3.7.7 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{3/2}}{x^{10}} \, dx\)

Optimal. Leaf size=210 \[ -\frac {a^2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+3 A b)}{8 x^8 (a+b x)}-\frac {3 a b \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{7 x^7 (a+b x)}-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (3 a B+A b)}{6 x^6 (a+b x)}-\frac {b^3 B \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac {a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)} \]

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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)}{8 x^8 (a+b x)}-\frac {3 a b \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{7 x^7 (a+b x)}-\frac {b^2 \sqrt {a^2+2 a b x+b^2 x^2} (3 a B+A b)}{6 x^6 (a+b x)}-\frac {a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac {b^3 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)^(3/2))/x^10,x]

[Out]

-(a^3*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*x^9*(a + b*x)) - (a^2*(3*A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(
8*x^8*(a + b*x)) - (3*a*b*(A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*x^7*(a + b*x)) - (b^2*(A*b + 3*a*B)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2])/(6*x^6*(a + b*x)) - (b^3*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 )^{3/2}}{x^{10}} \, 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^{10}} \, 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^{10}}+\frac {a^2 b^3 (3 A b+a B)}{x^9}+\frac {3 a b^4 (A b+a B)}{x^8}+\frac {b^5 (A b+3 a B)}{x^7}+\frac {b^6 B}{x^6}\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}}{9 x^9 (a+b x)}-\frac {a^2 (3 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)}-\frac {3 a b (A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac {b^2 (A b+3 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac {b^3 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 = 0.03, size = 87, normalized size = 0.41 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (35 a^3 (8 A+9 B x)+135 a^2 b x (7 A+8 B x)+180 a b^2 x^2 (6 A+7 B x)+84 b^3 x^3 (5 A+6 B x)\right )}{2520 x^9 (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^10,x]

[Out]

-1/2520*(Sqrt[(a + b*x)^2]*(84*b^3*x^3*(5*A + 6*B*x) + 180*a*b^2*x^2*(6*A + 7*B*x) + 135*a^2*b*x*(7*A + 8*B*x)
+ 35*a^3*(8*A + 9*B*x)))/(x^9*(a + b*x))

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IntegrateAlgebraic [B]  time = 2.90, size = 824, normalized size = 3.92 \begin {gather*} \frac {32 \sqrt {a^2+2 b x a+b^2 x^2} \left (-504 B x^{12} b^{12}-420 A x^{11} b^{12}-5292 a B x^{11} b^{11}-4440 a A x^{10} b^{11}-25272 a^2 B x^{10} b^{10}-21345 a^2 A x^9 b^{10}-72459 a^3 B x^9 b^9-61600 a^3 A x^8 b^9-138600 a^4 B x^8 b^8-118580 a^4 A x^7 b^8-185724 a^5 B x^7 b^7-159880 a^5 A x^6 b^7-177912 a^6 B x^6 b^6-154070 a^6 A x^5 b^6-121842 a^7 B x^5 b^5-106120 a^7 A x^4 b^5-58464 a^8 B x^4 b^4-51200 a^8 A x^3 b^4-18720 a^9 B x^3 b^3-16480 a^9 A x^2 b^3-3600 a^{10} B x^2 b^2-3185 a^{10} A x b^2-280 a^{11} A b-315 a^{11} B x b\right ) b^8+32 \sqrt {b^2} \left (504 b^{12} B x^{13}+420 A b^{12} x^{12}+5796 a b^{11} B x^{12}+4860 a A b^{11} x^{11}+30564 a^2 b^{10} B x^{11}+25785 a^2 A b^{10} x^{10}+97731 a^3 b^9 B x^{10}+82945 a^3 A b^9 x^9+211059 a^4 b^8 B x^9+180180 a^4 A b^8 x^8+324324 a^5 b^7 B x^8+278460 a^5 A b^7 x^7+363636 a^6 b^6 B x^7+313950 a^6 A b^6 x^6+299754 a^7 b^5 B x^6+260190 a^7 A b^5 x^5+180306 a^8 b^4 B x^5+157320 a^8 A b^4 x^4+77184 a^9 b^3 B x^4+67680 a^9 A b^3 x^3+22320 a^{10} b^2 B x^3+19665 a^{10} A b^2 x^2+3915 a^{11} b B x^2+3465 a^{11} A b x+315 a^{12} B x+280 a^{12} A\right ) b^8}{315 \sqrt {b^2} \sqrt {a^2+2 b x a+b^2 x^2} \left (-256 x^8 b^{16}-2048 a x^7 b^{15}-7168 a^2 x^6 b^{14}-14336 a^3 x^5 b^{13}-17920 a^4 x^4 b^{12}-14336 a^5 x^3 b^{11}-7168 a^6 x^2 b^{10}-2048 a^7 x b^9-256 a^8 b^8\right ) x^9+315 \left (256 x^9 b^{18}+2304 a x^8 b^{17}+9216 a^2 x^7 b^{16}+21504 a^3 x^6 b^{15}+32256 a^4 x^5 b^{14}+32256 a^5 x^4 b^{13}+21504 a^6 x^3 b^{12}+9216 a^7 x^2 b^{11}+2304 a^8 x b^{10}+256 a^9 b^9\right ) x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/x^10,x]

