)}-\frac {5 a b^3 (A b+2 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac {b^4 (A b+5 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 125, normalized size = 0.41 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (126 a^5 (10 A+11 B x)+770 a^4 b x (9 A+10 B x)+1925 a^3 b^2 x^2 (8 A+9 B x)+2475 a^2 b^3 x^3 (7 A+8 B x)+1650 a b^4 x^4 (6 A+7 B x)+462 b^5 x^5 (5 A+6 B x)\right )}{13860 x^{11} (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^12,x]

[Out]

-1/13860*(Sqrt[(a + b*x)^2]*(462*b^5*x^5*(5*A + 6*B*x) + 1650*a*b^4*x^4*(6*A + 7*B*x) + 2475*a^2*b^3*x^3*(7*A
+ 8*B*x) + 1925*a^3*b^2*x^2*(8*A + 9*B*x) + 770*a^4*b*x*(9*A + 10*B*x) + 126*a^5*(10*A + 11*B*x)))/(x^11*(a +
b*x))

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IntegrateAlgebraic [B]  time = 4.08, size = 1060, normalized size = 3.46 \begin {gather*} \frac {256 \sqrt {a^2+2 b x a+b^2 x^2} \left (-2772 B x^{16} b^{16}-2310 A x^{15} b^{16}-39270 a B x^{15} b^{15}-33000 a A x^{14} b^{15}-260040 a^2 B x^{14} b^{14}-220275 a^2 A x^{13} b^{14}-1067715 a^3 B x^{13} b^{13}-911350 a^3 A x^{12} b^{13}-3040070 a^4 B x^{12} b^{12}-2613655 a^4 A x^{11} b^{12}-6358055 a^5 B x^{11} b^{11}-5503680 a^5 A x^{10} b^{11}-10090080 a^6 B x^{10} b^{10}-8790600 a^6 A x^9 b^{10}-12372360 a^7 B x^9 b^9-10844400 a^7 A x^8 b^9-11817960 a^8 B x^8 b^8-10417500 a^8 A x^7 b^8-8793180 a^9 B x^7 b^7-7792560 a^9 A x^6 b^7-5054544 a^{10} B x^6 b^6-4501755 a^{10} A x^5 b^6-2204235 a^{11} B x^5 b^5-1972350 a^{11} A x^4 b^5-705870 a^{12} B x^4 b^4-634375 a^{12} A x^3 b^4-156695 a^{13} B x^3 b^3-141400 a^{13} A x^2 b^3-21560 a^{14} B x^2 b^2-19530 a^{14} A x b^2-1260 a^{15} A b-1386 a^{15} B x b\right ) b^{10}+256 \sqrt {b^2} \left (2772 b^{16} B x^{17}+2310 A b^{16} x^{16}+42042 a b^{15} B x^{16}+35310 a A b^{15} x^{15}+299310 a^2 b^{14} B x^{15}+253275 a^2 A b^{14} x^{14}+1327755 a^3 b^{13} B x^{14}+1131625 a^3 A b^{13} x^{13}+4107785 a^4 b^{12} B x^{13}+3525005 a^4 A b^{12} x^{12}+9398125 a^5 b^{11} B x^{12}+8117335 a^5 A b^{11} x^{11}[phone] a^6 b^{10} B x^{11}[phone] a^6 A b^{10} x^{10}[phone] a^7 b^9 B x^{10}[phone] a^7 A b^9 x^9+24190320 a^8 b^8 B x^9+21261900 a^8 A b^8 x^8+20611140 a^9 b^7 B x^8+18210060 a^9 A b^7 x^7+13847724 a^{10} b^6 B x^7+12294315 a^{10} A b^6 x^6+7258779 a^{11} b^5 B x^6+6474105 a^{11} A b^5 x^5+2910105 a^{12} b^4 B x^5+2606725 a^{12} A b^4 x^4+862565 a^{13} b^3 B x^4+775775 a^{13} A b^3 x^3+178255 a^{14} b^2 B x^3+160930 a^{14} A b^2 x^2+22946 a^{15} b B x^2+20790 a^{15} A b x+1386 a^{16} B x+1260 a^{16} A\right ) b^{10}}{3465 \sqrt {b^2} \sqrt {a^2+2 b x a+b^2 x^2} \left (-1024 x^{10} b^{20}-10240 a x^9 b^{19}-46080 a^2 x^8 b^{18}-122880 a^3 x^7 b^{17}-215040 a^4 x^6 b^{16}-258048 a^5 x^5 b^{15}-215040 a^6 x^4 b^{14}-122880 a^7 x^3 b^{13}-46080 a^8 x^2 b^{12}-10240 a^9 x b^{11}-1024 a^{10} b^{10}\right ) x^{11}+3465 \left (1024 x^{11} b^{22}+11264 a x^{10} b^{21}+56320 a^2 x^9 b^{20}+168960 a^3 x^8 b^{19}+337920 a^4 x^7 b^{18}+473088 a^5 x^6 b^{17}+473088 a^6 x^5 b^{16}+337920 a^7 x^4 b^{15}+168960 a^8 x^3 b^{14}+56320 a^9 x^2 b^{13}+11264 a^{10} x b^{12}+1024 a^{11} b^{11}\right ) x^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^12,x]

