3.699 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{x^{10}} \, dx\)

Optimal. Leaf size=304 \[ -\frac {5 a^2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{3 x^6 (a+b x)}-\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{4 x^4 (a+b x)}-\frac {a b^3 \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{x^5 (a+b x)}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac {a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac {a^4 \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{8 x^8 (a+b x)}-\frac {5 a^3 b \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{7 x^7 (a+b x)} \]

[Out]

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

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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} \[ -\frac {a^4 \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{8 x^8 (a+b x)}-\frac {5 a^3 b \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{7 x^7 (a+b x)}-\frac {5 a^2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{3 x^6 (a+b x)}-\frac {a b^3 \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{x^5 (a+b x)}-\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{4 x^4 (a+b x)}-\frac {a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^10,x]

[Out]

-(a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*x^9*(a + b*x)) - (a^4*(5*A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(
8*x^8*(a + b*x)) - (5*a^3*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^2*b^2*(A*b +
a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*x^6*(a + b*x)) - (a*b^3*(A*b + 2*a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(
x^5*(a + b*x)) - (b^4*(A*b + 5*a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*x^4*(a + b*x)) - (b^5*B*Sqrt[a^2 + 2*a*b
*x + b^2*x^2])/(3*x^3*(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^{10}} \, 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^{10}} \, 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^{10}}+\frac {a^4 b^5 (5 A b+a B)}{x^9}+\frac {5 a^3 b^6 (2 A b+a B)}{x^8}+\frac {10 a^2 b^7 (A b+a B)}{x^7}+\frac {5 a b^8 (A b+2 a B)}{x^6}+\frac {b^9 (A b+5 a B)}{x^5}+\frac {b^{10} B}{x^4}\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}}{9 x^9 (a+b x)}-\frac {a^4 (5 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)}-\frac {5 a^3 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^2 b^2 (A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^6 (a+b x)}-\frac {a b^3 (A b+2 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{x^5 (a+b x)}-\frac {b^4 (A b+5 a B) \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 125, normalized size = 0.41 \[ -\frac {\sqrt {(a+b x)^2} \left (7 a^5 (8 A+9 B x)+45 a^4 b x (7 A+8 B x)+120 a^3 b^2 x^2 (6 A+7 B x)+168 a^2 b^3 x^3 (5 A+6 B x)+126 a b^4 x^4 (4 A+5 B x)+42 b^5 x^5 (3 A+4 B x)\right )}{504 x^9 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^10,x]

[Out]

-1/504*(Sqrt[(a + b*x)^2]*(42*b^5*x^5*(3*A + 4*B*x) + 126*a*b^4*x^4*(4*A + 5*B*x) + 168*a^2*b^3*x^3*(5*A + 6*B
*x) + 120*a^3*b^2*x^2*(6*A + 7*B*x) + 45*a^4*b*x*(7*A + 8*B*x) + 7*a^5*(8*A + 9*B*x)))/(x^9*(a + b*x))

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fricas [A]  time = 0.86, size = 119, normalized size = 0.39 \[ -\frac {168 \, B b^{5} x^{6} + 56 \, A a^{5} + 126 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 504 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 840 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 360 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 63 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{504 \, x^{9}} \]

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^10,x, algorithm="fricas")

[Out]

-1/504*(168*B*b^5*x^6 + 56*A*a^5 + 126*(5*B*a*b^4 + A*b^5)*x^5 + 504*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 840*(B*a^3*
b^2 + A*a^2*b^3)*x^3 + 360*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 63*(B*a^5 + 5*A*a^4*b)*x)/x^9

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giac [A]  time = 0.17, size = 221, normalized size = 0.73 \[ -\frac {{\left (3 \, B a b^{8} - A b^{9}\right )} \mathrm {sgn}\left (b x + a\right )}{504 \, a^{4}} - \frac {168 \, B b^{5} x^{6} \mathrm {sgn}\left (b x + a\right ) + 630 \, B a b^{4} x^{5} \mathrm {sgn}\left (b x + a\right ) + 126 \, A b^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 1008 \, B a^{2} b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 504 \, A a b^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 840 \, B a^{3} b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 840 \, A a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 360 \, B a^{4} b x^{2} \mathrm {sgn}\left (b x + a\right ) + 720 \, A a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 63 \, B a^{5} x \mathrm {sgn}\left (b x + a\right ) + 315 \, A a^{4} b x \mathrm {sgn}\left (b x + a\right ) + 56 \, A a^{5} \mathrm {sgn}\left (b x + a\right )}{504 \, x^{9}} \]

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^10,x, algorithm="giac")

[Out]

-1/504*(3*B*a*b^8 - A*b^9)*sgn(b*x + a)/a^4 - 1/504*(168*B*b^5*x^6*sgn(b*x + a) + 630*B*a*b^4*x^5*sgn(b*x + a)
+ 126*A*b^5*x^5*sgn(b*x + a) + 1008*B*a^2*b^3*x^4*sgn(b*x + a) + 504*A*a*b^4*x^4*sgn(b*x + a) + 840*B*a^3*b^2
*x^3*sgn(b*x + a) + 840*A*a^2*b^3*x^3*sgn(b*x + a) + 360*B*a^4*b*x^2*sgn(b*x + a) + 720*A*a^3*b^2*x^2*sgn(b*x
+ a) + 63*B*a^5*x*sgn(b*x + a) + 315*A*a^4*b*x*sgn(b*x + a) + 56*A*a^5*sgn(b*x + a))/x^9

