3.2.38 \(\int \frac {(a^2+2 a b x+b^2 x^2)^{3/2}}{x^7} \, dx\)

Optimal. Leaf size=151 \[ -\frac {3 a^2 b \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac {3 a b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac {a^3 \sqrt {a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)} \]

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Rubi [A]  time = 0.03, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {646, 43} \begin {gather*} -\frac {a^3 \sqrt {a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac {3 a^2 b \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac {3 a b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^7,x]

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 )^{3/2}}{x^7} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^3}{x^7} \, 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 b^3}{x^7}+\frac {3 a^2 b^4}{x^6}+\frac {3 a b^5}{x^5}+\frac {b^6}{x^4}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac {a^3 \sqrt {a^2+2 a b x+b^2 x^2}}{6 x^6 (a+b x)}-\frac {3 a^2 b \sqrt {a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac {3 a b^2 \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {b^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 55, normalized size = 0.36 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (10 a^3+36 a^2 b x+45 a b^2 x^2+20 b^3 x^3\right )}{60 x^6 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^7,x]

[Out]

-1/60*(Sqrt[(a + b*x)^2]*(10*a^3 + 36*a^2*b*x + 45*a*b^2*x^2 + 20*b^3*x^3))/(x^6*(a + b*x))

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IntegrateAlgebraic [B]  time = 1.02, size = 388, normalized size = 2.57 \begin {gather*} \frac {8 b^5 \sqrt {a^2+2 a b x+b^2 x^2} \left (-10 a^8 b-86 a^7 b^2 x-325 a^6 b^3 x^2-705 a^5 b^4 x^3-960 a^4 b^5 x^4-840 a^3 b^6 x^5-461 a^2 b^7 x^6-145 a b^8 x^7-20 b^9 x^8\right )+8 \sqrt {b^2} b^5 \left (10 a^9+96 a^8 b x+411 a^7 b^2 x^2+1030 a^6 b^3 x^3+1665 a^5 b^4 x^4+1800 a^4 b^5 x^5+1301 a^3 b^6 x^6+606 a^2 b^7 x^7+165 a b^8 x^8+20 b^9 x^9\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^2 + 2*a*b*x + b^2*x^2)^(3/2)/x^7,x]

[Out]

(8*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(-10*a^8*b - 86*a^7*b^2*x - 325*a^6*b^3*x^2 - 705*a^5*b^4*x^3 - 960*a^4*b
^5*x^4 - 840*a^3*b^6*x^5 - 461*a^2*b^7*x^6 - 145*a*b^8*x^7 - 20*b^9*x^8) + 8*b^5*Sqrt[b^2]*(10*a^9 + 96*a^8*b*
x + 411*a^7*b^2*x^2 + 1030*a^6*b^3*x^3 + 1665*a^5*b^4*x^4 + 1800*a^4*b^5*x^5 + 1301*a^3*b^6*x^6 + 606*a^2*b^7*
x^7 + 165*a*b^8*x^8 + 20*b^9*x^9))/(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 + 48
0*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 = 0.39, size = 35, normalized size = 0.23 \begin {gather*} -\frac {20 \, b^{3} x^{3} + 45 \, a b^{2} x^{2} + 36 \, a^{2} b x + 10 \, a^{3}}{60 \, 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)^(3/2)/x^7,x, algorithm="fricas")

[Out]

-1/60*(20*b^3*x^3 + 45*a*b^2*x^2 + 36*a^2*b*x + 10*a^3)/x^6

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giac [A]  time = 0.16, size = 74, normalized size = 0.49 \begin {gather*} -\frac {b^{6} \mathrm {sgn}\left (b x + a\right )}{60 \, a^{3}} - \frac {20 \, b^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 45 \, a b^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 36 \, a^{2} b x \mathrm {sgn}\left (b x + a\right ) + 10 \, a^{3} \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^2*x^2+2*a*b*x+a^2)^(3/2)/x^7,x, algorithm="giac")

[Out]

-1/60*b^6*sgn(b*x + a)/a^3 - 1/60*(20*b^3*x^3*sgn(b*x + a) + 45*a*b^2*x^2*sgn(b*x + a) + 36*a^2*b*x*sgn(b*x +
a) + 10*a^3*sgn(b*x + a))/x^6

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maple [A]  time = 0.05, size = 52, normalized size = 0.34 \begin {gather*} -\frac {\left (20 b^{3} x^{3}+45 a \,b^{2} x^{2}+36 a^{2} b x +10 a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{60 \left (b x +a \right )^{3} 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)^(3/2)/x^7,x)

[Out]

-1/60*(20*b^3*x^3+45*a*b^2*x^2+36*a^2*b*x+10*a^3)*((b*x+a)^2)^(3/2)/x^6/(b*x+a)^3

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maxima [A]  time = 1.49, size = 196, normalized size = 1.30 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{6}}{4 \, a^{6}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} b^{5}}{4 \, a^{5} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{4}}{4 \, a^{6} x^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{3}}{4 \, a^{5} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b^{2}}{4 \, a^{4} x^{4}} + \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} b}{30 \, a^{3} x^{5}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{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)^(3/2)/x^7,x, algorithm="maxima")

[Out]

1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*b^6/a^6 + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*b^5/(a^5*x) - 1/4*(b^2*x^2 +
2*a*b*x + a^2)^(5/2)*b^4/(a^6*x^2) + 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*b^3/(a^5*x^3) - 1/4*(b^2*x^2 + 2*a*b
*x + a^2)^(5/2)*b^2/(a^4*x^4) + 7/30*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*b/(a^3*x^5) - 1/6*(b^2*x^2 + 2*a*b*x + a^
2)^(5/2)/(a^2*x^6)

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mupad [B]  time = 0.20, size = 135, normalized size = 0.89 \begin {gather*} -\frac {a^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{6\,x^6\,\left (a+b\,x\right )}-\frac {b^3\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{3\,x^3\,\left (a+b\,x\right )}-\frac {3\,a\,b^2\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{4\,x^4\,\left (a+b\,x\right )}-\frac {3\,a^2\,b\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{5\,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)^(3/2)/x^7,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (a + b x\right )^{2}\right )^{\frac {3}{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)**(3/2)/x**7,x)

[Out]

Integral(((a + b*x)**2)**(3/2)/x**7, x)

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