3.7.29 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{x^{13}} \, dx\)

Optimal. Leaf size=306 \[ -\frac {10 a^2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{9 x^9 (a+b x)}-\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{7 x^7 (a+b x)}-\frac {5 a b^3 \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{8 x^8 (a+b x)}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac {a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{12 x^{12} (a+b x)}-\frac {a^4 \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{11 x^{11} (a+b x)}-\frac {a^3 b \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{2 x^{10} (a+b x)} \]

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Rubi [A]  time = 0.11, 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)}{11 x^{11} (a+b x)}-\frac {a^3 b \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{2 x^{10} (a+b x)}-\frac {10 a^2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{9 x^9 (a+b x)}-\frac {5 a b^3 \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{8 x^8 (a+b x)}-\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{7 x^7 (a+b x)}-\frac {a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{12 x^{12} (a+b x)}-\frac {b^5 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)^(5/2))/x^13,x]

[Out]

-(a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(12*x^12*(a + b*x)) - (a^4*(5*A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(11*x^11*(a + b*x)) - (a^3*b*(2*A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*x^10*(a + b*x)) - (10*a^2*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*b^3*(A*b + 2*a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(8*x^8*(a + b*x)) - (b^4*(A*b + 5*a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*x^7*(a + b*x)) - (b^5*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 )^{5/2}}{x^{13}} \, 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^{13}} \, 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^{13}}+\frac {a^4 b^5 (5 A b+a B)}{x^{12}}+\frac {5 a^3 b^6 (2 A b+a B)}{x^{11}}+\frac {10 a^2 b^7 (A b+a B)}{x^{10}}+\frac {5 a b^8 (A b+2 a B)}{x^9}+\frac {b^9 (A b+5 a B)}{x^8}+\frac {b^{10} B}{x^7}\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}}{12 x^{12} (a+b x)}-\frac {a^4 (5 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac {a^3 b (2 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{2 x^{10} (a+b x)}-\frac {10 a^2 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 b^3 (A b+2 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)}-\frac {b^4 (A b+5 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac {b^5 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 = 0.04, size = 125, normalized size = 0.41 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (42 a^5 (11 A+12 B x)+252 a^4 b x (10 A+11 B x)+616 a^3 b^2 x^2 (9 A+10 B x)+770 a^2 b^3 x^3 (8 A+9 B x)+495 a b^4 x^4 (7 A+8 B x)+132 b^5 x^5 (6 A+7 B x)\right )}{5544 x^{12} (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^13,x]

[Out]

-1/5544*(Sqrt[(a + b*x)^2]*(132*b^5*x^5*(6*A + 7*B*x) + 495*a*b^4*x^4*(7*A + 8*B*x) + 770*a^2*b^3*x^3*(8*A + 9
*B*x) + 616*a^3*b^2*x^2*(9*A + 10*B*x) + 252*a^4*b*x*(10*A + 11*B*x) + 42*a^5*(11*A + 12*B*x)))/(x^12*(a + b*x
))

