3.7.30 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{x^{14}} \, dx\)

Optimal. Leaf size=304 \[ -\frac {a^2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{x^{10} (a+b x)}-\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{8 x^8 (a+b x)}-\frac {5 a b^3 \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{9 x^9 (a+b x)}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac {a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{13 x^{13} (a+b x)}-\frac {a^4 \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{12 x^{12} (a+b x)}-\frac {5 a^3 b \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{11 x^{11} (a+b x)} \]

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Rubi [A]  time = 0.12, antiderivative size = 304, 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)}{12 x^{12} (a+b x)}-\frac {5 a^3 b \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{11 x^{11} (a+b x)}-\frac {a^2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{x^{10} (a+b x)}-\frac {5 a b^3 \sqrt {a^2+2 a b x+b^2 x^2} (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)}{8 x^8 (a+b x)}-\frac {a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{13 x^{13} (a+b x)}-\frac {b^5 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)^(5/2))/x^14,x]

[Out]

-(a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*x^13*(a + b*x)) - (a^4*(5*A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(12*x^12*(a + b*x)) - (5*a^3*b*(2*A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*x^11*(a + b*x)) - (a^2*b^2*(A
*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(x^10*(a + b*x)) - (5*a*b^3*(A*b + 2*a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^
2])/(9*x^9*(a + b*x)) - (b^4*(A*b + 5*a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*x^8*(a + b*x)) - (b^5*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 )^{5/2}}{x^{14}} \, 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^{14}} \, 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^{14}}+\frac {a^4 b^5 (5 A b+a B)}{x^{13}}+\frac {5 a^3 b^6 (2 A b+a B)}{x^{12}}+\frac {10 a^2 b^7 (A b+a B)}{x^{11}}+\frac {5 a b^8 (A b+2 a B)}{x^{10}}+\frac {b^9 (A b+5 a B)}{x^9}+\frac {b^{10} B}{x^8}\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}}{13 x^{13} (a+b x)}-\frac {a^4 (5 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{12 x^{12} (a+b x)}-\frac {5 a^3 b (2 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac {a^2 b^2 (A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{x^{10} (a+b x)}-\frac {5 a b^3 (A b+2 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac {b^4 (A b+5 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)}-\frac {b^5 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 = 0.04, size = 125, normalized size = 0.41 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (462 a^5 (12 A+13 B x)+2730 a^4 b x (11 A+12 B x)+6552 a^3 b^2 x^2 (10 A+11 B x)+8008 a^2 b^3 x^3 (9 A+10 B x)+5005 a b^4 x^4 (8 A+9 B x)+1287 b^5 x^5 (7 A+8 B x)\right )}{72072 x^{13} (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^14,x]

[Out]

-1/72072*(Sqrt[(a + b*x)^2]*(1287*b^5*x^5*(7*A + 8*B*x) + 5005*a*b^4*x^4*(8*A + 9*B*x) + 8008*a^2*b^3*x^3*(9*A
+ 10*B*x) + 6552*a^3*b^2*x^2*(10*A + 11*B*x) + 2730*a^4*b*x*(11*A + 12*B*x) + 462*a^5*(12*A + 13*B*x)))/(x^13
*(a + b*x))

