3.2.53 \(\int \frac {(a^2+2 a b x+b^2 x^2)^{5/2}}{x^7} \, dx\)

Optimal. Leaf size=37 \[ -\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 a x^6} \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {646, 37} \begin {gather*} -\frac {(a+b x)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{6 a x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^7,x]

[Out]

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

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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [B]  time = 0.02, size = 75, normalized size = 2.03 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (a^5+6 a^4 b x+15 a^3 b^2 x^2+20 a^2 b^3 x^3+15 a b^4 x^4+6 b^5 x^5\right )}{6 x^6 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^7,x]

[Out]

-1/6*(Sqrt[(a + b*x)^2]*(a^5 + 6*a^4*b*x + 15*a^3*b^2*x^2 + 20*a^2*b^3*x^3 + 15*a*b^4*x^4 + 6*b^5*x^5))/(x^6*(
a + b*x))

________________________________________________________________________________________

IntegrateAlgebraic [B]  time = 2.64, size = 430, normalized size = 11.62 \begin {gather*} \frac {16 b^5 \sqrt {a^2+2 a b x+b^2 x^2} \left (-a^{10} b-11 a^9 b^2 x-55 a^8 b^3 x^2-165 a^7 b^4 x^3-330 a^6 b^5 x^4-462 a^5 b^6 x^5-461 a^4 b^7 x^6-325 a^3 b^8 x^7-155 a^2 b^9 x^8-45 a b^{10} x^9-6 b^{11} x^{10}\right )+16 \sqrt {b^2} b^5 \left (a^{11}+12 a^{10} b x+66 a^9 b^2 x^2+220 a^8 b^3 x^3+495 a^7 b^4 x^4+792 a^6 b^5 x^5+923 a^5 b^6 x^6+786 a^4 b^7 x^7+480 a^3 b^8 x^8+200 a^2 b^9 x^9+51 a b^{10} x^{10}+6 b^{11} x^{11}\right )}{3 \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 )+3 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^2 + 2*a*b*x + b^2*x^2)^(5/2)/x^7,x]

[Out]

(16*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-(a^10*b) - 11*a^9*b^2*x - 55*a^8*b^3*x^2 - 165*a^7*b^4*x^3 - 330*a^6*b
^5*x^4 - 462*a^5*b^6*x^5 - 461*a^4*b^7*x^6 - 325*a^3*b^8*x^7 - 155*a^2*b^9*x^8 - 45*a*b^10*x^9 - 6*b^11*x^10)
+ 16*b^5*Sqrt[b^2]*(a^11 + 12*a^10*b*x + 66*a^9*b^2*x^2 + 220*a^8*b^3*x^3 + 495*a^7*b^4*x^4 + 792*a^6*b^5*x^5
+ 923*a^5*b^6*x^6 + 786*a^4*b^7*x^7 + 480*a^3*b^8*x^8 + 200*a^2*b^9*x^9 + 51*a*b^10*x^10 + 6*b^11*x^11))/(3*Sq
rt[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 - 1
60*a*b^9*x^4 - 32*b^10*x^5) + 3*x^6*(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))

________________________________________________________________________________________

fricas [B]  time = 0.41, size = 55, normalized size = 1.49 \begin {gather*} -\frac {6 \, b^{5} x^{5} + 15 \, a b^{4} x^{4} + 20 \, a^{2} b^{3} x^{3} + 15 \, a^{3} b^{2} x^{2} + 6 \, a^{4} b x + a^{5}}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/x^7,x, algorithm="fricas")

[Out]

-1/6*(6*b^5*x^5 + 15*a*b^4*x^4 + 20*a^2*b^3*x^3 + 15*a^3*b^2*x^2 + 6*a^4*b*x + a^5)/x^6

________________________________________________________________________________________

giac [B]  time = 0.16, size = 107, normalized size = 2.89 \begin {gather*} -\frac {b^{6} \mathrm {sgn}\left (b x + a\right )}{6 \, a} - \frac {6 \, b^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 15 \, a b^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 20 \, a^{2} b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 15 \, a^{3} b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{4} b x \mathrm {sgn}\left (b x + a\right ) + a^{5} \mathrm {sgn}\left (b x + a\right )}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/x^7,x, algorithm="giac")

[Out]

-1/6*b^6*sgn(b*x + a)/a - 1/6*(6*b^5*x^5*sgn(b*x + a) + 15*a*b^4*x^4*sgn(b*x + a) + 20*a^2*b^3*x^3*sgn(b*x + a
) + 15*a^3*b^2*x^2*sgn(b*x + a) + 6*a^4*b*x*sgn(b*x + a) + a^5*sgn(b*x + a))/x^6

________________________________________________________________________________________

maple [B]  time = 0.05, size = 72, normalized size = 1.95 \begin {gather*} -\frac {\left (6 b^{5} x^{5}+15 a \,b^{4} x^{4}+20 a^{2} b^{3} x^{3}+15 a^{3} b^{2} x^{2}+6 a^{4} b x +a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{6 \left (b x +a \right )^{5} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^2+2*a*b*x+a^2)^(5/2)/x^7,x)

[Out]

-1/6*(6*b^5*x^5+15*a*b^4*x^4+20*a^2*b^3*x^3+15*a^3*b^2*x^2+6*a^4*b*x+a^5)*((b*x+a)^2)^(5/2)/x^6/(b*x+a)^5

________________________________________________________________________________________

maxima [B]  time = 1.58, size = 196, normalized size = 5.30 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{6}}{6 \, a^{6}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{5}}{6 \, a^{5} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{4}}{6 \, a^{6} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{3}}{6 \, a^{5} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b^{2}}{6 \, a^{4} x^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} b}{6 \, a^{3} x^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}}}{6 \, a^{2} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)^(5/2)/x^7,x, algorithm="maxima")

[Out]

1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*b^6/a^6 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*b^5/(a^5*x) - 1/6*(b^2*x^2 +
2*a*b*x + a^2)^(7/2)*b^4/(a^6*x^2) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*b^3/(a^5*x^3) - 1/6*(b^2*x^2 + 2*a*b
*x + a^2)^(7/2)*b^2/(a^4*x^4) + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*b/(a^3*x^5) - 1/6*(b^2*x^2 + 2*a*b*x + a^2
)^(7/2)/(a^2*x^6)

________________________________________________________________________________________

mupad [B]  time = 0.19, size = 207, normalized size = 5.59 \begin {gather*} -\frac {a^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{6\,x^6\,\left (a+b\,x\right )}-\frac {b^5\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x\,\left (a+b\,x\right )}-\frac {10\,a^2\,b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{3\,x^3\,\left (a+b\,x\right )}-\frac {5\,a^3\,b^2\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,x^4\,\left (a+b\,x\right )}-\frac {5\,a\,b^4\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{2\,x^2\,\left (a+b\,x\right )}-\frac {a^4\,b\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{x^5\,\left (a+b\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^2 + 2*a*b*x)^(5/2)/x^7,x)

[Out]

- (a^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(6*x^6*(a + b*x)) - (b^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(x*(a + b*x)
) - (10*a^2*b^3*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(3*x^3*(a + b*x)) - (5*a^3*b^2*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2
))/(2*x^4*(a + b*x)) - (5*a*b^4*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(2*x^2*(a + b*x)) - (a^4*b*(a^2 + b^2*x^2 + 2
*a*b*x)^(1/2))/(x^5*(a + b*x))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{x^{7}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**(5/2)/x**7,x)

[Out]

Integral(((a + b*x)**2)**(5/2)/x**7, x)

________________________________________________________________________________________