3.7.9 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{3/2}}{x^{12}} \, dx\)

Optimal. Leaf size=210 \[ -\frac {a^2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+3 A b)}{10 x^{10} (a+b x)}-\frac {a b \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{3 x^9 (a+b x)}-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (3 a B+A b)}{8 x^8 (a+b x)}-\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac {a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)} \]

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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)}{10 x^{10} (a+b x)}-\frac {a b \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{3 x^9 (a+b x)}-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (3 a B+A b)}{8 x^8 (a+b x)}-\frac {a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (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^12,x]

[Out]

-(a^3*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*x^11*(a + b*x)) - (a^2*(3*A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(10*x^10*(a + b*x)) - (a*b*(A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*x^9*(a + b*x)) - (b^2*(A*b + 3*a*B)*
Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*x^8*(a + b*x)) - (b^3*B*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*x^7*(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^{12}} \, 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^{12}} \, 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^{12}}+\frac {a^2 b^3 (3 A b+a B)}{x^{11}}+\frac {3 a b^4 (A b+a B)}{x^{10}}+\frac {b^5 (A b+3 a B)}{x^9}+\frac {b^6 B}{x^8}\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}}{11 x^{11} (a+b x)}-\frac {a^2 (3 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{10 x^{10} (a+b x)}-\frac {a b (A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^9 (a+b x)}-\frac {b^2 (A b+3 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)}-\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (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 (84 a^3 (10 A+11 B x)+308 a^2 b x (9 A+10 B x)+385 a b^2 x^2 (8 A+9 B x)+165 b^3 x^3 (7 A+8 B x)\right )}{9240 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)^(3/2))/x^12,x]

[Out]

-1/9240*(Sqrt[(a + b*x)^2]*(165*b^3*x^3*(7*A + 8*B*x) + 385*a*b^2*x^2*(8*A + 9*B*x) + 308*a^2*b*x*(9*A + 10*B*
x) + 84*a^3*(10*A + 11*B*x)))/(x^11*(a + b*x))

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IntegrateAlgebraic [B]  time = 3.81, size = 964, normalized size = 4.59 \begin {gather*} \frac {128 \sqrt {a^2+2 b x a+b^2 x^2} \left (-1320 B x^{14} b^{14}-1155 A x^{13} b^{14}-16665 a B x^{13} b^{13}-14630 a A x^{12} b^{13}-97130 a^2 B x^{12} b^{12}-85547 a^2 A x^{11} b^{12}-346049 a^3 B x^{11} b^{11}-305760 a^3 A x^{10} b^{11}-840840 a^4 B x^{10} b^{10}-745290 a^4 A x^9 b^{10}-1471470 a^5 B x^9 b^9-1308300 a^5 A x^8 b^9-1908060 a^6 B x^8 b^8-1701630 a^6 A x^7 b^8-1856250 a^7 B x^7 b^7-1660344 a^7 A x^6 b^7-1354848 a^8 B x^6 b^6-1215375 a^8 A x^5 b^6-732765 a^9 B x^5 b^5-659190 a^9 A x^4 b^5-285450 a^{10} B x^4 b^4-257495 a^{10} A x^3 b^4-75845 a^{11} B x^3 b^3-68600 a^{11} A x^2 b^3-12320 a^{12} B x^2 b^2-11172 a^{12} A x b^2-840 a^{13} A b-924 a^{13} B x b\right ) b^{10}+128 \sqrt {b^2} \left (1320 b^{14} B x^{15}+1155 A b^{14} x^{14}+17985 a b^{13} B x^{14}+15785 a A b^{13} x^{13}+113795 a^2 b^{12} B x^{13}+100177 a^2 A b^{12} x^{12}+443179 a^3 b^{11} B x^{12}+391307 a^3 A b^{11} x^{11}+1186889 a^4 b^{10} B x^{11}+1051050 a^4 A b^{10} x^{10}+2312310 a^5 b^9 B x^{10}+2053590 a^5 A b^9 x^9+3379530 a^6 b^8 B x^9+3009930 a^6 A b^8 x^8+3764310 a^7 b^7 B x^8+3361974 a^7 A b^7 x^7+3211098 a^8 b^6 B x^7+2875719 a^8 A b^6 x^6+2087613 a^9 b^5 B x^6+1874565 a^9 A b^5 x^5+1018215 a^{10} b^4 B x^5+916685 a^{10} A b^4 x^4+361295 a^{11} b^3 B x^4+326095 a^{11} A b^3 x^3+88165 a^{12} b^2 B x^3+79772 a^{12} A b^2 x^2+13244 a^{13} b B x^2+12012 a^{13} A b x+924 a^{14} B x+840 a^{14} A\right ) b^{10}}{1155 \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}+1155 \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)^(3/2))/x^12,x]

[Out]