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IntegrateAlgebraic [B]  time = 4.97, size = 1130, normalized size = 3.69 \begin {gather*} \frac {256 \sqrt {a^2+2 b x a+b^2 x^2} \left (-924 B x^{17} b^{17}-792 A x^{16} b^{17}-14124 a B x^{16} b^{16}-12177 a A x^{15} b^{16}-101310 a^2 B x^{15} b^{15}-87835 a^2 A x^{14} b^{15}-452650 a^3 B x^{14} b^{14}-394559 a^3 A x^{13} b^{14}-1410002 a^4 B x^{13} b^{13}-1235389 a^4 A x^{12} b^{13}-3246934 a^5 B x^{12} b^{12}-2858856 a^5 A x^{11} b^{12}-5717712 a^6 B x^{11} b^{11}-5057976 a^6 A x^{10} b^{11}-7854000 a^7 B x^{10} b^{10}-6978840 a^7 A x^9 b^{10}-8504760 a^8 B x^9 b^9-7589208 a^8 A x^8 b^9-7284024 a^9 B x^8 b^8-6526113 a^9 A x^7 b^8-4917726 a^{10} B x^7 b^7-4422891 a^{10} A x^6 b^7-2589642 a^{11} B x^6 b^6-2337511 a^{11} A x^5 b^6-1042690 a^{12} B x^5 b^5-944405 a^{12} A x^4 b^5-310310 a^{13} B x^4 b^4-281974 a^{13} A x^3 b^4-64372 a^{14} B x^3 b^3-58674 a^{14} A x^2 b^3-8316 a^{15} B x^2 b^2-7602 a^{15} A x b^2-462 a^{16} A b-504 a^{16} B x b\right ) b^{11}+256 \sqrt {b^2} \left (924 b^{17} B x^{18}+792 A b^{17} x^{17}+15048 a b^{16} B x^{17}+12969 a A b^{16} x^{16}+115434 a^2 b^{15} B x^{16}+100012 a^2 A b^{15} x^{15}+553960 a^3 b^{14} B x^{15}+482394 a^3 A b^{14} x^{14}+1862652 a^4 b^{13} B x^{14}+1629948 a^4 A b^{13} x^{13}+4656936 a^5 b^{12} B x^{13}+4094245 a^5 A b^{12} x^{12}+8964646 a^6 b^{11} B x^{12}+7916832 a^6 A b^{11} x^{11}[phone] a^7 b^{10} B x^{11}[phone] a^7 A b^{10} x^{10}[phone] a^8 b^9 B x^{10}[phone] a^8 A b^9 x^9+15788784 a^9 b^8 B x^9+14115321 a^9 A b^8 x^8+12201750 a^{10} b^7 B x^8+10949004 a^{10} A b^7 x^7+7507368 a^{11} b^6 B x^7+6760402 a^{11} A b^6 x^6+3632332 a^{12} b^5 B x^6+3281916 a^{12} A b^5 x^5+1353000 a^{13} b^4 B x^5+1226379 a^{13} A b^4 x^4+374682 a^{14} b^3 B x^4+340648 a^{14} A b^3 x^3+72688 a^{15} b^2 B x^3+66276 a^{15} A b^2 x^2+8820 a^{16} b B x^2+8064 a^{16} A b x+504 a^{17} B x+462 a^{17} A\right ) b^{11}}{693 \sqrt {b^2} \sqrt {a^2+2 b x a+b^2 x^2} \left (-2048 x^{11} b^{22}-22528 a x^{10} b^{21}-112640 a^2 x^9 b^{20}-337920 a^3 x^8 b^{19}-675840 a^4 x^7 b^{18}-946176 a^5 x^6 b^{17}-946176 a^6 x^5 b^{16}-675840 a^7 x^4 b^{15}-337920 a^8 x^3 b^{14}-112640 a^9 x^2 b^{13}-22528 a^{10} x b^{12}-2048 a^{11} b^{11}\right ) x^{12}+693 \left (2048 x^{12} b^{24}+24576 a x^{11} b^{23}+135168 a^2 x^{10} b^{22}+450560 a^3 x^9 b^{21}+1013760 a^4 x^8 b^{20}+1622016 a^5 x^7 b^{19}+1892352 a^6 x^6 b^{18}+1622016 a^7 x^5 b^{17}+1013760 a^8 x^4 b^{16}+450560 a^9 x^3 b^{15}+135168 a^{10} x^2 b^{14}+24576 a^{11} x b^{13}+2048 a^{12} b^{12}\right ) x^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^13,x]

[Out]

