3.701 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{x^{12}} \, dx\)

Optimal. Leaf size=306 \[ -\frac{a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{10 x^{10} (a+b x)}-\frac{5 a^3 b \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{9 x^9 (a+b x)}-\frac{5 a^2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{4 x^8 (a+b x)}-\frac{5 a b^3 \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{7 x^7 (a+b x)}-\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{6 x^6 (a+b x)}-\frac{a^5 A \sqrt{a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)} \]

[Out]

-(a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*x^11*(a + b*x)) - (a^4*(5*A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(10*x^10*(a + b*x)) - (5*a^3*b*(2*A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*x^9*(a + b*x)) - (5*a^2*b^2*(A
*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*x^8*(a + b*x)) - (5*a*b^3*(A*b + 2*a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(7*x^7*(a + b*x)) - (b^4*(A*b + 5*a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*x^6*(a + b*x)) - (b^5*B*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(5*x^5*(a + b*x))

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Rubi [A]  time = 0.117255, antiderivative size = 306, 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} \[ -\frac{a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{10 x^{10} (a+b x)}-\frac{5 a^3 b \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{9 x^9 (a+b x)}-\frac{5 a^2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{4 x^8 (a+b x)}-\frac{5 a b^3 \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{7 x^7 (a+b x)}-\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{6 x^6 (a+b x)}-\frac{a^5 A \sqrt{a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^12,x]

[Out]

-(a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*x^11*(a + b*x)) - (a^4*(5*A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(10*x^10*(a + b*x)) - (5*a^3*b*(2*A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*x^9*(a + b*x)) - (5*a^2*b^2*(A
*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*x^8*(a + b*x)) - (5*a*b^3*(A*b + 2*a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(7*x^7*(a + b*x)) - (b^4*(A*b + 5*a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*x^6*(a + b*x)) - (b^5*B*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(5*x^5*(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^{12}} \, 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^{12}} \, 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^{12}}+\frac{a^4 b^5 (5 A b+a B)}{x^{11}}+\frac{5 a^3 b^6 (2 A b+a B)}{x^{10}}+\frac{10 a^2 b^7 (A b+a B)}{x^9}+\frac{5 a b^8 (A b+2 a B)}{x^8}+\frac{b^9 (A b+5 a B)}{x^7}+\frac{b^{10} B}{x^6}\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}}{11 x^{11} (a+b x)}-\frac{a^4 (5 A b+a B) \sqrt{a^2+2 a b x+b^2 x^2}}{10 x^{10} (a+b x)}-\frac{5 a^3 b (2 A b+a B) \sqrt{a^2+2 a b x+b^2 x^2}}{9 x^9 (a+b x)}-\frac{5 a^2 b^2 (A b+a B) \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^8 (a+b x)}-\frac{5 a b^3 (A b+2 a B) \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac{b^4 (A b+5 a B) \sqrt{a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.0393652, size = 125, normalized size = 0.41 \[ -\frac{\sqrt{(a+b x)^2} \left (1925 a^3 b^2 x^2 (8 A+9 B x)+2475 a^2 b^3 x^3 (7 A+8 B x)+770 a^4 b x (9 A+10 B x)+126 a^5 (10 A+11 B x)+1650 a b^4 x^4 (6 A+7 B x)+462 b^5 x^5 (5 A+6 B x)\right )}{13860 x^{11} (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^12,x]

[Out]

-(Sqrt[(a + b*x)^2]*(462*b^5*x^5*(5*A + 6*B*x) + 1650*a*b^4*x^4*(6*A + 7*B*x) + 2475*a^2*b^3*x^3*(7*A + 8*B*x)
+ 1925*a^3*b^2*x^2*(8*A + 9*B*x) + 770*a^4*b*x*(9*A + 10*B*x) + 126*a^5*(10*A + 11*B*x)))/(13860*x^11*(a + b*
x))