(128*b^10*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-840*a^13*A*b - 11172*a^12*A*b^2*x - 924*a^13*b*B*x - 68600*a^11*A*b^
3*x^2 - 12320*a^12*b^2*B*x^2 - 257495*a^10*A*b^4*x^3 - 75845*a^11*b^3*B*x^3 - 659190*a^9*A*b^5*x^4 - 285450*a^
10*b^4*B*x^4 - 1215375*a^8*A*b^6*x^5 - 732765*a^9*b^5*B*x^5 - 1660344*a^7*A*b^7*x^6 - 1354848*a^8*b^6*B*x^6 -
1701630*a^6*A*b^8*x^7 - 1856250*a^7*b^7*B*x^7 - 1308300*a^5*A*b^9*x^8 - 1908060*a^6*b^8*B*x^8 - 745290*a^4*A*b
^10*x^9 - 1471470*a^5*b^9*B*x^9 - 305760*a^3*A*b^11*x^10 - 840840*a^4*b^10*B*x^10 - 85547*a^2*A*b^12*x^11 - 34
6049*a^3*b^11*B*x^11 - 14630*a*A*b^13*x^12 - 97130*a^2*b^12*B*x^12 - 1155*A*b^14*x^13 - 16665*a*b^13*B*x^13 -
1320*b^14*B*x^14) + 128*b^10*Sqrt[b^2]*(840*a^14*A + 12012*a^13*A*b*x + 924*a^14*B*x + 79772*a^12*A*b^2*x^2 +
13244*a^13*b*B*x^2 + 326095*a^11*A*b^3*x^3 + 88165*a^12*b^2*B*x^3 + 916685*a^10*A*b^4*x^4 + 361295*a^11*b^3*B*
x^4 + 1874565*a^9*A*b^5*x^5 + 1018215*a^10*b^4*B*x^5 + 2875719*a^8*A*b^6*x^6 + 2087613*a^9*b^5*B*x^6 + 3361974
*a^7*A*b^7*x^7 + 3211098*a^8*b^6*B*x^7 + 3009930*a^6*A*b^8*x^8 + 3764310*a^7*b^7*B*x^8 + 2053590*a^5*A*b^9*x^9
+ 3379530*a^6*b^8*B*x^9 + 1051050*a^4*A*b^10*x^10 + 2312310*a^5*b^9*B*x^10 + 391307*a^3*A*b^11*x^11 + 1186889
*a^4*b^10*B*x^11 + 100177*a^2*A*b^12*x^12 + 443179*a^3*b^11*B*x^12 + 15785*a*A*b^13*x^13 + 113795*a^2*b^12*B*x
^13 + 1155*A*b^14*x^14 + 17985*a*b^13*B*x^14 + 1320*b^14*B*x^15))/(1155*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) + 1155*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 + 33792
0*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 = 73, normalized size = 0.35 \begin {gather*} -\frac {1320 \, B b^{3} x^{4} + 840 \, A a^{3} + 1155 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 3080 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} + 924 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{9240 \, 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)^(3/2)/x^12,x, algorithm="fricas")

[Out]

-1/9240*(1320*B*b^3*x^4 + 840*A*a^3 + 1155*(3*B*a*b^2 + A*b^3)*x^3 + 3080*(B*a^2*b + A*a*b^2)*x^2 + 924*(B*a^3
+ 3*A*a^2*b)*x)/x^11

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giac [A]  time = 0.20, size = 149, normalized size = 0.71 \begin {gather*} -\frac {{\left (11 \, B a b^{10} - 7 \, A b^{11}\right )} \mathrm {sgn}\left (b x + a\right )}{9240 \, a^{8}} - \frac {1320 \, B b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 3465 \, B a b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 1155 \, A b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 3080 \, B a^{2} b x^{2} \mathrm {sgn}\left (b x + a\right ) + 3080 \, A a b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 924 \, B a^{3} x \mathrm {sgn}\left (b x + a\right ) + 2772 \, A a^{2} b x \mathrm {sgn}\left (b x + a\right ) + 840 \, A a^{3} \mathrm {sgn}\left (b x + a\right )}{9240 \, 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)^(3/2)/x^12,x, algorithm="giac")

[Out]

-1/9240*(11*B*a*b^10 - 7*A*b^11)*sgn(b*x + a)/a^8 - 1/9240*(1320*B*b^3*x^4*sgn(b*x + a) + 3465*B*a*b^2*x^3*sgn
(b*x + a) + 1155*A*b^3*x^3*sgn(b*x + a) + 3080*B*a^2*b*x^2*sgn(b*x + a) + 3080*A*a*b^2*x^2*sgn(b*x + a) + 924*
B*a^3*x*sgn(b*x + a) + 2772*A*a^2*b*x*sgn(b*x + a) + 840*A*a^3*sgn(b*x + a))/x^11