(256*b^11*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-462*a^16*A*b - 7602*a^15*A*b^2*x - 504*a^16*b*B*x - 58674*a^14*A*b^3
*x^2 - 8316*a^15*b^2*B*x^2 - 281974*a^13*A*b^4*x^3 - 64372*a^14*b^3*B*x^3 - 944405*a^12*A*b^5*x^4 - 310310*a^1
3*b^4*B*x^4 - 2337511*a^11*A*b^6*x^5 - 1042690*a^12*b^5*B*x^5 - 4422891*a^10*A*b^7*x^6 - 2589642*a^11*b^6*B*x^
6 - 6526113*a^9*A*b^8*x^7 - 4917726*a^10*b^7*B*x^7 - 7589208*a^8*A*b^9*x^8 - 7284024*a^9*b^8*B*x^8 - 6978840*a
^7*A*b^10*x^9 - 8504760*a^8*b^9*B*x^9 - 5057976*a^6*A*b^11*x^10 - 7854000*a^7*b^10*B*x^10 - 2858856*a^5*A*b^12
*x^11 - 5717712*a^6*b^11*B*x^11 - 1235389*a^4*A*b^13*x^12 - 3246934*a^5*b^12*B*x^12 - 394559*a^3*A*b^14*x^13 -
1410002*a^4*b^13*B*x^13 - 87835*a^2*A*b^15*x^14 - 452650*a^3*b^14*B*x^14 - 12177*a*A*b^16*x^15 - 101310*a^2*b
^15*B*x^15 - 792*A*b^17*x^16 - 14124*a*b^16*B*x^16 - 924*b^17*B*x^17) + 256*b^11*Sqrt[b^2]*(462*a^17*A + 8064*
a^16*A*b*x + 504*a^17*B*x + 66276*a^15*A*b^2*x^2 + 8820*a^16*b*B*x^2 + 340648*a^14*A*b^3*x^3 + 72688*a^15*b^2*
B*x^3 + 1226379*a^13*A*b^4*x^4 + 374682*a^14*b^3*B*x^4 + 3281916*a^12*A*b^5*x^5 + 1353000*a^13*b^4*B*x^5 + 676
0402*a^11*A*b^6*x^6 + 3632332*a^12*b^5*B*x^6 + 10949004*a^10*A*b^7*x^7 + 7507368*a^11*b^6*B*x^7 + 14115321*a^9
*A*b^8*x^8 + 12201750*a^10*b^7*B*x^8 + 14568048*a^8*A*b^9*x^9 + 15788784*a^9*b^8*B*x^9 + 12036816*a^7*A*b^10*x
^10 + 16358760*a^8*b^9*B*x^10 + 7916832*a^6*A*b^11*x^11 + 13571712*a^7*b^10*B*x^11 + 4094245*a^5*A*b^12*x^12 +
8964646*a^6*b^11*B*x^12 + 1629948*a^4*A*b^13*x^13 + 4656936*a^5*b^12*B*x^13 + 482394*a^3*A*b^14*x^14 + 186265
2*a^4*b^13*B*x^14 + 100012*a^2*A*b^15*x^15 + 553960*a^3*b^14*B*x^15 + 12969*a*A*b^16*x^16 + 115434*a^2*b^15*B*
x^16 + 792*A*b^17*x^17 + 15048*a*b^16*B*x^17 + 924*b^17*B*x^18))/(693*Sqrt[b^2]*x^12*Sqrt[a^2 + 2*a*b*x + b^2*
x^2]*(-2048*a^11*b^11 - 22528*a^10*b^12*x - 112640*a^9*b^13*x^2 - 337920*a^8*b^14*x^3 - 675840*a^7*b^15*x^4 -
946176*a^6*b^16*x^5 - 946176*a^5*b^17*x^6 - 675840*a^4*b^18*x^7 - 337920*a^3*b^19*x^8 - 112640*a^2*b^20*x^9 -
22528*a*b^21*x^10 - 2048*b^22*x^11) + 693*x^12*(2048*a^12*b^12 + 24576*a^11*b^13*x + 135168*a^10*b^14*x^2 + 45
0560*a^9*b^15*x^3 + 1013760*a^8*b^16*x^4 + 1622016*a^7*b^17*x^5 + 1892352*a^6*b^18*x^6 + 1622016*a^5*b^19*x^7
+ 1013760*a^4*b^20*x^8 + 450560*a^3*b^21*x^9 + 135168*a^2*b^22*x^10 + 24576*a*b^23*x^11 + 2048*b^24*x^12))

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fricas [A]  time = 0.41, size = 119, normalized size = 0.39 \begin {gather*} -\frac {924 \, B b^{5} x^{6} + 462 \, A a^{5} + 792 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 3465 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 6160 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 2772 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 504 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{5544 \, x^{12}} \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^13,x, algorithm="fricas")