[Out]

(32*b^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-280*a^11*A*b - 3185*a^10*A*b^2*x - 315*a^11*b*B*x - 16480*a^9*A*b^3*x^
2 - 3600*a^10*b^2*B*x^2 - 51200*a^8*A*b^4*x^3 - 18720*a^9*b^3*B*x^3 - 106120*a^7*A*b^5*x^4 - 58464*a^8*b^4*B*x
^4 - 154070*a^6*A*b^6*x^5 - 121842*a^7*b^5*B*x^5 - 159880*a^5*A*b^7*x^6 - 177912*a^6*b^6*B*x^6 - 118580*a^4*A*
b^8*x^7 - 185724*a^5*b^7*B*x^7 - 61600*a^3*A*b^9*x^8 - 138600*a^4*b^8*B*x^8 - 21345*a^2*A*b^10*x^9 - 72459*a^3
*b^9*B*x^9 - 4440*a*A*b^11*x^10 - 25272*a^2*b^10*B*x^10 - 420*A*b^12*x^11 - 5292*a*b^11*B*x^11 - 504*b^12*B*x^
12) + 32*b^8*Sqrt[b^2]*(280*a^12*A + 3465*a^11*A*b*x + 315*a^12*B*x + 19665*a^10*A*b^2*x^2 + 3915*a^11*b*B*x^2
+ 67680*a^9*A*b^3*x^3 + 22320*a^10*b^2*B*x^3 + 157320*a^8*A*b^4*x^4 + 77184*a^9*b^3*B*x^4 + 260190*a^7*A*b^5*
x^5 + 180306*a^8*b^4*B*x^5 + 313950*a^6*A*b^6*x^6 + 299754*a^7*b^5*B*x^6 + 278460*a^5*A*b^7*x^7 + 363636*a^6*b
^6*B*x^7 + 180180*a^4*A*b^8*x^8 + 324324*a^5*b^7*B*x^8 + 82945*a^3*A*b^9*x^9 + 211059*a^4*b^8*B*x^9 + 25785*a^
2*A*b^10*x^10 + 97731*a^3*b^9*B*x^10 + 4860*a*A*b^11*x^11 + 30564*a^2*b^10*B*x^11 + 420*A*b^12*x^12 + 5796*a*b
^11*B*x^12 + 504*b^12*B*x^13))/(315*Sqrt[b^2]*x^9*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-256*a^8*b^8 - 2048*a^7*b^9*x
- 7168*a^6*b^10*x^2 - 14336*a^5*b^11*x^3 - 17920*a^4*b^12*x^4 - 14336*a^3*b^13*x^5 - 7168*a^2*b^14*x^6 - 2048
*a*b^15*x^7 - 256*b^16*x^8) + 315*x^9*(256*a^9*b^9 + 2304*a^8*b^10*x + 9216*a^7*b^11*x^2 + 21504*a^6*b^12*x^3
+ 32256*a^5*b^13*x^4 + 32256*a^4*b^14*x^5 + 21504*a^3*b^15*x^6 + 9216*a^2*b^16*x^7 + 2304*a*b^17*x^8 + 256*b^1
8*x^9))

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fricas [A]  time = 0.42, size = 73, normalized size = 0.35 \begin {gather*} -\frac {504 \, B b^{3} x^{4} + 280 \, A a^{3} + 420 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 1080 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} + 315 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2520 \, x^{9}} \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^10,x, algorithm="fricas")

[Out]

-1/2520*(504*B*b^3*x^4 + 280*A*a^3 + 420*(3*B*a*b^2 + A*b^3)*x^3 + 1080*(B*a^2*b + A*a*b^2)*x^2 + 315*(B*a^3 +
3*A*a^2*b)*x)/x^9

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giac [A]  time = 0.18, size = 149, normalized size = 0.71 \begin {gather*} -\frac {{\left (9 \, B a b^{8} - 5 \, A b^{9}\right )} \mathrm {sgn}\left (b x + a\right )}{2520 \, a^{6}} - \frac {504 \, B b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 1260 \, B a b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 420 \, A b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 1080 \, B a^{2} b x^{2} \mathrm {sgn}\left (b x + a\right ) + 1080 \, A a b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 315 \, B a^{3} x \mathrm {sgn}\left (b x + a\right ) + 945 \, A a^{2} b x \mathrm {sgn}\left (b x + a\right ) + 280 \, A a^{3} \mathrm {sgn}\left (b x + a\right )}{2520 \, x^{9}} \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^10,x, algorithm="giac")

[Out]

