3.7.3 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{3/2}}{x^6} \, dx\)

Optimal. Leaf size=77 \[ \frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (A b-a B)}{4 a^2 x^4}-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 a^2 x^5} \]

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Rubi [A]  time = 0.05, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {769, 646, 37} \begin {gather*} \frac {(a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (A b-a B)}{4 a^2 x^4}-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 a^2 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/x^6,x]

[Out]

((A*b - a*B)*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*a^2*x^4) - (A*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(5*a
^2*x^5)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 646

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rule 769

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(-2*c*(e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(2*c*d - b*e)^2), x] + Dist[(2*c*f -
b*g)/(2*c*d - b*e), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x]
&& EqQ[b^2 - 4*a*c, 0] && EqQ[m + 2*p + 3, 0] && NeQ[2*c*f - b*g, 0] && NeQ[2*c*d - b*e, 0]

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^6} \, dx &=-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 a^2 x^5}-\frac {\left (2 A b^2-2 a b B\right ) \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^5} \, dx}{2 a b}\\ &=-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 a^2 x^5}-\frac {\left (\left (2 A b^2-2 a b B\right ) \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {\left (a b+b^2 x\right )^3}{x^5} \, dx}{2 a b^3 \left (a b+b^2 x\right )}\\ &=\frac {(A b-a B) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{4 a^2 x^4}-\frac {A \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 a^2 x^5}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 84, normalized size = 1.09 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (a^3 (4 A+5 B x)+5 a^2 b x (3 A+4 B x)+10 a b^2 x^2 (2 A+3 B x)+10 b^3 x^3 (A+2 B x)\right )}{20 x^5 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/x^6,x]

[Out]

-1/20*(Sqrt[(a + b*x)^2]*(10*b^3*x^3*(A + 2*B*x) + 10*a*b^2*x^2*(2*A + 3*B*x) + 5*a^2*b*x*(3*A + 4*B*x) + a^3*
(4*A + 5*B*x)))/(x^5*(a + b*x))

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IntegrateAlgebraic [B]  time = 1.93, size = 544, normalized size = 7.06 \begin {gather*} \frac {4 b^4 \sqrt {a^2+2 a b x+b^2 x^2} \left (-4 a^7 A b-5 a^7 b B x-31 a^6 A b^2 x-40 a^6 b^2 B x^2-104 a^5 A b^3 x^2-140 a^5 b^3 B x^3-196 a^4 A b^4 x^3-280 a^4 b^4 B x^4-224 a^3 A b^5 x^4-345 a^3 b^5 B x^5-155 a^2 A b^6 x^5-260 a^2 b^6 B x^6-60 a A b^7 x^6-110 a b^7 B x^7-10 A b^8 x^7-20 b^8 B x^8\right )+4 \sqrt {b^2} b^4 \left (4 a^8 A+5 a^8 B x+35 a^7 A b x+45 a^7 b B x^2+135 a^6 A b^2 x^2+180 a^6 b^2 B x^3+300 a^5 A b^3 x^3+420 a^5 b^3 B x^4+420 a^4 A b^4 x^4+625 a^4 b^4 B x^5+379 a^3 A b^5 x^5+605 a^3 b^5 B x^6+215 a^2 A b^6 x^6+370 a^2 b^6 B x^7+70 a A b^7 x^7+130 a b^7 B x^8+10 A b^8 x^8+20 b^8 B x^9\right )}{5 \sqrt {b^2} x^5 \sqrt {a^2+2 a b x+b^2 x^2} \left (-16 a^4 b^4-64 a^3 b^5 x-96 a^2 b^6 x^2-64 a b^7 x^3-16 b^8 x^4\right )+5 x^5 \left (16 a^5 b^5+80 a^4 b^6 x+160 a^3 b^7 x^2+160 a^2 b^8 x^3+80 a b^9 x^4+16 b^{10} x^5\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/x^6,x]

[Out]

(4*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-4*a^7*A*b - 31*a^6*A*b^2*x - 5*a^7*b*B*x - 104*a^5*A*b^3*x^2 - 40*a^6*b
^2*B*x^2 - 196*a^4*A*b^4*x^3 - 140*a^5*b^3*B*x^3 - 224*a^3*A*b^5*x^4 - 280*a^4*b^4*B*x^4 - 155*a^2*A*b^6*x^5 -
345*a^3*b^5*B*x^5 - 60*a*A*b^7*x^6 - 260*a^2*b^6*B*x^6 - 10*A*b^8*x^7 - 110*a*b^7*B*x^7 - 20*b^8*B*x^8) + 4*b
^4*Sqrt[b^2]*(4*a^8*A + 35*a^7*A*b*x + 5*a^8*B*x + 135*a^6*A*b^2*x^2 + 45*a^7*b*B*x^2 + 300*a^5*A*b^3*x^3 + 18
0*a^6*b^2*B*x^3 + 420*a^4*A*b^4*x^4 + 420*a^5*b^3*B*x^4 + 379*a^3*A*b^5*x^5 + 625*a^4*b^4*B*x^5 + 215*a^2*A*b^
6*x^6 + 605*a^3*b^5*B*x^6 + 70*a*A*b^7*x^7 + 370*a^2*b^6*B*x^7 + 10*A*b^8*x^8 + 130*a*b^7*B*x^8 + 20*b^8*B*x^9
))/(5*Sqrt[b^2]*x^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-16*a^4*b^4 - 64*a^3*b^5*x - 96*a^2*b^6*x^2 - 64*a*b^7*x^3
- 16*b^8*x^4) + 5*x^5*(16*a^5*b^5 + 80*a^4*b^6*x + 160*a^3*b^7*x^2 + 160*a^2*b^8*x^3 + 80*a*b^9*x^4 + 16*b^10*
x^5))

