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

Optimal. Leaf size=104 \[ \frac {c^2 \left (c+\frac {d}{x^2}\right )^{5/2} (b c-a d)}{5 d^4}+\frac {\left (c+\frac {d}{x^2}\right )^{9/2} (3 b c-a d)}{9 d^4}-\frac {c \left (c+\frac {d}{x^2}\right )^{7/2} (3 b c-2 a d)}{7 d^4}-\frac {b \left (c+\frac {d}{x^2}\right )^{11/2}}{11 d^4} \]

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Rubi [A]  time = 0.07, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {446, 77} \begin {gather*} \frac {c^2 \left (c+\frac {d}{x^2}\right )^{5/2} (b c-a d)}{5 d^4}+\frac {\left (c+\frac {d}{x^2}\right )^{9/2} (3 b c-a d)}{9 d^4}-\frac {c \left (c+\frac {d}{x^2}\right )^{7/2} (3 b c-2 a d)}{7 d^4}-\frac {b \left (c+\frac {d}{x^2}\right )^{11/2}}{11 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*(b*c - a*d)*(c + d/x^2)^(5/2))/(5*d^4) - (c*(3*b*c - 2*a*d)*(c + d/x^2)^(7/2))/(7*d^4) + ((3*b*c - a*d)*(
c + d/x^2)^(9/2))/(9*d^4) - (b*(c + d/x^2)^(11/2))/(11*d^4)

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {\left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2}}{x^7} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int x^2 (a+b x) (c+d x)^{3/2} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {c^2 (b c-a d) (c+d x)^{3/2}}{d^3}+\frac {c (3 b c-2 a d) (c+d x)^{5/2}}{d^3}+\frac {(-3 b c+a d) (c+d x)^{7/2}}{d^3}+\frac {b (c+d x)^{9/2}}{d^3}\right ) \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {c^2 (b c-a d) \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d^4}-\frac {c (3 b c-2 a d) \left (c+\frac {d}{x^2}\right )^{7/2}}{7 d^4}+\frac {(3 b c-a d) \left (c+\frac {d}{x^2}\right )^{9/2}}{9 d^4}-\frac {b \left (c+\frac {d}{x^2}\right )^{11/2}}{11 d^4}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 94, normalized size = 0.90 \begin {gather*} \frac {\sqrt {c+\frac {d}{x^2}} \left (c x^2+d\right )^2 \left (-11 a d x^2 \left (8 c^2 x^4-20 c d x^2+35 d^2\right )-3 b \left (-16 c^3 x^6+40 c^2 d x^4-70 c d^2 x^2+105 d^3\right )\right )}{3465 d^4 x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c + d/x^2]*(d + c*x^2)^2*(-11*a*d*x^2*(35*d^2 - 20*c*d*x^2 + 8*c^2*x^4) - 3*b*(105*d^3 - 70*c*d^2*x^2 +
40*c^2*d*x^4 - 16*c^3*x^6)))/(3465*d^4*x^10)

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IntegrateAlgebraic [A]  time = 0.08, size = 138, normalized size = 1.33 \begin {gather*} \frac {\sqrt {\frac {c x^2+d}{x^2}} \left (-88 a c^4 d x^{10}+44 a c^3 d^2 x^8-33 a c^2 d^3 x^6-550 a c d^4 x^4-385 a d^5 x^2+48 b c^5 x^{10}-24 b c^4 d x^8+18 b c^3 d^2 x^6-15 b c^2 d^3 x^4-420 b c d^4 x^2-315 b d^5\right )}{3465 d^4 x^{10}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((a + b/x^2)*(c + d/x^2)^(3/2))/x^7,x]

[Out]

(Sqrt[(d + c*x^2)/x^2]*(-315*b*d^5 - 420*b*c*d^4*x^2 - 385*a*d^5*x^2 - 15*b*c^2*d^3*x^4 - 550*a*c*d^4*x^4 + 18
*b*c^3*d^2*x^6 - 33*a*c^2*d^3*x^6 - 24*b*c^4*d*x^8 + 44*a*c^3*d^2*x^8 + 48*b*c^5*x^10 - 88*a*c^4*d*x^10))/(346
5*d^4*x^10)