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IntegrateAlgebraic [B]  time = 4.79, size = 1200, normalized size = 3.95 \begin {gather*} \frac {512 \sqrt {a^2+2 b x a+b^2 x^2} \left (-10296 B x^{18} b^{18}-9009 A x^{17} b^{18}-168597 a B x^{17} b^{17}-148148 a A x^{16} b^{17}-1300156 a^2 B x^{16} b^{16}-1147146 a^2 A x^{15} b^{16}-6271122 a^3 B x^{15} b^{15}-5555004 a^3 A x^{14} b^{15}-21189324 a^4 B x^{14} b^{14}-18841277 a^4 A x^{13} b^{14}-53225185 a^5 B x^{13} b^{13}-47500992 a^5 A x^{12} b^{13}-102918816 a^6 B x^{12} b^{12}-92174544 a^6 A x^{11} b^{12}-156478608 a^7 B x^{11} b^{11}-140618016 a^7 A x^{10} b^{11}-189384624 a^8 B x^{10} b^{10}-170742033 a^8 A x^9 b^{10}-183499173 a^9 B x^9 b^9-165951324 a^9 A x^8 b^9-142337052 a^{10} B x^8 b^8-129109442 a^{10} A x^7 b^8-87885226 a^{11} B x^7 b^7-79945404 a^{11} A x^6 b^7-42664908 a^{12} B x^6 b^6-38916339 a^{12} A x^5 b^6-15942927 a^{13} B x^5 b^5-14580104 a^{13} A x^4 b^5-4428424 a^{14} B x^4 b^4-4059972 a^{14} A x^3 b^4-861588 a^{15} B x^3 b^3-791784 a^{15} A x^2 b^3-104832 a^{16} B x^2 b^2-96558 a^{16} A x b^2-5544 a^{17} A b-6006 a^{17} B x b\right ) b^{12}+512 \sqrt {b^2} \left (10296 b^{18} B x^{19}+9009 A b^{18} x^{18}+178893 a b^{17} B x^{18}+157157 a A b^{17} x^{17}+1468753 a^2 b^{16} B x^{17}+1295294 a^2 A b^{16} x^{16}+7571278 a^3 b^{15} B x^{16}+6702150 a^3 A b^{15} x^{15}[phone] a^4 b^{14} B x^{15}[phone] a^4 A b^{14} x^{14}[phone] a^5 b^{13} B x^{14}[phone] a^5 A b^{13} x^{13}[phone] a^6 b^{12} B x^{13}[phone] a^6 A b^{12} x^{12}[phone] a^7 b^{11} B x^{12}[phone] a^7 A b^{11} x^{11}[phone] a^8 b^{10} B x^{11}[phone] a^8 A b^{10} x^{10}[phone] a^9 b^9 B x^{10}[phone] a^9 A b^9 x^9+325836225 a^{10} b^8 B x^9+295060766 a^{10} A b^8 x^8+230222278 a^{11} b^7 B x^8+209054846 a^{11} A b^7 x^7+130550134 a^{12} b^6 B x^7+118861743 a^{12} A b^6 x^6+58607835 a^{13} b^5 B x^6+53496443 a^{13} A b^5 x^5+20371351 a^{14} b^4 B x^5+18640076 a^{14} A b^4 x^4+5290012 a^{15} b^3 B x^4+4851756 a^{15} A b^3 x^3+966420 a^{16} b^2 B x^3+888342 a^{16} A b^2 x^2+110838 a^{17} b B x^2+102102 a^{17} A b x+6006 a^{18} B x+5544 a^{18} A\right ) b^{12}}{9009 \sqrt {b^2} \sqrt {a^2+2 b x a+b^2 x^2} \left (-4096 x^{12} b^{24}-49152 a x^{11} b^{23}-270336 a^2 x^{10} b^{22}-901120 a^3 x^9 b^{21}-2027520 a^4 x^8 b^{20}-3244032 a^5 x^7 b^{19}-3784704 a^6 x^6 b^{18}-3244032 a^7 x^5 b^{17}-2027520 a^8 x^4 b^{16}-901120 a^9 x^3 b^{15}-270336 a^{10} x^2 b^{14}-49152 a^{11} x b^{13}-4096 a^{12} b^{12}\right ) x^{13}+9009 \left (4096 x^{13} b^{26}+53248 a x^{12} b^{25}+319488 a^2 x^{11} b^{24}+1171456 a^3 x^{10} b^{23}+2928640 a^4 x^9 b^{22}+5271552 a^5 x^8 b^{21}+7028736 a^6 x^7 b^{20}+7028736 a^7 x^6 b^{19}+5271552 a^8 x^5 b^{18}+2928640 a^9 x^4 b^{17}+1171456 a^{10} x^3 b^{16}+319488 a^{11} x^2 b^{15}+53248 a^{12} x b^{14}+4096 a^{13} b^{13}\right ) x^{13}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^14,x]

[Out]