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maple [A]  time = 0.06, size = 92, normalized size = 0.44 \begin {gather*} -\frac {\left (1320 B \,b^{3} x^{4}+1155 A \,b^{3} x^{3}+3465 B a \,b^{2} x^{3}+3080 A a \,b^{2} x^{2}+3080 B \,a^{2} b \,x^{2}+2772 A \,a^{2} b x +924 B \,a^{3} x +840 A \,a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{9240 \left (b x +a \right )^{3} 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)^(3/2)/x^12,x)

[Out]

-1/9240*(1320*B*b^3*x^4+1155*A*b^3*x^3+3465*B*a*b^2*x^3+3080*A*a*b^2*x^2+3080*B*a^2*b*x^2+2772*A*a^2*b*x+924*B
*a^3*x+840*A*a^3)*((b*x+a)^2)^(3/2)/x^11/(b*x+a)^3

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maxima [B]  time = 0.65, size = 675, normalized size = 3.21 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b^{10}}{4 \, a^{10}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{11}}{4 \, a^{11}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b^{9}}{4 \, a^{9} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{10}}{4 \, a^{10} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{8}}{4 \, a^{10} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{9}}{4 \, a^{11} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{7}}{4 \, a^{9} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{8}}{4 \, a^{10} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{6}}{4 \, a^{8} x^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{7}}{4 \, a^{9} x^{4}} + \frac {209 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{5}}{840 \, a^{7} x^{5}} - \frac {329 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{6}}{1320 \, a^{8} x^{5}} - \frac {41 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{4}}{168 \, a^{6} x^{6}} + \frac {65 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{5}}{264 \, a^{7} x^{6}} + \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{3}}{56 \, a^{5} x^{7}} - \frac {21 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{4}}{88 \, a^{6} x^{7}} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{2}}{24 \, a^{4} x^{8}} + \frac {59 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{3}}{264 \, a^{5} x^{8}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b}{6 \, a^{3} x^{9}} - \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{2}}{66 \, a^{4} x^{9}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B}{10 \, a^{2} x^{10}} + \frac {17 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b}{110 \, a^{3} x^{10}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{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)^(3/2)/x^12,x, algorithm="maxima")

[Out]

1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*B*b^10/a^10 - 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*A*b^11/a^11 + 1/4*(b^2*x
^2 + 2*a*b*x + a^2)^(3/2)*B*b^9/(a^9*x) - 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*A*b^10/(a^10*x) - 1/4*(b^2*x^2 +
2*a*b*x + a^2)^(5/2)*B*b^8/(a^10*x^2) + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^9/(a^11*x^2) + 1/4*(b^2*x^2 +
2*a*b*x + a^2)^(5/2)*B*b^7/(a^9*x^3) - 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^8/(a^10*x^3) - 1/4*(b^2*x^2 +
2*a*b*x + a^2)^(5/2)*B*b^6/(a^8*x^4) + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^7/(a^9*x^4) + 209/840*(b^2*x^2
+ 2*a*b*x + a^2)^(5/2)*B*b^5/(a^7*x^5) - 329/1320*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^6/(a^8*x^5) - 41/168*(b^
2*x^2 + 2*a*b*x + a^2)^(5/2)*B*b^4/(a^6*x^6) + 65/264*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^5/(a^7*x^6) + 13/56*
(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*b^3/(a^5*x^7) - 21/88*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^4/(a^6*x^7) - 5/24
*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*b^2/(a^4*x^8) + 59/264*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^3/(a^5*x^8) + 1/
6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*b/(a^3*x^9) - 13/66*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^2/(a^4*x^9) - 1/10
*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B/(a^2*x^10) + 17/110*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b/(a^3*x^10) - 1/11*(
b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A/(a^2*x^11)

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mupad [B]  time = 1.16, size = 196, normalized size = 0.93 \begin {gather*} -\frac {\left (\frac {B\,a^3}{10}+\frac {3\,A\,b\,a^2}{10}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^{10}\,\left (a+b\,x\right )}-\frac {\left (\frac {A\,b^3}{8}+\frac {3\,B\,a\,b^2}{8}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^8\,\left (a+b\,x\right )}-\frac {A\,a^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{11\,x^{11}\,\left (a+b\,x\right )}-\frac {B\,b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{7\,x^7\,\left (a+b\,x\right )}-\frac {a\,b\,\left (A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{3\,x^9\,\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^12,x)

[Out]

- (((B*a^3)/10 + (3*A*a^2*b)/10)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(x^10*(a + b*x)) - (((A*b^3)/8 + (3*B*a*b^2)
/8)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(x^8*(a + b*x)) - (A*a^3*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(11*x^11*(a + b
*x)) - (B*b^3*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(7*x^7*(a + b*x)) - (a*b*(A*b + B*a)*(a^2 + b^2*x^2 + 2*a*b*x)^
(1/2))/(3*x^9*(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^{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)**(3/2)/x**12,x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(3/2)/x**12, x)

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