3.702 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{x^{13}} \, dx\)

Optimal. Leaf size=306 \[ -\frac{a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{11 x^{11} (a+b x)}-\frac{a^3 b \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{2 x^{10} (a+b x)}-\frac{10 a^2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{9 x^9 (a+b x)}-\frac{5 a b^3 \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{8 x^8 (a+b x)}-\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{7 x^7 (a+b x)}-\frac{a^5 A \sqrt{a^2+2 a b x+b^2 x^2}}{12 x^{12} (a+b x)}-\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)} \]

[Out]

-(a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(12*x^12*(a + b*x)) - (a^4*(5*A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(11*x^11*(a + b*x)) - (a^3*b*(2*A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*x^10*(a + b*x)) - (10*a^2*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*b^3*(A*b + 2*a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(8*x^8*(a + b*x)) - (b^4*(A*b + 5*a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*x^7*(a + b*x)) - (b^5*B*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(6*x^6*(a + b*x))

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Rubi [A]  time = 0.113304, 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} \[ -\frac{a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{11 x^{11} (a+b x)}-\frac{a^3 b \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{2 x^{10} (a+b x)}-\frac{10 a^2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{9 x^9 (a+b x)}-\frac{5 a b^3 \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{8 x^8 (a+b x)}-\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{7 x^7 (a+b x)}-\frac{a^5 A \sqrt{a^2+2 a b x+b^2 x^2}}{12 x^{12} (a+b x)}-\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^13,x]

[Out]

-(a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(12*x^12*(a + b*x)) - (a^4*(5*A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
/(11*x^11*(a + b*x)) - (a^3*b*(2*A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(2*x^10*(a + b*x)) - (10*a^2*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*b^3*(A*b + 2*a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(8*x^8*(a + b*x)) - (b^4*(A*b + 5*a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*x^7*(a + b*x)) - (b^5*B*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(6*x^6*(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^{13}} \, 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^{13}} \, 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^{13}}+\frac{a^4 b^5 (5 A b+a B)}{x^{12}}+\frac{5 a^3 b^6 (2 A b+a B)}{x^{11}}+\frac{10 a^2 b^7 (A b+a B)}{x^{10}}+\frac{5 a b^8 (A b+2 a B)}{x^9}+\frac{b^9 (A b+5 a B)}{x^8}+\frac{b^{10} B}{x^7}\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}}{12 x^{12} (a+b x)}-\frac{a^4 (5 A b+a B) \sqrt{a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac{a^3 b (2 A b+a B) \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^{10} (a+b x)}-\frac{10 a^2 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 b^3 (A b+2 a B) \sqrt{a^2+2 a b x+b^2 x^2}}{8 x^8 (a+b x)}-\frac{b^4 (A b+5 a B) \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)}-\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.0379048, size = 125, normalized size = 0.41 \[ -\frac{\sqrt{(a+b x)^2} \left (616 a^3 b^2 x^2 (9 A+10 B x)+770 a^2 b^3 x^3 (8 A+9 B x)+252 a^4 b x (10 A+11 B x)+42 a^5 (11 A+12 B x)+495 a b^4 x^4 (7 A+8 B x)+132 b^5 x^5 (6 A+7 B x)\right )}{5544 x^{12} (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^13,x]

[Out]

-(Sqrt[(a + b*x)^2]*(132*b^5*x^5*(6*A + 7*B*x) + 495*a*b^4*x^4*(7*A + 8*B*x) + 770*a^2*b^3*x^3*(8*A + 9*B*x) +
616*a^3*b^2*x^2*(9*A + 10*B*x) + 252*a^4*b*x*(10*A + 11*B*x) + 42*a^5*(11*A + 12*B*x)))/(5544*x^12*(a + b*x))