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maple [A]  time = 0.05, size = 140, normalized size = 0.46 \[ -\frac {\left (168 B \,b^{5} x^{6}+126 A \,b^{5} x^{5}+630 B a \,b^{4} x^{5}+504 A a \,b^{4} x^{4}+1008 B \,a^{2} b^{3} x^{4}+840 A \,a^{2} b^{3} x^{3}+840 B \,a^{3} b^{2} x^{3}+720 A \,a^{3} b^{2} x^{2}+360 B \,a^{4} b \,x^{2}+315 A \,a^{4} b x +63 B \,a^{5} x +56 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{504 \left (b x +a \right )^{5} x^{9}} \]

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^10,x)

[Out]

-1/504*(168*B*b^5*x^6+126*A*b^5*x^5+630*B*a*b^4*x^5+504*A*a*b^4*x^4+1008*B*a^2*b^3*x^4+840*A*a^2*b^3*x^3+840*B
*a^3*b^2*x^3+720*A*a^3*b^2*x^2+360*B*a^4*b*x^2+315*A*a^4*b*x+63*B*a^5*x+56*A*a^5)*((b*x+a)^2)^(5/2)/x^9/(b*x+a
)^5

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maxima [B]  time = 0.74, size = 555, normalized size = 1.83 \[ \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{8}}{6 \, a^{8}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{9}}{6 \, a^{9}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{7}}{6 \, a^{7} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{8}}{6 \, a^{8} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{6}}{6 \, a^{8} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{7}}{6 \, a^{9} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{5}}{6 \, a^{7} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{6}}{6 \, a^{8} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{4}}{6 \, a^{6} x^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{5}}{6 \, a^{7} x^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{3}}{6 \, a^{5} x^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{4}}{6 \, a^{6} x^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b^{2}}{6 \, a^{4} x^{6}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{3}}{6 \, a^{5} x^{6}} + \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b}{56 \, a^{3} x^{7}} - \frac {83 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{2}}{504 \, a^{4} x^{7}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B}{8 \, a^{2} x^{8}} + \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b}{72 \, a^{3} x^{8}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A}{9 \, a^{2} x^{9}} \]

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^10,x, algorithm="maxima")

[Out]

1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*b^8/a^8 - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^9/a^9 + 1/6*(b^2*x^2 +
2*a*b*x + a^2)^(5/2)*B*b^7/(a^7*x) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^8/(a^8*x) - 1/6*(b^2*x^2 + 2*a*b
*x + a^2)^(7/2)*B*b^6/(a^8*x^2) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^7/(a^9*x^2) + 1/6*(b^2*x^2 + 2*a*b*x
+ a^2)^(7/2)*B*b^5/(a^7*x^3) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^6/(a^8*x^3) - 1/6*(b^2*x^2 + 2*a*b*x +
a^2)^(7/2)*B*b^4/(a^6*x^4) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^5/(a^7*x^4) + 1/6*(b^2*x^2 + 2*a*b*x + a
^2)^(7/2)*B*b^3/(a^5*x^5) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^4/(a^6*x^5) - 1/6*(b^2*x^2 + 2*a*b*x + a^2
)^(7/2)*B*b^2/(a^4*x^6) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^3/(a^5*x^6) + 9/56*(b^2*x^2 + 2*a*b*x + a^2)
^(7/2)*B*b/(a^3*x^7) - 83/504*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^2/(a^4*x^7) - 1/8*(b^2*x^2 + 2*a*b*x + a^2)^
(7/2)*B/(a^2*x^8) + 11/72*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b/(a^3*x^8) - 1/9*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*
A/(a^2*x^9)

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mupad [B]  time = 1.30, size = 284, normalized size = 0.93 \[ -\frac {\left (\frac {B\,a^5}{8}+\frac {5\,A\,b\,a^4}{8}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^8\,\left (a+b\,x\right )}-\frac {\left (\frac {A\,b^5}{4}+\frac {5\,B\,a\,b^4}{4}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^4\,\left (a+b\,x\right )}-\frac {A\,a^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{9\,x^9\,\left (a+b\,x\right )}-\frac {B\,b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{3\,x^3\,\left (a+b\,x\right )}-\frac {a\,b^3\,\left (A\,b+2\,B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^5\,\left (a+b\,x\right )}-\frac {5\,a^3\,b\,\left (2\,A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{7\,x^7\,\left (a+b\,x\right )}-\frac {5\,a^2\,b^2\,\left (A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{3\,x^6\,\left (a+b\,x\right )} \]

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^10,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{x^{10}}\, dx \]

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**10,x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(5/2)/x**10, x)

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