-1/2520*(9*B*a*b^8 - 5*A*b^9)*sgn(b*x + a)/a^6 - 1/2520*(504*B*b^3*x^4*sgn(b*x + a) + 1260*B*a*b^2*x^3*sgn(b*x
+ a) + 420*A*b^3*x^3*sgn(b*x + a) + 1080*B*a^2*b*x^2*sgn(b*x + a) + 1080*A*a*b^2*x^2*sgn(b*x + a) + 315*B*a^3
*x*sgn(b*x + a) + 945*A*a^2*b*x*sgn(b*x + a) + 280*A*a^3*sgn(b*x + a))/x^9

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maple [A]  time = 0.05, size = 92, normalized size = 0.44 \begin {gather*} -\frac {\left (504 B \,b^{3} x^{4}+420 A \,b^{3} x^{3}+1260 B a \,b^{2} x^{3}+1080 A a \,b^{2} x^{2}+1080 B \,a^{2} b \,x^{2}+945 A \,a^{2} b x +315 B \,a^{3} x +280 A \,a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{2520 \left (b x +a \right )^{3} x^{9}} \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^10,x)

[Out]

-1/2520*(504*B*b^3*x^4+420*A*b^3*x^3+1260*B*a*b^2*x^3+1080*A*a*b^2*x^2+1080*B*a^2*b*x^2+945*A*a^2*b*x+315*B*a^
3*x+280*A*a^3)*((b*x+a)^2)^(3/2)/x^9/(b*x+a)^3

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maxima [B]  time = 0.57, size = 555, normalized size = 2.64 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b^{8}}{4 \, a^{8}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{9}}{4 \, a^{9}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b^{7}}{4 \, a^{7} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{8}}{4 \, a^{8} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{6}}{4 \, a^{8} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{7}}{4 \, a^{9} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{5}}{4 \, a^{7} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{6}}{4 \, a^{8} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{4}}{4 \, a^{6} x^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{5}}{4 \, a^{7} x^{4}} + \frac {69 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{3}}{280 \, a^{5} x^{5}} - \frac {125 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{4}}{504 \, a^{6} x^{5}} - \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{2}}{56 \, a^{4} x^{6}} + \frac {121 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{3}}{504 \, a^{5} x^{6}} + \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b}{56 \, a^{3} x^{7}} - \frac {37 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{2}}{168 \, a^{4} x^{7}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B}{8 \, a^{2} x^{8}} + \frac {13 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b}{72 \, a^{3} x^{8}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A}{9 \, a^{2} x^{9}} \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^10,x, algorithm="maxima")

[Out]

1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*B*b^8/a^8 - 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*A*b^9/a^9 + 1/4*(b^2*x^2 +
2*a*b*x + a^2)^(3/2)*B*b^7/(a^7*x) - 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*A*b^8/(a^8*x) - 1/4*(b^2*x^2 + 2*a*b
*x + a^2)^(5/2)*B*b^6/(a^8*x^2) + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^7/(a^9*x^2) + 1/4*(b^2*x^2 + 2*a*b*x
+ a^2)^(5/2)*B*b^5/(a^7*x^3) - 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^6/(a^8*x^3) - 1/4*(b^2*x^2 + 2*a*b*x +
a^2)^(5/2)*B*b^4/(a^6*x^4) + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^5/(a^7*x^4) + 69/280*(b^2*x^2 + 2*a*b*x
+ a^2)^(5/2)*B*b^3/(a^5*x^5) - 125/504*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^4/(a^6*x^5) - 13/56*(b^2*x^2 + 2*a*
b*x + a^2)^(5/2)*B*b^2/(a^4*x^6) + 121/504*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^3/(a^5*x^6) + 11/56*(b^2*x^2 +
2*a*b*x + a^2)^(5/2)*B*b/(a^3*x^7) - 37/168*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^2/(a^4*x^7) - 1/8*(b^2*x^2 + 2
*a*b*x + a^2)^(5/2)*B/(a^2*x^8) + 13/72*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b/(a^3*x^8) - 1/9*(b^2*x^2 + 2*a*b*x
+ a^2)^(5/2)*A/(a^2*x^9)

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mupad [B]  time = 1.17, size = 196, normalized size = 0.93 \begin {gather*} -\frac {\left (\frac {B\,a^3}{8}+\frac {3\,A\,b\,a^2}{8}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^8\,\left (a+b\,x\right )}-\frac {\left (\frac {A\,b^3}{6}+\frac {B\,a\,b^2}{2}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^6\,\left (a+b\,x\right )}-\frac {A\,a^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{9\,x^9\,\left (a+b\,x\right )}-\frac {B\,b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{5\,x^5\,\left (a+b\,x\right )}-\frac {3\,a\,b\,\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)^(3/2))/x^10,x)

[Out]

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

[Out]

Integral((A + B*x)*((a + b*x)**2)**(3/2)/x**10, x)

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