[Out]

(256*b^10*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-1260*a^15*A*b - 19530*a^14*A*b^2*x - 1386*a^15*b*B*x - 141400*a^13*A
*b^3*x^2 - 21560*a^14*b^2*B*x^2 - 634375*a^12*A*b^4*x^3 - 156695*a^13*b^3*B*x^3 - 1972350*a^11*A*b^5*x^4 - 705
870*a^12*b^4*B*x^4 - 4501755*a^10*A*b^6*x^5 - 2204235*a^11*b^5*B*x^5 - 7792560*a^9*A*b^7*x^6 - 5054544*a^10*b^
6*B*x^6 - 10417500*a^8*A*b^8*x^7 - 8793180*a^9*b^7*B*x^7 - 10844400*a^7*A*b^9*x^8 - 11817960*a^8*b^8*B*x^8 - 8
790600*a^6*A*b^10*x^9 - 12372360*a^7*b^9*B*x^9 - 5503680*a^5*A*b^11*x^10 - 10090080*a^6*b^10*B*x^10 - 2613655*
a^4*A*b^12*x^11 - 6358055*a^5*b^11*B*x^11 - 911350*a^3*A*b^13*x^12 - 3040070*a^4*b^12*B*x^12 - 220275*a^2*A*b^
14*x^13 - 1067715*a^3*b^13*B*x^13 - 33000*a*A*b^15*x^14 - 260040*a^2*b^14*B*x^14 - 2310*A*b^16*x^15 - 39270*a*
b^15*B*x^15 - 2772*b^16*B*x^16) + 256*b^10*Sqrt[b^2]*(1260*a^16*A + 20790*a^15*A*b*x + 1386*a^16*B*x + 160930*
a^14*A*b^2*x^2 + 22946*a^15*b*B*x^2 + 775775*a^13*A*b^3*x^3 + 178255*a^14*b^2*B*x^3 + 2606725*a^12*A*b^4*x^4 +
862565*a^13*b^3*B*x^4 + 6474105*a^11*A*b^5*x^5 + 2910105*a^12*b^4*B*x^5 + 12294315*a^10*A*b^6*x^6 + 7258779*a
^11*b^5*B*x^6 + 18210060*a^9*A*b^7*x^7 + 13847724*a^10*b^6*B*x^7 + 21261900*a^8*A*b^8*x^8 + 20611140*a^9*b^7*B
*x^8 + 19635000*a^7*A*b^9*x^9 + 24190320*a^8*b^8*B*x^9 + 14294280*a^6*A*b^10*x^10 + 22462440*a^7*b^9*B*x^10 +
8117335*a^5*A*b^11*x^11 + 16448135*a^6*b^10*B*x^11 + 3525005*a^4*A*b^12*x^12 + 9398125*a^5*b^11*B*x^12 + 11316
25*a^3*A*b^13*x^13 + 4107785*a^4*b^12*B*x^13 + 253275*a^2*A*b^14*x^14 + 1327755*a^3*b^13*B*x^14 + 35310*a*A*b^
15*x^15 + 299310*a^2*b^14*B*x^15 + 2310*A*b^16*x^16 + 42042*a*b^15*B*x^16 + 2772*b^16*B*x^17))/(3465*Sqrt[b^2]
*x^11*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-1024*a^10*b^10 - 10240*a^9*b^11*x - 46080*a^8*b^12*x^2 - 122880*a^7*b^13
*x^3 - 215040*a^6*b^14*x^4 - 258048*a^5*b^15*x^5 - 215040*a^4*b^16*x^6 - 122880*a^3*b^17*x^7 - 46080*a^2*b^18*
x^8 - 10240*a*b^19*x^9 - 1024*b^20*x^10) + 3465*x^11*(1024*a^11*b^11 + 11264*a^10*b^12*x + 56320*a^9*b^13*x^2
+ 168960*a^8*b^14*x^3 + 337920*a^7*b^15*x^4 + 473088*a^6*b^16*x^5 + 473088*a^5*b^17*x^6 + 337920*a^4*b^18*x^7
+ 168960*a^3*b^19*x^8 + 56320*a^2*b^20*x^9 + 11264*a*b^21*x^10 + 1024*b^22*x^11))