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fricas [A]  time = 0.54, size = 134, normalized size = 1.29 \begin {gather*} \frac {{\left (8 \, {\left (6 \, b c^{5} - 11 \, a c^{4} d\right )} x^{10} - 4 \, {\left (6 \, b c^{4} d - 11 \, a c^{3} d^{2}\right )} x^{8} + 3 \, {\left (6 \, b c^{3} d^{2} - 11 \, a c^{2} d^{3}\right )} x^{6} - 315 \, b d^{5} - 5 \, {\left (3 \, b c^{2} d^{3} + 110 \, a c d^{4}\right )} x^{4} - 35 \, {\left (12 \, b c d^{4} + 11 \, a d^{5}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{3465 \, d^{4} x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*(c+d/x^2)^(3/2)/x^7,x, algorithm="fricas")

[Out]

1/3465*(8*(6*b*c^5 - 11*a*c^4*d)*x^10 - 4*(6*b*c^4*d - 11*a*c^3*d^2)*x^8 + 3*(6*b*c^3*d^2 - 11*a*c^2*d^3)*x^6
- 315*b*d^5 - 5*(3*b*c^2*d^3 + 110*a*c*d^4)*x^4 - 35*(12*b*c*d^4 + 11*a*d^5)*x^2)*sqrt((c*x^2 + d)/x^2)/(d^4*x
^10)

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giac [B]  time = 3.28, size = 490, normalized size = 4.71 \begin {gather*} \frac {16 \, {\left (2310 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{16} a c^{\frac {9}{2}} \mathrm {sgn}\relax (x) + 6930 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{14} b c^{\frac {11}{2}} \mathrm {sgn}\relax (x) - 1155 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{14} a c^{\frac {9}{2}} d \mathrm {sgn}\relax (x) + 12474 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{12} b c^{\frac {11}{2}} d \mathrm {sgn}\relax (x) + 231 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{12} a c^{\frac {9}{2}} d^{2} \mathrm {sgn}\relax (x) + 15246 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{10} b c^{\frac {11}{2}} d^{2} \mathrm {sgn}\relax (x) - 4851 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{10} a c^{\frac {9}{2}} d^{3} \mathrm {sgn}\relax (x) + 4950 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} b c^{\frac {11}{2}} d^{3} \mathrm {sgn}\relax (x) + 2475 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} a c^{\frac {9}{2}} d^{4} \mathrm {sgn}\relax (x) + 990 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} b c^{\frac {11}{2}} d^{4} \mathrm {sgn}\relax (x) + 495 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} a c^{\frac {9}{2}} d^{5} \mathrm {sgn}\relax (x) - 330 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} b c^{\frac {11}{2}} d^{5} \mathrm {sgn}\relax (x) + 605 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} a c^{\frac {9}{2}} d^{6} \mathrm {sgn}\relax (x) + 66 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} b c^{\frac {11}{2}} d^{6} \mathrm {sgn}\relax (x) - 121 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} a c^{\frac {9}{2}} d^{7} \mathrm {sgn}\relax (x) - 6 \, b c^{\frac {11}{2}} d^{7} \mathrm {sgn}\relax (x) + 11 \, a c^{\frac {9}{2}} d^{8} \mathrm {sgn}\relax (x)\right )}}{3465 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} - d\right )}^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*(c+d/x^2)^(3/2)/x^7,x, algorithm="giac")

[Out]