(512*b^12*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-5544*a^17*A*b - 96558*a^16*A*b^2*x - 6006*a^17*b*B*x - 791784*a^15*A
*b^3*x^2 - 104832*a^16*b^2*B*x^2 - 4059972*a^14*A*b^4*x^3 - 861588*a^15*b^3*B*x^3 - 14580104*a^13*A*b^5*x^4 -
4428424*a^14*b^4*B*x^4 - 38916339*a^12*A*b^6*x^5 - 15942927*a^13*b^5*B*x^5 - 79945404*a^11*A*b^7*x^6 - 4266490
8*a^12*b^6*B*x^6 - 129109442*a^10*A*b^8*x^7 - 87885226*a^11*b^7*B*x^7 - 165951324*a^9*A*b^9*x^8 - 142337052*a^
10*b^8*B*x^8 - 170742033*a^8*A*b^10*x^9 - 183499173*a^9*b^9*B*x^9 - 140618016*a^7*A*b^11*x^10 - 189384624*a^8*
b^10*B*x^10 - 92174544*a^6*A*b^12*x^11 - 156478608*a^7*b^11*B*x^11 - 47500992*a^5*A*b^13*x^12 - 102918816*a^6*
b^12*B*x^12 - 18841277*a^4*A*b^14*x^13 - 53225185*a^5*b^13*B*x^13 - 5555004*a^3*A*b^15*x^14 - 21189324*a^4*b^1
4*B*x^14 - 1147146*a^2*A*b^16*x^15 - 6271122*a^3*b^15*B*x^15 - 148148*a*A*b^17*x^16 - 1300156*a^2*b^16*B*x^16
- 9009*A*b^18*x^17 - 168597*a*b^17*B*x^17 - 10296*b^18*B*x^18) + 512*b^12*Sqrt[b^2]*(5544*a^18*A + 102102*a^17
*A*b*x + 6006*a^18*B*x + 888342*a^16*A*b^2*x^2 + 110838*a^17*b*B*x^2 + 4851756*a^15*A*b^3*x^3 + 966420*a^16*b^
2*B*x^3 + 18640076*a^14*A*b^4*x^4 + 5290012*a^15*b^3*B*x^4 + 53496443*a^13*A*b^5*x^5 + 20371351*a^14*b^4*B*x^5
+ 118861743*a^12*A*b^6*x^6 + 58607835*a^13*b^5*B*x^6 + 209054846*a^11*A*b^7*x^7 + 130550134*a^12*b^6*B*x^7 +
295060766*a^10*A*b^8*x^8 + 230222278*a^11*b^7*B*x^8 + 336693357*a^9*A*b^9*x^9 + 325836225*a^10*b^8*B*x^9 + 311
360049*a^8*A*b^10*x^10 + 372883797*a^9*b^9*B*x^10 + 232792560*a^7*A*b^11*x^11 + 345863232*a^8*b^10*B*x^11 + 13
9675536*a^6*A*b^12*x^12 + 259397424*a^7*b^11*B*x^12 + 66342269*a^5*A*b^13*x^13 + 156144001*a^6*b^12*B*x^13 + 2
4396281*a^4*A*b^14*x^14 + 74414509*a^5*b^13*B*x^14 + 6702150*a^3*A*b^15*x^15 + 27460446*a^4*b^14*B*x^15 + 1295
294*a^2*A*b^16*x^16 + 7571278*a^3*b^15*B*x^16 + 157157*a*A*b^17*x^17 + 1468753*a^2*b^16*B*x^17 + 9009*A*b^18*x
^18 + 178893*a*b^17*B*x^18 + 10296*b^18*B*x^19))/(9009*Sqrt[b^2]*x^13*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-4096*a^1
2*b^12 - 49152*a^11*b^13*x - 270336*a^10*b^14*x^2 - 901120*a^9*b^15*x^3 - 2027520*a^8*b^16*x^4 - 3244032*a^7*b
^17*x^5 - 3784704*a^6*b^18*x^6 - 3244032*a^5*b^19*x^7 - 2027520*a^4*b^20*x^8 - 901120*a^3*b^21*x^9 - 270336*a^
2*b^22*x^10 - 49152*a*b^23*x^11 - 4096*b^24*x^12) + 9009*x^13*(4096*a^13*b^13 + 53248*a^12*b^14*x + 319488*a^1
1*b^15*x^2 + 1171456*a^10*b^16*x^3 + 2928640*a^9*b^17*x^4 + 5271552*a^8*b^18*x^5 + 7028736*a^7*b^19*x^6 + 7028
736*a^6*b^20*x^7 + 5271552*a^5*b^21*x^8 + 2928640*a^4*b^22*x^9 + 1171456*a^3*b^23*x^10 + 319488*a^2*b^24*x^11
+ 53248*a*b^25*x^12 + 4096*b^26*x^13))