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fricas [A]  time = 0.41, size = 119, normalized size = 0.39 \begin {gather*} -\frac {2772 \, B b^{5} x^{6} + 1260 \, A a^{5} + 2310 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 9900 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 17325 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 7700 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 1386 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{13860 \, x^{11}} \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)^(5/2)/x^12,x, algorithm="fricas")

[Out]

-1/13860*(2772*B*b^5*x^6 + 1260*A*a^5 + 2310*(5*B*a*b^4 + A*b^5)*x^5 + 9900*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 1732
5*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 7700*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 1386*(B*a^5 + 5*A*a^4*b)*x)/x^11

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giac [A]  time = 0.20, size = 221, normalized size = 0.72 \begin {gather*} -\frac {{\left (11 \, B a b^{10} - 5 \, A b^{11}\right )} \mathrm {sgn}\left (b x + a\right )}{13860 \, a^{6}} - \frac {2772 \, B b^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) + 11550 \, B a b^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + 2310 \, A b^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 19800 \, B a^{2} b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 9900 \, A a b^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 17325 \, B a^{3} b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 17325 \, A a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 7700 \, B a^{4} b x^{2} \mathrm {sgn}\left (b x + a\right ) + 15400 \, A a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 1386 \, B a^{5} x \mathrm {sgn}\left (b x + a\right ) + 6930 \, A a^{4} b x \mathrm {sgn}\left (b x + a\right ) + 1260 \, A a^{5} \mathrm {sgn}\left (b x + a\right )}{13860 \, x^{11}} \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)^(5/2)/x^12,x, algorithm="giac")

[Out]

-1/13860*(11*B*a*b^10 - 5*A*b^11)*sgn(b*x + a)/a^6 - 1/13860*(2772*B*b^5*x^6*sgn(b*x + a) + 11550*B*a*b^4*x^5*
sgn(b*x + a) + 2310*A*b^5*x^5*sgn(b*x + a) + 19800*B*a^2*b^3*x^4*sgn(b*x + a) + 9900*A*a*b^4*x^4*sgn(b*x + a)
+ 17325*B*a^3*b^2*x^3*sgn(b*x + a) + 17325*A*a^2*b^3*x^3*sgn(b*x + a) + 7700*B*a^4*b*x^2*sgn(b*x + a) + 15400*
A*a^3*b^2*x^2*sgn(b*x + a) + 1386*B*a^5*x*sgn(b*x + a) + 6930*A*a^4*b*x*sgn(b*x + a) + 1260*A*a^5*sgn(b*x + a)
)/x^11

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maple [A]  time = 0.06, size = 140, normalized size = 0.46 \begin {gather*} -\frac {\left (2772 B \,b^{5} x^{6}+2310 A \,b^{5} x^{5}+11550 B a \,b^{4} x^{5}+9900 A a \,b^{4} x^{4}+19800 B \,a^{2} b^{3} x^{4}+17325 A \,a^{2} b^{3} x^{3}+17325 B \,a^{3} b^{2} x^{3}+15400 A \,a^{3} b^{2} x^{2}+7700 B \,a^{4} b \,x^{2}+6930 A \,a^{4} b x +1386 B \,a^{5} x +1260 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{13860 \left (b x +a \right )^{5} x^{11}} \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)^(5/2)/x^12,x)

[Out]

-1/13860*(2772*B*b^5*x^6+2310*A*b^5*x^5+11550*B*a*b^4*x^5+9900*A*a*b^4*x^4+19800*B*a^2*b^3*x^4+17325*A*a^2*b^3
*x^3+17325*B*a^3*b^2*x^3+15400*A*a^3*b^2*x^2+7700*B*a^4*b*x^2+6930*A*a^4*b*x+1386*B*a^5*x+1260*A*a^5)*((b*x+a)
^2)^(5/2)/x^11/(b*x+a)^5