16/3465*(2310*(sqrt(c)*x - sqrt(c*x^2 + d))^16*a*c^(9/2)*sgn(x) + 6930*(sqrt(c)*x - sqrt(c*x^2 + d))^14*b*c^(1
1/2)*sgn(x) - 1155*(sqrt(c)*x - sqrt(c*x^2 + d))^14*a*c^(9/2)*d*sgn(x) + 12474*(sqrt(c)*x - sqrt(c*x^2 + d))^1
2*b*c^(11/2)*d*sgn(x) + 231*(sqrt(c)*x - sqrt(c*x^2 + d))^12*a*c^(9/2)*d^2*sgn(x) + 15246*(sqrt(c)*x - sqrt(c*
x^2 + d))^10*b*c^(11/2)*d^2*sgn(x) - 4851*(sqrt(c)*x - sqrt(c*x^2 + d))^10*a*c^(9/2)*d^3*sgn(x) + 4950*(sqrt(c
)*x - sqrt(c*x^2 + d))^8*b*c^(11/2)*d^3*sgn(x) + 2475*(sqrt(c)*x - sqrt(c*x^2 + d))^8*a*c^(9/2)*d^4*sgn(x) + 9
90*(sqrt(c)*x - sqrt(c*x^2 + d))^6*b*c^(11/2)*d^4*sgn(x) + 495*(sqrt(c)*x - sqrt(c*x^2 + d))^6*a*c^(9/2)*d^5*s
gn(x) - 330*(sqrt(c)*x - sqrt(c*x^2 + d))^4*b*c^(11/2)*d^5*sgn(x) + 605*(sqrt(c)*x - sqrt(c*x^2 + d))^4*a*c^(9
/2)*d^6*sgn(x) + 66*(sqrt(c)*x - sqrt(c*x^2 + d))^2*b*c^(11/2)*d^6*sgn(x) - 121*(sqrt(c)*x - sqrt(c*x^2 + d))^
2*a*c^(9/2)*d^7*sgn(x) - 6*b*c^(11/2)*d^7*sgn(x) + 11*a*c^(9/2)*d^8*sgn(x))/((sqrt(c)*x - sqrt(c*x^2 + d))^2 -
d)^11

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maple [A]  time = 0.05, size = 94, normalized size = 0.90 \begin {gather*} -\frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} \left (88 a \,c^{2} d \,x^{6}-48 b \,c^{3} x^{6}-220 a c \,d^{2} x^{4}+120 b \,c^{2} d \,x^{4}+385 a \,d^{3} x^{2}-210 b c \,d^{2} x^{2}+315 b \,d^{3}\right ) \left (c \,x^{2}+d \right )}{3465 d^{4} x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x^2)*(c+d/x^2)^(3/2)/x^7,x)

[Out]

-1/3465*((c*x^2+d)/x^2)^(3/2)*(88*a*c^2*d*x^6-48*b*c^3*x^6-220*a*c*d^2*x^4+120*b*c^2*d*x^4+385*a*d^3*x^2-210*b
*c*d^2*x^2+315*b*d^3)*(c*x^2+d)/d^4/x^8

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maxima [A]  time = 0.58, size = 118, normalized size = 1.13 \begin {gather*} -\frac {1}{315} \, {\left (\frac {35 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {9}{2}}}{d^{3}} - \frac {90 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}} c}{d^{3}} + \frac {63 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} c^{2}}{d^{3}}\right )} a - \frac {1}{1155} \, {\left (\frac {105 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {11}{2}}}{d^{4}} - \frac {385 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {9}{2}} c}{d^{4}} + \frac {495 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}} c^{2}}{d^{4}} - \frac {231 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} c^{3}}{d^{4}}\right )} b \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*(c+d/x^2)^(3/2)/x^7,x, algorithm="maxima")

[Out]

-1/315*(35*(c + d/x^2)^(9/2)/d^3 - 90*(c + d/x^2)^(7/2)*c/d^3 + 63*(c + d/x^2)^(5/2)*c^2/d^3)*a - 1/1155*(105*
(c + d/x^2)^(11/2)/d^4 - 385*(c + d/x^2)^(9/2)*c/d^4 + 495*(c + d/x^2)^(7/2)*c^2/d^4 - 231*(c + d/x^2)^(5/2)*c
^3/d^4)*b

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mupad [B]  time = 6.31, size = 206, normalized size = 1.98 \begin {gather*} \frac {16\,b\,c^5\,\sqrt {c+\frac {d}{x^2}}}{1155\,d^4}-\frac {8\,a\,c^4\,\sqrt {c+\frac {d}{x^2}}}{315\,d^3}-\frac {10\,a\,c\,\sqrt {c+\frac {d}{x^2}}}{63\,x^6}-\frac {a\,d\,\sqrt {c+\frac {d}{x^2}}}{9\,x^8}-\frac {4\,b\,c\,\sqrt {c+\frac {d}{x^2}}}{33\,x^8}-\frac {b\,d\,\sqrt {c+\frac {d}{x^2}}}{11\,x^{10}}-\frac {a\,c^2\,\sqrt {c+\frac {d}{x^2}}}{105\,d\,x^4}+\frac {4\,a\,c^3\,\sqrt {c+\frac {d}{x^2}}}{315\,d^2\,x^2}-\frac {b\,c^2\,\sqrt {c+\frac {d}{x^2}}}{231\,d\,x^6}+\frac {2\,b\,c^3\,\sqrt {c+\frac {d}{x^2}}}{385\,d^2\,x^4}-\frac {8\,b\,c^4\,\sqrt {c+\frac {d}{x^2}}}{1155\,d^3\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b/x^2)*(c + d/x^2)^(3/2))/x^7,x)