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fricas [A]  time = 0.39, size = 119, normalized size = 0.39 \begin {gather*} -\frac {10296 \, B b^{5} x^{6} + 5544 \, A a^{5} + 9009 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 40040 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 72072 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 32760 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 6006 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{72072 \, x^{13}} \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^14,x, algorithm="fricas")

[Out]

-1/72072*(10296*B*b^5*x^6 + 5544*A*a^5 + 9009*(5*B*a*b^4 + A*b^5)*x^5 + 40040*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 72
072*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 32760*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 6006*(B*a^5 + 5*A*a^4*b)*x)/x^13

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giac [A]  time = 0.20, size = 221, normalized size = 0.73 \begin {gather*} -\frac {{\left (13 \, B a b^{12} - 7 \, A b^{13}\right )} \mathrm {sgn}\left (b x + a\right )}{72072 \, a^{8}} - \frac {10296 \, B b^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) + 45045 \, B a b^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + 9009 \, A b^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 80080 \, B a^{2} b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 40040 \, A a b^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 72072 \, B a^{3} b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 72072 \, A a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 32760 \, B a^{4} b x^{2} \mathrm {sgn}\left (b x + a\right ) + 65520 \, A a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 6006 \, B a^{5} x \mathrm {sgn}\left (b x + a\right ) + 30030 \, A a^{4} b x \mathrm {sgn}\left (b x + a\right ) + 5544 \, A a^{5} \mathrm {sgn}\left (b x + a\right )}{72072 \, x^{13}} \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^14,x, algorithm="giac")

[Out]

-1/72072*(13*B*a*b^12 - 7*A*b^13)*sgn(b*x + a)/a^8 - 1/72072*(10296*B*b^5*x^6*sgn(b*x + a) + 45045*B*a*b^4*x^5
*sgn(b*x + a) + 9009*A*b^5*x^5*sgn(b*x + a) + 80080*B*a^2*b^3*x^4*sgn(b*x + a) + 40040*A*a*b^4*x^4*sgn(b*x + a
) + 72072*B*a^3*b^2*x^3*sgn(b*x + a) + 72072*A*a^2*b^3*x^3*sgn(b*x + a) + 32760*B*a^4*b*x^2*sgn(b*x + a) + 655
20*A*a^3*b^2*x^2*sgn(b*x + a) + 6006*B*a^5*x*sgn(b*x + a) + 30030*A*a^4*b*x*sgn(b*x + a) + 5544*A*a^5*sgn(b*x
+ a))/x^13

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maple [A]  time = 0.06, size = 140, normalized size = 0.46 \begin {gather*} -\frac {\left (10296 B \,b^{5} x^{6}+9009 A \,b^{5} x^{5}+45045 B a \,b^{4} x^{5}+40040 A a \,b^{4} x^{4}+80080 B \,a^{2} b^{3} x^{4}+72072 A \,a^{2} b^{3} x^{3}+72072 B \,a^{3} b^{2} x^{3}+65520 A \,a^{3} b^{2} x^{2}+32760 B \,a^{4} b \,x^{2}+30030 A \,a^{4} b x +6006 B \,a^{5} x +5544 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{72072 \left (b x +a \right )^{5} x^{13}} \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^14,x)

[Out]

-1/72072*(10296*B*b^5*x^6+9009*A*b^5*x^5+45045*B*a*b^4*x^5+40040*A*a*b^4*x^4+80080*B*a^2*b^3*x^4+72072*A*a^2*b
^3*x^3+72072*B*a^3*b^2*x^3+65520*A*a^3*b^2*x^2+32760*B*a^4*b*x^2+30030*A*a^4*b*x+6006*B*a^5*x+5544*A*a^5)*((b*
x+a)^2)^(5/2)/x^13/(b*x+a)^5

