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

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

[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)

_______________________________________________________________________________________

Rubi [A]  time = 0.230749, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{c^2 \left (c+\frac{d}{x^2}\right )^{5/2} (b c-a d)}{5 d^4}+\frac{\left (c+\frac{d}{x^2}\right )^{9/2} (3 b c-a d)}{9 d^4}-\frac{c \left (c+\frac{d}{x^2}\right )^{7/2} (3 b c-2 a d)}{7 d^4}-\frac{b \left (c+\frac{d}{x^2}\right )^{11/2}}{11 d^4} \]

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)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 23.1413, size = 92, normalized size = 0.88 \[ - \frac{b \left (c + \frac{d}{x^{2}}\right )^{\frac{11}{2}}}{11 d^{4}} - \frac{c^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}} \left (a d - b c\right )}{5 d^{4}} + \frac{c \left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}} \left (2 a d - 3 b c\right )}{7 d^{4}} - \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{9}{2}} \left (a d - 3 b c\right )}{9 d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.105346, size = 94, normalized size = 0.9 \[ \frac{\sqrt{c+\frac{d}{x^2}} \left (c x^2+d\right )^2 \left (-11 a d x^2 \left (8 c^2 x^4-20 c d x^2+35 d^2\right )-3 b \left (-16 c^3 x^6+40 c^2 d x^4-70 c d^2 x^2+105 d^3\right )\right )}{3465 d^4 x^{10}} \]

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)

_______________________________________________________________________________________

Maple [A]  time = 0.012, size = 94, normalized size = 0.9 \[ -{\frac{ \left ( 88\,a{c}^{2}d{x}^{6}-48\,b{c}^{3}{x}^{6}-220\,ac{d}^{2}{x}^{4}+120\,b{c}^{2}d{x}^{4}+385\,a{d}^{3}{x}^{2}-210\,bc{d}^{2}{x}^{2}+315\,b{d}^{3} \right ) \left ( c{x}^{2}+d \right ) }{3465\,{d}^{4}{x}^{8}} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{{\frac{3}{2}}}} \]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.37257, size = 159, normalized size = 1.53 \[ -\frac{1}{315} \,{\left (\frac{35 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{9}{2}}}{d^{3}} - \frac{90 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}} c}{d^{3}} + \frac{63 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} c^{2}}{d^{3}}\right )} a - \frac{1}{1155} \,{\left (\frac{105 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{11}{2}}}{d^{4}} - \frac{385 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{9}{2}} c}{d^{4}} + \frac{495 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{7}{2}} c^{2}}{d^{4}} - \frac{231 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} c^{3}}{d^{4}}\right )} b \]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.488112, size = 181, normalized size = 1.74 \[ \frac{{\left (8 \,{\left (6 \, b c^{5} - 11 \, a c^{4} d\right )} x^{10} - 4 \,{\left (6 \, b c^{4} d - 11 \, a c^{3} d^{2}\right )} x^{8} + 3 \,{\left (6 \, b c^{3} d^{2} - 11 \, a c^{2} d^{3}\right )} x^{6} - 315 \, b d^{5} - 5 \,{\left (3 \, b c^{2} d^{3} + 110 \, a c d^{4}\right )} x^{4} - 35 \,{\left (12 \, b c d^{4} + 11 \, a d^{5}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{3465 \, d^{4} x^{10}} \]

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)

