3.7.27 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{x^{11}} \, dx\)

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

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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)}{9 x^9 (a+b x)}-\frac {5 a^3 b \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{8 x^8 (a+b x)}-\frac {10 a^2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (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)}{6 x^6 (a+b x)}-\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{5 x^5 (a+b x)}-\frac {a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{10 x^{10} (a+b x)}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (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^11,x]

[Out]

-(a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(10*x^10*(a + b*x)) - (a^4*(5*A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(9*x^9*(a + b*x)) - (5*a^3*b*(2*A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(8*x^8*(a + b*x)) - (10*a^2*b^2*(A*
b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*x^7*(a + b*x)) - (5*a*b^3*(A*b + 2*a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^
2])/(6*x^6*(a + b*x)) - (b^4*(A*b + 5*a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*x^5*(a + b*x)) - (b^5*B*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(4*x^4*(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^{11}} \, 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^{11}} \, 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^{11}}+\frac {a^4 b^5 (5 A b+a B)}{x^{10}}+\frac {5 a^3 b^6 (2 A b+a B)}{x^9}+\frac {10 a^2 b^7 (A b+a B)}{x^8}+\frac {5 a b^8 (A b+2 a B)}{x^7}+\frac {b^9 (A b+5 a B)}{x^6}+\frac {b^{10} B}{x^5}\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}}{10 x^{10} (a+b x)}-\frac {a^4 (5 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac {5 a^3 b (2 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)}-\frac {10 a^2 b^2 (A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac {5 a b^3 (A b+2 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac {b^4 (A b+5 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (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 (28 a^5 (9 A+10 B x)+175 a^4 b x (8 A+9 B x)+450 a^3 b^2 x^2 (7 A+8 B x)+600 a^2 b^3 x^3 (6 A+7 B x)+420 a b^4 x^4 (5 A+6 B x)+126 b^5 x^5 (4 A+5 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)^(5/2))/x^11,x]

[Out]

-1/2520*(Sqrt[(a + b*x)^2]*(126*b^5*x^5*(4*A + 5*B*x) + 420*a*b^4*x^4*(5*A + 6*B*x) + 600*a^2*b^3*x^3*(6*A + 7
*B*x) + 450*a^3*b^2*x^2*(7*A + 8*B*x) + 175*a^4*b*x*(8*A + 9*B*x) + 28*a^5*(9*A + 10*B*x)))/(x^10*(a + b*x))

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IntegrateAlgebraic [B]  time = 3.80, size = 990, normalized size = 3.24 \begin {gather*} \frac {64 \sqrt {a^2+2 b x a+b^2 x^2} \left (-630 B x^{15} b^{15}-504 A x^{14} b^{15}-8190 a B x^{14} b^{14}-6636 a A x^{13} b^{14}-49560 a^2 B x^{13} b^{13}-40644 a^2 A x^{12} b^{13}-185040 a^3 B x^{12} b^{12}-153486 a^3 A x^{11} b^{12}-476235 a^4 B x^{11} b^{11}-399254 a^4 A x^{10} b^{11}-893755 a^5 B x^{10} b^{10}-756756 a^5 A x^9 b^{10}-1261260 a^6 B x^9 b^9-1077804 a^6 A x^8 b^9-1359540 a^7 B x^8 b^8-1171716 a^7 A x^7 b^8-1124760 a^8 B x^7 b^7-977004 a^8 A x^6 b^7-710640 a^9 B x^6 b^6-621756 a^9 A x^5 b^6-337500 a^{10} B x^5 b^5-297252 a^{10} A x^4 b^5-116820 a^{11} B x^4 b^4-103518 a^{11} A x^3 b^4-27855 a^{12} B x^3 b^3-24822 a^{12} A x^2 b^3-4095 a^{13} B x^2 b^2-3668 a^{13} A x b^2-252 a^{14} A b-280 a^{14} B x b\right ) b^9+64 \sqrt {b^2} \left (630 b^{15} B x^{16}+504 A b^{15} x^{15}+8820 a b^{14} B x^{15}+7140 a A b^{14} x^{14}+57750 a^2 b^{13} B x^{14}+47280 a^2 A b^{13} x^{13}+234600 a^3 b^{12} B x^{13}+194130 a^3 A b^{12} x^{12}+661275 a^4 b^{11} B x^{12}+552740 a^4 A b^{11} x^{11}+1369990 a^5 b^{10} B x^{11}+1156010 a^5 A b^{10} x^{10}+2155015 a^6 b^9 B x^{10}+1834560 a^6 A b^9 x^9+2620800 a^7 b^8 B x^9+2249520 a^7 A b^8 x^8+2484300 a^8 b^7 B x^8+2148720 a^8 A b^7 x^7+1835400 a^9 b^6 B x^7+1598760 a^9 A b^6 x^6+1048140 a^{10} b^5 B x^6+919008 a^{10} A b^5 x^5+454320 a^{11} b^4 B x^5+400770 a^{11} A b^4 x^4+144675 a^{12} b^3 B x^4+128340 a^{12} A b^3 x^3+31950 a^{13} b^2 B x^3+28490 a^{13} A b^2 x^2+4375 a^{14} b B x^2+3920 a^{14} A b x+280 a^{15} B x+252 a^{15} 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)^(5/2))/x^11,x]