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maxima [B]  time = 0.74, size = 795, normalized size = 2.62 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{12}}{6 \, a^{12}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{13}}{6 \, a^{13}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{11}}{6 \, a^{11} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{12}}{6 \, a^{12} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{10}}{6 \, a^{12} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{11}}{6 \, a^{13} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{9}}{6 \, a^{11} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{10}}{6 \, a^{12} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{8}}{6 \, a^{10} x^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{9}}{6 \, a^{11} x^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{7}}{6 \, a^{9} x^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{8}}{6 \, a^{10} x^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{6}}{6 \, a^{8} x^{6}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{7}}{6 \, a^{9} x^{6}} + \frac {923 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{5}}{5544 \, a^{7} x^{7}} - \frac {1715 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{6}}{10296 \, a^{8} x^{7}} - \frac {131 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{4}}{792 \, a^{6} x^{8}} + \frac {1709 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{5}}{10296 \, a^{7} x^{8}} + \frac {16 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{3}}{99 \, a^{5} x^{9}} - \frac {211 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{4}}{1287 \, a^{6} x^{9}} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{2}}{33 \, a^{4} x^{10}} + \frac {68 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{3}}{429 \, a^{5} x^{10}} + \frac {17 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b}{132 \, a^{3} x^{11}} - \frac {251 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{2}}{1716 \, a^{4} x^{11}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B}{12 \, a^{2} x^{12}} + \frac {19 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b}{156 \, a^{3} x^{12}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A}{13 \, a^{2} x^{13}} \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^14,x, algorithm="maxima")

[Out]

1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*b^12/a^12 - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^13/a^13 + 1/6*(b^2*x
^2 + 2*a*b*x + a^2)^(5/2)*B*b^11/(a^11*x) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^12/(a^12*x) - 1/6*(b^2*x^2
+ 2*a*b*x + a^2)^(7/2)*B*b^10/(a^12*x^2) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^11/(a^13*x^2) + 1/6*(b^2*x
^2 + 2*a*b*x + a^2)^(7/2)*B*b^9/(a^11*x^3) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^10/(a^12*x^3) - 1/6*(b^2*
x^2 + 2*a*b*x + a^2)^(7/2)*B*b^8/(a^10*x^4) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^9/(a^11*x^4) + 1/6*(b^2*
x^2 + 2*a*b*x + a^2)^(7/2)*B*b^7/(a^9*x^5) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^8/(a^10*x^5) - 1/6*(b^2*x
^2 + 2*a*b*x + a^2)^(7/2)*B*b^6/(a^8*x^6) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^7/(a^9*x^6) + 923/5544*(b^
2*x^2 + 2*a*b*x + a^2)^(7/2)*B*b^5/(a^7*x^7) - 1715/10296*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^6/(a^8*x^7) - 13
1/792*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*b^4/(a^6*x^8) + 1709/10296*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^5/(a^7*
x^8) + 16/99*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*b^3/(a^5*x^9) - 211/1287*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^4/
(a^6*x^9) - 5/33*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*b^2/(a^4*x^10) + 68/429*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b
^3/(a^5*x^10) + 17/132*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*b/(a^3*x^11) - 251/1716*(b^2*x^2 + 2*a*b*x + a^2)^(7/
2)*A*b^2/(a^4*x^11) - 1/12*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B/(a^2*x^12) + 19/156*(b^2*x^2 + 2*a*b*x + a^2)^(7/
2)*A*b/(a^3*x^12) - 1/13*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A/(a^2*x^13)

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mupad [B]  time = 1.23, size = 284, normalized size = 0.93 \begin {gather*} -\frac {\left (\frac {B\,a^5}{12}+\frac {5\,A\,b\,a^4}{12}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^{12}\,\left (a+b\,x\right )}-\frac {\left (\frac {A\,b^5}{8}+\frac {5\,B\,a\,b^4}{8}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^8\,\left (a+b\,x\right )}-\frac {A\,a^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{13\,x^{13}\,\left (a+b\,x\right )}-\frac {B\,b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{7\,x^7\,\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}}{9\,x^9\,\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}}{11\,x^{11}\,\left (a+b\,x\right )}-\frac {a^2\,b^2\,\left (A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^{10}\,\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^14,x)

[Out]

- (((B*a^5)/12 + (5*A*a^4*b)/12)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(x^12*(a + b*x)) - (((A*b^5)/8 + (5*B*a*b^4)
/8)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(x^8*(a + b*x)) - (A*a^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(13*x^13*(a + b
*x)) - (B*b^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(7*x^7*(a + b*x)) - (5*a*b^3*(A*b + 2*B*a)*(a^2 + b^2*x^2 + 2*a
*b*x)^(1/2))/(9*x^9*(a + b*x)) - (5*a^3*b*(2*A*b + B*a)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(11*x^11*(a + b*x)) -
(a^2*b^2*(A*b + B*a)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(x^10*(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^{14}}\, 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**14,x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(5/2)/x**14, x)

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