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fricas [A]  time = 0.42, size = 73, normalized size = 0.95 \begin {gather*} -\frac {20 \, B b^{3} x^{4} + 4 \, A a^{3} + 10 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 20 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} + 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x}{20 \, x^{5}} \end {gather*}

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^6,x, algorithm="fricas")

[Out]

-1/20*(20*B*b^3*x^4 + 4*A*a^3 + 10*(3*B*a*b^2 + A*b^3)*x^3 + 20*(B*a^2*b + A*a*b^2)*x^2 + 5*(B*a^3 + 3*A*a^2*b
)*x)/x^5

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giac [B]  time = 0.16, size = 149, normalized size = 1.94 \begin {gather*} -\frac {{\left (5 \, B a b^{4} - A b^{5}\right )} \mathrm {sgn}\left (b x + a\right )}{20 \, a^{2}} - \frac {20 \, B b^{3} x^{4} \mathrm {sgn}\left (b x + a\right ) + 30 \, B a b^{2} x^{3} \mathrm {sgn}\left (b x + a\right ) + 10 \, A b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 20 \, B a^{2} b x^{2} \mathrm {sgn}\left (b x + a\right ) + 20 \, A a b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 5 \, B a^{3} x \mathrm {sgn}\left (b x + a\right ) + 15 \, A a^{2} b x \mathrm {sgn}\left (b x + a\right ) + 4 \, A a^{3} \mathrm {sgn}\left (b x + a\right )}{20 \, x^{5}} \end {gather*}

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^6,x, algorithm="giac")

[Out]

-1/20*(5*B*a*b^4 - A*b^5)*sgn(b*x + a)/a^2 - 1/20*(20*B*b^3*x^4*sgn(b*x + a) + 30*B*a*b^2*x^3*sgn(b*x + a) + 1
0*A*b^3*x^3*sgn(b*x + a) + 20*B*a^2*b*x^2*sgn(b*x + a) + 20*A*a*b^2*x^2*sgn(b*x + a) + 5*B*a^3*x*sgn(b*x + a)
+ 15*A*a^2*b*x*sgn(b*x + a) + 4*A*a^3*sgn(b*x + a))/x^5

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maple [A]  time = 0.05, size = 92, normalized size = 1.19 \begin {gather*} -\frac {\left (20 B \,b^{3} x^{4}+10 A \,b^{3} x^{3}+30 B a \,b^{2} x^{3}+20 A a \,b^{2} x^{2}+20 B \,a^{2} b \,x^{2}+15 A \,a^{2} b x +5 B \,a^{3} x +4 A \,a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{20 \left (b x +a \right )^{3} x^{5}} \end {gather*}

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^6,x)

[Out]

-1/20*(20*B*b^3*x^4+10*A*b^3*x^3+30*B*a*b^2*x^3+20*A*a*b^2*x^2+20*B*a^2*b*x^2+15*A*a^2*b*x+5*B*a^3*x+4*A*a^3)*
((b*x+a)^2)^(3/2)/x^5/(b*x+a)^3

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maxima [B]  time = 0.81, size = 315, normalized size = 4.09 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b^{4}}{4 \, a^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{5}}{4 \, a^{5}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b^{3}}{4 \, a^{3} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{4}}{4 \, a^{4} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{2}}{4 \, a^{4} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{3}}{4 \, a^{5} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b}{4 \, a^{3} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{2}}{4 \, a^{4} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B}{4 \, a^{2} x^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b}{4 \, a^{3} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A}{5 \, a^{2} x^{5}} \end {gather*}

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^6,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.17, size = 196, normalized size = 2.55 \begin {gather*} -\frac {\left (\frac {B\,a^3}{4}+\frac {3\,A\,b\,a^2}{4}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^4\,\left (a+b\,x\right )}-\frac {\left (\frac {A\,b^3}{2}+\frac {3\,B\,a\,b^2}{2}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^2\,\left (a+b\,x\right )}-\frac {A\,a^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{5\,x^5\,\left (a+b\,x\right )}-\frac {B\,b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x\,\left (a+b\,x\right )}-\frac {a\,b\,\left (A\,b+B\,a\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^3\,\left (a+b\,x\right )} \end {gather*}

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)^(3/2))/x^6,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{x^{6}}\, dx \end {gather*}

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**6,x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(3/2)/x**6, x)

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