[Out]

-1/5544*(924*B*b^5*x^6 + 462*A*a^5 + 792*(5*B*a*b^4 + A*b^5)*x^5 + 3465*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 6160*(B*
a^3*b^2 + A*a^2*b^3)*x^3 + 2772*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 504*(B*a^5 + 5*A*a^4*b)*x)/x^12

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giac [A]  time = 0.22, size = 221, normalized size = 0.72 \begin {gather*} \frac {{\left (2 \, B a b^{11} - A b^{12}\right )} \mathrm {sgn}\left (b x + a\right )}{5544 \, a^{7}} - \frac {924 \, B b^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) + 3960 \, B a b^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + 792 \, A b^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 6930 \, B a^{2} b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 3465 \, A a b^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 6160 \, B a^{3} b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 6160 \, A a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 2772 \, B a^{4} b x^{2} \mathrm {sgn}\left (b x + a\right ) + 5544 \, A a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 504 \, B a^{5} x \mathrm {sgn}\left (b x + a\right ) + 2520 \, A a^{4} b x \mathrm {sgn}\left (b x + a\right ) + 462 \, A a^{5} \mathrm {sgn}\left (b x + a\right )}{5544 \, x^{12}} \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^13,x, algorithm="giac")

[Out]

1/5544*(2*B*a*b^11 - A*b^12)*sgn(b*x + a)/a^7 - 1/5544*(924*B*b^5*x^6*sgn(b*x + a) + 3960*B*a*b^4*x^5*sgn(b*x
+ a) + 792*A*b^5*x^5*sgn(b*x + a) + 6930*B*a^2*b^3*x^4*sgn(b*x + a) + 3465*A*a*b^4*x^4*sgn(b*x + a) + 6160*B*a
^3*b^2*x^3*sgn(b*x + a) + 6160*A*a^2*b^3*x^3*sgn(b*x + a) + 2772*B*a^4*b*x^2*sgn(b*x + a) + 5544*A*a^3*b^2*x^2
*sgn(b*x + a) + 504*B*a^5*x*sgn(b*x + a) + 2520*A*a^4*b*x*sgn(b*x + a) + 462*A*a^5*sgn(b*x + a))/x^12

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maple [A]  time = 0.06, size = 140, normalized size = 0.46 \begin {gather*} -\frac {\left (924 B \,b^{5} x^{6}+792 A \,b^{5} x^{5}+3960 B a \,b^{4} x^{5}+3465 A a \,b^{4} x^{4}+6930 B \,a^{2} b^{3} x^{4}+6160 A \,a^{2} b^{3} x^{3}+6160 B \,a^{3} b^{2} x^{3}+5544 A \,a^{3} b^{2} x^{2}+2772 B \,a^{4} b \,x^{2}+2520 A \,a^{4} b x +504 B \,a^{5} x +462 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{5544 \left (b x +a \right )^{5} x^{12}} \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^13,x)

[Out]

-1/5544*(924*B*b^5*x^6+792*A*b^5*x^5+3960*B*a*b^4*x^5+3465*A*a*b^4*x^4+6930*B*a^2*b^3*x^4+6160*A*a^2*b^3*x^3+6
160*B*a^3*b^2*x^3+5544*A*a^3*b^2*x^2+2772*B*a^4*b*x^2+2520*A*a^4*b*x+504*B*a^5*x+462*A*a^5)*((b*x+a)^2)^(5/2)/
x^12/(b*x+a)^5