[Out]

(16*b*c^5*(c + d/x^2)^(1/2))/(1155*d^4) - (8*a*c^4*(c + d/x^2)^(1/2))/(315*d^3) - (10*a*c*(c + d/x^2)^(1/2))/(
63*x^6) - (a*d*(c + d/x^2)^(1/2))/(9*x^8) - (4*b*c*(c + d/x^2)^(1/2))/(33*x^8) - (b*d*(c + d/x^2)^(1/2))/(11*x
^10) - (a*c^2*(c + d/x^2)^(1/2))/(105*d*x^4) + (4*a*c^3*(c + d/x^2)^(1/2))/(315*d^2*x^2) - (b*c^2*(c + d/x^2)^
(1/2))/(231*d*x^6) + (2*b*c^3*(c + d/x^2)^(1/2))/(385*d^2*x^4) - (8*b*c^4*(c + d/x^2)^(1/2))/(1155*d^3*x^2)

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sympy [B]  time = 17.96, size = 262, normalized size = 2.52 \begin {gather*} - \frac {a c \left (\frac {c^{2} \left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3} - \frac {2 c \left (c + \frac {d}{x^{2}}\right )^{\frac {5}{2}}}{5} + \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {7}{2}}}{7}\right )}{d^{3}} - \frac {a \left (- \frac {c^{3} \left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3} + \frac {3 c^{2} \left (c + \frac {d}{x^{2}}\right )^{\frac {5}{2}}}{5} - \frac {3 c \left (c + \frac {d}{x^{2}}\right )^{\frac {7}{2}}}{7} + \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {9}{2}}}{9}\right )}{d^{3}} - \frac {b c \left (- \frac {c^{3} \left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3} + \frac {3 c^{2} \left (c + \frac {d}{x^{2}}\right )^{\frac {5}{2}}}{5} - \frac {3 c \left (c + \frac {d}{x^{2}}\right )^{\frac {7}{2}}}{7} + \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {9}{2}}}{9}\right )}{d^{4}} - \frac {b \left (\frac {c^{4} \left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3} - \frac {4 c^{3} \left (c + \frac {d}{x^{2}}\right )^{\frac {5}{2}}}{5} + \frac {6 c^{2} \left (c + \frac {d}{x^{2}}\right )^{\frac {7}{2}}}{7} - \frac {4 c \left (c + \frac {d}{x^{2}}\right )^{\frac {9}{2}}}{9} + \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {11}{2}}}{11}\right )}{d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**2)*(c+d/x**2)**(3/2)/x**7,x)

[Out]

-a*c*(c**2*(c + d/x**2)**(3/2)/3 - 2*c*(c + d/x**2)**(5/2)/5 + (c + d/x**2)**(7/2)/7)/d**3 - a*(-c**3*(c + d/x
**2)**(3/2)/3 + 3*c**2*(c + d/x**2)**(5/2)/5 - 3*c*(c + d/x**2)**(7/2)/7 + (c + d/x**2)**(9/2)/9)/d**3 - b*c*(
-c**3*(c + d/x**2)**(3/2)/3 + 3*c**2*(c + d/x**2)**(5/2)/5 - 3*c*(c + d/x**2)**(7/2)/7 + (c + d/x**2)**(9/2)/9
)/d**4 - b*(c**4*(c + d/x**2)**(3/2)/3 - 4*c**3*(c + d/x**2)**(5/2)/5 + 6*c**2*(c + d/x**2)**(7/2)/7 - 4*c*(c
+ d/x**2)**(9/2)/9 + (c + d/x**2)**(11/2)/11)/d**4

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