[Out]

(64*b^9*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-252*a^14*A*b - 3668*a^13*A*b^2*x - 280*a^14*b*B*x - 24822*a^12*A*b^3*x
^2 - 4095*a^13*b^2*B*x^2 - 103518*a^11*A*b^4*x^3 - 27855*a^12*b^3*B*x^3 - 297252*a^10*A*b^5*x^4 - 116820*a^11*
b^4*B*x^4 - 621756*a^9*A*b^6*x^5 - 337500*a^10*b^5*B*x^5 - 977004*a^8*A*b^7*x^6 - 710640*a^9*b^6*B*x^6 - 11717
16*a^7*A*b^8*x^7 - 1124760*a^8*b^7*B*x^7 - 1077804*a^6*A*b^9*x^8 - 1359540*a^7*b^8*B*x^8 - 756756*a^5*A*b^10*x
^9 - 1261260*a^6*b^9*B*x^9 - 399254*a^4*A*b^11*x^10 - 893755*a^5*b^10*B*x^10 - 153486*a^3*A*b^12*x^11 - 476235
*a^4*b^11*B*x^11 - 40644*a^2*A*b^13*x^12 - 185040*a^3*b^12*B*x^12 - 6636*a*A*b^14*x^13 - 49560*a^2*b^13*B*x^13
- 504*A*b^15*x^14 - 8190*a*b^14*B*x^14 - 630*b^15*B*x^15) + 64*b^9*Sqrt[b^2]*(252*a^15*A + 3920*a^14*A*b*x +
280*a^15*B*x + 28490*a^13*A*b^2*x^2 + 4375*a^14*b*B*x^2 + 128340*a^12*A*b^3*x^3 + 31950*a^13*b^2*B*x^3 + 40077
0*a^11*A*b^4*x^4 + 144675*a^12*b^3*B*x^4 + 919008*a^10*A*b^5*x^5 + 454320*a^11*b^4*B*x^5 + 1598760*a^9*A*b^6*x
^6 + 1048140*a^10*b^5*B*x^6 + 2148720*a^8*A*b^7*x^7 + 1835400*a^9*b^6*B*x^7 + 2249520*a^7*A*b^8*x^8 + 2484300*
a^8*b^7*B*x^8 + 1834560*a^6*A*b^9*x^9 + 2620800*a^7*b^8*B*x^9 + 1156010*a^5*A*b^10*x^10 + 2155015*a^6*b^9*B*x^
10 + 552740*a^4*A*b^11*x^11 + 1369990*a^5*b^10*B*x^11 + 194130*a^3*A*b^12*x^12 + 661275*a^4*b^11*B*x^12 + 4728
0*a^2*A*b^13*x^13 + 234600*a^3*b^12*B*x^13 + 7140*a*A*b^14*x^14 + 57750*a^2*b^13*B*x^14 + 504*A*b^15*x^15 + 88
20*a*b^14*B*x^15 + 630*b^15*B*x^16))/(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 = 0.41, size = 119, normalized size = 0.39 \begin {gather*} -\frac {630 \, B b^{5} x^{6} + 252 \, A a^{5} + 504 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 2100 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 3600 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 1575 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 280 \, {\left (B a^{5} + 5 \, A a^{4} 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)^(5/2)/x^11,x, algorithm="fricas")

