3.6.91 \(\int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^7} \, dx\)

Optimal. Leaf size=114 \[ -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{5 x^5 (a+b x)}-\frac {a A \sqrt {a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac {b B \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)} \]

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Rubi [A]  time = 0.04, antiderivative size = 114, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {\sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{5 x^5 (a+b x)}-\frac {a A \sqrt {a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac {b B \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/x^7,x]

[Out]

-(a*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*x^6*(a + b*x)) - ((A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*x^5*(a
+ b*x)) - (b*B*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*x^4*(a + b*x))

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])

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]

Rubi steps

\begin {align*} \int \frac {(A+B x) \sqrt {a^2+2 a b x+b^2 x^2}}{x^7} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right ) (A+B x)}{x^7} \, dx}{a b+b^2 x}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a A b}{x^7}+\frac {b (A b+a B)}{x^6}+\frac {b^2 B}{x^5}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {a A \sqrt {a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac {(A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac {b B \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 49, normalized size = 0.43 \begin {gather*} -\frac {\sqrt {(a+b x)^2} (2 a (5 A+6 B x)+3 b x (4 A+5 B x))}{60 x^6 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/x^7,x]

[Out]

-1/60*(Sqrt[(a + b*x)^2]*(3*b*x*(4*A + 5*B*x) + 2*a*(5*A + 6*B*x)))/(x^6*(a + b*x))

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IntegrateAlgebraic [B]  time = 1.78, size = 518, normalized size = 4.54 \begin {gather*} \frac {8 b^5 \sqrt {a^2+2 a b x+b^2 x^2} \left (-10 a^6 A b-12 a^6 b B x-62 a^5 A b^2 x-75 a^5 b^2 B x^2-160 a^4 A b^3 x^2-195 a^4 b^3 B x^3-220 a^3 A b^4 x^3-270 a^3 b^4 B x^4-170 a^2 A b^5 x^4-210 a^2 b^5 B x^5-70 a A b^6 x^5-87 a b^6 B x^6-12 A b^7 x^6-15 b^7 B x^7\right )+8 \sqrt {b^2} b^5 \left (10 a^7 A+12 a^7 B x+72 a^6 A b x+87 a^6 b B x^2+222 a^5 A b^2 x^2+270 a^5 b^2 B x^3+380 a^4 A b^3 x^3+465 a^4 b^3 B x^4+390 a^3 A b^4 x^4+480 a^3 b^4 B x^5+240 a^2 A b^5 x^5+297 a^2 b^5 B x^6+82 a A b^6 x^6+102 a b^6 B x^7+12 A b^7 x^7+15 b^7 B x^8\right )}{15 \sqrt {b^2} x^6 \sqrt {a^2+2 a b x+b^2 x^2} \left (-32 a^5 b^5-160 a^4 b^6 x-320 a^3 b^7 x^2-320 a^2 b^8 x^3-160 a b^9 x^4-32 b^{10} x^5\right )+15 x^6 \left (32 a^6 b^6+192 a^5 b^7 x+480 a^4 b^8 x^2+640 a^3 b^9 x^3+480 a^2 b^{10} x^4+192 a b^{11} x^5+32 b^{12} x^6\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/x^7,x]

[Out]

(8*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-10*a^6*A*b - 62*a^5*A*b^2*x - 12*a^6*b*B*x - 160*a^4*A*b^3*x^2 - 75*a^5
*b^2*B*x^2 - 220*a^3*A*b^4*x^3 - 195*a^4*b^3*B*x^3 - 170*a^2*A*b^5*x^4 - 270*a^3*b^4*B*x^4 - 70*a*A*b^6*x^5 -
210*a^2*b^5*B*x^5 - 12*A*b^7*x^6 - 87*a*b^6*B*x^6 - 15*b^7*B*x^7) + 8*b^5*Sqrt[b^2]*(10*a^7*A + 72*a^6*A*b*x +
12*a^7*B*x + 222*a^5*A*b^2*x^2 + 87*a^6*b*B*x^2 + 380*a^4*A*b^3*x^3 + 270*a^5*b^2*B*x^3 + 390*a^3*A*b^4*x^4 +
465*a^4*b^3*B*x^4 + 240*a^2*A*b^5*x^5 + 480*a^3*b^4*B*x^5 + 82*a*A*b^6*x^6 + 297*a^2*b^5*B*x^6 + 12*A*b^7*x^7
+ 102*a*b^6*B*x^7 + 15*b^7*B*x^8))/(15*Sqrt[b^2]*x^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-32*a^5*b^5 - 160*a^4*b^6
*x - 320*a^3*b^7*x^2 - 320*a^2*b^8*x^3 - 160*a*b^9*x^4 - 32*b^10*x^5) + 15*x^6*(32*a^6*b^6 + 192*a^5*b^7*x + 4
80*a^4*b^8*x^2 + 640*a^3*b^9*x^3 + 480*a^2*b^10*x^4 + 192*a*b^11*x^5 + 32*b^12*x^6))