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Maple [A]  time = 0.007, size = 140, normalized size = 0.5 \begin{align*} -{\frac{2772\,B{b}^{5}{x}^{6}+2310\,A{x}^{5}{b}^{5}+11550\,B{x}^{5}a{b}^{4}+9900\,A{x}^{4}a{b}^{4}+19800\,B{x}^{4}{a}^{2}{b}^{3}+17325\,A{x}^{3}{a}^{2}{b}^{3}+17325\,B{x}^{3}{a}^{3}{b}^{2}+15400\,A{x}^{2}{a}^{3}{b}^{2}+7700\,B{x}^{2}{a}^{4}b+6930\,A{a}^{4}bx+1386\,B{a}^{5}x+1260\,A{a}^{5}}{13860\,{x}^{11} \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^12,x)

[Out]

-1/13860*(2772*B*b^5*x^6+2310*A*b^5*x^5+11550*B*a*b^4*x^5+9900*A*a*b^4*x^4+19800*B*a^2*b^3*x^4+17325*A*a^2*b^3
*x^3+17325*B*a^3*b^2*x^3+15400*A*a^3*b^2*x^2+7700*B*a^4*b*x^2+6930*A*a^4*b*x+1386*B*a^5*x+1260*A*a^5)*((b*x+a)
^2)^(5/2)/x^11/(b*x+a)^5

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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^12,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.61718, size = 285, normalized size = 0.93 \begin{align*} -\frac{2772 \, B b^{5} x^{6} + 1260 \, A a^{5} + 2310 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 9900 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 17325 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 7700 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 1386 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{13860 \, x^{11}} \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^12,x, algorithm="fricas")

[Out]

-1/13860*(2772*B*b^5*x^6 + 1260*A*a^5 + 2310*(5*B*a*b^4 + A*b^5)*x^5 + 9900*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 1732
5*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 7700*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 1386*(B*a^5 + 5*A*a^4*b)*x)/x^11

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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^{12}}\, 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**12,x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(5/2)/x**12, x)

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Giac [A]  time = 1.19919, size = 298, normalized size = 0.97 \begin{align*} -\frac{{\left (11 \, B a b^{10} - 5 \, A b^{11}\right )} \mathrm{sgn}\left (b x + a\right )}{13860 \, a^{6}} - \frac{2772 \, B b^{5} x^{6} \mathrm{sgn}\left (b x + a\right ) + 11550 \, B a b^{4} x^{5} \mathrm{sgn}\left (b x + a\right ) + 2310 \, A b^{5} x^{5} \mathrm{sgn}\left (b x + a\right ) + 19800 \, B a^{2} b^{3} x^{4} \mathrm{sgn}\left (b x + a\right ) + 9900 \, A a b^{4} x^{4} \mathrm{sgn}\left (b x + a\right ) + 17325 \, B a^{3} b^{2} x^{3} \mathrm{sgn}\left (b x + a\right ) + 17325 \, A a^{2} b^{3} x^{3} \mathrm{sgn}\left (b x + a\right ) + 7700 \, B a^{4} b x^{2} \mathrm{sgn}\left (b x + a\right ) + 15400 \, A a^{3} b^{2} x^{2} \mathrm{sgn}\left (b x + a\right ) + 1386 \, B a^{5} x \mathrm{sgn}\left (b x + a\right ) + 6930 \, A a^{4} b x \mathrm{sgn}\left (b x + a\right ) + 1260 \, A a^{5} \mathrm{sgn}\left (b x + a\right )}{13860 \, x^{11}} \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^12,x, algorithm="giac")

[Out]

-1/13860*(11*B*a*b^10 - 5*A*b^11)*sgn(b*x + a)/a^6 - 1/13860*(2772*B*b^5*x^6*sgn(b*x + a) + 11550*B*a*b^4*x^5*
sgn(b*x + a) + 2310*A*b^5*x^5*sgn(b*x + a) + 19800*B*a^2*b^3*x^4*sgn(b*x + a) + 9900*A*a*b^4*x^4*sgn(b*x + a)
+ 17325*B*a^3*b^2*x^3*sgn(b*x + a) + 17325*A*a^2*b^3*x^3*sgn(b*x + a) + 7700*B*a^4*b*x^2*sgn(b*x + a) + 15400*
A*a^3*b^2*x^2*sgn(b*x + a) + 1386*B*a^5*x*sgn(b*x + a) + 6930*A*a^4*b*x*sgn(b*x + a) + 1260*A*a^5*sgn(b*x + a)
)/x^11