[Out]

-1/2520*(630*B*b^5*x^6 + 252*A*a^5 + 504*(5*B*a*b^4 + A*b^5)*x^5 + 2100*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 3600*(B*
a^3*b^2 + A*a^2*b^3)*x^3 + 1575*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 280*(B*a^5 + 5*A*a^4*b)*x)/x^10

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giac [A]  time = 0.22, size = 221, normalized size = 0.72 \begin {gather*} \frac {{\left (5 \, B a b^{9} - 2 \, A b^{10}\right )} \mathrm {sgn}\left (b x + a\right )}{2520 \, a^{5}} - \frac {630 \, B b^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) + 2520 \, B a b^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + 504 \, A b^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 4200 \, B a^{2} b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 2100 \, A a b^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 3600 \, B a^{3} b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 3600 \, A a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 1575 \, B a^{4} b x^{2} \mathrm {sgn}\left (b x + a\right ) + 3150 \, A a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 280 \, B a^{5} x \mathrm {sgn}\left (b x + a\right ) + 1400 \, A a^{4} b x \mathrm {sgn}\left (b x + a\right ) + 252 \, A a^{5} \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)^(5/2)/x^11,x, algorithm="giac")

[Out]

1/2520*(5*B*a*b^9 - 2*A*b^10)*sgn(b*x + a)/a^5 - 1/2520*(630*B*b^5*x^6*sgn(b*x + a) + 2520*B*a*b^4*x^5*sgn(b*x
+ a) + 504*A*b^5*x^5*sgn(b*x + a) + 4200*B*a^2*b^3*x^4*sgn(b*x + a) + 2100*A*a*b^4*x^4*sgn(b*x + a) + 3600*B*
a^3*b^2*x^3*sgn(b*x + a) + 3600*A*a^2*b^3*x^3*sgn(b*x + a) + 1575*B*a^4*b*x^2*sgn(b*x + a) + 3150*A*a^3*b^2*x^
2*sgn(b*x + a) + 280*B*a^5*x*sgn(b*x + a) + 1400*A*a^4*b*x*sgn(b*x + a) + 252*A*a^5*sgn(b*x + a))/x^10

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maple [A]  time = 0.05, size = 140, normalized size = 0.46 \begin {gather*} -\frac {\left (630 B \,b^{5} x^{6}+504 A \,b^{5} x^{5}+2520 B a \,b^{4} x^{5}+2100 A a \,b^{4} x^{4}+4200 B \,a^{2} b^{3} x^{4}+3600 A \,a^{2} b^{3} x^{3}+3600 B \,a^{3} b^{2} x^{3}+3150 A \,a^{3} b^{2} x^{2}+1575 B \,a^{4} b \,x^{2}+1400 A \,a^{4} b x +280 B \,a^{5} x +252 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{2520 \left (b x +a \right )^{5} 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)^(5/2)/x^11,x)

[Out]