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fricas [A]  time = 0.40, size = 27, normalized size = 0.24 \begin {gather*} -\frac {15 \, B b x^{2} + 10 \, A a + 12 \, {\left (B a + A b\right )} x}{60 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*((b*x+a)^2)^(1/2)/x^7,x, algorithm="fricas")

[Out]

-1/60*(15*B*b*x^2 + 10*A*a + 12*(B*a + A*b)*x)/x^6

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giac [A]  time = 0.16, size = 77, normalized size = 0.68 \begin {gather*} \frac {{\left (3 \, B a b^{5} - 2 \, A b^{6}\right )} \mathrm {sgn}\left (b x + a\right )}{60 \, a^{5}} - \frac {15 \, B b x^{2} \mathrm {sgn}\left (b x + a\right ) + 12 \, B a x \mathrm {sgn}\left (b x + a\right ) + 12 \, A b x \mathrm {sgn}\left (b x + a\right ) + 10 \, A a \mathrm {sgn}\left (b x + a\right )}{60 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*((b*x+a)^2)^(1/2)/x^7,x, algorithm="giac")

[Out]

1/60*(3*B*a*b^5 - 2*A*b^6)*sgn(b*x + a)/a^5 - 1/60*(15*B*b*x^2*sgn(b*x + a) + 12*B*a*x*sgn(b*x + a) + 12*A*b*x
*sgn(b*x + a) + 10*A*a*sgn(b*x + a))/x^6

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maple [A]  time = 0.05, size = 44, normalized size = 0.39 \begin {gather*} -\frac {\left (15 B b \,x^{2}+12 A b x +12 B a x +10 A a \right ) \sqrt {\left (b x +a \right )^{2}}}{60 \left (b x +a \right ) x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*((b*x+a)^2)^(1/2)/x^7,x)

[Out]

-1/60*(15*B*b*x^2+12*A*b*x+12*B*a*x+10*A*a)*((b*x+a)^2)^(1/2)/x^6/(b*x+a)

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maxima [B]  time = 0.61, size = 375, normalized size = 3.29 \begin {gather*} -\frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B b^{5}}{2 \, a^{5}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{6}}{2 \, a^{6}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B b^{4}}{2 \, a^{4} x} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{5}}{2 \, a^{5} x} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b^{3}}{2 \, a^{5} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{4}}{2 \, a^{6} x^{2}} - \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b^{2}}{20 \, a^{4} x^{3}} + \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{3}}{15 \, a^{5} x^{3}} + \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b}{20 \, a^{3} x^{4}} - \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{2}}{5 \, a^{4} x^{4}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B}{5 \, a^{2} x^{5}} + \frac {3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b}{10 \, a^{3} x^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A}{6 \, a^{2} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*((b*x+a)^2)^(1/2)/x^7,x, algorithm="maxima")

[Out]

-1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B*b^5/a^5 + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*A*b^6/a^6 - 1/2*sqrt(b^2*x^2
+ 2*a*b*x + a^2)*B*b^4/(a^4*x) + 1/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*A*b^5/(a^5*x) + 1/2*(b^2*x^2 + 2*a*b*x + a^
2)^(3/2)*B*b^3/(a^5*x^2) - 1/2*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*A*b^4/(a^6*x^2) - 9/20*(b^2*x^2 + 2*a*b*x + a^2
)^(3/2)*B*b^2/(a^4*x^3) + 7/15*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*A*b^3/(a^5*x^3) + 7/20*(b^2*x^2 + 2*a*b*x + a^2
)^(3/2)*B*b/(a^3*x^4) - 2/5*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*A*b^2/(a^4*x^4) - 1/5*(b^2*x^2 + 2*a*b*x + a^2)^(3
/2)*B/(a^2*x^5) + 3/10*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*A*b/(a^3*x^5) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*A/(
a^2*x^6)

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mupad [B]  time = 1.15, size = 43, normalized size = 0.38 \begin {gather*} -\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (10\,A\,a+12\,A\,b\,x+12\,B\,a\,x+15\,B\,b\,x^2\right )}{60\,x^6\,\left (a+b\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((a + b*x)^2)^(1/2)*(A + B*x))/x^7,x)

[Out]

-(((a + b*x)^2)^(1/2)*(10*A*a + 12*A*b*x + 12*B*a*x + 15*B*b*x^2))/(60*x^6*(a + b*x))

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sympy [A]  time = 0.51, size = 31, normalized size = 0.27 \begin {gather*} \frac {- 10 A a - 15 B b x^{2} + x \left (- 12 A b - 12 B a\right )}{60 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*((b*x+a)**2)**(1/2)/x**7,x)

[Out]

(-10*A*a - 15*B*b*x**2 + x*(-12*A*b - 12*B*a))/(60*x**6)

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