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maxima [B]  time = 0.83, size = 735, normalized size = 2.40 \begin {gather*} -\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{11}}{6 \, a^{11}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{12}}{6 \, a^{12}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{10}}{6 \, a^{10} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{11}}{6 \, a^{11} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{9}}{6 \, a^{11} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{10}}{6 \, a^{12} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{8}}{6 \, a^{10} x^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{9}}{6 \, a^{11} x^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{7}}{6 \, a^{9} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{8}}{6 \, a^{10} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{6}}{6 \, a^{8} x^{5}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{7}}{6 \, a^{9} x^{5}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{5}}{6 \, a^{7} x^{6}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{6}}{6 \, a^{8} x^{6}} - \frac {461 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{4}}{2772 \, a^{6} x^{7}} + \frac {923 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{5}}{5544 \, a^{7} x^{7}} + \frac {65 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{3}}{396 \, a^{5} x^{8}} - \frac {131 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{4}}{792 \, a^{6} x^{8}} - \frac {31 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{2}}{198 \, a^{4} x^{9}} + \frac {16 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{3}}{99 \, a^{5} x^{9}} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b}{22 \, a^{3} x^{10}} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{2}}{33 \, a^{4} x^{10}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B}{11 \, a^{2} x^{11}} + \frac {17 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b}{132 \, a^{3} x^{11}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A}{12 \, a^{2} x^{12}} \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^13,x, algorithm="maxima")

[Out]

-1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*b^11/a^11 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^12/a^12 - 1/6*(b^2*
x^2 + 2*a*b*x + a^2)^(5/2)*B*b^10/(a^10*x) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^11/(a^11*x) + 1/6*(b^2*x^
2 + 2*a*b*x + a^2)^(7/2)*B*b^9/(a^11*x^2) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^10/(a^12*x^2) - 1/6*(b^2*x
^2 + 2*a*b*x + a^2)^(7/2)*B*b^8/(a^10*x^3) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^9/(a^11*x^3) + 1/6*(b^2*x
^2 + 2*a*b*x + a^2)^(7/2)*B*b^7/(a^9*x^4) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^8/(a^10*x^4) - 1/6*(b^2*x^
2 + 2*a*b*x + a^2)^(7/2)*B*b^6/(a^8*x^5) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^7/(a^9*x^5) + 1/6*(b^2*x^2
+ 2*a*b*x + a^2)^(7/2)*B*b^5/(a^7*x^6) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^6/(a^8*x^6) - 461/2772*(b^2*x
^2 + 2*a*b*x + a^2)^(7/2)*B*b^4/(a^6*x^7) + 923/5544*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^5/(a^7*x^7) + 65/396*
(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*b^3/(a^5*x^8) - 131/792*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^4/(a^6*x^8) - 31
/198*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*b^2/(a^4*x^9) + 16/99*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^3/(a^5*x^9) +
3/22*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*b/(a^3*x^10) - 5/33*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^2/(a^4*x^10) -
1/11*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B/(a^2*x^11) + 17/132*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b/(a^3*x^11) - 1
/12*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A/(a^2*x^12)

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mupad [B]  time = 2.19, size = 284, normalized size = 0.93 \begin {gather*} -\frac {\left (\frac {B\,a^5}{11}+\frac {5\,A\,b\,a^4}{11}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^{11}\,\left (a+b\,x\right )}-\frac {\left (\frac {A\,b^5}{7}+\frac {5\,B\,a\,b^4}{7}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^7\,\left (a+b\,x\right )}-\frac {A\,a^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{12\,x^{12}\,\left (a+b\,x\right )}-\frac {B\,b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{6\,x^6\,\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}}{8\,x^8\,\left (a+b\,x\right )}-\frac {a^3\,b\,\left (2\,A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,x^{10}\,\left (a+b\,x\right )}-\frac {10\,a^2\,b^2\,\left (A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{9\,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)^(5/2))/x^13,x)

[Out]

- (((B*a^5)/11 + (5*A*a^4*b)/11)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(x^11*(a + b*x)) - (((A*b^5)/7 + (5*B*a*b^4)
/7)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(x^7*(a + b*x)) - (A*a^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(12*x^12*(a + b
*x)) - (B*b^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(6*x^6*(a + b*x)) - (5*a*b^3*(A*b + 2*B*a)*(a^2 + b^2*x^2 + 2*a
*b*x)^(1/2))/(8*x^8*(a + b*x)) - (a^3*b*(2*A*b + B*a)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(2*x^10*(a + b*x)) - (1
0*a^2*b^2*(A*b + B*a)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(9*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 {5}{2}}}{x^{13}}\, 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**13,x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(5/2)/x**13, x)

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