3.682 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^{3/2}}{x^{12}} \, dx\)

Optimal. Leaf size=210 \[ -\frac{a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{10 x^{10} (a+b x)}-\frac{a b \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{3 x^9 (a+b x)}-\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{8 x^8 (a+b x)}-\frac{a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.0786117, antiderivative size = 210, 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} \[ -\frac{a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{10 x^{10} (a+b x)}-\frac{a b \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{3 x^9 (a+b x)}-\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{8 x^8 (a+b x)}-\frac{a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{11 x^{11} (a+b x)}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/x^12,x]

[Out]

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

Mathematica [A]  time = 0.0291052, size = 87, normalized size = 0.41 \[ -\frac{\sqrt{(a+b x)^2} \left (308 a^2 b x (9 A+10 B x)+84 a^3 (10 A+11 B x)+385 a b^2 x^2 (8 A+9 B x)+165 b^3 x^3 (7 A+8 B x)\right )}{9240 x^{11} (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/x^12,x]

[Out]

-(Sqrt[(a + b*x)^2]*(165*b^3*x^3*(7*A + 8*B*x) + 385*a*b^2*x^2*(8*A + 9*B*x) + 308*a^2*b*x*(9*A + 10*B*x) + 84
*a^3*(10*A + 11*B*x)))/(9240*x^11*(a + b*x))

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Maple [A]  time = 0.007, size = 92, normalized size = 0.4 \begin{align*} -{\frac{1320\,B{x}^{4}{b}^{3}+1155\,A{b}^{3}{x}^{3}+3465\,B{x}^{3}a{b}^{2}+3080\,A{x}^{2}a{b}^{2}+3080\,B{x}^{2}{a}^{2}b+2772\,A{a}^{2}bx+924\,{a}^{3}Bx+840\,A{a}^{3}}{9240\,{x}^{11} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{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)^(3/2)/x^12,x)

[Out]

-1/9240*(1320*B*b^3*x^4+1155*A*b^3*x^3+3465*B*a*b^2*x^3+3080*A*a*b^2*x^2+3080*B*a^2*b*x^2+2772*A*a^2*b*x+924*B
*a^3*x+840*A*a^3)*((b*x+a)^2)^(3/2)/x^11/(b*x+a)^3

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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)^(3/2)/x^12,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.26959, size = 180, normalized size = 0.86 \begin{align*} -\frac{1320 \, B b^{3} x^{4} + 840 \, A a^{3} + 1155 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 3080 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 924 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{9240 \, 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)^(3/2)/x^12,x, algorithm="fricas")

[Out]

-1/9240*(1320*B*b^3*x^4 + 840*A*a^3 + 1155*(3*B*a*b^2 + A*b^3)*x^3 + 3080*(B*a^2*b + A*a*b^2)*x^2 + 924*(B*a^3
+ 3*A*a^2*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{3}{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)**(3/2)/x**12,x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(3/2)/x**12, x)

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Giac [A]  time = 1.21673, size = 201, normalized size = 0.96 \begin{align*} -\frac{{\left (11 \, B a b^{10} - 7 \, A b^{11}\right )} \mathrm{sgn}\left (b x + a\right )}{9240 \, a^{8}} - \frac{1320 \, B b^{3} x^{4} \mathrm{sgn}\left (b x + a\right ) + 3465 \, B a b^{2} x^{3} \mathrm{sgn}\left (b x + a\right ) + 1155 \, A b^{3} x^{3} \mathrm{sgn}\left (b x + a\right ) + 3080 \, B a^{2} b x^{2} \mathrm{sgn}\left (b x + a\right ) + 3080 \, A a b^{2} x^{2} \mathrm{sgn}\left (b x + a\right ) + 924 \, B a^{3} x \mathrm{sgn}\left (b x + a\right ) + 2772 \, A a^{2} b x \mathrm{sgn}\left (b x + a\right ) + 840 \, A a^{3} \mathrm{sgn}\left (b x + a\right )}{9240 \, 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)^(3/2)/x^12,x, algorithm="giac")

[Out]

-1/9240*(11*B*a*b^10 - 7*A*b^11)*sgn(b*x + a)/a^8 - 1/9240*(1320*B*b^3*x^4*sgn(b*x + a) + 3465*B*a*b^2*x^3*sgn
(b*x + a) + 1155*A*b^3*x^3*sgn(b*x + a) + 3080*B*a^2*b*x^2*sgn(b*x + a) + 3080*A*a*b^2*x^2*sgn(b*x + a) + 924*
B*a^3*x*sgn(b*x + a) + 2772*A*a^2*b*x*sgn(b*x + a) + 840*A*a^3*sgn(b*x + a))/x^11