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maxima [B]  time = 0.74, size = 675, normalized size = 2.21 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{10}}{6 \, a^{10}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{11}}{6 \, a^{11}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{9}}{6 \, a^{9} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{10}}{6 \, a^{10} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{8}}{6 \, a^{10} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{9}}{6 \, a^{11} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{7}}{6 \, a^{9} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{8}}{6 \, a^{10} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{6}}{6 \, a^{8} x^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{7}}{6 \, a^{9} x^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{5}}{6 \, a^{7} x^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{6}}{6 \, a^{8} x^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{4}}{6 \, a^{6} x^{6}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{5}}{6 \, a^{7} x^{6}} + \frac {209 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{3}}{1260 \, a^{5} x^{7}} - \frac {461 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{4}}{2772 \, a^{6} x^{7}} - \frac {29 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{2}}{180 \, a^{4} x^{8}} + \frac {65 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{3}}{396 \, a^{5} x^{8}} + \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b}{90 \, a^{3} x^{9}} - \frac {31 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{2}}{198 \, a^{4} x^{9}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B}{10 \, a^{2} x^{10}} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b}{22 \, a^{3} x^{10}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A}{11 \, a^{2} x^{11}} \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)^(5/2)/x^12,x, algorithm="maxima")

[Out]

1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*b^10/a^10 - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^11/a^11 + 1/6*(b^2*x
^2 + 2*a*b*x + a^2)^(5/2)*B*b^9/(a^9*x) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^10/(a^10*x) - 1/6*(b^2*x^2 +
2*a*b*x + a^2)^(7/2)*B*b^8/(a^10*x^2) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^9/(a^11*x^2) + 1/6*(b^2*x^2 +
2*a*b*x + a^2)^(7/2)*B*b^7/(a^9*x^3) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^8/(a^10*x^3) - 1/6*(b^2*x^2 +
2*a*b*x + a^2)^(7/2)*B*b^6/(a^8*x^4) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^7/(a^9*x^4) + 1/6*(b^2*x^2 + 2*
a*b*x + a^2)^(7/2)*B*b^5/(a^7*x^5) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^6/(a^8*x^5) - 1/6*(b^2*x^2 + 2*a*
b*x + a^2)^(7/2)*B*b^4/(a^6*x^6) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^5/(a^7*x^6) + 209/1260*(b^2*x^2 + 2
*a*b*x + a^2)^(7/2)*B*b^3/(a^5*x^7) - 461/2772*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^4/(a^6*x^7) - 29/180*(b^2*x
^2 + 2*a*b*x + a^2)^(7/2)*B*b^2/(a^4*x^8) + 65/396*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^3/(a^5*x^8) + 13/90*(b^
2*x^2 + 2*a*b*x + a^2)^(7/2)*B*b/(a^3*x^9) - 31/198*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^2/(a^4*x^9) - 1/10*(b^
2*x^2 + 2*a*b*x + a^2)^(7/2)*B/(a^2*x^10) + 3/22*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b/(a^3*x^10) - 1/11*(b^2*x^
2 + 2*a*b*x + a^2)^(7/2)*A/(a^2*x^11)

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mupad [B]  time = 1.39, size = 284, normalized size = 0.93 \begin {gather*} -\frac {\left (\frac {B\,a^5}{10}+\frac {A\,b\,a^4}{2}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^{10}\,\left (a+b\,x\right )}-\frac {\left (\frac {A\,b^5}{6}+\frac {5\,B\,a\,b^4}{6}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^6\,\left (a+b\,x\right )}-\frac {A\,a^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{11\,x^{11}\,\left (a+b\,x\right )}-\frac {B\,b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{5\,x^5\,\left (a+b\,x\right )}-\frac {5\,a\,b^3\,\left (A\,b+2\,B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{7\,x^7\,\left (a+b\,x\right )}-\frac {5\,a^3\,b\,\left (2\,A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{9\,x^9\,\left (a+b\,x\right )}-\frac {5\,a^2\,b^2\,\left (A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,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)^(5/2))/x^12,x)

[Out]

- (((B*a^5)/10 + (A*a^4*b)/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(x^10*(a + b*x)) - (((A*b^5)/6 + (5*B*a*b^4)/6)
*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(x^6*(a + b*x)) - (A*a^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(11*x^11*(a + b*x)
) - (B*b^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(5*x^5*(a + b*x)) - (5*a*b^3*(A*b + 2*B*a)*(a^2 + b^2*x^2 + 2*a*b*
x)^(1/2))/(7*x^7*(a + b*x)) - (5*a^3*b*(2*A*b + B*a)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(9*x^9*(a + b*x)) - (5*a
^2*b^2*(A*b + B*a)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(4*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 {5}{2}}}{x^{12}}\, 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)**(5/2)/x**12,x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(5/2)/x**12, x)

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