_______________________________________________________________________________________

Sympy [A]  time = 9.23717, size = 262, normalized size = 2.52 \[ - \frac{a c \left (\frac{c^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} - \frac{2 c \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}}}{7}\right )}{d^{3}} - \frac{a \left (- \frac{c^{3} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} + \frac{3 c^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5} - \frac{3 c \left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}}}{7} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{9}{2}}}{9}\right )}{d^{3}} - \frac{b c \left (- \frac{c^{3} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} + \frac{3 c^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5} - \frac{3 c \left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}}}{7} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{9}{2}}}{9}\right )}{d^{4}} - \frac{b \left (\frac{c^{4} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} - \frac{4 c^{3} \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5} + \frac{6 c^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{7}{2}}}{7} - \frac{4 c \left (c + \frac{d}{x^{2}}\right )^{\frac{9}{2}}}{9} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{11}{2}}}{11}\right )}{d^{4}} \]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 1.24163, size = 662, normalized size = 6.37 \[ \frac{16 \,{\left (2310 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{16} a c^{\frac{9}{2}}{\rm sign}\left (x\right ) + 6930 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{14} b c^{\frac{11}{2}}{\rm sign}\left (x\right ) - 1155 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{14} a c^{\frac{9}{2}} d{\rm sign}\left (x\right ) + 12474 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{12} b c^{\frac{11}{2}} d{\rm sign}\left (x\right ) + 231 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{12} a c^{\frac{9}{2}} d^{2}{\rm sign}\left (x\right ) + 15246 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{10} b c^{\frac{11}{2}} d^{2}{\rm sign}\left (x\right ) - 4851 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{10} a c^{\frac{9}{2}} d^{3}{\rm sign}\left (x\right ) + 4950 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{8} b c^{\frac{11}{2}} d^{3}{\rm sign}\left (x\right ) + 2475 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{8} a c^{\frac{9}{2}} d^{4}{\rm sign}\left (x\right ) + 990 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} b c^{\frac{11}{2}} d^{4}{\rm sign}\left (x\right ) + 495 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} a c^{\frac{9}{2}} d^{5}{\rm sign}\left (x\right ) - 330 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} b c^{\frac{11}{2}} d^{5}{\rm sign}\left (x\right ) + 605 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} a c^{\frac{9}{2}} d^{6}{\rm sign}\left (x\right ) + 66 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} b c^{\frac{11}{2}} d^{6}{\rm sign}\left (x\right ) - 121 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} a c^{\frac{9}{2}} d^{7}{\rm sign}\left (x\right ) - 6 \, b c^{\frac{11}{2}} d^{7}{\rm sign}\left (x\right ) + 11 \, a c^{\frac{9}{2}} d^{8}{\rm sign}\left (x\right )\right )}}{3465 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} - d\right )}^{11}} \]

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)*sign(x) + 6930*(sqrt(c)
*x - sqrt(c*x^2 + d))^14*b*c^(11/2)*sign(x) - 1155*(sqrt(c)*x - sqrt(c*x^2 + d))
^14*a*c^(9/2)*d*sign(x) + 12474*(sqrt(c)*x - sqrt(c*x^2 + d))^12*b*c^(11/2)*d*si
gn(x) + 231*(sqrt(c)*x - sqrt(c*x^2 + d))^12*a*c^(9/2)*d^2*sign(x) + 15246*(sqrt
(c)*x - sqrt(c*x^2 + d))^10*b*c^(11/2)*d^2*sign(x) - 4851*(sqrt(c)*x - sqrt(c*x^
2 + d))^10*a*c^(9/2)*d^3*sign(x) + 4950*(sqrt(c)*x - sqrt(c*x^2 + d))^8*b*c^(11/
2)*d^3*sign(x) + 2475*(sqrt(c)*x - sqrt(c*x^2 + d))^8*a*c^(9/2)*d^4*sign(x) + 99
0*(sqrt(c)*x - sqrt(c*x^2 + d))^6*b*c^(11/2)*d^4*sign(x) + 495*(sqrt(c)*x - sqrt
(c*x^2 + d))^6*a*c^(9/2)*d^5*sign(x) - 330*(sqrt(c)*x - sqrt(c*x^2 + d))^4*b*c^(
11/2)*d^5*sign(x) + 605*(sqrt(c)*x - sqrt(c*x^2 + d))^4*a*c^(9/2)*d^6*sign(x) +
66*(sqrt(c)*x - sqrt(c*x^2 + d))^2*b*c^(11/2)*d^6*sign(x) - 121*(sqrt(c)*x - sqr
t(c*x^2 + d))^2*a*c^(9/2)*d^7*sign(x) - 6*b*c^(11/2)*d^7*sign(x) + 11*a*c^(9/2)*
d^8*sign(x))/((sqrt(c)*x - sqrt(c*x^2 + d))^2 - d)^11