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Maple [A]  time = 0.006, size = 140, normalized size = 0.5 \begin{align*} -{\frac{924\,B{b}^{5}{x}^{6}+792\,A{x}^{5}{b}^{5}+3960\,B{x}^{5}a{b}^{4}+3465\,A{x}^{4}a{b}^{4}+6930\,B{x}^{4}{a}^{2}{b}^{3}+6160\,A{x}^{3}{a}^{2}{b}^{3}+6160\,B{x}^{3}{a}^{3}{b}^{2}+5544\,A{x}^{2}{a}^{3}{b}^{2}+2772\,B{x}^{2}{a}^{4}b+2520\,A{a}^{4}bx+504\,B{a}^{5}x+462\,A{a}^{5}}{5544\,{x}^{12} \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^13,x)

[Out]

-1/5544*(924*B*b^5*x^6+792*A*b^5*x^5+3960*B*a*b^4*x^5+3465*A*a*b^4*x^4+6930*B*a^2*b^3*x^4+6160*A*a^2*b^3*x^3+6
160*B*a^3*b^2*x^3+5544*A*a^3*b^2*x^2+2772*B*a^4*b*x^2+2520*A*a^4*b*x+504*B*a^5*x+462*A*a^5)*((b*x+a)^2)^(5/2)/
x^12/(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^13,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.61413, size = 277, normalized size = 0.91 \begin{align*} -\frac{924 \, B b^{5} x^{6} + 462 \, A a^{5} + 792 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 3465 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 6160 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 2772 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 504 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{5544 \, x^{12}} \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^13,x, algorithm="fricas")

[Out]

-1/5544*(924*B*b^5*x^6 + 462*A*a^5 + 792*(5*B*a*b^4 + A*b^5)*x^5 + 3465*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 6160*(B*
a^3*b^2 + A*a^2*b^3)*x^3 + 2772*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 504*(B*a^5 + 5*A*a^4*b)*x)/x^12

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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^{13}}\, 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**13,x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(5/2)/x**13, x)

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Giac [A]  time = 1.17574, size = 298, normalized size = 0.97 \begin{align*} \frac{{\left (2 \, B a b^{11} - A b^{12}\right )} \mathrm{sgn}\left (b x + a\right )}{5544 \, a^{7}} - \frac{924 \, B b^{5} x^{6} \mathrm{sgn}\left (b x + a\right ) + 3960 \, B a b^{4} x^{5} \mathrm{sgn}\left (b x + a\right ) + 792 \, A b^{5} x^{5} \mathrm{sgn}\left (b x + a\right ) + 6930 \, B a^{2} b^{3} x^{4} \mathrm{sgn}\left (b x + a\right ) + 3465 \, A a b^{4} x^{4} \mathrm{sgn}\left (b x + a\right ) + 6160 \, B a^{3} b^{2} x^{3} \mathrm{sgn}\left (b x + a\right ) + 6160 \, A a^{2} b^{3} x^{3} \mathrm{sgn}\left (b x + a\right ) + 2772 \, B a^{4} b x^{2} \mathrm{sgn}\left (b x + a\right ) + 5544 \, A a^{3} b^{2} x^{2} \mathrm{sgn}\left (b x + a\right ) + 504 \, B a^{5} x \mathrm{sgn}\left (b x + a\right ) + 2520 \, A a^{4} b x \mathrm{sgn}\left (b x + a\right ) + 462 \, A a^{5} \mathrm{sgn}\left (b x + a\right )}{5544 \, x^{12}} \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^13,x, algorithm="giac")

[Out]

1/5544*(2*B*a*b^11 - A*b^12)*sgn(b*x + a)/a^7 - 1/5544*(924*B*b^5*x^6*sgn(b*x + a) + 3960*B*a*b^4*x^5*sgn(b*x
+ a) + 792*A*b^5*x^5*sgn(b*x + a) + 6930*B*a^2*b^3*x^4*sgn(b*x + a) + 3465*A*a*b^4*x^4*sgn(b*x + a) + 6160*B*a
^3*b^2*x^3*sgn(b*x + a) + 6160*A*a^2*b^3*x^3*sgn(b*x + a) + 2772*B*a^4*b*x^2*sgn(b*x + a) + 5544*A*a^3*b^2*x^2
*sgn(b*x + a) + 504*B*a^5*x*sgn(b*x + a) + 2520*A*a^4*b*x*sgn(b*x + a) + 462*A*a^5*sgn(b*x + a))/x^12