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

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

[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))

________________________________________________________________________________________

Rubi [A]  time = 0.116621, antiderivative size = 304, normalized size of antiderivative = 1., 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} \[ -\frac{a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{12 x^{12} (a+b x)}-\frac{5 a^3 b \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{11 x^{11} (a+b x)}-\frac{a^2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{x^{10} (a+b x)}-\frac{5 a b^3 \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{9 x^9 (a+b x)}-\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{8 x^8 (a+b x)}-\frac{a^5 A \sqrt{a^2+2 a b x+b^2 x^2}}{13 x^{13} (a+b x)}-\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)} \]

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 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]

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])

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*}

Mathematica [A]  time = 0.0417822, size = 125, normalized size = 0.41 \[ -\frac{\sqrt{(a+b x)^2} \left (6552 a^3 b^2 x^2 (10 A+11 B x)+8008 a^2 b^3 x^3 (9 A+10 B x)+2730 a^4 b x (11 A+12 B x)+462 a^5 (12 A+13 B x)+5005 a b^4 x^4 (8 A+9 B x)+1287 b^5 x^5 (7 A+8 B x)\right )}{72072 x^{13} (a+b x)} \]

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]

-(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)))/(72072*x^13*(
a + b*x))

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 140, normalized size = 0.5 \begin{align*} -{\frac{10296\,B{b}^{5}{x}^{6}+9009\,A{x}^{5}{b}^{5}+45045\,B{x}^{5}a{b}^{4}+40040\,A{x}^{4}a{b}^{4}+80080\,B{x}^{4}{a}^{2}{b}^{3}+72072\,A{x}^{3}{a}^{2}{b}^{3}+72072\,B{x}^{3}{a}^{3}{b}^{2}+65520\,A{x}^{2}{a}^{3}{b}^{2}+32760\,B{x}^{2}{a}^{4}b+30030\,A{a}^{4}bx+6006\,B{a}^{5}x+5544\,A{a}^{5}}{72072\,{x}^{13} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.57874, size = 289, normalized size = 0.95 \begin{align*} -\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{align*}

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}{x^{14}}\, dx \end{align*}

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)

________________________________________________________________________________________

Giac [A]  time = 1.31883, size = 298, normalized size = 0.98 \begin{align*} -\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{align*}

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