-1/2520*(630*B*b^5*x^6+504*A*b^5*x^5+2520*B*a*b^4*x^5+2100*A*a*b^4*x^4+4200*B*a^2*b^3*x^4+3600*A*a^2*b^3*x^3+3
600*B*a^3*b^2*x^3+3150*A*a^3*b^2*x^2+1575*B*a^4*b*x^2+1400*A*a^4*b*x+280*B*a^5*x+252*A*a^5)*((b*x+a)^2)^(5/2)/
x^10/(b*x+a)^5

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maxima [B]  time = 0.66, size = 615, normalized size = 2.01 \begin {gather*} -\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{9}}{6 \, a^{9}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{10}}{6 \, a^{10}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{8}}{6 \, a^{8} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{9}}{6 \, a^{9} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{7}}{6 \, a^{9} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{8}}{6 \, a^{10} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{6}}{6 \, a^{8} x^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{7}}{6 \, a^{9} x^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{5}}{6 \, a^{7} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{6}}{6 \, a^{8} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{4}}{6 \, a^{6} x^{5}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{5}}{6 \, a^{7} x^{5}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{3}}{6 \, a^{5} x^{6}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{4}}{6 \, a^{6} x^{6}} - \frac {83 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{2}}{504 \, a^{4} x^{7}} + \frac {209 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{3}}{1260 \, a^{5} x^{7}} + \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b}{72 \, a^{3} x^{8}} - \frac {29 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{2}}{180 \, a^{4} x^{8}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B}{9 \, a^{2} x^{9}} + \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b}{90 \, a^{3} x^{9}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{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)^(5/2)/x^11,x, algorithm="maxima")

[Out]

-1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*b^9/a^9 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^10/a^10 - 1/6*(b^2*x^
2 + 2*a*b*x + a^2)^(5/2)*B*b^8/(a^8*x) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^9/(a^9*x) + 1/6*(b^2*x^2 + 2*
a*b*x + a^2)^(7/2)*B*b^7/(a^9*x^2) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^8/(a^10*x^2) - 1/6*(b^2*x^2 + 2*a
*b*x + a^2)^(7/2)*B*b^6/(a^8*x^3) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^7/(a^9*x^3) + 1/6*(b^2*x^2 + 2*a*b
*x + a^2)^(7/2)*B*b^5/(a^7*x^4) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^6/(a^8*x^4) - 1/6*(b^2*x^2 + 2*a*b*x
+ a^2)^(7/2)*B*b^4/(a^6*x^5) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^5/(a^7*x^5) + 1/6*(b^2*x^2 + 2*a*b*x +
a^2)^(7/2)*B*b^3/(a^5*x^6) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^4/(a^6*x^6) - 83/504*(b^2*x^2 + 2*a*b*x
+ a^2)^(7/2)*B*b^2/(a^4*x^7) + 209/1260*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^3/(a^5*x^7) + 11/72*(b^2*x^2 + 2*a
*b*x + a^2)^(7/2)*B*b/(a^3*x^8) - 29/180*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^2/(a^4*x^8) - 1/9*(b^2*x^2 + 2*a*
b*x + a^2)^(7/2)*B/(a^2*x^9) + 13/90*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b/(a^3*x^9) - 1/10*(b^2*x^2 + 2*a*b*x +
a^2)^(7/2)*A/(a^2*x^10)

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mupad [B]  time = 1.31, size = 283, normalized size = 0.92 \begin {gather*} -\frac {\left (\frac {B\,a^5}{9}+\frac {5\,A\,b\,a^4}{9}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^9\,\left (a+b\,x\right )}-\frac {\left (\frac {A\,b^5}{5}+B\,a\,b^4\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^5\,\left (a+b\,x\right )}-\frac {A\,a^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{10\,x^{10}\,\left (a+b\,x\right )}-\frac {B\,b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,x^4\,\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}}{6\,x^6\,\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}}{8\,x^8\,\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}}{7\,x^7\,\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^11,x)

[Out]

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

[Out]

Integral((A + B*x)*((a + b*x)**2)**(5/2)/x**11, x)

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